src/HOL/Library/GCD.thy
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(*  Title:      HOL/GCD.thy
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    ID:         $Id$
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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 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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  "is_gcd p m n \<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 m a b \<Longrightarrow> is_gcd n a b \<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 m a b \<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 k m n = is_gcd k n m"
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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 \<times> 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:
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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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apply (rule gcd.induct [of "split P" "(m, n)", unfolded Product_Type.split])
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apply (case_tac "n = 0")
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apply simp_all
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using assms apply simp_all
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done
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lemma gcd_0 [simp]: "gcd (m, 0) = m"
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  by simp
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lemma gcd_0_left [simp]: "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]: "gcd (m, Suc 0) = 1"
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  by simp
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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]: "gcd (m, n) dvd m"
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  and gcd_dvd2 [iff]: "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 (gcd (m, n)) 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]: "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: "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]: "gcd (Suc 0, m) = 1"
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  by (simp add: gcd_commute)
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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]: "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]: "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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  apply (blast intro: relprime_dvd_mult dvd_trans)
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  done
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lemma gcd_mult_cancel: "gcd (k, n) = 1 ==> gcd (k * m, n) = gcd (m, n)"
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   158
  apply (rule dvd_anti_sym)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   159
   apply (rule gcd_greatest)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   160
    apply (rule_tac n = k in relprime_dvd_mult)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   161
     apply (simp add: gcd_assoc)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   162
     apply (simp add: gcd_commute)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   163
    apply (simp_all add: mult_commute)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   164
  apply (blast intro: dvd_trans)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   165
  done
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   166
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   167
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   168
text {* \medskip Addition laws *}
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   169
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   170
lemma gcd_add1 [simp]: "gcd (m + n, n) = gcd (m, n)"
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   171
  apply (case_tac "n = 0")
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   172
   apply (simp_all add: gcd_non_0)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   173
  done
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   174
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   175
lemma gcd_add2 [simp]: "gcd (m, m + n) = gcd (m, n)"
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   176
proof -
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   177
  have "gcd (m, m + n) = gcd (m + n, m)" by (rule gcd_commute)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   178
  also have "... = gcd (n + m, m)" by (simp add: add_commute)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   179
  also have "... = gcd (n, m)" by simp
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   180
  also have  "... = gcd (m, n)" by (rule gcd_commute)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   181
  finally show ?thesis .
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   182
qed
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   183
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   184
lemma gcd_add2' [simp]: "gcd (m, n + m) = gcd (m, n)"
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   185
  apply (subst add_commute)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   186
  apply (rule gcd_add2)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   187
  done
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   188
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   189
lemma gcd_add_mult: "gcd (m, k * m + n) = gcd (m, n)"
21263
wenzelm
parents: 21256
diff changeset
   190
  by (induct k) (simp_all add: add_assoc)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   191
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   192
lemma gcd_dvd_prod: "gcd (m, n) dvd m * n"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   193
  using mult_dvd_mono [of 1] by auto
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   194
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   195
text {*
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   196
  \medskip Division by gcd yields rrelatively primes.
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   197
*}
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   198
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   199
lemma div_gcd_relprime:
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   200
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   201
  shows "gcd (a div gcd(a,b), b div gcd(a,b)) = 1"
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   202
proof -
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   203
  let ?g = "gcd (a, b)"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   204
  let ?a' = "a div ?g"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   205
  let ?b' = "b div ?g"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   206
  let ?g' = "gcd (?a', ?b')"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   207
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   208
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   209
  from dvdg dvdg' obtain ka kb ka' kb' where
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   210
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   211
    unfolding dvd_def by blast
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   212
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'" by simp_all
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   213
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   214
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   215
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   216
  have "?g \<noteq> 0" using nz by (simp add: gcd_zero)
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   217
  then have gp: "?g > 0" by simp
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   218
  from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   219
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   220
qed
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   221
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   222
subsection {* LCM defined by GCD *}
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   223
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   224
definition
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   225
  lcm :: "nat \<times> nat \<Rightarrow> nat"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   226
where
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   227
  "lcm = (\<lambda>(m, n). m * n div gcd (m, n))"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   228
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   229
lemma lcm_def:
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   230
  "lcm (m, n) = m * n div gcd (m, n)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   231
  unfolding lcm_def by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   232
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   233
lemma prod_gcd_lcm:
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   234
  "m * n = gcd (m, n) * lcm (m, n)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   235
  unfolding lcm_def by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod])
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   236
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   237
lemma lcm_0 [simp]: "lcm (m, 0) = 0"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   238
  unfolding lcm_def by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   239
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   240
lemma lcm_1 [simp]: "lcm (m, 1) = m"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   241
  unfolding lcm_def by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   242
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   243
lemma lcm_0_left [simp]: "lcm (0, n) = 0"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   244
  unfolding lcm_def by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   245
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   246
lemma lcm_1_left [simp]: "lcm (1, m) = m"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   247
  unfolding lcm_def by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   248
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   249
lemma dvd_pos:
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   250
  fixes n m :: nat
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   251
  assumes "n > 0" and "m dvd n"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   252
  shows "m > 0"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   253
using assms by (cases m) auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   254
23951
b188cac107ad renamed lcm_lowest to lcm_least
haftmann
parents: 23687
diff changeset
   255
lemma lcm_least:
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   256
  assumes "m dvd k" and "n dvd k"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   257
  shows "lcm (m, n) dvd k"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   258
proof (cases k)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   259
  case 0 then show ?thesis by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   260
next
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   261
  case (Suc _) then have pos_k: "k > 0" by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   262
  from assms dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   263
  with gcd_zero [of m n] have pos_gcd: "gcd (m, n) > 0" by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   264
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   265
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   266
  from pos_k k_m have pos_p: "p > 0" by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   267
  from pos_k k_n have pos_q: "q > 0" by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   268
  have "k * k * gcd (q, p) = k * gcd (k * q, k * p)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   269
    by (simp add: mult_ac gcd_mult_distrib2)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   270
  also have "\<dots> = k * gcd (m * p * q, n * q * p)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   271
    by (simp add: k_m [symmetric] k_n [symmetric])
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   272
  also have "\<dots> = k * p * q * gcd (m, n)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   273
    by (simp add: mult_ac gcd_mult_distrib2)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   274
  finally have "(m * p) * (n * q) * gcd (q, p) = k * p * q * gcd (m, n)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   275
    by (simp only: k_m [symmetric] k_n [symmetric])
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   276
  then have "p * q * m * n * gcd (q, p) = p * q * k * gcd (m, n)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   277
    by (simp add: mult_ac)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   278
  with pos_p pos_q have "m * n * gcd (q, p) = k * gcd (m, n)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   279
    by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   280
  with prod_gcd_lcm [of m n]
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   281
  have "lcm (m, n) * gcd (q, p) * gcd (m, n) = k * gcd (m, n)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   282
    by (simp add: mult_ac)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   283
  with pos_gcd have "lcm (m, n) * gcd (q, p) = k" by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   284
  then show ?thesis using dvd_def by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   285
qed
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   286
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   287
lemma lcm_dvd1 [iff]:
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   288
  "m dvd lcm (m, n)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   289
proof (cases m)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   290
  case 0 then show ?thesis by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   291
next
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   292
  case (Suc _)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   293
  then have mpos: "m > 0" by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   294
  show ?thesis
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   295
  proof (cases n)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   296
    case 0 then show ?thesis by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   297
  next
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   298
    case (Suc _)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   299
    then have npos: "n > 0" by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   300
    have "gcd (m, n) dvd n" by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   301
    then obtain k where "n = gcd (m, n) * k" using dvd_def by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   302
    then have "m * n div gcd (m, n) = m * (gcd (m, n) * k) div gcd (m, n)" by (simp add: mult_ac)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   303
    also have "\<dots> = m * k" using mpos npos gcd_zero by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   304
    finally show ?thesis by (simp add: lcm_def)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   305
  qed
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   306
qed
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   307
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   308
lemma lcm_dvd2 [iff]: 
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   309
  "n dvd lcm (m, n)"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   310
proof (cases n)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   311
  case 0 then show ?thesis by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   312
next
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   313
  case (Suc _)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   314
  then have npos: "n > 0" by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   315
  show ?thesis
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   316
  proof (cases m)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   317
    case 0 then show ?thesis by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   318
  next
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   319
    case (Suc _)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   320
    then have mpos: "m > 0" by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   321
    have "gcd (m, n) dvd m" by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   322
    then obtain k where "m = gcd (m, n) * k" using dvd_def by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   323
    then have "m * n div gcd (m, n) = (gcd (m, n) * k) * n div gcd (m, n)" by (simp add: mult_ac)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   324
    also have "\<dots> = n * k" using mpos npos gcd_zero by simp
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   325
    finally show ?thesis by (simp add: lcm_def)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   326
  qed
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   327
qed
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   328
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   329
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   330
subsection {* GCD and LCM on integers *}
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   331
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   332
definition
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   333
  igcd :: "int \<Rightarrow> int \<Rightarrow> int" where
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   334
  "igcd i j = int (gcd (nat (abs i), nat (abs j)))"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   335
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   336
lemma igcd_dvd1 [simp]: "igcd i j dvd i"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   337
  by (simp add: igcd_def int_dvd_iff)
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   338
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   339
lemma igcd_dvd2 [simp]: "igcd i j dvd j"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   340
  by (simp add: igcd_def int_dvd_iff)
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   341
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   342
lemma igcd_pos: "igcd i j \<ge> 0"
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   343
  by (simp add: igcd_def)
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   344
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   345
lemma igcd0 [simp]: "(igcd i j = 0) = (i = 0 \<and> j = 0)"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   346
  by (simp add: igcd_def gcd_zero) arith
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   347
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   348
lemma igcd_commute: "igcd i j = igcd j i"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   349
  unfolding igcd_def by (simp add: gcd_commute)
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   350
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   351
lemma igcd_neg1 [simp]: "igcd (- i) j = igcd i j"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   352
  unfolding igcd_def by simp
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   353
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   354
lemma igcd_neg2 [simp]: "igcd i (- j) = igcd i j"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   355
  unfolding igcd_def by simp
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   356
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   357
lemma zrelprime_dvd_mult: "igcd i j = 1 \<Longrightarrow> i dvd k * j \<Longrightarrow> i dvd k"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   358
  unfolding igcd_def
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   359
proof -
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   360
  assume "int (gcd (nat \<bar>i\<bar>, nat \<bar>j\<bar>)) = 1" "i dvd k * j"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   361
  then have g: "gcd (nat \<bar>i\<bar>, nat \<bar>j\<bar>) = 1" by simp
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   362
  from `i dvd k * j` obtain h where h: "k*j = i*h" unfolding dvd_def by blast
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   363
  have th: "nat \<bar>i\<bar> dvd nat \<bar>k\<bar> * nat \<bar>j\<bar>"
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   364
    unfolding dvd_def
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   365
    by (rule_tac x= "nat \<bar>h\<bar>" in exI, simp add: h nat_abs_mult_distrib [symmetric])
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   366
  from relprime_dvd_mult [OF g th] obtain h' where h': "nat \<bar>k\<bar> = nat \<bar>i\<bar> * h'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   367
    unfolding dvd_def by blast
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   368
  from h' have "int (nat \<bar>k\<bar>) = int (nat \<bar>i\<bar> * h')" by simp
23431
25ca91279a9b change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents: 23365
diff changeset
   369
  then have "\<bar>k\<bar> = \<bar>i\<bar> * int h'" by (simp add: int_mult)
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   370
  then show ?thesis
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   371
    apply (subst zdvd_abs1 [symmetric])
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   372
    apply (subst zdvd_abs2 [symmetric])
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   373
    apply (unfold dvd_def)
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   374
    apply (rule_tac x = "int h'" in exI, simp)
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   375
    done
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   376
qed
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   377
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   378
lemma int_nat_abs: "int (nat (abs x)) = abs x"  by arith
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   379
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   380
lemma igcd_greatest:
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   381
  assumes "k dvd m" and "k dvd n"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   382
  shows "k dvd igcd m n"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   383
proof -
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   384
  let ?k' = "nat \<bar>k\<bar>"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   385
  let ?m' = "nat \<bar>m\<bar>"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   386
  let ?n' = "nat \<bar>n\<bar>"
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   387
  from `k dvd m` and `k dvd n` have dvd': "?k' dvd ?m'" "?k' dvd ?n'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   388
    unfolding zdvd_int by (simp_all only: int_nat_abs zdvd_abs1 zdvd_abs2)
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   389
  from gcd_greatest [OF dvd'] have "int (nat \<bar>k\<bar>) dvd igcd m n"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   390
    unfolding igcd_def by (simp only: zdvd_int)
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   391
  then have "\<bar>k\<bar> dvd igcd m n" by (simp only: int_nat_abs)
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   392
  then show "k dvd igcd m n" by (simp add: zdvd_abs1)
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   393
qed
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   394
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   395
lemma div_igcd_relprime:
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   396
  assumes nz: "a \<noteq> 0 \<or> b \<noteq> 0"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   397
  shows "igcd (a div (igcd a b)) (b div (igcd a b)) = 1"
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   398
proof -
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   399
  from nz have nz': "nat \<bar>a\<bar> \<noteq> 0 \<or> nat \<bar>b\<bar> \<noteq> 0" by simp
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   400
  let ?g = "igcd a b"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   401
  let ?a' = "a div ?g"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   402
  let ?b' = "b div ?g"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   403
  let ?g' = "igcd ?a' ?b'"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   404
  have dvdg: "?g dvd a" "?g dvd b" by (simp_all add: igcd_dvd1 igcd_dvd2)
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   405
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by (simp_all add: igcd_dvd1 igcd_dvd2)
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   406
  from dvdg dvdg' obtain ka kb ka' kb' where
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   407
   kab: "a = ?g*ka" "b = ?g*kb" "?a' = ?g'*ka'" "?b' = ?g' * kb'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   408
    unfolding dvd_def by blast
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   409
  then have "?g* ?a' = (?g * ?g') * ka'" "?g* ?b' = (?g * ?g') * kb'" by simp_all
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   410
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   411
    by (auto simp add: zdvd_mult_div_cancel [OF dvdg(1)]
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   412
      zdvd_mult_div_cancel [OF dvdg(2)] dvd_def)
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   413
  have "?g \<noteq> 0" using nz by simp
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   414
  then have gp: "?g \<noteq> 0" using igcd_pos[where i="a" and j="b"] by arith
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   415
  from igcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   416
  with zdvd_mult_cancel1 [OF gp] have "\<bar>?g'\<bar> = 1" by simp
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   417
  with igcd_pos show "?g' = 1" by simp
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   418
qed
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   419
23244
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   420
definition "ilcm = (\<lambda>i j. int (lcm(nat(abs i),nat(abs j))))"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   421
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   422
(* ilcm_dvd12 are needed later *)
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   423
lemma ilcm_dvd1: 
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   424
assumes anz: "a \<noteq> 0" 
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   425
  and bnz: "b \<noteq> 0"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   426
  shows "a dvd (ilcm a b)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   427
proof-
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   428
  let ?na = "nat (abs a)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   429
  let ?nb = "nat (abs b)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   430
  have nap: "?na >0" using anz by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   431
  have nbp: "?nb >0" using bnz by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   432
  from nap nbp have "?na dvd lcm(?na,?nb)" using lcm_dvd1 by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   433
  thus ?thesis by (simp add: ilcm_def dvd_int_iff)
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   434
qed
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   435
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   436
lemma ilcm_dvd2: 
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   437
assumes anz: "a \<noteq> 0" 
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   438
  and bnz: "b \<noteq> 0"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   439
  shows "b dvd (ilcm a b)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   440
proof-
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   441
  let ?na = "nat (abs a)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   442
  let ?nb = "nat (abs b)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   443
  have nap: "?na >0" using anz by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   444
  have nbp: "?nb >0" using bnz by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   445
  from nap nbp have "?nb dvd lcm(?na,?nb)" using lcm_dvd2 by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   446
  thus ?thesis by (simp add: ilcm_def dvd_int_iff)
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   447
qed
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   448
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   449
lemma zdvd_self_abs1: "(d::int) dvd (abs d)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   450
by (case_tac "d <0", simp_all)
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   451
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   452
lemma zdvd_self_abs2: "(abs (d::int)) dvd d"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   453
by (case_tac "d<0", simp_all)
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   454
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   455
lemma zdvd_abs1: "((d::int) dvd t) = ((abs d) dvd t)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   456
 by (cases "d < 0") simp_all
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   457
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   458
(* lcm a b is positive for positive a and b *)
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   459
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   460
lemma lcm_pos: 
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   461
  assumes mpos: "m > 0"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   462
  and npos: "n>0"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   463
  shows "lcm (m,n) > 0"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   464
proof(rule ccontr, simp add: lcm_def gcd_zero)
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   465
assume h:"m*n div gcd(m,n) = 0"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   466
from mpos npos have "gcd (m,n) \<noteq> 0" using gcd_zero by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   467
hence gcdp: "gcd(m,n) > 0" by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   468
with h
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   469
have "m*n < gcd(m,n)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   470
  by (cases "m * n < gcd (m, n)") (auto simp add: div_if[OF gcdp, where m="m*n"])
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   471
moreover 
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   472
have "gcd(m,n) dvd m" by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   473
 with mpos dvd_imp_le have t1:"gcd(m,n) \<le> m" by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   474
 with npos have t1:"gcd(m,n)*n \<le> m*n" by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   475
 have "gcd(m,n) \<le> gcd(m,n)*n" using npos by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   476
 with t1 have "gcd(m,n) \<le> m*n" by arith
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   477
ultimately show "False" by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   478
qed
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   479
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   480
lemma ilcm_pos: 
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   481
  assumes apos: " 0 < a"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   482
  and bpos: "0 < b" 
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   483
  shows "0 < ilcm  a b"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   484
proof-
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   485
  let ?na = "nat (abs a)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   486
  let ?nb = "nat (abs b)"
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   487
  have nap: "?na >0" using apos by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   488
  have nbp: "?nb >0" using bpos by simp
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   489
  have "0 < lcm (?na,?nb)" by (rule lcm_pos[OF nap nbp])
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   490
  thus ?thesis by (simp add: ilcm_def)
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   491
qed
1630951f0512 added lcm, ilcm (lcm for integers) and some lemmas about them;
chaieb
parents: 22367
diff changeset
   492
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   493
end