src/HOL/Rings.thy
author haftmann
Mon, 26 Sep 2016 07:56:54 +0200
changeset 63947 559f0882d6a6
parent 63924 f91766530e13
child 63950 cdc1e59aa513
permissions -rw-r--r--
more lemmas
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(*  Title:      HOL/Rings.thy
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    Author:     Gertrud Bauer
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    Author:     Steven Obua
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Markus Wenzel
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    Author:     Jeremy Avigad
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*)
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section \<open>Rings\<close>
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theory Rings
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  imports Groups Set
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begin
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c"
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  assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
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begin
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text \<open>For the \<open>combine_numerals\<close> simproc\<close>
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lemma combine_common_factor: "a * e + (b * e + c) = (a + b) * e + c"
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  by (simp add: distrib_right ac_simps)
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end
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "0 * a = 0"
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  assumes mult_zero_right [simp]: "a * 0 = 0"
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begin
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lemma mult_not_zero: "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
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  by auto
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end
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0"
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    by (simp add: distrib_right [symmetric])
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  then show "0 * a = 0"
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    by (simp only: add_left_cancel)
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  have "a * 0 + a * 0 = a * 0 + 0"
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    by (simp add: distrib_left [symmetric])
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  then show "a * 0 = 0"
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    by (simp only: add_left_cancel)
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qed
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end
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a + b) * c = a * c + b * c"
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begin
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subclass semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c"
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    by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a"
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    by (simp add: ac_simps)
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  also have "\<dots> = b * a + c * a"
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    by (simp only: distrib)
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  also have "\<dots> = a * b + a * c"
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    by (simp add: ac_simps)
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  finally show "a * (b + c) = a * b + a * c"
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    by blast
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qed
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end
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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begin
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subclass semiring_0 ..
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end
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class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0_cancel ..
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subclass comm_semiring_0 ..
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end
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "0 \<noteq> 1"
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begin
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lemma one_neq_zero [simp]: "1 \<noteq> 0"
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  by (rule not_sym) (rule zero_neq_one)
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definition of_bool :: "bool \<Rightarrow> 'a"
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  where "of_bool p = (if p then 1 else 0)"
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lemma of_bool_eq [simp, code]:
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  "of_bool False = 0"
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  "of_bool True = 1"
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  by (simp_all add: of_bool_def)
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lemma of_bool_eq_iff: "of_bool p = of_bool q \<longleftrightarrow> p = q"
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  by (simp add: of_bool_def)
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lemma split_of_bool [split]: "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
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  by (cases p) simp_all
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lemma split_of_bool_asm: "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
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  by (cases p) simp_all
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24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
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end
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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text \<open>Abstract divisibility\<close>
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class dvd = times
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begin
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definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50)
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  where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
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lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
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  unfolding dvd_def ..
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding dvd_def by blast
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end
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context comm_monoid_mult
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begin
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subclass dvd .
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lemma dvd_refl [simp]: "a dvd a"
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proof
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  show "a = a * 1" by simp
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qed
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lemma dvd_trans [trans]:
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  assumes "a dvd b" and "b dvd c"
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  shows "a dvd c"
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proof -
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  from assms obtain v where "b = a * v"
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    by (auto elim!: dvdE)
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  moreover from assms obtain w where "c = b * w"
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    by (auto elim!: dvdE)
d0e2bad67bd4 misc tuning and modernization;
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   156
  ultimately have "c = a * (v * w)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   157
    by (simp add: mult.assoc)
28559
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   158
  then show ?thesis ..
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   159
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   160
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   161
lemma subset_divisors_dvd: "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
62366
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   162
  by (auto simp add: subset_iff intro: dvd_trans)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   163
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   164
lemma strict_subset_divisors_dvd: "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
62366
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   165
  by (auto simp add: subset_iff intro: dvd_trans)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   166
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   167
lemma one_dvd [simp]: "1 dvd a"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   168
  by (auto intro!: dvdI)
28559
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   169
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   170
lemma dvd_mult [simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   171
  by (auto intro!: mult.left_commute dvdI elim!: dvdE)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   172
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   173
lemma dvd_mult2 [simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   174
  using dvd_mult [of a b c] by (simp add: ac_simps)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   175
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   176
lemma dvd_triv_right [simp]: "a dvd b * a"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   177
  by (rule dvd_mult) (rule dvd_refl)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   178
63325
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   179
lemma dvd_triv_left [simp]: "a dvd a * b"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   180
  by (rule dvd_mult2) (rule dvd_refl)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   181
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   182
lemma mult_dvd_mono:
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   183
  assumes "a dvd b"
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   184
    and "c dvd d"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   185
  shows "a * c dvd b * d"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   186
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   187
  from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   188
  moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" ..
63588
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parents: 63456
diff changeset
   189
  ultimately have "b * d = (a * c) * (b' * d')"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   190
    by (simp add: ac_simps)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   191
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   192
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   193
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diff changeset
   194
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   195
  by (simp add: dvd_def mult.assoc) blast
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   196
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   197
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   198
  using dvd_mult_left [of b a c] by (simp add: ac_simps)
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   199
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   200
end
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   201
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   202
class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   203
begin
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   204
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   205
subclass semiring_1 ..
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   206
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   207
lemma dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   208
  by (auto intro: dvd_refl elim!: dvdE)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   209
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   210
lemma dvd_0_right [iff]: "a dvd 0"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   211
proof
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   212
  show "0 = a * 0" by simp
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   213
qed
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   214
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diff changeset
   215
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   216
  by simp
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   217
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   218
lemma dvd_add [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   219
  assumes "a dvd b" and "a dvd c"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   220
  shows "a dvd (b + c)"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   221
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   222
  from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   223
  moreover from \<open>a dvd c\<close> obtain c' where "c = a * c'" ..
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   224
  ultimately have "b + c = a * (b' + c')"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   225
    by (simp add: distrib_left)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   226
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   227
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   228
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   229
end
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   230
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   231
class semiring_1_cancel = semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   232
  + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   233
begin
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   234
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   235
subclass semiring_0_cancel ..
25512
4134f7c782e2 using intro_locales instead of unfold_locales if appropriate
haftmann
parents: 25450
diff changeset
   236
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   237
subclass semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   238
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   239
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   240
63325
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diff changeset
   241
class comm_semiring_1_cancel =
1086d56cde86 misc tuning and modernization;
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diff changeset
   242
  comm_semiring + cancel_comm_monoid_add + zero_neq_one + comm_monoid_mult +
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   243
  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   244
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   245
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   246
subclass semiring_1_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   247
subclass comm_semiring_0_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   248
subclass comm_semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   249
63325
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diff changeset
   250
lemma left_diff_distrib' [algebra_simps]: "(b - c) * a = b * a - c * a"
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   251
  by (simp add: algebra_simps)
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   252
63325
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diff changeset
   253
lemma dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b"
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   254
proof -
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   255
  have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   256
  proof
63325
1086d56cde86 misc tuning and modernization;
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parents: 63040
diff changeset
   257
    assume ?Q
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   258
    then show ?P by simp
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   259
  next
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   260
    assume ?P
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   261
    then obtain d where "a * c + b = a * d" ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   262
    then have "a * c + b - a * c = a * d - a * c" by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   263
    then have "b = a * d - a * c" by simp
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   264
    then have "b = a * (d - c)" by (simp add: algebra_simps)
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   265
    then show ?Q ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   266
  qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   267
  then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   268
qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   269
63325
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diff changeset
   270
lemma dvd_add_times_triv_right_iff [simp]: "a dvd b + c * a \<longleftrightarrow> a dvd b"
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   271
  using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   272
63325
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diff changeset
   273
lemma dvd_add_triv_left_iff [simp]: "a dvd a + b \<longleftrightarrow> a dvd b"
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   274
  using dvd_add_times_triv_left_iff [of a 1 b] by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   275
63325
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diff changeset
   276
lemma dvd_add_triv_right_iff [simp]: "a dvd b + a \<longleftrightarrow> a dvd b"
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   277
  using dvd_add_times_triv_right_iff [of a b 1] by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   278
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   279
lemma dvd_add_right_iff:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   280
  assumes "a dvd b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   281
  shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   282
proof
63325
1086d56cde86 misc tuning and modernization;
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parents: 63040
diff changeset
   283
  assume ?P
1086d56cde86 misc tuning and modernization;
wenzelm
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diff changeset
   284
  then obtain d where "b + c = a * d" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   285
  moreover from \<open>a dvd b\<close> obtain e where "b = a * e" ..
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   286
  ultimately have "a * e + c = a * d" by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   287
  then have "a * e + c - a * e = a * d - a * e" by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   288
  then have "c = a * d - a * e" by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   289
  then have "c = a * (d - e)" by (simp add: algebra_simps)
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   290
  then show ?Q ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   291
next
63325
1086d56cde86 misc tuning and modernization;
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parents: 63040
diff changeset
   292
  assume ?Q
1086d56cde86 misc tuning and modernization;
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parents: 63040
diff changeset
   293
  with assms show ?P by simp
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   294
qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   295
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   296
lemma dvd_add_left_iff: "a dvd c \<Longrightarrow> a dvd b + c \<longleftrightarrow> a dvd b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   297
  using dvd_add_right_iff [of a c b] by (simp add: ac_simps)
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   298
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   299
end
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   300
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   301
class ring = semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   302
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   303
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   304
subclass semiring_0_cancel ..
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   305
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   306
text \<open>Distribution rules\<close>
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   307
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   308
lemma minus_mult_left: "- (a * b) = - a * b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   309
  by (rule minus_unique) (simp add: distrib_right [symmetric])
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   310
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   311
lemma minus_mult_right: "- (a * b) = a * - b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   312
  by (rule minus_unique) (simp add: distrib_left [symmetric])
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   313
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   314
text \<open>Extract signs from products\<close>
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   315
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   316
lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
29407
5ef7e97fd9e4 move lemmas mult_minus{left,right} inside class ring
huffman
parents: 29406
diff changeset
   317
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   318
lemma minus_mult_minus [simp]: "- a * - b = a * b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   319
  by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   320
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   321
lemma minus_mult_commute: "- a * b = a * - b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   322
  by simp
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   323
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   324
lemma right_diff_distrib [algebra_simps]: "a * (b - c) = a * b - a * c"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   325
  using distrib_left [of a b "-c "] by simp
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   326
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   327
lemma left_diff_distrib [algebra_simps]: "(a - b) * c = a * c - b * c"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   328
  using distrib_right [of a "- b" c] by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   329
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   330
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   331
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   332
lemma eq_add_iff1: "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   333
  by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   334
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   335
lemma eq_add_iff2: "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   336
  by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   337
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   338
end
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   339
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   340
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   341
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   342
class comm_ring = comm_semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   343
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   344
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   345
subclass ring ..
28141
193c3ea0f63b instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents: 27651
diff changeset
   346
subclass comm_semiring_0_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   347
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   348
lemma square_diff_square_factored: "x * x - y * y = (x + y) * (x - y)"
44350
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   349
  by (simp add: algebra_simps)
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   350
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   351
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   352
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   353
class ring_1 = ring + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   354
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   355
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   356
subclass semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   357
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   358
lemma square_diff_one_factored: "x * x - 1 = (x + 1) * (x - 1)"
44346
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   359
  by (simp add: algebra_simps)
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   360
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   361
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   362
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   363
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   364
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   365
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   366
subclass ring_1 ..
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   367
subclass comm_semiring_1_cancel
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   368
  by unfold_locales (simp add: algebra_simps)
58647
fce800afeec7 more facts about abstract divisibility
haftmann
parents: 58198
diff changeset
   369
29465
b2cfb5d0a59e change dvd_minus_iff, minus_dvd_iff from [iff] to [simp] (due to problems with Library/Primes.thy)
huffman
parents: 29461
diff changeset
   370
lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
29408
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   371
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   372
  assume "x dvd - y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   373
  then have "x dvd - 1 * - y" by (rule dvd_mult)
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   374
  then show "x dvd y" by simp
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   375
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   376
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   377
  then have "x dvd - 1 * y" by (rule dvd_mult)
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   378
  then show "x dvd - y" by simp
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   379
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   380
29465
b2cfb5d0a59e change dvd_minus_iff, minus_dvd_iff from [iff] to [simp] (due to problems with Library/Primes.thy)
huffman
parents: 29461
diff changeset
   381
lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
29408
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   382
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   383
  assume "- x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   384
  then obtain k where "y = - x * k" ..
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   385
  then have "y = x * - k" by simp
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   386
  then show "x dvd y" ..
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   387
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   388
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   389
  then obtain k where "y = x * k" ..
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   390
  then have "y = - x * - k" by simp
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   391
  then show "- x dvd y" ..
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   392
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   393
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   394
lemma dvd_diff [simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   395
  using dvd_add [of x y "- z"] by simp
29409
f0a8fe83bc07 add lemma dvd_diff to class comm_ring_1
huffman
parents: 29408
diff changeset
   396
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   397
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   398
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   399
class semiring_no_zero_divisors = semiring_0 +
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   400
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   401
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   402
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   403
lemma divisors_zero:
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   404
  assumes "a * b = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   405
  shows "a = 0 \<or> b = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   406
proof (rule classical)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   407
  assume "\<not> ?thesis"
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   408
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   409
  with no_zero_divisors have "a * b \<noteq> 0" by blast
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   410
  with assms show ?thesis by simp
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   411
qed
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   412
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   413
lemma mult_eq_0_iff [simp]: "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   414
proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   415
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   416
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   417
    then show ?thesis using no_zero_divisors by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   418
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   419
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   420
  then show ?thesis by auto
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   421
qed
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   422
58952
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   423
end
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   424
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   425
class semiring_1_no_zero_divisors = semiring_1 + semiring_no_zero_divisors
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   426
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   427
class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   428
  assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   429
    and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
58952
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   430
begin
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   431
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   432
lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   433
  by simp
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   434
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   435
lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   436
  by simp
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   437
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   438
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   439
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   440
class ring_no_zero_divisors = ring + semiring_no_zero_divisors
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   441
begin
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   442
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   443
subclass semiring_no_zero_divisors_cancel
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   444
proof
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   445
  fix a b c
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   446
  have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   447
    by (simp add: algebra_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   448
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   449
    by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   450
  finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   451
  have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   452
    by (simp add: algebra_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   453
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   454
    by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   455
  finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   456
qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   457
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   458
end
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   459
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   460
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   461
begin
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   462
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   463
subclass semiring_1_no_zero_divisors ..
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   464
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   465
lemma square_eq_1_iff: "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
36821
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   466
proof -
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   467
  have "(x - 1) * (x + 1) = x * x - 1"
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   468
    by (simp add: algebra_simps)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   469
  then have "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
36821
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   470
    by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   471
  then show ?thesis
36821
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   472
    by (simp add: eq_neg_iff_add_eq_0)
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   473
qed
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   474
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   475
lemma mult_cancel_right1 [simp]: "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   476
  using mult_cancel_right [of 1 c b] by auto
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   477
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   478
lemma mult_cancel_right2 [simp]: "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   479
  using mult_cancel_right [of a c 1] by simp
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   480
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   481
lemma mult_cancel_left1 [simp]: "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   482
  using mult_cancel_left [of c 1 b] by force
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   483
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   484
lemma mult_cancel_left2 [simp]: "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   485
  using mult_cancel_left [of c a 1] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   486
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   487
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   488
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   489
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   490
begin
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   491
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   492
subclass semiring_1_no_zero_divisors ..
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   493
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   494
end
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   495
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   496
class idom = comm_ring_1 + semiring_no_zero_divisors
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   497
begin
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   498
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   499
subclass semidom ..
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   500
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   501
subclass ring_1_no_zero_divisors ..
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   502
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   503
lemma dvd_mult_cancel_right [simp]: "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   504
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   505
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   506
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   507
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   508
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   509
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   510
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   511
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   512
lemma dvd_mult_cancel_left [simp]: "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   513
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   514
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   515
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   516
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   517
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   518
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   519
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   520
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   521
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   522
proof
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   523
  assume "a * a = b * b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   524
  then have "(a - b) * (a + b) = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   525
    by (simp add: algebra_simps)
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   526
  then show "a = b \<or> a = - b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   527
    by (simp add: eq_neg_iff_add_eq_0)
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   528
next
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   529
  assume "a = b \<or> a = - b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   530
  then show "a * a = b * b" by auto
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   531
qed
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   532
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   533
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   534
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   535
text \<open>
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   536
  The theory of partially ordered rings is taken from the books:
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   537
    \<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   538
    \<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   539
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   540
  Most of the used notions can also be looked up in
63680
6e1e8b5abbfa more symbols;
wenzelm
parents: 63588
diff changeset
   541
    \<^item> \<^url>\<open>http://www.mathworld.com\<close> by Eric Weisstein et. al.
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   542
    \<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   543
\<close>
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   544
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   545
class divide =
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   546
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   547
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   548
setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   549
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   550
context semiring
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   551
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   552
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   553
lemma [field_simps]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   554
  shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   555
    and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   556
  by (rule distrib_left distrib_right)+
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   557
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   558
end
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   559
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   560
context ring
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   561
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   562
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   563
lemma [field_simps]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   564
  shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   565
    and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   566
  by (rule left_diff_distrib right_diff_distrib)+
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   567
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   568
end
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   569
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   570
setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   571
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   572
class semidom_divide = semidom + divide +
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   573
  assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   574
  assumes divide_zero [simp]: "a div 0 = 0"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   575
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   576
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   577
lemma nonzero_mult_divide_cancel_left [simp]: "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   578
  using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   579
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   580
subclass semiring_no_zero_divisors_cancel
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   581
proof
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   582
  show *: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" for a b c
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   583
  proof (cases "c = 0")
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   584
    case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   585
    then show ?thesis by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   586
  next
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   587
    case False
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   588
    have "a = b" if "a * c = b * c"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   589
    proof -
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   590
      from that have "a * c div c = b * c div c"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   591
        by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   592
      with False show ?thesis
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   593
        by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   594
    qed
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   595
    then show ?thesis by auto
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   596
  qed
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   597
  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" for a b c
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   598
    using * [of a c b] by (simp add: ac_simps)
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   599
qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   600
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   601
lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   602
  using nonzero_mult_divide_cancel_left [of a 1] by simp
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   603
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   604
lemma divide_zero_left [simp]: "0 div a = 0"
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   605
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   606
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   607
  then show ?thesis by simp
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   608
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   609
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   610
  then have "a * 0 div a = 0"
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   611
    by (rule nonzero_mult_divide_cancel_left)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   612
  then show ?thesis by simp
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   613
qed
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   614
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   615
lemma divide_1 [simp]: "a div 1 = a"
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   616
  using nonzero_mult_divide_cancel_left [of 1 a] by simp
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   617
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   618
end
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   619
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   620
class idom_divide = idom + semidom_divide
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   621
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   622
class algebraic_semidom = semidom_divide
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   623
begin
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   624
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   625
text \<open>
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   626
  Class @{class algebraic_semidom} enriches a integral domain
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   627
  by notions from algebra, like units in a ring.
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   628
  It is a separate class to avoid spoiling fields with notions
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   629
  which are degenerated there.
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   630
\<close>
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   631
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   632
lemma dvd_times_left_cancel_iff [simp]:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   633
  assumes "a \<noteq> 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   634
  shows "a * b dvd a * c \<longleftrightarrow> b dvd c"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   635
    (is "?lhs \<longleftrightarrow> ?rhs")
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   636
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   637
  assume ?lhs
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   638
  then obtain d where "a * c = a * b * d" ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   639
  with assms have "c = b * d" by (simp add: ac_simps)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   640
  then show ?rhs ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   641
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   642
  assume ?rhs
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   643
  then obtain d where "c = b * d" ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   644
  then have "a * c = a * b * d" by (simp add: ac_simps)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   645
  then show ?lhs ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   646
qed
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   647
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   648
lemma dvd_times_right_cancel_iff [simp]:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   649
  assumes "a \<noteq> 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   650
  shows "b * a dvd c * a \<longleftrightarrow> b dvd c"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   651
  using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   652
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   653
lemma div_dvd_iff_mult:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   654
  assumes "b \<noteq> 0" and "b dvd a"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   655
  shows "a div b dvd c \<longleftrightarrow> a dvd c * b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   656
proof -
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   657
  from \<open>b dvd a\<close> obtain d where "a = b * d" ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   658
  with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps)
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   659
qed
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   660
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   661
lemma dvd_div_iff_mult:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   662
  assumes "c \<noteq> 0" and "c dvd b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   663
  shows "a dvd b div c \<longleftrightarrow> a * c dvd b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   664
proof -
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   665
  from \<open>c dvd b\<close> obtain d where "b = c * d" ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   666
  with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a])
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   667
qed
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   668
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   669
lemma div_dvd_div [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   670
  assumes "a dvd b" and "a dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   671
  shows "b div a dvd c div a \<longleftrightarrow> b dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   672
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   673
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   674
  with assms show ?thesis by simp
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   675
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   676
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   677
  moreover from assms obtain k l where "b = a * k" and "c = a * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   678
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   679
  ultimately show ?thesis by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   680
qed
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   681
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   682
lemma div_add [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   683
  assumes "c dvd a" and "c dvd b"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   684
  shows "(a + b) div c = a div c + b div c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   685
proof (cases "c = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   686
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   687
  then show ?thesis by simp
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   688
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   689
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   690
  moreover from assms obtain k l where "a = c * k" and "b = c * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   691
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   692
  moreover have "c * k + c * l = c * (k + l)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   693
    by (simp add: algebra_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   694
  ultimately show ?thesis
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   695
    by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   696
qed
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   697
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   698
lemma div_mult_div_if_dvd:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   699
  assumes "b dvd a" and "d dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   700
  shows "(a div b) * (c div d) = (a * c) div (b * d)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   701
proof (cases "b = 0 \<or> c = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   702
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   703
  with assms show ?thesis by auto
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   704
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   705
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   706
  moreover from assms obtain k l where "a = b * k" and "c = d * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   707
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   708
  moreover have "b * k * (d * l) div (b * d) = (b * d) * (k * l) div (b * d)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   709
    by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   710
  ultimately show ?thesis by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   711
qed
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   712
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   713
lemma dvd_div_eq_mult:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   714
  assumes "a \<noteq> 0" and "a dvd b"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   715
  shows "b div a = c \<longleftrightarrow> b = c * a"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   716
    (is "?lhs \<longleftrightarrow> ?rhs")
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   717
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   718
  assume ?rhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   719
  then show ?lhs by (simp add: assms)
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   720
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   721
  assume ?lhs
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   722
  then have "b div a * a = c * a" by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   723
  moreover from assms have "b div a * a = b"
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   724
    by (auto elim!: dvdE simp add: ac_simps)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   725
  ultimately show ?rhs by simp
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   726
qed
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   727
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   728
lemma dvd_div_mult_self [simp]: "a dvd b \<Longrightarrow> b div a * a = b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   729
  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   730
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   731
lemma dvd_mult_div_cancel [simp]: "a dvd b \<Longrightarrow> a * (b div a) = b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   732
  using dvd_div_mult_self [of a b] by (simp add: ac_simps)
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   733
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   734
lemma div_mult_swap:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   735
  assumes "c dvd b"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   736
  shows "a * (b div c) = (a * b) div c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   737
proof (cases "c = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   738
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   739
  then show ?thesis by simp
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   740
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   741
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   742
  from assms obtain d where "b = c * d" ..
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   743
  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   744
    by simp
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   745
  ultimately show ?thesis by (simp add: ac_simps)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   746
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   747
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   748
lemma dvd_div_mult: "c dvd b \<Longrightarrow> b div c * a = (b * a) div c"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   749
  using div_mult_swap [of c b a] by (simp add: ac_simps)
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   750
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   751
lemma dvd_div_mult2_eq:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   752
  assumes "b * c dvd a"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   753
  shows "a div (b * c) = a div b div c"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   754
proof -
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   755
  from assms obtain k where "a = b * c * k" ..
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   756
  then show ?thesis
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   757
    by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   758
qed
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   759
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   760
lemma dvd_div_div_eq_mult:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   761
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   762
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   763
    (is "?lhs \<longleftrightarrow> ?rhs")
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   764
proof -
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   765
  from assms have "a * c \<noteq> 0" by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   766
  then have "?lhs \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)"
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   767
    by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   768
  also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   769
    by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   770
  also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   771
    using assms by (simp add: div_mult_swap)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   772
  also have "\<dots> \<longleftrightarrow> ?rhs"
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   773
    using assms by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   774
  finally show ?thesis .
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   775
qed
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   776
63359
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   777
lemma dvd_mult_imp_div:
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   778
  assumes "a * c dvd b"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   779
  shows "a dvd b div c"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   780
proof (cases "c = 0")
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   781
  case True then show ?thesis by simp
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   782
next
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   783
  case False
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   784
  from \<open>a * c dvd b\<close> obtain d where "b = a * c * d" ..
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   785
  with False show ?thesis
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   786
    by (simp add: mult.commute [of a] mult.assoc)
63359
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   787
qed
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   788
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   789
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   790
text \<open>Units: invertible elements in a ring\<close>
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   791
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   792
abbreviation is_unit :: "'a \<Rightarrow> bool"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   793
  where "is_unit a \<equiv> a dvd 1"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   794
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   795
lemma not_is_unit_0 [simp]: "\<not> is_unit 0"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   796
  by simp
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   797
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   798
lemma unit_imp_dvd [dest]: "is_unit b \<Longrightarrow> b dvd a"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   799
  by (rule dvd_trans [of _ 1]) simp_all
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   800
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   801
lemma unit_dvdE:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   802
  assumes "is_unit a"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   803
  obtains c where "a \<noteq> 0" and "b = a * c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   804
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   805
  from assms have "a dvd b" by auto
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   806
  then obtain c where "b = a * c" ..
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   807
  moreover from assms have "a \<noteq> 0" by auto
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   808
  ultimately show thesis using that by blast
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   809
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   810
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   811
lemma dvd_unit_imp_unit: "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   812
  by (rule dvd_trans)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   813
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   814
lemma unit_div_1_unit [simp, intro]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   815
  assumes "is_unit a"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   816
  shows "is_unit (1 div a)"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   817
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   818
  from assms have "1 = 1 div a * a" by simp
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   819
  then show "is_unit (1 div a)" by (rule dvdI)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   820
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   821
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   822
lemma is_unitE [elim?]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   823
  assumes "is_unit a"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   824
  obtains b where "a \<noteq> 0" and "b \<noteq> 0"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   825
    and "is_unit b" and "1 div a = b" and "1 div b = a"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   826
    and "a * b = 1" and "c div a = c * b"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   827
proof (rule that)
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62626
diff changeset
   828
  define b where "b = 1 div a"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   829
  then show "1 div a = b" by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   830
  from assms b_def show "is_unit b" by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   831
  with assms show "a \<noteq> 0" and "b \<noteq> 0" by auto
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   832
  from assms b_def show "a * b = 1" by simp
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   833
  then have "1 = a * b" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   834
  with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   835
  from assms have "a dvd c" ..
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   836
  then obtain d where "c = a * d" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   837
  with \<open>a \<noteq> 0\<close> \<open>a * b = 1\<close> show "c div a = c * b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   838
    by (simp add: mult.assoc mult.left_commute [of a])
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   839
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   840
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   841
lemma unit_prod [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   842
  by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   843
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   844
lemma is_unit_mult_iff: "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b"
62366
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   845
  by (auto dest: dvd_mult_left dvd_mult_right)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   846
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   847
lemma unit_div [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   848
  by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   849
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   850
lemma mult_unit_dvd_iff:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   851
  assumes "is_unit b"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   852
  shows "a * b dvd c \<longleftrightarrow> a dvd c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   853
proof
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   854
  assume "a * b dvd c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   855
  with assms show "a dvd c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   856
    by (simp add: dvd_mult_left)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   857
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   858
  assume "a dvd c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   859
  then obtain k where "c = a * k" ..
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   860
  with assms have "c = (a * b) * (1 div b * k)"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   861
    by (simp add: mult_ac)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   862
  then show "a * b dvd c" by (rule dvdI)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   863
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   864
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
   865
lemma mult_unit_dvd_iff': "is_unit a \<Longrightarrow> (a * b) dvd c \<longleftrightarrow> b dvd c"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
   866
  using mult_unit_dvd_iff [of a b c] by (simp add: ac_simps)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
   867
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   868
lemma dvd_mult_unit_iff:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   869
  assumes "is_unit b"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   870
  shows "a dvd c * b \<longleftrightarrow> a dvd c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   871
proof
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   872
  assume "a dvd c * b"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   873
  with assms have "c * b dvd c * (b * (1 div b))"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   874
    by (subst mult_assoc [symmetric]) simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   875
  also from assms have "b * (1 div b) = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   876
    by (rule is_unitE) simp
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   877
  finally have "c * b dvd c" by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   878
  with \<open>a dvd c * b\<close> show "a dvd c" by (rule dvd_trans)
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   879
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   880
  assume "a dvd c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   881
  then show "a dvd c * b" by simp
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   882
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   883
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
   884
lemma dvd_mult_unit_iff': "is_unit b \<Longrightarrow> a dvd b * c \<longleftrightarrow> a dvd c"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
   885
  using dvd_mult_unit_iff [of b a c] by (simp add: ac_simps)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
   886
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   887
lemma div_unit_dvd_iff: "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   888
  by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   889
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   890
lemma dvd_div_unit_iff: "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   891
  by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   892
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
   893
lemmas unit_dvd_iff = mult_unit_dvd_iff mult_unit_dvd_iff'
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
   894
  dvd_mult_unit_iff dvd_mult_unit_iff' 
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
   895
  div_unit_dvd_iff dvd_div_unit_iff (* FIXME consider named_theorems *)
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   896
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   897
lemma unit_mult_div_div [simp]: "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   898
  by (erule is_unitE [of _ b]) simp
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   899
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   900
lemma unit_div_mult_self [simp]: "is_unit a \<Longrightarrow> b div a * a = b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   901
  by (rule dvd_div_mult_self) auto
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   902
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   903
lemma unit_div_1_div_1 [simp]: "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   904
  by (erule is_unitE) simp
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   905
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   906
lemma unit_div_mult_swap: "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   907
  by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   908
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   909
lemma unit_div_commute: "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   910
  using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   911
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   912
lemma unit_eq_div1: "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   913
  by (auto elim: is_unitE)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   914
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   915
lemma unit_eq_div2: "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   916
  using unit_eq_div1 [of b c a] by auto
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   917
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   918
lemma unit_mult_left_cancel: "is_unit a \<Longrightarrow> a * b = a * c \<longleftrightarrow> b = c"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   919
  using mult_cancel_left [of a b c] by auto
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   920
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   921
lemma unit_mult_right_cancel: "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   922
  using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   923
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   924
lemma unit_div_cancel:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   925
  assumes "is_unit a"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   926
  shows "b div a = c div a \<longleftrightarrow> b = c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   927
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   928
  from assms have "is_unit (1 div a)" by simp
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   929
  then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   930
    by (rule unit_mult_right_cancel)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   931
  with assms show ?thesis by simp
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   932
qed
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   933
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   934
lemma is_unit_div_mult2_eq:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   935
  assumes "is_unit b" and "is_unit c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   936
  shows "a div (b * c) = a div b div c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   937
proof -
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   938
  from assms have "is_unit (b * c)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   939
    by (simp add: unit_prod)
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   940
  then have "b * c dvd a"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   941
    by (rule unit_imp_dvd)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   942
  then show ?thesis
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   943
    by (rule dvd_div_mult2_eq)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   944
qed
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   945
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   946
lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   947
  dvd_div_unit_iff unit_div_mult_swap unit_div_commute
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   948
  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   949
  unit_eq_div1 unit_eq_div2
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   950
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   951
lemma is_unit_divide_mult_cancel_left:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   952
  assumes "a \<noteq> 0" and "is_unit b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   953
  shows "a div (a * b) = 1 div b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   954
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   955
  from assms have "a div (a * b) = a div a div b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   956
    by (simp add: mult_unit_dvd_iff dvd_div_mult2_eq)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   957
  with assms show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   958
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   959
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   960
lemma is_unit_divide_mult_cancel_right:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   961
  assumes "a \<noteq> 0" and "is_unit b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   962
  shows "a div (b * a) = 1 div b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   963
  using assms is_unit_divide_mult_cancel_left [of a b] by (simp add: ac_simps)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   964
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   965
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   966
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   967
class normalization_semidom = algebraic_semidom +
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   968
  fixes normalize :: "'a \<Rightarrow> 'a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   969
    and unit_factor :: "'a \<Rightarrow> 'a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   970
  assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   971
    and normalize_0 [simp]: "normalize 0 = 0"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   972
    and unit_factor_0 [simp]: "unit_factor 0 = 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   973
    and is_unit_normalize: "is_unit a  \<Longrightarrow> normalize a = 1"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   974
    and unit_factor_is_unit [iff]: "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   975
    and unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   976
begin
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   977
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   978
text \<open>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   979
  Class @{class normalization_semidom} cultivates the idea that each integral
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   980
  domain can be split into equivalence classes whose representants are
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   981
  associated, i.e. divide each other. @{const normalize} specifies a canonical
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   982
  representant for each equivalence class. The rationale behind this is that
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   983
  it is easier to reason about equality than equivalences, hence we prefer to
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   984
  think about equality of normalized values rather than associated elements.
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   985
\<close>
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   986
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   987
lemma unit_factor_dvd [simp]: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   988
  by (rule unit_imp_dvd) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   989
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   990
lemma unit_factor_self [simp]: "unit_factor a dvd a"
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   991
  by (cases "a = 0") simp_all
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   992
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   993
lemma normalize_mult_unit_factor [simp]: "normalize a * unit_factor a = a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   994
  using unit_factor_mult_normalize [of a] by (simp add: ac_simps)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   995
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   996
lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   997
  (is "?lhs \<longleftrightarrow> ?rhs")
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   998
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   999
  assume ?lhs
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1000
  moreover have "unit_factor a * normalize a = a" by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1001
  ultimately show ?rhs by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1002
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1003
  assume ?rhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1004
  then show ?lhs by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1005
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1006
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1007
lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1008
  (is "?lhs \<longleftrightarrow> ?rhs")
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1009
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1010
  assume ?lhs
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1011
  moreover have "unit_factor a * normalize a = a" by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1012
  ultimately show ?rhs by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1013
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1014
  assume ?rhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1015
  then show ?lhs by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1016
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1017
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1018
lemma is_unit_unit_factor:
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1019
  assumes "is_unit a"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1020
  shows "unit_factor a = a"
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
  1021
proof -
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1022
  from assms have "normalize a = 1" by (rule is_unit_normalize)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1023
  moreover from unit_factor_mult_normalize have "unit_factor a * normalize a = a" .
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1024
  ultimately show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1025
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1026
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1027
lemma unit_factor_1 [simp]: "unit_factor 1 = 1"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1028
  by (rule is_unit_unit_factor) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1029
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1030
lemma normalize_1 [simp]: "normalize 1 = 1"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1031
  by (rule is_unit_normalize) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1032
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1033
lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1034
  (is "?lhs \<longleftrightarrow> ?rhs")
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1035
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1036
  assume ?rhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1037
  then show ?lhs by (rule is_unit_normalize)
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1038
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1039
  assume ?lhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1040
  then have "unit_factor a * normalize a = unit_factor a * 1"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1041
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1042
  then have "unit_factor a = a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1043
    by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1044
  moreover
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1045
  from \<open>?lhs\<close> have "a \<noteq> 0" by auto
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1046
  then have "is_unit (unit_factor a)" by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1047
  ultimately show ?rhs by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1048
qed
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
  1049
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1050
lemma div_normalize [simp]: "a div normalize a = unit_factor a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1051
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1052
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1053
  then show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1054
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1055
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1056
  then have "normalize a \<noteq> 0" by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1057
  with nonzero_mult_divide_cancel_right
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1058
  have "unit_factor a * normalize a div normalize a = unit_factor a" by blast
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1059
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1060
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1061
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1062
lemma div_unit_factor [simp]: "a div unit_factor a = normalize a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1063
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1064
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1065
  then show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1066
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1067
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1068
  then have "unit_factor a \<noteq> 0" by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1069
  with nonzero_mult_divide_cancel_left
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1070
  have "unit_factor a * normalize a div unit_factor a = normalize a"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1071
    by blast
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1072
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1073
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1074
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1075
lemma normalize_div [simp]: "normalize a div a = 1 div unit_factor a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1076
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1077
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1078
  then show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1079
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1080
  case False
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1081
  have "normalize a div a = normalize a div (unit_factor a * normalize a)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1082
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1083
  also have "\<dots> = 1 div unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1084
    using False by (subst is_unit_divide_mult_cancel_right) simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1085
  finally show ?thesis .
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1086
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1087
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1088
lemma mult_one_div_unit_factor [simp]: "a * (1 div unit_factor b) = a div unit_factor b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1089
  by (cases "b = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1090
63947
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1091
lemma inv_unit_factor_eq_0_iff [simp]:
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1092
  "1 div unit_factor a = 0 \<longleftrightarrow> a = 0"
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1093
  (is "?lhs \<longleftrightarrow> ?rhs")
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1094
proof
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1095
  assume ?lhs
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1096
  then have "a * (1 div unit_factor a) = a * 0"
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1097
    by simp
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1098
  then show ?rhs
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1099
    by simp
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1100
next
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1101
  assume ?rhs
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1102
  then show ?lhs by simp
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1103
qed
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1104
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1105
lemma normalize_mult: "normalize (a * b) = normalize a * normalize b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1106
proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1107
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1108
  then show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1109
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1110
  case False
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1111
  have "unit_factor (a * b) * normalize (a * b) = a * b"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1112
    by (rule unit_factor_mult_normalize)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1113
  then have "normalize (a * b) = a * b div unit_factor (a * b)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1114
    by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1115
  also have "\<dots> = a * b div unit_factor (b * a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1116
    by (simp add: ac_simps)
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1117
  also have "\<dots> = a * b div unit_factor b div unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1118
    using False by (simp add: unit_factor_mult is_unit_div_mult2_eq [symmetric])
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1119
  also have "\<dots> = a * (b div unit_factor b) div unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1120
    using False by (subst unit_div_mult_swap) simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1121
  also have "\<dots> = normalize a * normalize b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1122
    using False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1123
    by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1124
  finally show ?thesis .
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1125
qed
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
  1126
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1127
lemma unit_factor_idem [simp]: "unit_factor (unit_factor a) = unit_factor a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1128
  by (cases "a = 0") (auto intro: is_unit_unit_factor)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1129
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1130
lemma normalize_unit_factor [simp]: "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1131
  by (rule is_unit_normalize) simp
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
  1132
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1133
lemma normalize_idem [simp]: "normalize (normalize a) = normalize a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1134
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1135
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1136
  then show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1137
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1138
  case False
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1139
  have "normalize a = normalize (unit_factor a * normalize a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1140
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1141
  also have "\<dots> = normalize (unit_factor a) * normalize (normalize a)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1142
    by (simp only: normalize_mult)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1143
  finally show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1144
    using False by simp_all
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1145
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1146
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1147
lemma unit_factor_normalize [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1148
  assumes "a \<noteq> 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1149
  shows "unit_factor (normalize a) = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1150
proof -
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1151
  from assms have *: "normalize a \<noteq> 0"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1152
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1153
  have "unit_factor (normalize a) * normalize (normalize a) = normalize a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1154
    by (simp only: unit_factor_mult_normalize)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1155
  then have "unit_factor (normalize a) * normalize a = normalize a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1156
    by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1157
  with * have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1158
    by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1159
  with * show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1160
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1161
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1162
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1163
lemma dvd_unit_factor_div:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1164
  assumes "b dvd a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1165
  shows "unit_factor (a div b) = unit_factor a div unit_factor b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1166
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1167
  from assms have "a = a div b * b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1168
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1169
  then have "unit_factor a = unit_factor (a div b * b)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1170
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1171
  then show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1172
    by (cases "b = 0") (simp_all add: unit_factor_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1173
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1174
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1175
lemma dvd_normalize_div:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1176
  assumes "b dvd a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1177
  shows "normalize (a div b) = normalize a div normalize b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1178
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1179
  from assms have "a = a div b * b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1180
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1181
  then have "normalize a = normalize (a div b * b)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1182
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1183
  then show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1184
    by (cases "b = 0") (simp_all add: normalize_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1185
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1186
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1187
lemma normalize_dvd_iff [simp]: "normalize a dvd b \<longleftrightarrow> a dvd b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1188
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1189
  have "normalize a dvd b \<longleftrightarrow> unit_factor a * normalize a dvd b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1190
    using mult_unit_dvd_iff [of "unit_factor a" "normalize a" b]
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1191
      by (cases "a = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1192
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1193
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1194
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1195
lemma dvd_normalize_iff [simp]: "a dvd normalize b \<longleftrightarrow> a dvd b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1196
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1197
  have "a dvd normalize  b \<longleftrightarrow> a dvd normalize b * unit_factor b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1198
    using dvd_mult_unit_iff [of "unit_factor b" a "normalize b"]
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1199
      by (cases "b = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1200
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1201
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1202
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1203
text \<open>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1204
  We avoid an explicit definition of associated elements but prefer explicit
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1205
  normalisation instead. In theory we could define an abbreviation like @{prop
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1206
  "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is counterproductive
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1207
  without suggestive infix syntax, which we do not want to sacrifice for this
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1208
  purpose here.
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1209
\<close>
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1210
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1211
lemma associatedI:
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1212
  assumes "a dvd b" and "b dvd a"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1213
  shows "normalize a = normalize b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1214
proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1215
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1216
  with assms show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1217
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1218
  case False
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1219
  from \<open>a dvd b\<close> obtain c where b: "b = a * c" ..
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1220
  moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1221
  ultimately have "b * 1 = b * (c * d)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1222
    by (simp add: ac_simps)
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1223
  with False have "1 = c * d"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1224
    unfolding mult_cancel_left by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1225
  then have "is_unit c" and "is_unit d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1226
    by auto
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1227
  with a b show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1228
    by (simp add: normalize_mult is_unit_normalize)
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1229
qed
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1230
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1231
lemma associatedD1: "normalize a = normalize b \<Longrightarrow> a dvd b"
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1232
  using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric]
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1233
  by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1234
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1235
lemma associatedD2: "normalize a = normalize b \<Longrightarrow> b dvd a"
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1236
  using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric]
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1237
  by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1238
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1239
lemma associated_unit: "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1240
  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1241
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1242
lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1243
  (is "?lhs \<longleftrightarrow> ?rhs")
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1244
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1245
  assume ?rhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1246
  then show ?lhs by (auto intro!: associatedI)
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1247
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1248
  assume ?lhs
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1249
  then have "unit_factor a * normalize a = unit_factor a * normalize b"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1250
    by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1251
  then have *: "normalize b * unit_factor a = a"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1252
    by (simp add: ac_simps)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1253
  show ?rhs
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1254
  proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1255
    case True
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1256
    with \<open>?lhs\<close> show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1257
  next
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
  1258
    case False
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1259
    then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1260
      by (simp_all add: mult_unit_dvd_iff dvd_mult_unit_iff)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1261
    with * show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1262
  qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1263
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1264
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1265
lemma associated_eqI:
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1266
  assumes "a dvd b" and "b dvd a"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1267
  assumes "normalize a = a" and "normalize b = b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1268
  shows "a = b"
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1269
proof -
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1270
  from assms have "normalize a = normalize b"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1271
    unfolding associated_iff_dvd by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1272
  with \<open>normalize a = a\<close> have "a = normalize b"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1273
    by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1274
  with \<open>normalize b = b\<close> show "a = b"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1275
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1276
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1277
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
ha