src/HOL/Rings.thy
author haftmann
Tue, 01 Mar 2016 10:36:19 +0100
changeset 62481 b5d8e57826df
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child 62608 19f87fa0cfcb
permissions -rw-r--r--
tuned bootstrap order to provide type classes in a more sensible order
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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:
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  "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:
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  "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" by (simp add: distrib_right [symmetric])
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  thus "0 * a = 0" by (simp only: add_left_cancel)
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next
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  fix a :: 'a
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  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
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  thus "a * 0 = 0" 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" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: ac_simps)
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  finally show "a * (b + c) = a * b + a * c" 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
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  "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:
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  "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]:
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  "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:
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  "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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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) where
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  "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]:
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  "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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   152
  from assms obtain v where "b = a * v" by (auto elim!: dvdE)
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   153
  moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56544
diff changeset
   154
  ultimately have "c = a * (v * w)" by (simp add: mult.assoc)
28559
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   155
  then show ?thesis ..
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   156
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   157
62366
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   158
lemma subset_divisors_dvd:
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   159
  "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   160
  by (auto simp add: subset_iff intro: dvd_trans)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   161
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   162
lemma strict_subset_divisors_dvd:
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   163
  "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   164
  by (auto simp add: subset_iff intro: dvd_trans)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   165
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   166
lemma one_dvd [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   167
  "1 dvd a"
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
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   170
lemma dvd_mult [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   171
  "a dvd c \<Longrightarrow> a dvd (b * c)"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   172
  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
   173
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   174
lemma dvd_mult2 [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   175
  "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
   176
  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
   177
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   178
lemma dvd_triv_right [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   179
  "a dvd b * a"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   180
  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
   181
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   182
lemma dvd_triv_left [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   183
  "a dvd a * b"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   184
  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
   185
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   186
lemma mult_dvd_mono:
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   187
  assumes "a dvd b"
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   188
    and "c dvd d"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   189
  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
   190
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   191
  from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   192
  moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" ..
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   193
  ultimately have "b * d = (a * c) * (b' * d')" 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
   194
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   195
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   196
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   197
lemma dvd_mult_left:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   198
  "a * b dvd c \<Longrightarrow> a dvd c"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   199
  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
   200
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   201
lemma dvd_mult_right:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   202
  "a * b dvd c \<Longrightarrow> b dvd c"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   203
  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
   204
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   205
end
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   206
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   207
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
   208
begin
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   209
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   210
subclass semiring_1 ..
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   211
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   212
lemma dvd_0_left_iff [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   213
  "0 dvd a \<longleftrightarrow> a = 0"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   214
  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
   215
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   216
lemma dvd_0_right [iff]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   217
  "a dvd 0"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   218
proof
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   219
  show "0 = a * 0" by simp
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   220
qed
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   221
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   222
lemma dvd_0_left:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   223
  "0 dvd a \<Longrightarrow> a = 0"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   224
  by simp
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   225
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   226
lemma dvd_add [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   227
  assumes "a dvd b" and "a dvd c"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   228
  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
   229
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   230
  from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   231
  moreover from \<open>a dvd c\<close> obtain c' where "c = a * c'" ..
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   232
  ultimately have "b + c = a * (b' + c')" 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
   233
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   234
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   235
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   236
end
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   237
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   238
class semiring_1_cancel = semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   239
  + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   240
begin
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   241
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   242
subclass semiring_0_cancel ..
25512
4134f7c782e2 using intro_locales instead of unfold_locales if appropriate
haftmann
parents: 25450
diff changeset
   243
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   244
subclass semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   245
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   246
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   247
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
   248
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add +
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
   249
                               zero_neq_one + comm_monoid_mult +
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
   250
  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   251
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   252
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   253
subclass semiring_1_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   254
subclass comm_semiring_0_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   255
subclass comm_semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   256
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   257
lemma left_diff_distrib' [algebra_simps]:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   258
  "(b - c) * a = b * a - c * a"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   259
  by (simp add: algebra_simps)
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   260
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   261
lemma dvd_add_times_triv_left_iff [simp]:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   262
  "a dvd c * a + b \<longleftrightarrow> a dvd b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   263
proof -
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   264
  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
   265
  proof
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   266
    assume ?Q then show ?P by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   267
  next
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   268
    assume ?P
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   269
    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
   270
    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
   271
    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
   272
    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
   273
    then show ?Q ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   274
  qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   275
  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
   276
qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   277
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   278
lemma dvd_add_times_triv_right_iff [simp]:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   279
  "a dvd b + c * a \<longleftrightarrow> a dvd b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   280
  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
   281
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   282
lemma dvd_add_triv_left_iff [simp]:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   283
  "a dvd a + b \<longleftrightarrow> a dvd b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   284
  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
   285
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   286
lemma dvd_add_triv_right_iff [simp]:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   287
  "a dvd b + a \<longleftrightarrow> a dvd b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   288
  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
   289
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   290
lemma dvd_add_right_iff:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   291
  assumes "a dvd b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   292
  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
   293
proof
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   294
  assume ?P then obtain d where "b + c = a * d" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   295
  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
   296
  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
   297
  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
   298
  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
   299
  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
   300
  then show ?Q ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   301
next
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   302
  assume ?Q with assms show ?P by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   303
qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   304
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   305
lemma dvd_add_left_iff:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   306
  assumes "a dvd c"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   307
  shows "a dvd b + c \<longleftrightarrow> a dvd b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   308
  using assms dvd_add_right_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
   309
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   310
end
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   311
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   312
class ring = semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   313
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   314
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   315
subclass semiring_0_cancel ..
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   316
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   317
text \<open>Distribution rules\<close>
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   318
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   319
lemma minus_mult_left: "- (a * b) = - 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
   320
by (rule minus_unique) (simp add: distrib_right [symmetric])
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   321
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   322
lemma minus_mult_right: "- (a * b) = 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
   323
by (rule minus_unique) (simp add: distrib_left [symmetric])
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   324
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   325
text\<open>Extract signs from products\<close>
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   326
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   327
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
   328
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   329
lemma minus_mult_minus [simp]: "- a * - b = a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   330
by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   331
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   332
lemma minus_mult_commute: "- a * b = a * - b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   333
by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   334
58776
95e58e04e534 use NO_MATCH-simproc for distribution rules in field_simps, otherwise field_simps on '(a / (c + d)) * (e + f)' can be non-terminating
hoelzl
parents: 58649
diff changeset
   335
lemma right_diff_distrib [algebra_simps]:
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   336
  "a * (b - c) = a * b - a * c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   337
  using distrib_left [of a b "-c "] by simp
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   338
58776
95e58e04e534 use NO_MATCH-simproc for distribution rules in field_simps, otherwise field_simps on '(a / (c + d)) * (e + f)' can be non-terminating
hoelzl
parents: 58649
diff changeset
   339
lemma left_diff_distrib [algebra_simps]:
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   340
  "(a - b) * c = a * c - b * c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   341
  using distrib_right [of a "- b" c] by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   342
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   343
lemmas ring_distribs =
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   344
  distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   345
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   346
lemma eq_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   347
  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   348
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   349
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   350
lemma eq_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   351
  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   352
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   353
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   354
end
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   355
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   356
lemmas ring_distribs =
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   357
  distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   358
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   359
class comm_ring = comm_semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   360
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   361
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   362
subclass ring ..
28141
193c3ea0f63b instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents: 27651
diff changeset
   363
subclass comm_semiring_0_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   364
44350
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   365
lemma square_diff_square_factored:
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   366
  "x * x - y * y = (x + y) * (x - y)"
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   367
  by (simp add: algebra_simps)
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   368
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   369
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   370
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   371
class ring_1 = ring + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   372
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   373
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   374
subclass semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   375
44346
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   376
lemma square_diff_one_factored:
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   377
  "x * x - 1 = (x + 1) * (x - 1)"
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   378
  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
   379
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   380
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   381
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   382
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   383
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   384
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   385
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
   386
subclass comm_semiring_1_cancel
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   387
  by unfold_locales (simp add: algebra_simps)
58647
fce800afeec7 more facts about abstract divisibility
haftmann
parents: 58198
diff changeset
   388
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
   389
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
   390
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   391
  assume "x dvd - y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   392
  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
   393
  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
   394
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   395
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   396
  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
   397
  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
   398
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   399
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
   400
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
   401
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   402
  assume "- x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   403
  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
   404
  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
   405
  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
   406
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   407
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   408
  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
   409
  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
   410
  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
   411
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   412
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   413
lemma dvd_diff [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   414
  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   415
  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
   416
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   417
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   418
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   419
class semiring_no_zero_divisors = semiring_0 +
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   420
  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
   421
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   422
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   423
lemma divisors_zero:
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   424
  assumes "a * b = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   425
  shows "a = 0 \<or> b = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   426
proof (rule classical)
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   427
  assume "\<not> (a = 0 \<or> b = 0)"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   428
  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
   429
  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
   430
  with assms show ?thesis by simp
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   431
qed
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   432
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   433
lemma mult_eq_0_iff [simp]:
58952
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   434
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   435
proof (cases "a = 0 \<or> b = 0")
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   436
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   437
    then show ?thesis using no_zero_divisors by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   438
next
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   439
  case True then show ?thesis by auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   440
qed
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   441
58952
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   442
end
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   443
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   444
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
   445
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   446
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
   447
  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
   448
    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
   449
begin
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   450
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   451
lemma mult_left_cancel:
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   452
  "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
   453
  by simp
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   454
58952
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   455
lemma mult_right_cancel:
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   456
  "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
   457
  by simp
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   458
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   459
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   460
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   461
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
   462
begin
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   463
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   464
subclass semiring_no_zero_divisors_cancel
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   465
proof
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   466
  fix a b c
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   467
  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
   468
    by (simp add: algebra_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   469
  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
   470
    by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   471
  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
   472
  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
   473
    by (simp add: algebra_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   474
  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
   475
    by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   476
  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
   477
qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   478
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   479
end
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   480
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   481
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   482
begin
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   483
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   484
subclass semiring_1_no_zero_divisors ..
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   485
36970
fb3fdb4b585e remove simp attribute from square_eq_1_iff
huffman
parents: 36821
diff changeset
   486
lemma square_eq_1_iff:
36821
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   487
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   488
proof -
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   489
  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
   490
    by (simp add: algebra_simps)
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   491
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   492
    by simp
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   493
  thus ?thesis
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   494
    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
   495
qed
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   496
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   497
lemma mult_cancel_right1 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   498
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   499
by (insert mult_cancel_right [of 1 c b], force)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   500
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   501
lemma mult_cancel_right2 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   502
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   503
by (insert mult_cancel_right [of a c 1], 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
   504
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   505
lemma mult_cancel_left1 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   506
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   507
by (insert mult_cancel_left [of c 1 b], force)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   508
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   509
lemma mult_cancel_left2 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   510
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   511
by (insert mult_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   512
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   513
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   514
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
   515
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
   516
begin
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   517
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   518
subclass semiring_1_no_zero_divisors ..
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   519
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   520
end
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   521
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   522
class idom = comm_ring_1 + semiring_no_zero_divisors
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   523
begin
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   524
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   525
subclass semidom ..
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   526
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   527
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
   528
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   529
lemma dvd_mult_cancel_right [simp]:
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   530
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   531
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   532
  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
   533
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   534
  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
   535
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   536
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   537
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   538
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   539
lemma dvd_mult_cancel_left [simp]:
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   540
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   541
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   542
  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
   543
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   544
  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
   545
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   546
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   547
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   548
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   549
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
   550
proof
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   551
  assume "a * a = b * b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   552
  then have "(a - b) * (a + b) = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   553
    by (simp add: algebra_simps)
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   554
  then show "a = b \<or> a = - b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   555
    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
   556
next
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   557
  assume "a = b \<or> a = - b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   558
  then show "a * a = b * b" by auto
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   559
qed
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   560
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   561
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   562
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   563
text \<open>
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   564
  The theory of partially ordered rings is taken from the books:
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   565
  \begin{itemize}
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
   566
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   567
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   568
  \end{itemize}
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
   569
  Most of the used notions can also be looked up in
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   570
  \begin{itemize}
54703
499f92dc6e45 more antiquotations;
wenzelm
parents: 54489
diff changeset
   571
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   572
  \item \emph{Algebra I} by van der Waerden, Springer.
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   573
  \end{itemize}
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   574
\<close>
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   575
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   576
class divide =
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   577
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   578
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   579
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
   580
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   581
context semiring
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   582
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   583
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   584
lemma [field_simps]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   585
  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
   586
    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
   587
  by (rule distrib_left distrib_right)+
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   588
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   589
end
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   590
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   591
context ring
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   592
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   593
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   594
lemma [field_simps]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   595
  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
   596
    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
   597
  by (rule left_diff_distrib right_diff_distrib)+
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   598
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   599
end
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   600
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   601
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
   602
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   603
class semidom_divide = semidom + divide +
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   604
  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
   605
  assumes divide_zero [simp]: "a div 0 = 0"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   606
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   607
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   608
lemma nonzero_mult_divide_cancel_left [simp]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   609
  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   610
  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
   611
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   612
subclass semiring_no_zero_divisors_cancel
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   613
proof
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   614
  fix a b c
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   615
  { fix a b c
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   616
    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
   617
    proof (cases "c = 0")
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   618
      case True then show ?thesis by simp
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   619
    next
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   620
      case False
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   621
      { assume "a * c = b * c"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   622
        then have "a * c div c = b * c div c"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   623
          by simp
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   624
        with False have "a = b"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   625
          by simp
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   626
      } then show ?thesis by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   627
    qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   628
  }
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   629
  from this [of a c b]
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   630
  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
   631
    by (simp add: ac_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   632
qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   633
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   634
lemma div_self [simp]:
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   635
  assumes "a \<noteq> 0"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   636
  shows "a div a = 1"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   637
  using assms nonzero_mult_divide_cancel_left [of a 1] by simp
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   638
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   639
lemma divide_zero_left [simp]:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   640
  "0 div a = 0"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   641
proof (cases "a = 0")
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   642
  case True then show ?thesis by simp
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   643
next
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   644
  case False then have "a * 0 div a = 0"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   645
    by (rule nonzero_mult_divide_cancel_left)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   646
  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
   647
qed
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   648
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   649
lemma divide_1 [simp]:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   650
  "a div 1 = a"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   651
  using nonzero_mult_divide_cancel_left [of 1 a] by simp
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   652
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   653
end
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   654
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   655
class idom_divide = idom + semidom_divide
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   656
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   657
class algebraic_semidom = semidom_divide
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   658
begin
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   659
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   660
text \<open>
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   661
  Class @{class algebraic_semidom} enriches a integral domain
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   662
  by notions from algebra, like units in a ring.
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   663
  It is a separate class to avoid spoiling fields with notions
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   664
  which are degenerated there.
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   665
\<close>
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   666
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   667
lemma dvd_times_left_cancel_iff [simp]:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   668
  assumes "a \<noteq> 0"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   669
  shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   670
proof
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   671
  assume ?P then obtain d where "a * c = a * b * d" ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   672
  with assms have "c = b * d" by (simp add: ac_simps)
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   673
  then show ?Q ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   674
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
   675
  assume ?Q then obtain d where "c = b * d" ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   676
  then have "a * c = a * b * d" by (simp add: ac_simps)
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   677
  then show ?P ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   678
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
   679
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   680
lemma dvd_times_right_cancel_iff [simp]:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   681
  assumes "a \<noteq> 0"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   682
  shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   683
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
   684
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   685
lemma div_dvd_iff_mult:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   686
  assumes "b \<noteq> 0" and "b dvd a"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   687
  shows "a div b dvd c \<longleftrightarrow> a dvd c * b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   688
proof -
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   689
  from \<open>b dvd a\<close> obtain d where "a = b * d" ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   690
  with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps)
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   691
qed
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   692
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   693
lemma dvd_div_iff_mult:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   694
  assumes "c \<noteq> 0" and "c dvd b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   695
  shows "a dvd b div c \<longleftrightarrow> a * c dvd b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   696
proof -
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   697
  from \<open>c dvd b\<close> obtain d where "b = c * d" ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   698
  with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a])
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   699
qed
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   700
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   701
lemma div_dvd_div [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   702
  assumes "a dvd b" and "a dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   703
  shows "b div a dvd c div a \<longleftrightarrow> b dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   704
proof (cases "a = 0")
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   705
  case True with assms show ?thesis by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   706
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   707
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   708
  moreover from assms obtain k l where "b = a * k" and "c = a * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   709
    by (auto elim!: dvdE)
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
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   712
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   713
lemma div_add [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   714
  assumes "c dvd a" and "c dvd b"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   715
  shows "(a + b) div c = a div c + b div c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   716
proof (cases "c = 0")
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   717
  case True then show ?thesis by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   718
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   719
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   720
  moreover from assms obtain k l where "a = c * k" and "b = c * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   721
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   722
  moreover have "c * k + c * l = c * (k + l)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   723
    by (simp add: algebra_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   724
  ultimately show ?thesis
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   725
    by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   726
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
   727
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   728
lemma div_mult_div_if_dvd:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   729
  assumes "b dvd a" and "d dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   730
  shows "(a div b) * (c div d) = (a * c) div (b * d)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   731
proof (cases "b = 0 \<or> c = 0")
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   732
  case True with assms show ?thesis by auto
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   733
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   734
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   735
  moreover from assms obtain k l where "a = b * k" and "c = d * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   736
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   737
  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
   738
    by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   739
  ultimately show ?thesis by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   740
qed
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   741
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   742
lemma dvd_div_eq_mult:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   743
  assumes "a \<noteq> 0" and "a dvd b"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   744
  shows "b div a = c \<longleftrightarrow> b = c * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   745
proof
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   746
  assume "b = c * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   747
  then show "b div a = c" by (simp add: assms)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   748
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   749
  assume "b div a = c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   750
  then have "b div a * a = c * a" by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   751
  moreover from \<open>a \<noteq> 0\<close> \<open>a dvd b\<close> have "b div a * a = b"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   752
    by (auto elim!: dvdE simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   753
  ultimately show "b = c * a" by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   754
qed
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   755
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   756
lemma dvd_div_mult_self [simp]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   757
  "a dvd b \<Longrightarrow> b div a * a = b"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   758
  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
   759
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   760
lemma dvd_mult_div_cancel [simp]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   761
  "a dvd b \<Longrightarrow> a * (b div a) = b"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   762
  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
   763
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   764
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
   765
  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
   766
  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
   767
proof (cases "c = 0")
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   768
  case True then 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
   769
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   770
  case False from assms obtain d where "b = c * d" ..
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   771
  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
   772
    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
   773
  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
   774
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   775
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   776
lemma dvd_div_mult:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   777
  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
   778
  shows "b div c * a = (b * a) 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
   779
  using assms div_mult_swap [of c b 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
   780
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   781
lemma dvd_div_mult2_eq:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   782
  assumes "b * c dvd a"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   783
  shows "a div (b * c) = a div b div c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   784
using assms proof
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   785
  fix k
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   786
  assume "a = b * c * k"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   787
  then show ?thesis
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   788
    by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   789
qed
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   790
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   791
lemma dvd_div_div_eq_mult:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   792
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   793
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d" (is "?P \<longleftrightarrow> ?Q")
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   794
proof -
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   795
  from assms have "a * c \<noteq> 0" by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   796
  then have "?P \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   797
    by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   798
  also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   799
    by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   800
  also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   801
    using assms by (simp add: div_mult_swap)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   802
  also have "\<dots> \<longleftrightarrow> ?Q"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   803
    using assms by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   804
  finally show ?thesis .
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   805
qed
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   806
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
   807
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   808
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
   809
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   810
abbreviation is_unit :: "'a \<Rightarrow> bool"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   811
where
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   812
  "is_unit a \<equiv> a dvd 1"
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 not_is_unit_0 [simp]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   815
  "\<not> is_unit 0"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   816
  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
   817
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
   818
lemma unit_imp_dvd [dest]:
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   819
  "is_unit b \<Longrightarrow> b dvd a"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   820
  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
   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 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
   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 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
   825
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   826
  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
   827
  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
   828
  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
   829
  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
   830
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   831
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   832
lemma dvd_unit_imp_unit:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   833
  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> 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
   834
  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
   835
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   836
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
   837
  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
   838
  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
   839
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   840
  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
   841
  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
   842
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   843
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   844
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
   845
  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
   846
  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
   847
    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
   848
    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
   849
proof (rule that)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   850
  def b \<equiv> "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
   851
  then show "1 div a = b" by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   852
  from b_def \<open>is_unit a\<close> show "is_unit b" by simp
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   853
  from \<open>is_unit a\<close> and \<open>is_unit b\<close> show "a \<noteq> 0" and "b \<noteq> 0" by auto
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   854
  from b_def \<open>is_unit a\<close> 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
   855
  then have "1 = a * b" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   856
  with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   857
  from \<open>is_unit a\<close> 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
   858
  then obtain d where "c = a * d" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   859
  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
   860
    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
   861
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   862
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   863
lemma unit_prod [intro]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   864
  "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
   865
  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
   866
62366
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   867
lemma is_unit_mult_iff:
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   868
  "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b" (is "?P \<longleftrightarrow> ?Q")
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   869
  by (auto dest: dvd_mult_left dvd_mult_right)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   870
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   871
lemma unit_div [intro]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   872
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a 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
   873
  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
   874
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   875
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
   876
  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
   877
  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
   878
proof
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   879
  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
   880
  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
   881
    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
   882
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   883
  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
   884
  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
   885
  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
   886
    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
   887
  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
   888
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   889
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   890
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
   891
  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
   892
  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
   893
proof
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   894
  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
   895
  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
   896
    by (subst mult_assoc [symmetric]) simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   897
  also from \<open>is_unit b\<close> have "b * (1 div b) = 1" 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
   898
  finally have "c * b dvd c" by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   899
  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
   900
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   901
  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
   902
  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
   903
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   904
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   905
lemma div_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
   906
  "is_unit b \<Longrightarrow> a div 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
   907
  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
   908
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   909
lemma dvd_div_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
   910
  "is_unit b \<Longrightarrow> a dvd c div 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
   911
  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
   912
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   913
lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
   914
  dvd_mult_unit_iff dvd_div_unit_iff \<comment> \<open>FIXME consider fact collection\<close>
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   915
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   916
lemma unit_mult_div_div [simp]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   917
  "is_unit a \<Longrightarrow> b * (1 div a) = b 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
   918
  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
   919
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   920
lemma unit_div_mult_self [simp]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   921
  "is_unit a \<Longrightarrow> b div a * a = b"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   922
  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
   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_1_div_1 [simp]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   925
  "is_unit a \<Longrightarrow> 1 div (1 div a) = 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
  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
   927
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   928
lemma unit_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
   929
  "is_unit c \<Longrightarrow> 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
   930
  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
   931
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   932
lemma unit_div_commute:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   933
  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) 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
   934
  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
   935
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   936
lemma unit_eq_div1:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   937
  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> 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
   938
  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
   939
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   940
lemma 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
   941
  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * 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
   942
  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
   943
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   944
lemma unit_mult_left_cancel:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   945
  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
   946
  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
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
   947
  using assms 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
   948
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   949
lemma 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
   950
  "is_unit a \<Longrightarrow> b * a = c * 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
   951
  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
   952
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   953
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
   954
  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
   955
  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
   956
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   957
  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
   958
  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
   959
    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
   960
  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
   961
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
   962
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   963
lemma is_unit_div_mult2_eq:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   964
  assumes "is_unit b" and "is_unit c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   965
  shows "a div (b * c) = a div b div c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   966
proof -
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   967
  from assms have "is_unit (b * c)" by (simp add: unit_prod)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   968
  then have "b * c dvd a"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   969
    by (rule unit_imp_dvd)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   970
  then show ?thesis
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   971
    by (rule dvd_div_mult2_eq)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   972
qed
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   973
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
   974
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
   975
  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
   976
  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
   977
  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
   978
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   979
lemma is_unit_divide_mult_cancel_left:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   980
  assumes "a \<noteq> 0" and "is_unit b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   981
  shows "a div (a * b) = 1 div b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   982
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   983
  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
   984
    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
   985
  with assms show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   986
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   987
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   988
lemma is_unit_divide_mult_cancel_right:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   989
  assumes "a \<noteq> 0" and "is_unit b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   990
  shows "a div (b * a) = 1 div b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   991
  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
   992
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   993
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   994
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   995
class normalization_semidom = algebraic_semidom +
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   996
  fixes normalize :: "'a \<Rightarrow> 'a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   997
    and unit_factor :: "'a \<Rightarrow> 'a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   998
  assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   999
  assumes normalize_0 [simp]: "normalize 0 = 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1000
    and unit_factor_0 [simp]: "unit_factor 0 = 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1001
  assumes is_unit_normalize:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1002
    "is_unit a  \<Longrightarrow> normalize a = 1"
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
  1003
  assumes unit_factor_is_unit [iff]:
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1004
    "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1005
  assumes unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1006
begin
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1007
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1008
text \<open>
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1009
  Class @{class normalization_semidom} cultivates the idea that
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1010
  each integral domain can be split into equivalence classes
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1011
  whose representants are associated, i.e. divide each other.
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1012
  @{const normalize} specifies a canonical representant for each equivalence
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1013
  class.  The rationale behind this is that it is easier to reason about equality
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1014
  than equivalences, hence we prefer to think about equality of normalized
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1015
  values rather than associated elements.
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1016
\<close>
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1017
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1018
lemma unit_factor_dvd [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1019
  "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1020
  by (rule unit_imp_dvd) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1021
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1022
lemma unit_factor_self [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1023
  "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
  1024
  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
  1025
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1026
lemma normalize_mult_unit_factor [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1027
  "normalize a * unit_factor a = a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1028
  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
  1029
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1030
lemma normalize_eq_0_iff [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1031
  "normalize a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1032
proof
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1033
  assume ?P
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1034
  moreover have "unit_factor a * normalize a = a" 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
  1035
  ultimately show ?Q by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1036
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1037
  assume ?Q then show ?P by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1038
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1039
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1040
lemma unit_factor_eq_0_iff [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1041
  "unit_factor a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1042
proof
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1043
  assume ?P
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1044
  moreover have "unit_factor a * normalize a = a" 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
  1045
  ultimately show ?Q by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1046
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1047
  assume ?Q then show ?P by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1048
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1049
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1050
lemma is_unit_unit_factor:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1051
  assumes "is_unit a" 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
  1052
proof -
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1053
  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
  1054
  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
  1055
  ultimately show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1056
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1057
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1058
lemma unit_factor_1 [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1059
  "unit_factor 1 = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1060
  by (rule is_unit_unit_factor) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1061
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1062
lemma normalize_1 [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1063
  "normalize 1 = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1064
  by (rule is_unit_normalize) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1065
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1066
lemma normalize_1_iff:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1067
  "normalize a = 1 \<longleftrightarrow> is_unit a" (is "?P \<longleftrightarrow> ?Q")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1068
proof
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1069
  assume ?Q then show ?P by (rule is_unit_normalize)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1070
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1071
  assume ?P
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1072
  then have "a \<noteq> 0" by auto
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1073
  from \<open>?P\<close> have "unit_factor a * normalize a = unit_factor a * 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1074
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1075
  then have "unit_factor a = a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1076
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1077
  moreover have "is_unit (unit_factor a)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1078
    using \<open>a \<noteq> 0\<close> by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1079
  ultimately show ?Q by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1080
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
  1081
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1082
lemma div_normalize [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1083
  "a div normalize a = unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1084
proof (cases "a = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1085
  case True then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1086
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
  1087
  case False then have "normalize a \<noteq> 0" by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1088
  with nonzero_mult_divide_cancel_right
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1089
  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
  1090
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1091
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1092
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1093
lemma div_unit_factor [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1094
  "a div unit_factor a = normalize a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1095
proof (cases "a = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1096
  case True then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1097
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
  1098
  case False 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
  1099
  with nonzero_mult_divide_cancel_left
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1100
  have "unit_factor a * normalize a div unit_factor a = normalize a" by blast
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1101
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1102
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1103
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1104
lemma normalize_div [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1105
  "normalize a div a = 1 div unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1106
proof (cases "a = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1107
  case True then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1108
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1109
  case False
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1110
  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
  1111
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1112
  also have "\<dots> = 1 div unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1113
    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
  1114
  finally show ?thesis .
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1115
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1116
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1117
lemma mult_one_div_unit_factor [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1118
  "a * (1 div unit_factor b) = a div unit_factor b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1119
  by (cases "b = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1120
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1121
lemma normalize_mult:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1122
  "normalize (a * b) = normalize a * normalize b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1123
proof (cases "a = 0 \<or> b = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1124
  case True then show ?thesis by auto
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1125
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1126
  case False
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1127
  from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" .
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1128
  then have "normalize (a * b) = a * b div unit_factor (a * b)" by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1129
  also have "\<dots> = a * b div unit_factor (b * a)" by (simp add: ac_simps)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1130
  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
  1131
    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
  1132
  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
  1133
    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
  1134
  also have "\<dots> = normalize a * normalize b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1135
    using False by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1136
  finally show ?thesis .
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1137
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
  1138
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1139
lemma unit_factor_idem [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1140
  "unit_factor (unit_factor a) = unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1141
  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
  1142
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1143
lemma normalize_unit_factor [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1144
  "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1145
  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
  1146
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1147
lemma normalize_idem [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1148
  "normalize (normalize a) = normalize a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1149
proof (cases "a = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1150
  case True then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1151
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1152
  case False
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1153
  have "normalize a = normalize (unit_factor a * normalize a)" by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1154
  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
  1155
    by (simp only: normalize_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1156
  finally show ?thesis using False by simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1157
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1158
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1159
lemma unit_factor_normalize [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1160
  assumes "a \<noteq> 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1161
  shows "unit_factor (normalize a) = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1162
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1163
  from assms have "normalize a \<noteq> 0" by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1164
  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
  1165
    by (simp only: unit_factor_mult_normalize)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1166
  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
  1167
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1168
  with \<open>normalize a \<noteq> 0\<close>
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1169
  have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
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
  with \<open>normalize a \<noteq> 0\<close>
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1172
  show ?thesis by simp
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_unit_factor_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 "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
  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 "unit_factor a = unit_factor (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: unit_factor_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
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1187
lemma dvd_normalize_div:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1188
  assumes "b dvd a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1189
  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
  1190
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1191
  from assms have "a = a div b * b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1192
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1193
  then have "normalize a = normalize (a div b * b)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1194
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1195
  then show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1196
    by (cases "b = 0") (simp_all add: normalize_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1197
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1198
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1199
lemma normalize_dvd_iff [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1200
  "normalize a dvd b \<longleftrightarrow> a dvd b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1201
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1202
  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
  1203
    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
  1204
      by (cases "a = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1205
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1206
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1207
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1208
lemma dvd_normalize_iff [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1209
  "a dvd normalize b \<longleftrightarrow> a dvd b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1210
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1211
  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
  1212
    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
  1213
      by (cases "b = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1214
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1215
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1216
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1217
text \<open>
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1218
  We avoid an explicit definition of associated elements but prefer
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1219
  explicit normalisation instead.  In theory we could define an abbreviation
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1220
  like @{prop "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1221
  counterproductive without suggestive infix syntax, which we do not want
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1222
  to sacrifice for this purpose here.
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1223
\<close>
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1224
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1225
lemma associatedI:
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1226
  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
  1227
  shows "normalize a = normalize b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1228
proof (cases "a = 0 \<or> b = 0")
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1229
  case True with assms show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1230
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1231
  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
  1232
  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
  1233
  moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1234
  ultimately have "b * 1 = b * (c * d)" by (simp add: ac_simps)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1235
  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
  1236
    unfolding mult_cancel_left by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1237
  then have "is_unit c" and "is_unit d" by auto
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1238
  with a b show ?thesis by (simp add: normalize_mult is_unit_normalize)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1239
qed
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1240
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1241
lemma associatedD1:
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1242
  "normalize a = normalize b \<Longrightarrow> 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
  1243
  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
  1244
  by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1245
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1246
lemma associatedD2:
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1247
  "normalize a = normalize b \<Longrightarrow> 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
  1248
  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
  1249
  by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1250
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1251
lemma associated_unit:
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1252
  "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1253
  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
  1254
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1255
lemma associated_iff_dvd:
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1256
  "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a" (is "?P \<longleftrightarrow> ?Q")
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1257
proof
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1258
  assume ?Q then show ?P by (auto intro!: associatedI)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1259
next
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1260
  assume ?P
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1261
  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
  1262
    by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1263
  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
  1264
    by (simp add: ac_simps)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1265
  show ?Q
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1266
  proof (cases "a = 0 \<or> b = 0")
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1267
    case True with \<open>?P\<close> show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1268
  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
  1269
    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
  1270
    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
  1271
      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
  1272
    with * show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1273
  qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1274
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1275
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1276
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
  1277
  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
  1278
  assumes "normalize a = a" and "normalize b = b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1279
  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
  1280
proof -
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1281
  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
  1282
    unfolding associated_iff_dvd by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1283
  with \<open>normalize a = a\<close> have "a = normalize b" by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1284
  with \<open>normalize b = b\<close> show "a = b" by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615