src/HOL/Rings.thy
author haftmann
Wed, 08 Jul 2015 14:01:34 +0200
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moved normalization and unit_factor into Main HOL corpus
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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 {* Rings *}
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theory Rings
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imports Groups
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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{*For the @{text combine_numerals} simproc*}
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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 {* Abstract divisibility *}
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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:
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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
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   158
lemma one_dvd [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   159
  "1 dvd a"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   160
  by (auto intro!: dvdI)
28559
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   161
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   162
lemma dvd_mult [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   163
  "a dvd c \<Longrightarrow> a dvd (b * c)"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   164
  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
   165
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   166
lemma dvd_mult2 [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   167
  "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
   168
  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
   169
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   170
lemma dvd_triv_right [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   171
  "a dvd b * a"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   172
  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
   173
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   174
lemma dvd_triv_left [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   175
  "a dvd a * b"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   176
  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
   177
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   178
lemma mult_dvd_mono:
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   179
  assumes "a dvd b"
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   180
    and "c dvd d"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   181
  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
   182
proof -
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   183
  from `a dvd b` obtain b' where "b = a * b'" ..
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   184
  moreover from `c dvd d` 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
   185
  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
   186
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   187
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   188
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   189
lemma dvd_mult_left:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   190
  "a * b dvd c \<Longrightarrow> a dvd c"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   191
  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
   192
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   193
lemma dvd_mult_right:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   194
  "a * b dvd c \<Longrightarrow> b dvd c"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   195
  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
   196
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   197
end
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   198
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   199
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
   200
begin
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   201
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   202
subclass semiring_1 ..
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   203
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   204
lemma dvd_0_left_iff [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   205
  "0 dvd a \<longleftrightarrow> a = 0"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   206
  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
   207
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   208
lemma dvd_0_right [iff]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   209
  "a dvd 0"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   210
proof
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   211
  show "0 = a * 0" by simp
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   212
qed
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   213
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   214
lemma dvd_0_left:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   215
  "0 dvd a \<Longrightarrow> a = 0"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   216
  by simp
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   217
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   218
lemma dvd_add [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   219
  assumes "a dvd b" and "a dvd c"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   220
  shows "a dvd (b + c)"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   221
proof -
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29915
diff changeset
   222
  from `a dvd b` obtain b' where "b = a * b'" ..
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29915
diff changeset
   223
  moreover from `a dvd c` obtain c' where "c = a * c'" ..
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   224
  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
   225
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   226
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   227
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   228
end
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   229
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   230
class semiring_1_cancel = semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   231
  + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   232
begin
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   233
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   234
subclass semiring_0_cancel ..
25512
4134f7c782e2 using intro_locales instead of unfold_locales if appropriate
haftmann
parents: 25450
diff changeset
   235
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   236
subclass semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   237
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   238
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   239
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
   240
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
   241
                               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
   242
  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   243
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   244
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   245
subclass semiring_1_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   246
subclass comm_semiring_0_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   247
subclass comm_semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   248
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   249
lemma left_diff_distrib' [algebra_simps]:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   250
  "(b - c) * a = b * a - c * a"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   251
  by (simp add: algebra_simps)
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   252
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   253
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
   254
  "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
   255
proof -
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   256
  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
   257
  proof
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   258
    assume ?Q then show ?P by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   259
  next
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   260
    assume ?P
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   261
    then obtain d where "a * c + b = a * d" ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   262
    then have "a * c + b - a * c = a * d - a * c" by simp
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   263
    then have "b = a * d - a * c" by simp
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   264
    then have "b = a * (d - c)" by (simp add: algebra_simps)
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   265
    then show ?Q ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   266
  qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   267
  then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   268
qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   269
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   270
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
   271
  "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
   272
  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
   273
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   274
lemma dvd_add_triv_left_iff [simp]:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   275
  "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
   276
  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
   277
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   278
lemma dvd_add_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 + 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_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
   281
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   282
lemma dvd_add_right_iff:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   283
  assumes "a dvd b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   284
  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
   285
proof
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   286
  assume ?P then obtain d where "b + c = a * d" ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   287
  moreover from `a dvd b` obtain e where "b = a * e" ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   288
  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
   289
  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
   290
  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
   291
  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
   292
  then show ?Q ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   293
next
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   294
  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
   295
qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   296
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   297
lemma dvd_add_left_iff:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   298
  assumes "a dvd c"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   299
  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
   300
  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
   301
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   302
end
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   303
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   304
class ring = semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   305
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   306
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   307
subclass semiring_0_cancel ..
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   308
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   309
text {* Distribution rules *}
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   310
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   311
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
   312
by (rule minus_unique) (simp add: distrib_right [symmetric])
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   313
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   314
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
   315
by (rule minus_unique) (simp add: distrib_left [symmetric])
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   316
29407
5ef7e97fd9e4 move lemmas mult_minus{left,right} inside class ring
huffman
parents: 29406
diff changeset
   317
text{*Extract signs from products*}
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   318
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   319
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
   320
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   321
lemma minus_mult_minus [simp]: "- a * - b = a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   322
by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   323
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   324
lemma minus_mult_commute: "- a * b = a * - b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   325
by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   326
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
   327
lemma right_diff_distrib [algebra_simps]:
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   328
  "a * (b - c) = a * b - a * c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   329
  using distrib_left [of a b "-c "] by simp
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   330
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
   331
lemma left_diff_distrib [algebra_simps]:
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   332
  "(a - b) * c = a * c - b * c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   333
  using distrib_right [of a "- b" c] by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   334
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   335
lemmas ring_distribs =
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   336
  distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   337
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   338
lemma eq_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   339
  "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
   340
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   341
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   342
lemma eq_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   343
  "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
   344
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   345
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   346
end
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   347
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   348
lemmas ring_distribs =
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   349
  distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   350
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   351
class comm_ring = comm_semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   352
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   353
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   354
subclass ring ..
28141
193c3ea0f63b instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents: 27651
diff changeset
   355
subclass comm_semiring_0_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   356
44350
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   357
lemma square_diff_square_factored:
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   358
  "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
   359
  by (simp add: algebra_simps)
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   360
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   361
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   362
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   363
class ring_1 = ring + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   364
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   365
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   366
subclass semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   367
44346
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   368
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
   369
  "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
   370
  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
   371
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   372
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   373
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   374
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   375
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   376
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   377
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
   378
subclass comm_semiring_1_cancel
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   379
  by unfold_locales (simp add: algebra_simps)
58647
fce800afeec7 more facts about abstract divisibility
haftmann
parents: 58198
diff changeset
   380
29465
b2cfb5d0a59e change dvd_minus_iff, minus_dvd_iff from [iff] to [simp] (due to problems with Library/Primes.thy)
huffman
parents: 29461
diff changeset
   381
lemma 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
   382
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   383
  assume "x dvd - y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   384
  then 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
   385
  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
   386
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   387
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   388
  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
   389
  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
   390
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   391
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
   392
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
   393
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   394
  assume "- x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   395
  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
   396
  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
   397
  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
   398
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   399
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   400
  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
   401
  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
   402
  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
   403
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   404
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   405
lemma dvd_diff [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   406
  "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
   407
  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
   408
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   409
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   410
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   411
class semiring_no_zero_divisors = semiring_0 +
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   412
  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
   413
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   414
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   415
lemma divisors_zero:
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   416
  assumes "a * b = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   417
  shows "a = 0 \<or> b = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   418
proof (rule classical)
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   419
  assume "\<not> (a = 0 \<or> b = 0)"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   420
  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
   421
  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
   422
  with assms show ?thesis by simp
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   423
qed
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   424
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   425
lemma mult_eq_0_iff [simp]:
58952
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   426
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   427
proof (cases "a = 0 \<or> b = 0")
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   428
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   429
    then show ?thesis using no_zero_divisors by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   430
next
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   431
  case True then show ?thesis by auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   432
qed
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   433
58952
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   434
end
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   435
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   436
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
   437
  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
   438
    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
   439
begin
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   440
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   441
lemma mult_left_cancel:
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   442
  "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
   443
  by simp
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   444
58952
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   445
lemma mult_right_cancel:
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   446
  "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
   447
  by simp
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   448
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   449
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   450
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   451
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
   452
begin
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   453
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   454
subclass semiring_no_zero_divisors_cancel
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   455
proof
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   456
  fix a b c
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   457
  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
   458
    by (simp add: algebra_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   459
  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
   460
    by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   461
  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
   462
  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
   463
    by (simp add: algebra_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   464
  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
   465
    by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   466
  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
   467
qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   468
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   469
end
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   470
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   471
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   472
begin
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   473
36970
fb3fdb4b585e remove simp attribute from square_eq_1_iff
huffman
parents: 36821
diff changeset
   474
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
   475
  "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
   476
proof -
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   477
  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
   478
    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
   479
  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
   480
    by simp
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   481
  thus ?thesis
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   482
    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
   483
qed
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   484
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   485
lemma mult_cancel_right1 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   486
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   487
by (insert mult_cancel_right [of 1 c b], force)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   488
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   489
lemma mult_cancel_right2 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   490
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   491
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
   492
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   493
lemma mult_cancel_left1 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   494
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   495
by (insert mult_cancel_left [of c 1 b], force)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   496
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   497
lemma mult_cancel_left2 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   498
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   499
by (insert mult_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   500
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   501
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   502
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
   503
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   504
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   505
class idom = comm_ring_1 + semiring_no_zero_divisors
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   506
begin
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   507
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   508
subclass semidom ..
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   509
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   510
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
   511
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   512
lemma dvd_mult_cancel_right [simp]:
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   513
  "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
   514
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   515
  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
   516
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   517
  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
   518
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   519
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   520
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   521
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   522
lemma dvd_mult_cancel_left [simp]:
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   523
  "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
   524
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   525
  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
   526
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   527
  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
   528
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   529
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   530
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   531
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   532
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
   533
proof
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   534
  assume "a * a = b * b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   535
  then have "(a - b) * (a + b) = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   536
    by (simp add: algebra_simps)
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   537
  then show "a = b \<or> a = - b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   538
    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
   539
next
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   540
  assume "a = b \<or> a = - b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   541
  then show "a * a = b * b" by auto
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   542
qed
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   543
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   544
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   545
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   546
text {*
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   547
  The theory of partially ordered rings is taken from the books:
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   548
  \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
   549
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   550
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   551
  \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
   552
  Most of the used notions can also be looked up in
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   553
  \begin{itemize}
54703
499f92dc6e45 more antiquotations;
wenzelm
parents: 54489
diff changeset
   554
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   555
  \item \emph{Algebra I} by van der Waerden, Springer.
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   556
  \end{itemize}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   557
*}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   558
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   559
class divide =
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   560
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   561
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   562
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   563
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   564
context semiring
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   565
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   566
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   567
lemma [field_simps]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   568
  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
   569
    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
   570
  by (rule distrib_left distrib_right)+
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   571
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   572
end
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   573
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   574
context ring
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   575
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   576
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   577
lemma [field_simps]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   578
  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
   579
    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
   580
  by (rule left_diff_distrib right_diff_distrib)+
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   581
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   582
end
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
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   585
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   586
class semidom_divide = semidom + divide +
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   587
  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
   588
  assumes divide_zero [simp]: "a div 0 = 0"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   589
begin
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
lemma nonzero_mult_divide_cancel_left [simp]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   592
  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   593
  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
   594
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   595
subclass semiring_no_zero_divisors_cancel
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   596
proof
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   597
  fix a b c
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   598
  { fix a b c
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   599
    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
   600
    proof (cases "c = 0")
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   601
      case True then show ?thesis by simp
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   602
    next
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   603
      case False
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   604
      { assume "a * c = b * c"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   605
        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
   606
          by simp
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   607
        with False have "a = b"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   608
          by simp
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   609
      } then show ?thesis by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   610
    qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   611
  }
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   612
  from this [of a c b]
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   613
  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
   614
    by (simp add: ac_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   615
qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   616
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   617
lemma div_self [simp]:
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   618
  assumes "a \<noteq> 0"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   619
  shows "a div a = 1"
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   620
  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
   621
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   622
lemma divide_zero_left [simp]:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   623
  "0 div a = 0"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   624
proof (cases "a = 0")
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   625
  case True then show ?thesis by simp
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   626
next
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   627
  case False then have "a * 0 div a = 0"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   628
    by (rule nonzero_mult_divide_cancel_left)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   629
  then show ?thesis by simp
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   630
qed 
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   631
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   632
end
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   633
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   634
class idom_divide = idom + semidom_divide
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   635
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   636
class algebraic_semidom = semidom_divide
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   637
begin
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   638
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   639
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
   640
  "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
   641
  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
   642
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   643
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
   644
  "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
   645
  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
   646
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   647
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
   648
  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
   649
  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
   650
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
   651
  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
   652
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   653
  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
   654
  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
   655
    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
   656
  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
   657
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   658
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   659
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
   660
  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
   661
  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
   662
  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
   663
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   664
lemma dvd_div_mult2_eq:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   665
  assumes "b * c dvd a"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   666
  shows "a div (b * c) = a div b div c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   667
using assms proof
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   668
  fix k
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   669
  assume "a = b * c * k"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   670
  then show ?thesis
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   671
    by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   672
qed
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   673
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
   674
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   675
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
   676
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   677
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
   678
where
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   679
  "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
   680
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   681
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
   682
  "\<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
   683
  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
   684
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
   685
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
   686
  "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
   687
  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
   688
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   689
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
   690
  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
   691
  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
   692
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   693
  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
   694
  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
   695
  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
   696
  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
   697
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   698
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   699
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
   700
  "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
   701
  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
   702
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   703
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
   704
  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
   705
  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
   706
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   707
  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
   708
  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
   709
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   710
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   711
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
   712
  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
   713
  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
   714
    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
   715
    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
   716
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
   717
  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
   718
  then show "1 div a = 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
   719
  from b_def `is_unit a` show "is_unit 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
   720
  from `is_unit a` and `is_unit b` show "a \<noteq> 0" and "b \<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
   721
  from b_def `is_unit a` show "a * b = 1" 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
   722
  then have "1 = 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
   723
  with b_def `b \<noteq> 0` show "1 div b = 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
   724
  from `is_unit a` have "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
   725
  then obtain d where "c = a * d" ..
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   726
  with `a \<noteq> 0` `a * b = 1` show "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
   727
    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
   728
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   729
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   730
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
   731
  "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
   732
  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
   733
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   734
lemma 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
   735
  "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
   736
  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
   737
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   738
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
   739
  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
   740
  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
   741
proof
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   742
  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
   743
  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
   744
    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
   745
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   746
  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
   747
  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
   748
  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
   749
    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
   750
  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
   751
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   752
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   753
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
   754
  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
   755
  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
   756
proof
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   757
  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
   758
  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
   759
    by (subst mult_assoc [symmetric]) simp
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   760
  also from `is_unit b` have "b * (1 div b) = 1" by (rule 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
   761
  finally have "c * b dvd c" 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
   762
  with `a dvd c * b` show "a dvd c" 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
   763
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   764
  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
   765
  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
   766
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   767
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   768
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
   769
  "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
   770
  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
   771
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   772
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
   773
  "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
   774
  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
   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
lemmas unit_dvd_iff = mult_unit_dvd_iff 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
   777
  dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   778
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   779
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
   780
  "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
   781
  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
   782
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   783
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
   784
  "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
   785
  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
   786
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   787
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
   788
  "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
   789
  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
   790
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   791
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
   792
  "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
   793
  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
   794
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   795
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
   796
  "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
   797
  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
   798
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   799
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
   800
  "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
   801
  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
   802
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   803
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
   804
  "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
   805
  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
   806
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   807
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
   808
  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
   809
  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
   810
  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
   811
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   812
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
   813
  "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
   814
  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
   815
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   816
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
   817
  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
   818
  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
   819
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   820
  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
   821
  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
   822
    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
   823
  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
   824
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
   825
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   826
lemma is_unit_div_mult2_eq:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   827
  assumes "is_unit b" and "is_unit c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   828
  shows "a div (b * c) = a div b div c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   829
proof -
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   830
  from assms have "is_unit (b * c)" by (simp add: unit_prod)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   831
  then have "b * c dvd a"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   832
    by (rule unit_imp_dvd)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   833
  then show ?thesis
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   834
    by (rule dvd_div_mult2_eq)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   835
qed
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   836
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   837
60529
24c2ef12318b avoid suspicious Unicode;
wenzelm
parents: 60517
diff changeset
   838
text \<open>Associated elements in a ring --- an equivalence relation induced
24c2ef12318b avoid suspicious Unicode;
wenzelm
parents: 60517
diff changeset
   839
  by the quasi-order divisibility.\<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
   840
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
   841
definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   842
where
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   843
  "associated a b \<longleftrightarrow> a dvd b \<and> 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
   844
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   845
lemma associatedI:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   846
  "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated 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
   847
  by (simp add: associated_def)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   848
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   849
lemma associatedD1:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   850
  "associated a b \<Longrightarrow> a 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
   851
  by (simp add: associated_def)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   852
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   853
lemma associatedD2:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   854
  "associated a 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
   855
  by (simp add: associated_def)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   856
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   857
lemma associated_refl [simp]:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   858
  "associated 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
   859
  by (auto intro: associatedI)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   860
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   861
lemma associated_sym:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   862
  "associated b a \<longleftrightarrow> associated 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
   863
  by (auto intro: associatedI dest: associatedD1 associatedD2)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   864
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   865
lemma associated_trans:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   866
  "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated 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
   867
  by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   868
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   869
lemma associated_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
   870
  "associated 0 b \<longleftrightarrow> b = 0"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   871
  "associated a 0 \<longleftrightarrow> a = 0"
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   872
  by (auto dest: associatedD1 associatedD2)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   873
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   874
lemma associated_unit:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   875
  "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> 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
   876
  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   877
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   878
lemma is_unit_associatedI:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   879
  assumes "is_unit c" and "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
   880
  shows "associated 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
   881
proof (rule associatedI)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   882
  from `a = c * b` show "b dvd 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
   883
  from `is_unit c` obtain d where "c * d = 1" by (rule 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
   884
  moreover from `a = c * b` have "d * a = d * (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
   885
  ultimately have "b = a * d" 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
   886
  then show "a 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
   887
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   888
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   889
lemma associated_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
   890
  assumes "associated 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
   891
  obtains c where "is_unit c" and "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
   892
proof (cases "b = 0")
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   893
  case True with assms have "is_unit 1" and "a = 1 * b" by 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
   894
  with that show thesis .
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   895
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   896
  case False
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   897
  from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   898
  then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   899
  then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all 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
   900
  with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] 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
   901
  then have "is_unit c" 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
   902
  with `a = c * b` that show thesis 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
   903
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
   904
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
   905
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
   906
  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
   907
  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
   908
  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
   909
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   910
end
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   911
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   912
context algebraic_semidom
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   913
begin
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   914
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   915
lemma is_unit_divide_mult_cancel_left:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   916
  assumes "a \<noteq> 0" and "is_unit b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   917
  shows "a div (a * b) = 1 div b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   918
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   919
  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
   920
    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
   921
  with assms show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   922
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   923
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   924
lemma is_unit_divide_mult_cancel_right:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   925
  assumes "a \<noteq> 0" and "is_unit b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   926
  shows "a div (b * a) = 1 div b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   927
  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
   928
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   929
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   930
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   931
class normalization_semidom = algebraic_semidom +
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   932
  fixes normalize :: "'a \<Rightarrow> 'a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   933
    and unit_factor :: "'a \<Rightarrow> 'a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   934
  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
   935
  assumes normalize_0 [simp]: "normalize 0 = 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   936
    and unit_factor_0 [simp]: "unit_factor 0 = 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   937
  assumes is_unit_normalize:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   938
    "is_unit a  \<Longrightarrow> normalize a = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   939
  assumes unit_factor_is_unit [iff]: 
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   940
    "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   941
  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
   942
begin
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   943
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   944
lemma unit_factor_dvd [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   945
  "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   946
  by (rule unit_imp_dvd) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   947
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   948
lemma unit_factor_self [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   949
  "unit_factor a dvd a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   950
  by (cases "a = 0") simp_all 
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   951
  
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   952
lemma normalize_mult_unit_factor [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   953
  "normalize a * unit_factor a = a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   954
  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
   955
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   956
lemma normalize_eq_0_iff [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   957
  "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
   958
proof
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   959
  assume ?P
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   960
  moreover have "unit_factor a * normalize a = a" by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   961
  ultimately show ?Q by simp 
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   962
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   963
  assume ?Q then show ?P by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   964
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   965
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   966
lemma unit_factor_eq_0_iff [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   967
  "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
   968
proof
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   969
  assume ?P
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   970
  moreover have "unit_factor a * normalize a = a" by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   971
  ultimately show ?Q by simp 
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   972
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   973
  assume ?Q then show ?P by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   974
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   975
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   976
lemma is_unit_unit_factor:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   977
  assumes "is_unit a" shows "unit_factor a = a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   978
proof - 
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   979
  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
   980
  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
   981
  ultimately show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   982
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   983
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   984
lemma unit_factor_1 [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   985
  "unit_factor 1 = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   986
  by (rule is_unit_unit_factor) simp
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 normalize_1 [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   989
  "normalize 1 = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   990
  by (rule is_unit_normalize) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   991
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   992
lemma normalize_1_iff:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   993
  "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
   994
proof
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   995
  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
   996
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   997
  assume ?P
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   998
  then have "a \<noteq> 0" by auto
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   999
  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
  1000
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1001
  then have "unit_factor a = a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1002
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1003
  moreover have "is_unit (unit_factor a)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1004
    using \<open>a \<noteq> 0\<close> by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1005
  ultimately show ?Q by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1006
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1007
  
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1008
lemma div_normalize [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1009
  "a div normalize a = unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1010
proof (cases "a = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1011
  case True then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1012
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1013
  case False then have "normalize a \<noteq> 0" by simp 
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1014
  with nonzero_mult_divide_cancel_right
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1015
  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
  1016
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1017
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1018
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1019
lemma div_unit_factor [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1020
  "a div unit_factor a = normalize a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1021
proof (cases "a = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1022
  case True then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1023
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1024
  case False then have "unit_factor a \<noteq> 0" by simp 
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1025
  with nonzero_mult_divide_cancel_left
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1026
  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
  1027
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1028
qed
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_div [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1031
  "normalize a div a = 1 div unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1032
proof (cases "a = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1033
  case True then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1034
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1035
  case False
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1036
  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
  1037
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1038
  also have "\<dots> = 1 div unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1039
    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
  1040
  finally show ?thesis .
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1041
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1042
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1043
lemma mult_one_div_unit_factor [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1044
  "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
  1045
  by (cases "b = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1046
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1047
lemma normalize_mult:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1048
  "normalize (a * b) = normalize a * normalize b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1049
proof (cases "a = 0 \<or> b = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1050
  case True then show ?thesis by auto
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1051
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1052
  case False
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1053
  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
  1054
  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
  1055
  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
  1056
  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
  1057
    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
  1058
  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
  1059
    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
  1060
  also have "\<dots> = normalize a * normalize b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1061
    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
  1062
  finally show ?thesis .
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1063
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1064
 
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1065
lemma unit_factor_idem [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1066
  "unit_factor (unit_factor a) = unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1067
  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
  1068
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1069
lemma normalize_unit_factor [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1070
  "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1071
  by (rule is_unit_normalize) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1072
  
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1073
lemma normalize_idem [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1074
  "normalize (normalize a) = normalize a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1075
proof (cases "a = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1076
  case True then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1077
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1078
  case False
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1079
  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
  1080
  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
  1081
    by (simp only: normalize_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1082
  finally show ?thesis using False by simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1083
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1084
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1085
lemma unit_factor_normalize [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1086
  assumes "a \<noteq> 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1087
  shows "unit_factor (normalize a) = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1088
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1089
  from assms have "normalize a \<noteq> 0" by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1090
  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
  1091
    by (simp only: unit_factor_mult_normalize)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1092
  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
  1093
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1094
  with \<open>normalize a \<noteq> 0\<close>
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1095
  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
  1096
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1097
  with \<open>normalize a \<noteq> 0\<close>
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1098
  show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1099
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1100
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1101
lemma dvd_unit_factor_div:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1102
  assumes "b dvd a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1103
  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
  1104
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1105
  from assms have "a = a div b * b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1106
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1107
  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
  1108
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1109
  then show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1110
    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
  1111
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1112
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1113
lemma dvd_normalize_div:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1114
  assumes "b dvd a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1115
  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
  1116
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1117
  from assms have "a = a div b * b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1118
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1119
  then have "normalize a = normalize (a div b * b)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1120
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1121
  then show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1122
    by (cases "b = 0") (simp_all add: normalize_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1123
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1124
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1125
lemma normalize_dvd_iff [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1126
  "normalize a dvd b \<longleftrightarrow> a dvd b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1127
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1128
  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
  1129
    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
  1130
      by (cases "a = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1131
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1132
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1133
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1134
lemma dvd_normalize_iff [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1135
  "a dvd normalize b \<longleftrightarrow> a dvd b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1136
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1137
  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
  1138
    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
  1139
      by (cases "b = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1140
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1141
qed
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 associated_normalize_left [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1144
  "associated (normalize a) b \<longleftrightarrow> associated a b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1145
  by (auto simp add: associated_def)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1146
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1147
lemma associated_normalize_right [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1148
  "associated a (normalize b) \<longleftrightarrow> associated a b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1149
  by (auto simp add: associated_def)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1150
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1151
lemma associated_iff_normalize:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1152
  "associated a b \<longleftrightarrow> normalize a = normalize b" (is "?P \<longleftrightarrow> ?Q")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1153
proof (cases "a = 0 \<or> b = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1154
  case True then show ?thesis by auto
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1155
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1156
  case False
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1157
  show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1158
  proof
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1159
    assume ?P then show ?Q
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1160
      by (rule associated_is_unitE) (simp add: normalize_mult is_unit_normalize)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1161
  next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1162
    from False have *: "is_unit (unit_factor a div unit_factor b)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1163
      by auto
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1164
    assume ?Q then have "unit_factor a * normalize a = unit_factor a * normalize b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1165
      by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1166
    then have "a = unit_factor a * (b div unit_factor b)"
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 False have "a = (unit_factor a div unit_factor b) * b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1169
      by (simp add: unit_div_commute unit_div_mult_swap [symmetric])
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1170
    with * show ?P 
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1171
      by (rule is_unit_associatedI)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1172
  qed
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 associated_eqI:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1176
  assumes "associated a b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1177
  assumes "a \<noteq> 0 \<Longrightarrow> unit_factor a = 1" and "b \<noteq> 0 \<Longrightarrow> unit_factor b = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1178
  shows "a = b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1179
proof (cases "a = 0")
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1180
  case True with assms show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1181
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1182
  case False with assms have "b \<noteq> 0" by auto
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1183
  with False assms have "unit_factor a = unit_factor b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1184
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1185
  moreover from assms have "normalize a = normalize b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1186
    by (simp add: associated_iff_normalize)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1187
  ultimately have "unit_factor a * normalize a = unit_factor b * normalize b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1188
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1189
  then show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1190
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1191
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1192
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1193
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1194
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  1195
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  1196
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  1197
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1198
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1199
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1200
lemma mult_mono:
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  1201
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1202
apply (erule mult_right_mono [THEN order_trans], assumption)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1203
apply (erule mult_left_mono, assumption)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1204
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1205
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1206
lemma mult_mono':
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  1207
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1208
apply (rule mult_mono)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1209
apply (fast intro: order_trans)+
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1210
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1211
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1212
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
  1213
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  1214
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  1215
begin
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1216
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
  1217
subclass semiring_0_cancel ..
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
  1218
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56480
diff changeset
  1219
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1220
using mult_left_mono [of 0 b a] by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1221
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1222
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1223
using mult_left_mono [of b 0 a] by simp
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  1224
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  1225
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1226
using mult_right_mono [of a 0 b] by simp
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  1227
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  1228
text {* Legacy - use @{text mult_nonpos_nonneg} *}
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
  1229
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1230
by (drule mult_right_mono [of b 0], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1231
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
  1232
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1233
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1234
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1235
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1236
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  1237
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  1238
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1239
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1240
subclass ordered_cancel_semiring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1241
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1242
subclass ordered_comm_monoid_add ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1243
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1244
lemma mult_left_less_imp_less:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1245
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1246
by (force simp add: mult_left_mono not_le [symmetric])
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
  1247
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1248
lemma mult_right_less_imp_less:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1249
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1250
by (force simp add: mult_right_mono not_le [symmetric])
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
  1251
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
  1252
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
  1253
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
  1254
class linordered_semiring_1 = linordered_semiring + semiring_1
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1255
begin
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1256
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1257
lemma convex_bound_le:
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1258
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1259
  shows "u * x + v * y \<le> a"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1260
proof-
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1261
  from assms have "u * x + v * y \<le> u * a + v * a"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1262
    by (simp add: add_mono mult_left_mono)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
  1263
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1264
qed
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1265
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1266
end
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
  1267
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
  1268
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1269
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1270
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  1271
begin
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
  1272
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
  1273
subclass semiring_0_cancel ..
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
  1274
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1275
subclass linordered_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
  1276
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
  1277
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
  1278
  assume A: "a \<le> b" "0 \<le> c"
d4f1d6ef119c convert instance proofs to Isar style
huffman