src/HOL/Rings.thy
author Manuel Eberl <eberlm@in.tum.de>
Fri, 01 Jul 2016 08:35:15 +0200
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permissions -rw-r--r--
More lemmas on Gcd/Lcm
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(*  Title:      HOL/Rings.thy
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    Author:     Gertrud Bauer
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    Author:     Steven Obua
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Markus Wenzel
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    Author:     Jeremy Avigad
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*)
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section \<open>Rings\<close>
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theory Rings
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imports Groups Set
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begin
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c"
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  assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
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begin
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text \<open>For the \<open>combine_numerals\<close> simproc\<close>
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lemma combine_common_factor: "a * e + (b * e + c) = (a + b) * e + c"
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  by (simp add: distrib_right ac_simps)
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end
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "0 * a = 0"
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  assumes mult_zero_right [simp]: "a * 0 = 0"
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begin
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lemma mult_not_zero: "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
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  by auto
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end
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
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  then show "0 * a = 0" by (simp only: add_left_cancel)
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  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
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  then show "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 "\<dots> = b * a + c * a" by (simp only: distrib)
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  also have "\<dots> = 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 "of_bool p = (if p then 1 else 0)"
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lemma of_bool_eq [simp, code]:
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  "of_bool False = 0"
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  "of_bool True = 1"
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  by (simp_all add: of_bool_def)
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lemma of_bool_eq_iff: "of_bool p = of_bool q \<longleftrightarrow> p = q"
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  by (simp add: of_bool_def)
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lemma split_of_bool [split]: "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
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  by (cases p) simp_all
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lemma split_of_bool_asm: "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
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  by (cases p) simp_all
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end
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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text \<open>Abstract divisibility\<close>
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class dvd = times
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begin
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definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50)
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  where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
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lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
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  unfolding dvd_def ..
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding dvd_def by blast
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end
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context comm_monoid_mult
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begin
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subclass dvd .
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lemma dvd_refl [simp]: "a dvd a"
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proof
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  show "a = a * 1" by simp
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qed
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lemma dvd_trans [trans]:
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  assumes "a dvd b" and "b dvd c"
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  shows "a dvd c"
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proof -
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  from assms obtain v where "b = a * v" by (auto elim!: dvdE)
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  moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
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  ultimately have "c = a * (v * w)" by (simp add: mult.assoc)
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  then show ?thesis ..
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qed
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lemma subset_divisors_dvd: "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
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  by (auto simp add: subset_iff intro: dvd_trans)
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parents: 62349
diff changeset
   151
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   152
lemma strict_subset_divisors_dvd: "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
62366
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haftmann
parents: 62349
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   153
  by (auto simp add: subset_iff intro: dvd_trans)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   154
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   155
lemma one_dvd [simp]: "1 dvd a"
59009
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haftmann
parents: 59000
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   156
  by (auto intro!: dvdI)
28559
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   157
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   158
lemma dvd_mult [simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   159
  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
   160
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   161
lemma dvd_mult2 [simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   162
  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
   163
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   164
lemma dvd_triv_right [simp]: "a dvd b * a"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   165
  by (rule dvd_mult) (rule dvd_refl)
27651
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haftmann
parents: 27516
diff changeset
   166
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   167
lemma dvd_triv_left [simp]: "a dvd a * b"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   168
  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
   169
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
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   170
lemma mult_dvd_mono:
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   171
  assumes "a dvd b"
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
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   172
    and "c dvd d"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   173
  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
   174
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
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parents: 60690
diff changeset
   175
  from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   176
  moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" ..
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   177
  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
   178
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   179
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   180
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   181
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
59009
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haftmann
parents: 59000
diff changeset
   182
  by (simp add: dvd_def mult.assoc) blast
27651
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haftmann
parents: 27516
diff changeset
   183
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   184
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
59009
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haftmann
parents: 59000
diff changeset
   185
  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
   186
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   187
end
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   188
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   189
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
   190
begin
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   191
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   192
subclass semiring_1 ..
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   193
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diff changeset
   194
lemma dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   195
  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
   196
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   197
lemma dvd_0_right [iff]: "a dvd 0"
59009
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haftmann
parents: 59000
diff changeset
   198
proof
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   199
  show "0 = a * 0" by simp
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   200
qed
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   201
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   202
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
59009
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haftmann
parents: 59000
diff changeset
   203
  by simp
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   204
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   205
lemma dvd_add [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   206
  assumes "a dvd b" and "a dvd c"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   207
  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
   208
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   209
  from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   210
  moreover from \<open>a dvd c\<close> obtain c' where "c = a * c'" ..
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   211
  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
   212
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   213
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   214
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   215
end
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   216
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   217
class semiring_1_cancel = semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   218
  + zero_neq_one + monoid_mult
25267
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haftmann
parents: 25238
diff changeset
   219
begin
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   220
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   221
subclass semiring_0_cancel ..
25512
4134f7c782e2 using intro_locales instead of unfold_locales if appropriate
haftmann
parents: 25450
diff changeset
   222
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   223
subclass semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
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parents: 25238
diff changeset
   224
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   225
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   226
63325
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diff changeset
   227
class comm_semiring_1_cancel =
1086d56cde86 misc tuning and modernization;
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   228
  comm_semiring + cancel_comm_monoid_add + zero_neq_one + comm_monoid_mult +
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   229
  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
25267
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haftmann
parents: 25238
diff changeset
   230
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   231
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   232
subclass semiring_1_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   233
subclass comm_semiring_0_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   234
subclass comm_semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   235
63325
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diff changeset
   236
lemma left_diff_distrib' [algebra_simps]: "(b - c) * a = b * a - c * a"
59816
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haftmann
parents: 59557
diff changeset
   237
  by (simp add: algebra_simps)
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   238
63325
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diff changeset
   239
lemma dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b"
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   240
proof -
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   241
  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
   242
  proof
63325
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parents: 63040
diff changeset
   243
    assume ?Q
1086d56cde86 misc tuning and modernization;
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diff changeset
   244
    then show ?P by simp
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   245
  next
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   246
    assume ?P
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   247
    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
   248
    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
   249
    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
   250
    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
   251
    then show ?Q ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   252
  qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   253
  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
   254
qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   255
63325
1086d56cde86 misc tuning and modernization;
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parents: 63040
diff changeset
   256
lemma dvd_add_times_triv_right_iff [simp]: "a dvd b + c * a \<longleftrightarrow> a dvd b"
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   257
  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
   258
63325
1086d56cde86 misc tuning and modernization;
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diff changeset
   259
lemma dvd_add_triv_left_iff [simp]: "a dvd a + b \<longleftrightarrow> a dvd b"
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   260
  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
   261
63325
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parents: 63040
diff changeset
   262
lemma dvd_add_triv_right_iff [simp]: "a dvd b + a \<longleftrightarrow> a dvd b"
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   263
  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
   264
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   265
lemma dvd_add_right_iff:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   266
  assumes "a dvd b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   267
  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
   268
proof
63325
1086d56cde86 misc tuning and modernization;
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parents: 63040
diff changeset
   269
  assume ?P
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   270
  then obtain d where "b + c = a * d" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   271
  moreover from \<open>a dvd b\<close> obtain e where "b = a * e" ..
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   272
  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
   273
  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
   274
  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
   275
  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
   276
  then show ?Q ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   277
next
63325
1086d56cde86 misc tuning and modernization;
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diff changeset
   278
  assume ?Q
1086d56cde86 misc tuning and modernization;
wenzelm
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diff changeset
   279
  with assms show ?P by simp
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   280
qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   281
63325
1086d56cde86 misc tuning and modernization;
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diff changeset
   282
lemma dvd_add_left_iff: "a dvd c \<Longrightarrow> a dvd b + c \<longleftrightarrow> a dvd b"
1086d56cde86 misc tuning and modernization;
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parents: 63040
diff changeset
   283
  using dvd_add_right_iff [of a c b] by (simp add: ac_simps)
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   284
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   285
end
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   286
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   287
class ring = semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   288
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   289
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   290
subclass semiring_0_cancel ..
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   291
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   292
text \<open>Distribution rules\<close>
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   293
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   294
lemma minus_mult_left: "- (a * b) = - a * b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   295
  by (rule minus_unique) (simp add: distrib_right [symmetric])
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   296
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   297
lemma minus_mult_right: "- (a * b) = a * - b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   298
  by (rule minus_unique) (simp add: distrib_left [symmetric])
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   299
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   300
text \<open>Extract signs from products\<close>
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   301
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   302
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
   303
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   304
lemma minus_mult_minus [simp]: "- a * - b = a * b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   305
  by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   306
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   307
lemma minus_mult_commute: "- a * b = a * - b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   308
  by simp
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   309
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   310
lemma right_diff_distrib [algebra_simps]: "a * (b - c) = a * b - a * c"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   311
  using distrib_left [of a b "-c "] by simp
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   312
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   313
lemma left_diff_distrib [algebra_simps]: "(a - b) * c = a * c - b * c"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   314
  using distrib_right [of a "- b" c] by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   315
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   316
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   317
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   318
lemma eq_add_iff1: "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   319
  by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   320
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   321
lemma eq_add_iff2: "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   322
  by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   323
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   324
end
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   325
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   326
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   327
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   328
class comm_ring = comm_semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   329
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   330
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   331
subclass ring ..
28141
193c3ea0f63b instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents: 27651
diff changeset
   332
subclass comm_semiring_0_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   333
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   334
lemma square_diff_square_factored: "x * x - y * y = (x + y) * (x - y)"
44350
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   335
  by (simp add: algebra_simps)
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   336
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   337
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   338
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   339
class ring_1 = ring + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   340
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   341
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   342
subclass semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   343
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   344
lemma square_diff_one_factored: "x * x - 1 = (x + 1) * (x - 1)"
44346
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   345
  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
   346
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   347
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   348
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   349
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   350
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   351
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   352
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
   353
subclass comm_semiring_1_cancel
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   354
  by unfold_locales (simp add: algebra_simps)
58647
fce800afeec7 more facts about abstract divisibility
haftmann
parents: 58198
diff changeset
   355
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
   356
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
   357
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   358
  assume "x dvd - y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   359
  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
   360
  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
   361
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   362
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   363
  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
   364
  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
   365
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   366
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
   367
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
   368
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   369
  assume "- x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   370
  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
   371
  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
   372
  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
   373
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   374
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   375
  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
   376
  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
   377
  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
   378
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   379
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   380
lemma dvd_diff [simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   381
  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
   382
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   383
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   384
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   385
class semiring_no_zero_divisors = semiring_0 +
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   386
  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
   387
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   388
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   389
lemma divisors_zero:
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   390
  assumes "a * b = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   391
  shows "a = 0 \<or> b = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   392
proof (rule classical)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   393
  assume "\<not> ?thesis"
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   394
  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
   395
  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
   396
  with assms show ?thesis by simp
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   397
qed
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   398
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   399
lemma mult_eq_0_iff [simp]: "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   400
proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   401
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   402
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   403
    then show ?thesis using no_zero_divisors by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   404
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   405
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   406
  then show ?thesis by auto
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   407
qed
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   408
58952
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   409
end
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   410
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   411
class semiring_1_no_zero_divisors = semiring_1 + semiring_no_zero_divisors
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   412
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   413
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
   414
  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
   415
    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
   416
begin
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   417
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   418
lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   419
  by simp
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   420
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   421
lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   422
  by simp
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   423
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   424
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   425
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   426
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
   427
begin
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   428
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   429
subclass semiring_no_zero_divisors_cancel
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   430
proof
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   431
  fix a b c
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   432
  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
   433
    by (simp add: algebra_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   434
  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
   435
    by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   436
  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
   437
  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
   438
    by (simp add: algebra_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   439
  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
   440
    by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   441
  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
   442
qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   443
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   444
end
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   445
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   446
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   447
begin
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   448
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   449
subclass semiring_1_no_zero_divisors ..
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   450
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   451
lemma square_eq_1_iff: "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
36821
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   452
proof -
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   453
  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
   454
    by (simp add: algebra_simps)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   455
  then have "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
36821
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   456
    by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   457
  then show ?thesis
36821
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   458
    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
   459
qed
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   460
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   461
lemma mult_cancel_right1 [simp]: "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   462
  using mult_cancel_right [of 1 c b] by auto
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   463
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   464
lemma mult_cancel_right2 [simp]: "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   465
  using mult_cancel_right [of a c 1] by simp
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   466
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   467
lemma mult_cancel_left1 [simp]: "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   468
  using mult_cancel_left [of c 1 b] by force
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   469
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   470
lemma mult_cancel_left2 [simp]: "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   471
  using mult_cancel_left [of c a 1] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   472
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   473
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   474
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
   475
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   476
begin
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   477
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   478
subclass semiring_1_no_zero_divisors ..
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   479
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   480
end
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   481
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   482
class idom = comm_ring_1 + semiring_no_zero_divisors
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   483
begin
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   484
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   485
subclass semidom ..
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   486
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   487
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
   488
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   489
lemma dvd_mult_cancel_right [simp]: "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   490
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   491
  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
   492
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   493
  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
   494
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   495
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   496
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   497
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   498
lemma dvd_mult_cancel_left [simp]: "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   499
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   500
  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
   501
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   502
  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
   503
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   504
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   505
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   506
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   507
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
   508
proof
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   509
  assume "a * a = b * b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   510
  then have "(a - b) * (a + b) = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   511
    by (simp add: algebra_simps)
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   512
  then show "a = b \<or> a = - b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   513
    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
   514
next
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   515
  assume "a = b \<or> a = - b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   516
  then show "a * a = b * b" by auto
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   517
qed
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   518
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   519
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   520
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   521
text \<open>
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   522
  The theory of partially ordered rings is taken from the books:
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   523
    \<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   524
    \<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   525
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
   526
  Most of the used notions can also be looked up in
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   527
    \<^item> @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   528
    \<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   529
\<close>
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   530
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   531
class divide =
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   532
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   533
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   534
setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   535
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   536
context semiring
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   537
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   538
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   539
lemma [field_simps]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   540
  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
   541
    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
   542
  by (rule distrib_left distrib_right)+
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   543
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   544
end
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   545
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   546
context ring
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   547
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   548
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   549
lemma [field_simps]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   550
  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
   551
    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
   552
  by (rule left_diff_distrib right_diff_distrib)+
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   553
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   554
end
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   555
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   556
setup \<open>Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"})\<close>
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   557
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   558
class semidom_divide = semidom + divide +
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   559
  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
   560
  assumes divide_zero [simp]: "a div 0 = 0"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   561
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   562
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   563
lemma nonzero_mult_divide_cancel_left [simp]: "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   564
  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
   565
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   566
subclass semiring_no_zero_divisors_cancel
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   567
proof
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   568
  show *: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" for a b c
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   569
  proof (cases "c = 0")
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   570
    case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   571
    then show ?thesis by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   572
  next
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   573
    case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   574
    {
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   575
      assume "a * c = b * c"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   576
      then have "a * c div c = b * c div c"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   577
        by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   578
      with False have "a = b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   579
        by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   580
    }
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   581
    then show ?thesis by auto
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   582
  qed
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   583
  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" for a b c
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   584
    using * [of a c b] by (simp add: ac_simps)
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   585
qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   586
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   587
lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   588
  using nonzero_mult_divide_cancel_left [of a 1] by simp
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   589
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   590
lemma divide_zero_left [simp]: "0 div a = 0"
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   591
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   592
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   593
  then show ?thesis by simp
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   594
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   595
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   596
  then have "a * 0 div a = 0"
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   597
    by (rule nonzero_mult_divide_cancel_left)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   598
  then show ?thesis by simp
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   599
qed
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   600
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   601
lemma divide_1 [simp]: "a div 1 = a"
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   602
  using nonzero_mult_divide_cancel_left [of 1 a] by simp
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   603
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   604
end
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   605
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   606
class idom_divide = idom + semidom_divide
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   607
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   608
class algebraic_semidom = semidom_divide
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   609
begin
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   610
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   611
text \<open>
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   612
  Class @{class algebraic_semidom} enriches a integral domain
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   613
  by notions from algebra, like units in a ring.
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   614
  It is a separate class to avoid spoiling fields with notions
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   615
  which are degenerated there.
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   616
\<close>
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   617
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   618
lemma dvd_times_left_cancel_iff [simp]:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   619
  assumes "a \<noteq> 0"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   620
  shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   621
proof
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   622
  assume ?P
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   623
  then obtain d where "a * c = a * b * d" ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   624
  with assms have "c = b * d" by (simp add: ac_simps)
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   625
  then show ?Q ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   626
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   627
  assume ?Q
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   628
  then obtain d where "c = b * d" ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   629
  then have "a * c = a * b * d" by (simp add: ac_simps)
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   630
  then show ?P ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   631
qed
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   632
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   633
lemma dvd_times_right_cancel_iff [simp]:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   634
  assumes "a \<noteq> 0"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   635
  shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   636
  using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   637
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   638
lemma div_dvd_iff_mult:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   639
  assumes "b \<noteq> 0" and "b dvd a"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   640
  shows "a div b dvd c \<longleftrightarrow> a dvd c * b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   641
proof -
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   642
  from \<open>b dvd a\<close> obtain d where "a = b * d" ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   643
  with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps)
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   644
qed
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   645
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   646
lemma dvd_div_iff_mult:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   647
  assumes "c \<noteq> 0" and "c dvd b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   648
  shows "a dvd b div c \<longleftrightarrow> a * c dvd b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   649
proof -
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   650
  from \<open>c dvd b\<close> obtain d where "b = c * d" ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   651
  with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a])
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   652
qed
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   653
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   654
lemma div_dvd_div [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   655
  assumes "a dvd b" and "a dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   656
  shows "b div a dvd c div a \<longleftrightarrow> b dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   657
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   658
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   659
  with assms show ?thesis by simp
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   660
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   661
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   662
  moreover from assms obtain k l where "b = a * k" and "c = a * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   663
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   664
  ultimately show ?thesis by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   665
qed
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   666
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   667
lemma div_add [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   668
  assumes "c dvd a" and "c dvd b"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   669
  shows "(a + b) div c = a div c + b div c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   670
proof (cases "c = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   671
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   672
  then show ?thesis by simp
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   673
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   674
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   675
  moreover from assms obtain k l where "a = c * k" and "b = c * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   676
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   677
  moreover have "c * k + c * l = c * (k + l)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   678
    by (simp add: algebra_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   679
  ultimately show ?thesis
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   680
    by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   681
qed
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   682
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   683
lemma div_mult_div_if_dvd:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   684
  assumes "b dvd a" and "d dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   685
  shows "(a div b) * (c div d) = (a * c) div (b * d)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   686
proof (cases "b = 0 \<or> c = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   687
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   688
  with assms show ?thesis by auto
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   689
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   690
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   691
  moreover from assms obtain k l where "a = b * k" and "c = d * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   692
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   693
  moreover have "b * k * (d * l) div (b * d) = (b * d) * (k * l) div (b * d)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   694
    by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   695
  ultimately show ?thesis by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   696
qed
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   697
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   698
lemma dvd_div_eq_mult:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   699
  assumes "a \<noteq> 0" and "a dvd b"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   700
  shows "b div a = c \<longleftrightarrow> b = c * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   701
proof
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   702
  assume "b = c * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   703
  then show "b div a = c" by (simp add: assms)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   704
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   705
  assume "b div a = c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   706
  then have "b div a * a = c * a" by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   707
  moreover from assms have "b div a * a = b"
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   708
    by (auto elim!: dvdE simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   709
  ultimately show "b = c * a" by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   710
qed
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   711
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   712
lemma dvd_div_mult_self [simp]: "a dvd b \<Longrightarrow> b div a * a = b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   713
  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
   714
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   715
lemma dvd_mult_div_cancel [simp]: "a dvd b \<Longrightarrow> a * (b div a) = b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   716
  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
   717
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   718
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
   719
  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
   720
  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
   721
proof (cases "c = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   722
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   723
  then show ?thesis by simp
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   724
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   725
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   726
  from assms obtain d where "b = c * d" ..
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   727
  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
   728
    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
   729
  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
   730
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   731
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   732
lemma dvd_div_mult: "c dvd b \<Longrightarrow> b div c * a = (b * a) div c"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   733
  using div_mult_swap [of c b a] by (simp add: ac_simps)
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   734
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   735
lemma dvd_div_mult2_eq:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   736
  assumes "b * c dvd a"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   737
  shows "a div (b * c) = a div b div c"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   738
proof -
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   739
  from assms obtain k where "a = b * c * k" ..
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   740
  then show ?thesis
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   741
    by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   742
qed
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   743
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   744
lemma dvd_div_div_eq_mult:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   745
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   746
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d" (is "?P \<longleftrightarrow> ?Q")
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   747
proof -
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   748
  from assms have "a * c \<noteq> 0" by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   749
  then have "?P \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   750
    by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   751
  also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   752
    by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   753
  also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   754
    using assms by (simp add: div_mult_swap)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   755
  also have "\<dots> \<longleftrightarrow> ?Q"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   756
    using assms by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   757
  finally show ?thesis .
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   758
qed
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   759
63359
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   760
lemma dvd_mult_imp_div:
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   761
  assumes "a * c dvd b"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   762
  shows "a dvd b div c"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   763
proof (cases "c = 0")
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   764
  case True then show ?thesis by simp
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   765
next
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   766
  case False
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   767
  from \<open>a * c dvd b\<close> obtain d where "b = a * c * d" ..
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   768
  with False show ?thesis by (simp add: mult.commute [of a] mult.assoc)
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   769
qed
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   770
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
   771
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   772
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
   773
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   774
abbreviation is_unit :: "'a \<Rightarrow> bool"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   775
  where "is_unit a \<equiv> a dvd 1"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   776
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   777
lemma not_is_unit_0 [simp]: "\<not> is_unit 0"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   778
  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
   779
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   780
lemma unit_imp_dvd [dest]: "is_unit b \<Longrightarrow> b dvd a"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   781
  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
   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_dvdE:
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   784
  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
   785
  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
   786
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   787
  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
   788
  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
   789
  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
   790
  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
   791
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   792
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   793
lemma dvd_unit_imp_unit: "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   794
  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
   795
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   796
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
   797
  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
   798
  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
   799
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   800
  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
   801
  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
   802
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   803
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   804
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
   805
  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
   806
  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
   807
    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
   808
    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
   809
proof (rule that)
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62626
diff changeset
   810
  define b where "b = 1 div a"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   811
  then show "1 div a = b" by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   812
  from assms b_def show "is_unit b" by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   813
  with assms show "a \<noteq> 0" and "b \<noteq> 0" by auto
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   814
  from assms b_def show "a * b = 1" by simp
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   815
  then have "1 = a * b" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   816
  with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   817
  from assms have "a dvd c" ..
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   818
  then obtain d where "c = a * d" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   819
  with \<open>a \<noteq> 0\<close> \<open>a * b = 1\<close> show "c div a = c * b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   820
    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
   821
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   822
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   823
lemma unit_prod [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
   824
  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
   825
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   826
lemma is_unit_mult_iff: "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b"
62366
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   827
  by (auto dest: dvd_mult_left dvd_mult_right)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   828
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   829
lemma unit_div [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   830
  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
   831
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   832
lemma 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
   833
  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
   834
  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
   835
proof
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   836
  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
   837
  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
   838
    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
   839
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   840
  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
   841
  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
   842
  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
   843
    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
   844
  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
   845
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   846
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   847
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
   848
  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
   849
  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
   850
proof
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   851
  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
   852
  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
   853
    by (subst mult_assoc [symmetric]) simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   854
  also from assms have "b * (1 div b) = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   855
    by (rule is_unitE) simp
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   856
  finally have "c * b dvd c" by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   857
  with \<open>a dvd c * b\<close> show "a dvd c" by (rule dvd_trans)
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   858
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   859
  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
   860
  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
   861
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   862
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   863
lemma div_unit_dvd_iff: "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   864
  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
   865
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   866
lemma dvd_div_unit_iff: "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   867
  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
   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
lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   870
  dvd_mult_unit_iff dvd_div_unit_iff  (* FIXME consider named_theorems *)
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   871
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   872
lemma unit_mult_div_div [simp]: "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   873
  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
   874
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   875
lemma unit_div_mult_self [simp]: "is_unit a \<Longrightarrow> b div a * a = b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   876
  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
   877
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   878
lemma unit_div_1_div_1 [simp]: "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   879
  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
   880
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   881
lemma unit_div_mult_swap: "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   882
  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
   883
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   884
lemma unit_div_commute: "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   885
  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
   886
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   887
lemma unit_eq_div1: "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   888
  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
   889
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   890
lemma unit_eq_div2: "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   891
  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
   892
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   893
lemma unit_mult_left_cancel: "is_unit a \<Longrightarrow> a * b = a * c \<longleftrightarrow> b = c"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   894
  using mult_cancel_left [of a b c] by auto
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   895
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   896
lemma unit_mult_right_cancel: "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   897
  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
   898
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   899
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
   900
  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
   901
  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
   902
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   903
  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
   904
  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
   905
    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
   906
  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
   907
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
   908
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   909
lemma is_unit_div_mult2_eq:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   910
  assumes "is_unit b" and "is_unit c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   911
  shows "a div (b * c) = a div b div c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   912
proof -
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   913
  from assms have "is_unit (b * c)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   914
    by (simp add: unit_prod)
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   915
  then have "b * c dvd a"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   916
    by (rule unit_imp_dvd)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   917
  then show ?thesis
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   918
    by (rule dvd_div_mult2_eq)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   919
qed
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   920
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
   921
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
   922
  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
   923
  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
   924
  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
   925
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   926
lemma is_unit_divide_mult_cancel_left:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   927
  assumes "a \<noteq> 0" and "is_unit b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   928
  shows "a div (a * b) = 1 div b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   929
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   930
  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
   931
    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
   932
  with assms show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   933
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   934
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   935
lemma is_unit_divide_mult_cancel_right:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   936
  assumes "a \<noteq> 0" and "is_unit b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   937
  shows "a div (b * a) = 1 div b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   938
  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
   939
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   940
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   941
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   942
class normalization_semidom = algebraic_semidom +
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   943
  fixes normalize :: "'a \<Rightarrow> 'a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   944
    and unit_factor :: "'a \<Rightarrow> 'a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   945
  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
   946
  assumes normalize_0 [simp]: "normalize 0 = 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   947
    and unit_factor_0 [simp]: "unit_factor 0 = 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   948
  assumes is_unit_normalize:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   949
    "is_unit a  \<Longrightarrow> normalize a = 1"
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   950
  assumes unit_factor_is_unit [iff]:
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   951
    "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   952
  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
   953
begin
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   954
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   955
text \<open>
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   956
  Class @{class normalization_semidom} cultivates the idea that
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   957
  each integral domain can be split into equivalence classes
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   958
  whose representants are associated, i.e. divide each other.
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   959
  @{const normalize} specifies a canonical representant for each equivalence
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   960
  class.  The rationale behind this is that it is easier to reason about equality
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   961
  than equivalences, hence we prefer to think about equality of normalized
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   962
  values rather than associated elements.
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   963
\<close>
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   964
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   965
lemma unit_factor_dvd [simp]: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   966
  by (rule unit_imp_dvd) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   967
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   968
lemma unit_factor_self [simp]: "unit_factor a dvd a"
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   969
  by (cases "a = 0") simp_all
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   970
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   971
lemma normalize_mult_unit_factor [simp]: "normalize a * unit_factor a = a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   972
  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
   973
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   974
lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   975
  (is "?P \<longleftrightarrow> ?Q")
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   976
proof
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   977
  assume ?P
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   978
  moreover have "unit_factor a * normalize a = a" by simp
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   979
  ultimately show ?Q by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   980
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   981
  assume ?Q
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   982
  then show ?P by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   983
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   984
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   985
lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   986
  (is "?P \<longleftrightarrow> ?Q")
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   987
proof
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   988
  assume ?P
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   989
  moreover have "unit_factor a * normalize a = a" by simp
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   990
  ultimately show ?Q by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   991
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   992
  assume ?Q
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   993
  then show ?P by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   994
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   995
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
   996
lemma is_unit_unit_factor:
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   997
  assumes "is_unit a"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   998
  shows "unit_factor a = a"
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
   999
proof -
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1000
  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
  1001
  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
  1002
  ultimately show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1003
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1004
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1005
lemma unit_factor_1 [simp]: "unit_factor 1 = 1"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1006
  by (rule is_unit_unit_factor) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1007
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1008
lemma normalize_1 [simp]: "normalize 1 = 1"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1009
  by (rule is_unit_normalize) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1010
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1011
lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1012
  (is "?P \<longleftrightarrow> ?Q")
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1013
proof
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1014
  assume ?Q
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1015
  then show ?P by (rule is_unit_normalize)
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1016
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1017
  assume ?P
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1018
  then have "a \<noteq> 0" by auto
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1019
  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
  1020
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1021
  then have "unit_factor a = a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1022
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1023
  moreover have "is_unit (unit_factor a)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1024
    using \<open>a \<noteq> 0\<close> by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1025
  ultimately show ?Q by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1026
qed
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
  1027
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1028
lemma div_normalize [simp]: "a div normalize a = unit_factor a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1029
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1030
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1031
  then show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1032
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1033
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1034
  then have "normalize a \<noteq> 0" by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1035
  with nonzero_mult_divide_cancel_right
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1036
  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
  1037
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1038
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1039
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1040
lemma div_unit_factor [simp]: "a div unit_factor a = normalize a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1041
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1042
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1043
  then show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1044
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1045
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1046
  then have "unit_factor a \<noteq> 0" by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1047
  with nonzero_mult_divide_cancel_left
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1048
  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
  1049
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1050
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1051
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1052
lemma normalize_div [simp]: "normalize a div a = 1 div unit_factor a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1053
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1054
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1055
  then show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1056
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1057
  case False
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1058
  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
  1059
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1060
  also have "\<dots> = 1 div unit_factor a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1061
    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
  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
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1065
lemma mult_one_div_unit_factor [simp]: "a * (1 div unit_factor b) = a div unit_factor b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1066
  by (cases "b = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1067
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1068
lemma normalize_mult: "normalize (a * b) = normalize a * normalize b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1069
proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1070
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1071
  then show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1072
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1073
  case False
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1074
  from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" .
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1075
  then have "normalize (a * b) = a * b div unit_factor (a * b)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1076
    by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1077
  also have "\<dots> = a * b div unit_factor (b * a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1078
    by (simp add: ac_simps)
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1079
  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
  1080
    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
  1081
  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
  1082
    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
  1083
  also have "\<dots> = normalize a * normalize b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1084
    using False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1085
    by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1086
  finally show ?thesis .
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1087
qed
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
  1088
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1089
lemma unit_factor_idem [simp]: "unit_factor (unit_factor a) = unit_factor a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1090
  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
  1091
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1092
lemma normalize_unit_factor [simp]: "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1093
  by (rule is_unit_normalize) simp
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
  1094
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1095
lemma normalize_idem [simp]: "normalize (normalize a) = normalize a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1096
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1097
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1098
  then show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1099
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1100
  case False
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1101
  have "normalize a = normalize (unit_factor a * normalize a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1102
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1103
  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
  1104
    by (simp only: normalize_mult)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1105
  finally show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1106
    using False by simp_all
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1107
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1108
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1109
lemma unit_factor_normalize [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1110
  assumes "a \<noteq> 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1111
  shows "unit_factor (normalize a) = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1112
proof -
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1113
  from assms have *: "normalize a \<noteq> 0"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1114
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1115
  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
  1116
    by (simp only: unit_factor_mult_normalize)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1117
  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
  1118
    by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1119
  with * have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1120
    by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1121
  with * show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1122
    by simp
60685
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 dvd_unit_factor_div:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1126
  assumes "b dvd a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1127
  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
  1128
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1129
  from assms have "a = a div b * b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1130
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1131
  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
  1132
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1133
  then show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1134
    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
  1135
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1136
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1137
lemma dvd_normalize_div:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1138
  assumes "b dvd a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1139
  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
  1140
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1141
  from assms have "a = a div b * b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1142
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1143
  then have "normalize a = normalize (a div b * b)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1144
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1145
  then show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1146
    by (cases "b = 0") (simp_all add: normalize_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1147
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1148
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1149
lemma normalize_dvd_iff [simp]: "normalize a dvd b \<longleftrightarrow> a dvd b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1150
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1151
  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
  1152
    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
  1153
      by (cases "a = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1154
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1155
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1156
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1157
lemma dvd_normalize_iff [simp]: "a dvd normalize b \<longleftrightarrow> a dvd b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1158
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1159
  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
  1160
    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
  1161
      by (cases "b = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1162
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1163
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1164
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1165
text \<open>
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1166
  We avoid an explicit definition of associated elements but prefer
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1167
  explicit normalisation instead.  In theory we could define an abbreviation
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1168
  like @{prop "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1169
  counterproductive without suggestive infix syntax, which we do not want
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1170
  to sacrifice for this purpose here.
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1171
\<close>
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1172
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1173
lemma associatedI:
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1174
  assumes "a dvd b" and "b dvd a"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1175
  shows "normalize a = normalize b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1176
proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1177
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1178
  with assms show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1179
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1180
  case False
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1181
  from \<open>a dvd b\<close> obtain c where b: "b = a * c" ..
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1182
  moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1183
  ultimately have "b * 1 = b * (c * d)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1184
    by (simp add: ac_simps)
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1185
  with False have "1 = c * d"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1186
    unfolding mult_cancel_left by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1187
  then have "is_unit c" and "is_unit d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1188
    by auto
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1189
  with a b show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1190
    by (simp add: normalize_mult is_unit_normalize)
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1191
qed
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1192
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1193
lemma associatedD1: "normalize a = normalize b \<Longrightarrow> a dvd b"
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1194
  using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric]
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1195
  by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1196
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1197
lemma associatedD2: "normalize a = normalize b \<Longrightarrow> b dvd a"
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1198
  using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric]
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1199
  by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1200
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1201
lemma associated_unit: "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1202
  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1203
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1204
lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1205
  (is "?P \<longleftrightarrow> ?Q")
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1206
proof
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1207
  assume ?Q
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1208
  then show ?P by (auto intro!: associatedI)
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1209
next
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1210
  assume ?P
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1211
  then have "unit_factor a * normalize a = unit_factor a * normalize b"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1212
    by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1213
  then have *: "normalize b * unit_factor a = a"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1214
    by (simp add: ac_simps)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1215
  show ?Q
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1216
  proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1217
    case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1218
    with \<open>?P\<close> show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1219
  next
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
  1220
    case False
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1221
    then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1222
      by (simp_all add: mult_unit_dvd_iff dvd_mult_unit_iff)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1223
    with * show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1224
  qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1225
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1226
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1227
lemma associated_eqI:
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1228
  assumes "a dvd b" and "b dvd a"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1229
  assumes "normalize a = a" and "normalize b = b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1230
  shows "a = b"
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1231
proof -
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1232
  from assms have "normalize a = normalize b"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1233
    unfolding associated_iff_dvd by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1234
  with \<open>normalize a = a\<close> have "a = normalize b" by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1235
  with \<open>normalize b = b\<close> show "a = b" by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1236
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1237
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1238
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1239
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62366
diff changeset
  1240
class ordered_semiring = semiring + ordered_comm_monoid_add +
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  1241
  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
  1242
  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
  1243
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1244
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1245
lemma mult_mono: "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1246
  apply (erule (1) mult_right_mono [THEN order_trans])
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1247
  apply (erule (1) mult_left_mono)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1248
  done
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1249
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1250
lemma mult_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1251
  apply (rule mult_mono)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1252
  apply (fast intro: order_trans)+
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1253
  done
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1254
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1255
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
  1256
62377
ace69956d018 moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents: 62376
diff changeset
  1257
class ordered_semiring_0 = semiring_0 + ordered_semiring
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  1258
begin
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1259
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1260
lemma mult_nonneg_nonneg [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1261
  using mult_left_mono [of 0 b a] by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1262
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1263
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1264
  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
  1265
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  1266
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1267
  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
  1268
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61762
diff changeset
  1269
text \<open>Legacy - use \<open>mult_nonpos_nonneg\<close>\<close>
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
  1270
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1271
  apply (drule mult_right_mono [of b 0])
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1272
  apply auto
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1273
  done
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1274