src/HOL/Rings.thy
author wenzelm
Fri, 05 Apr 2019 17:05:32 +0200
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auxiliary operation for common uses of 'compile_generated_files';
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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 Fun
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begin
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c"
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  assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
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begin
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text \<open>For the \<open>combine_numerals\<close> simproc\<close>
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lemma combine_common_factor: "a * e + (b * e + c) = (a + b) * e + c"
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  by (simp add: distrib_right ac_simps)
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end
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "0 * a = 0"
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  assumes mult_zero_right [simp]: "a * 0 = 0"
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begin
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lemma mult_not_zero: "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
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  by auto
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end
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0"
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    by (simp add: distrib_right [symmetric])
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  then show "0 * a = 0"
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    by (simp only: add_left_cancel)
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  have "a * 0 + a * 0 = a * 0 + 0"
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    by (simp add: distrib_left [symmetric])
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  then show "a * 0 = 0"
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    by (simp only: add_left_cancel)
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qed
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end
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a + b) * c = a * c + b * c"
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begin
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subclass semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c"
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    by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a"
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    by (simp add: ac_simps)
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  also have "\<dots> = b * a + c * a"
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    by (simp only: distrib)
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  also have "\<dots> = a * b + a * c"
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    by (simp add: ac_simps)
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  finally show "a * (b + c) = a * b + a * c"
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    by blast
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qed
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end
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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begin
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subclass semiring_0 ..
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end
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class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0_cancel ..
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subclass comm_semiring_0 ..
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end
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "0 \<noteq> 1"
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begin
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lemma one_neq_zero [simp]: "1 \<noteq> 0"
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  by (rule not_sym) (rule zero_neq_one)
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definition of_bool :: "bool \<Rightarrow> 'a"
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  where "of_bool p = (if p then 1 else 0)"
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lemma of_bool_eq [simp, code]:
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  "of_bool False = 0"
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  "of_bool True = 1"
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  by (simp_all add: of_bool_def)
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lemma of_bool_eq_iff: "of_bool p = of_bool q \<longleftrightarrow> p = q"
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  by (simp add: of_bool_def)
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lemma split_of_bool [split]: "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
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  by (cases p) simp_all
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lemma split_of_bool_asm: "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
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  by (cases p) simp_all
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end
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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begin
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lemma (in semiring_1) of_bool_conj:
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  "of_bool (P \<and> Q) = of_bool P * of_bool Q"
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  by auto
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end
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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"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   157
  shows "a dvd c"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   158
proof -
63588
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wenzelm
parents: 63456
diff changeset
   159
  from assms obtain v where "b = a * v"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   160
    by (auto elim!: dvdE)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   161
  moreover from assms obtain w where "c = b * w"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   162
    by (auto elim!: dvdE)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   163
  ultimately have "c = a * (v * w)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   164
    by (simp add: mult.assoc)
28559
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   165
  then show ?thesis ..
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   166
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   167
63325
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   168
lemma subset_divisors_dvd: "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
62366
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   169
  by (auto simp add: subset_iff intro: dvd_trans)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   170
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   171
lemma strict_subset_divisors_dvd: "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
62366
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   172
  by (auto simp add: subset_iff intro: dvd_trans)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
   173
63325
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   174
lemma one_dvd [simp]: "1 dvd a"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   175
  by (auto intro!: dvdI)
28559
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   176
63325
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diff changeset
   177
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
   178
  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
   179
63325
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diff changeset
   180
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
   181
  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
   182
63325
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diff changeset
   183
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
   184
  by (rule dvd_mult) (rule dvd_refl)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   185
63325
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diff changeset
   186
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
   187
  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
   188
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   189
lemma mult_dvd_mono:
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   190
  assumes "a dvd b"
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   191
    and "c dvd d"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   192
  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
   193
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   194
  from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   195
  moreover from \<open>c dvd d\<close> obtain d' where "d = c * d'" ..
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   196
  ultimately have "b * d = (a * c) * (b' * d')"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   197
    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
   198
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   199
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   200
63325
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diff changeset
   201
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   202
  by (simp add: dvd_def mult.assoc) blast
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   203
63325
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diff changeset
   204
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   205
  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
   206
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   207
end
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   208
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   209
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
   210
begin
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   211
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   212
subclass semiring_1 ..
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   213
63325
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diff changeset
   214
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
   215
  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
   216
63325
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diff changeset
   217
lemma dvd_0_right [iff]: "a dvd 0"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   218
proof
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   219
  show "0 = a * 0" by simp
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   220
qed
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   221
63325
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diff changeset
   222
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
59009
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   223
  by simp
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   224
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   225
lemma dvd_add [simp]:
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   226
  assumes "a dvd b" and "a dvd c"
348561aa3869 generalized lemmas (particularly concerning dvd) as far as appropriate
haftmann
parents: 59000
diff changeset
   227
  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
   228
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   229
  from \<open>a dvd b\<close> obtain b' where "b = a * b'" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   230
  moreover from \<open>a dvd c\<close> obtain c' where "c = a * c'" ..
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   231
  ultimately have "b + c = a * (b' + c')"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   232
    by (simp add: distrib_left)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   233
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   234
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   235
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   236
end
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   237
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   238
class semiring_1_cancel = semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   239
  + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   240
begin
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   241
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   242
subclass semiring_0_cancel ..
25512
4134f7c782e2 using intro_locales instead of unfold_locales if appropriate
haftmann
parents: 25450
diff changeset
   243
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   244
subclass semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   245
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   246
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   247
63325
1086d56cde86 misc tuning and modernization;
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diff changeset
   248
class comm_semiring_1_cancel =
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   249
  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
   250
  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   251
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   252
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   253
subclass semiring_1_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   254
subclass comm_semiring_0_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   255
subclass comm_semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   256
63325
1086d56cde86 misc tuning and modernization;
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diff changeset
   257
lemma left_diff_distrib' [algebra_simps]: "(b - c) * a = b * a - c * a"
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   258
  by (simp add: algebra_simps)
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   259
63325
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diff changeset
   260
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
   261
proof -
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   262
  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
   263
  proof
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   264
    assume ?Q
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   265
    then show ?P by simp
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   266
  next
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   267
    assume ?P
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   268
    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
   269
    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
   270
    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
   271
    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
   272
    then show ?Q ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   273
  qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   274
  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
   275
qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   276
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   277
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
   278
  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
   279
63325
1086d56cde86 misc tuning and modernization;
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parents: 63040
diff changeset
   280
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
   281
  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
   282
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   283
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
   284
  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
   285
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   286
lemma dvd_add_right_iff:
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   287
  assumes "a dvd b"
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   288
  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
   289
proof
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   290
  assume ?P
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   291
  then obtain d where "b + c = a * d" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   292
  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
   293
  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
   294
  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
   295
  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
   296
  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
   297
  then show ?Q ..
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   298
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   299
  assume ?Q
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   300
  with assms show ?P by simp
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   301
qed
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   302
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   303
lemma dvd_add_left_iff: "a dvd c \<Longrightarrow> a dvd b + c \<longleftrightarrow> a dvd b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   304
  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
   305
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   306
end
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   307
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   308
class ring = semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   309
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   310
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   311
subclass semiring_0_cancel ..
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   312
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   313
text \<open>Distribution rules\<close>
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   314
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   315
lemma minus_mult_left: "- (a * b) = - a * b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   316
  by (rule minus_unique) (simp add: distrib_right [symmetric])
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   317
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   318
lemma minus_mult_right: "- (a * b) = a * - b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   319
  by (rule minus_unique) (simp add: distrib_left [symmetric])
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   320
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   321
text \<open>Extract signs from products\<close>
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   322
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   323
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
   324
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   325
lemma minus_mult_minus [simp]: "- a * - b = a * b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   326
  by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   327
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   328
lemma minus_mult_commute: "- a * b = a * - b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   329
  by simp
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   330
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   331
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
   332
  using distrib_left [of a b "-c "] by simp
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   333
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   334
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
   335
  using distrib_right [of a "- b" c] by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   336
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   337
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   338
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   339
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
   340
  by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   341
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   342
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
   343
  by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   344
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   345
end
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   346
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   347
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   348
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   349
class comm_ring = comm_semiring + ab_group_add
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 ..
28141
193c3ea0f63b instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents: 27651
diff changeset
   353
subclass comm_semiring_0_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   354
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   355
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
   356
  by (simp add: algebra_simps)
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   357
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   358
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   359
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   360
class ring_1 = ring + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   361
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   362
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   363
subclass semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   364
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   365
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
   366
  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
   367
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   368
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   369
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   370
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   371
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   372
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   373
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
   374
subclass comm_semiring_1_cancel
59816
034b13f4efae distributivity of partial minus establishes desired properties of dvd in semirings
haftmann
parents: 59557
diff changeset
   375
  by unfold_locales (simp add: algebra_simps)
58647
fce800afeec7 more facts about abstract divisibility
haftmann
parents: 58198
diff changeset
   376
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
   377
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
   378
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   379
  assume "x dvd - y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   380
  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
   381
  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
   382
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   383
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   384
  then have "x dvd - 1 * y" by (rule dvd_mult)
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   385
  then show "x dvd - y" by simp
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   386
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   387
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
   388
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
   389
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   390
  assume "- x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   391
  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
   392
  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
   393
  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
   394
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   395
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   396
  then 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
   397
  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
   398
  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
   399
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   400
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   401
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
   402
  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
   403
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   404
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   405
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   406
class semiring_no_zero_divisors = semiring_0 +
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   407
  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
   408
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   409
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   410
lemma divisors_zero:
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   411
  assumes "a * b = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   412
  shows "a = 0 \<or> b = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   413
proof (rule classical)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   414
  assume "\<not> ?thesis"
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   415
  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
   416
  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
   417
  with assms show ?thesis by simp
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   418
qed
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   419
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   420
lemma mult_eq_0_iff [simp]: "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   421
proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   422
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   423
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   424
    then show ?thesis using no_zero_divisors by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   425
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   426
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   427
  then show ?thesis by auto
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   428
qed
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   429
58952
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   430
end
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   431
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   432
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
   433
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   434
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
   435
  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
   436
    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
   437
begin
5d82cdef6c1b equivalence rules for structures without zero divisors
haftmann
parents: 58889
diff changeset
   438
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   439
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
   440
  by simp
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   441
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   442
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
   443
  by simp
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   444
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   445
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   446
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   447
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
   448
begin
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   449
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   450
subclass semiring_no_zero_divisors_cancel
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   451
proof
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   452
  fix a b c
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   453
  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
   454
    by (simp add: algebra_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   455
  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
   456
    by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   457
  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
   458
  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
   459
    by (simp add: algebra_simps)
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   460
  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
   461
    by auto
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   462
  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
   463
qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   464
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   465
end
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   466
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   467
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   468
begin
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   469
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   470
subclass semiring_1_no_zero_divisors ..
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   471
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   472
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
   473
proof -
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   474
  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
   475
    by (simp add: algebra_simps)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   476
  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
   477
    by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   478
  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
   479
    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
   480
qed
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   481
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   482
lemma mult_cancel_right1 [simp]: "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   483
  using mult_cancel_right [of 1 c b] by auto
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   484
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   485
lemma mult_cancel_right2 [simp]: "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   486
  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
   487
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   488
lemma mult_cancel_left1 [simp]: "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   489
  using mult_cancel_left [of c 1 b] by force
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   490
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   491
lemma mult_cancel_left2 [simp]: "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   492
  using mult_cancel_left [of c a 1] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   493
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   494
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   495
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
   496
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
   497
begin
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   498
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   499
subclass semiring_1_no_zero_divisors ..
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   500
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62390
diff changeset
   501
end
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   502
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   503
class idom = comm_ring_1 + semiring_no_zero_divisors
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   504
begin
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   505
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   506
subclass semidom ..
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   507
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   508
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
   509
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   510
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
   511
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   512
  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
   513
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   514
  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
   515
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   516
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   517
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   518
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   519
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
   520
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   521
  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
   522
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   523
  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
   524
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   525
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   526
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   527
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   528
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
   529
proof
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   530
  assume "a * a = b * b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   531
  then have "(a - b) * (a + b) = 0"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   532
    by (simp add: algebra_simps)
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   533
  then show "a = b \<or> a = - b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   534
    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
   535
next
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   536
  assume "a = b \<or> a = - b"
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   537
  then show "a * a = b * b" by auto
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   538
qed
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
   539
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   540
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   541
64290
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   542
class idom_abs_sgn = idom + abs + sgn +
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   543
  assumes sgn_mult_abs: "sgn a * \<bar>a\<bar> = a"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   544
    and sgn_sgn [simp]: "sgn (sgn a) = sgn a"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   545
    and abs_abs [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   546
    and abs_0 [simp]: "\<bar>0\<bar> = 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   547
    and sgn_0 [simp]: "sgn 0 = 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   548
    and sgn_1 [simp]: "sgn 1 = 1"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   549
    and sgn_minus_1: "sgn (- 1) = - 1"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   550
    and sgn_mult: "sgn (a * b) = sgn a * sgn b"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   551
begin
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   552
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   553
lemma sgn_eq_0_iff:
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   554
  "sgn a = 0 \<longleftrightarrow> a = 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   555
proof -
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   556
  { assume "sgn a = 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   557
    then have "sgn a * \<bar>a\<bar> = 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   558
      by simp
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   559
    then have "a = 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   560
      by (simp add: sgn_mult_abs)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   561
  } then show ?thesis
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   562
    by auto
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   563
qed
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   564
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   565
lemma abs_eq_0_iff:
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   566
  "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   567
proof -
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   568
  { assume "\<bar>a\<bar> = 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   569
    then have "sgn a * \<bar>a\<bar> = 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   570
      by simp
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   571
    then have "a = 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   572
      by (simp add: sgn_mult_abs)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   573
  } then show ?thesis
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   574
    by auto
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   575
qed
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   576
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   577
lemma abs_mult_sgn:
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   578
  "\<bar>a\<bar> * sgn a = a"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   579
  using sgn_mult_abs [of a] by (simp add: ac_simps)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   580
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   581
lemma abs_1 [simp]:
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   582
  "\<bar>1\<bar> = 1"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   583
  using sgn_mult_abs [of 1] by simp
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   584
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   585
lemma sgn_abs [simp]:
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   586
  "\<bar>sgn a\<bar> = of_bool (a \<noteq> 0)"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   587
  using sgn_mult_abs [of "sgn a"] mult_cancel_left [of "sgn a" "\<bar>sgn a\<bar>" 1]
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   588
  by (auto simp add: sgn_eq_0_iff)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   589
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   590
lemma abs_sgn [simp]:
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   591
  "sgn \<bar>a\<bar> = of_bool (a \<noteq> 0)"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   592
  using sgn_mult_abs [of "\<bar>a\<bar>"] mult_cancel_right [of "sgn \<bar>a\<bar>" "\<bar>a\<bar>" 1]
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   593
  by (auto simp add: abs_eq_0_iff)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   594
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   595
lemma abs_mult:
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   596
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   597
proof (cases "a = 0 \<or> b = 0")
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   598
  case True
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   599
  then show ?thesis
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   600
    by auto
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   601
next
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   602
  case False
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   603
  then have *: "sgn (a * b) \<noteq> 0"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   604
    by (simp add: sgn_eq_0_iff)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   605
  from abs_mult_sgn [of "a * b"] abs_mult_sgn [of a] abs_mult_sgn [of b]
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   606
  have "\<bar>a * b\<bar> * sgn (a * b) = \<bar>a\<bar> * sgn a * \<bar>b\<bar> * sgn b"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   607
    by (simp add: ac_simps)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   608
  then have "\<bar>a * b\<bar> * sgn (a * b) = \<bar>a\<bar> * \<bar>b\<bar> * sgn (a * b)"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   609
    by (simp add: sgn_mult ac_simps)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   610
  with * show ?thesis
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   611
    by simp
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   612
qed
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   613
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   614
lemma sgn_minus [simp]:
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   615
  "sgn (- a) = - sgn a"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   616
proof -
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   617
  from sgn_minus_1 have "sgn (- 1 * a) = - 1 * sgn a"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   618
    by (simp only: sgn_mult)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   619
  then show ?thesis
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   620
    by simp
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   621
qed
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   622
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   623
lemma abs_minus [simp]:
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   624
  "\<bar>- a\<bar> = \<bar>a\<bar>"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   625
proof -
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   626
  have [simp]: "\<bar>- 1\<bar> = 1"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   627
    using sgn_mult_abs [of "- 1"] by simp
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   628
  then have "\<bar>- 1 * a\<bar> = 1 * \<bar>a\<bar>"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   629
    by (simp only: abs_mult)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   630
  then show ?thesis
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   631
    by simp
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   632
qed
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   633
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   634
end
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
   635
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   636
text \<open>
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   637
  The theory of partially ordered rings is taken from the books:
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   638
    \<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   639
    \<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   640
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
   641
  Most of the used notions can also be looked up in
63680
6e1e8b5abbfa more symbols;
wenzelm
parents: 63588
diff changeset
   642
    \<^item> \<^url>\<open>http://www.mathworld.com\<close> by Eric Weisstein et. al.
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   643
    \<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
   644
\<close>
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   645
63950
cdc1e59aa513 syntactic type class for operation mod named after mod;
haftmann
parents: 63947
diff changeset
   646
text \<open>Syntactic division operator\<close>
cdc1e59aa513 syntactic type class for operation mod named after mod;
haftmann
parents: 63947
diff changeset
   647
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   648
class divide =
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   649
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   650
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 68253
diff changeset
   651
setup \<open>Sign.add_const_constraint (\<^const_name>\<open>divide\<close>, SOME \<^typ>\<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>)\<close>
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   652
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   653
context semiring
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   654
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   655
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   656
lemma [field_simps]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   657
  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
   658
    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
   659
  by (rule distrib_left distrib_right)+
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   660
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   661
end
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   662
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   663
context ring
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   664
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   665
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   666
lemma [field_simps]:
60429
d3d1e185cd63 uniform _ div _ as infix syntax for ring division
haftmann
parents: 60353
diff changeset
   667
  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
   668
    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
   669
  by (rule left_diff_distrib right_diff_distrib)+
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   670
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   671
end
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   672
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 68253
diff changeset
   673
setup \<open>Sign.add_const_constraint (\<^const_name>\<open>divide\<close>, SOME \<^typ>\<open>'a::divide \<Rightarrow> 'a \<Rightarrow> 'a\<close>)\<close>
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   674
63950
cdc1e59aa513 syntactic type class for operation mod named after mod;
haftmann
parents: 63947
diff changeset
   675
text \<open>Algebraic classes with division\<close>
cdc1e59aa513 syntactic type class for operation mod named after mod;
haftmann
parents: 63947
diff changeset
   676
  
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   677
class semidom_divide = semidom + divide +
64240
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
   678
  assumes nonzero_mult_div_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
   679
  assumes div_by_0 [simp]: "a div 0 = 0"
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   680
begin
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   681
64240
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
   682
lemma nonzero_mult_div_cancel_left [simp]: "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
   683
  using nonzero_mult_div_cancel_right [of a b] by (simp add: ac_simps)
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   684
60516
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   685
subclass semiring_no_zero_divisors_cancel
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   686
proof
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   687
  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
   688
  proof (cases "c = 0")
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   689
    case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   690
    then show ?thesis by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   691
  next
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   692
    case False
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   693
    have "a = b" if "a * c = b * c"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   694
    proof -
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   695
      from that have "a * c div c = b * c div c"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   696
        by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   697
      with False show ?thesis
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   698
        by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   699
    qed
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   700
    then show ?thesis by auto
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   701
  qed
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   702
  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
   703
    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
   704
qed
0826b7025d07 generalized some theorems about integral domains and moved to HOL theories
haftmann
parents: 60429
diff changeset
   705
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   706
lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
64240
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
   707
  using nonzero_mult_div_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
   708
64240
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
   709
lemma div_0 [simp]: "0 div a = 0"
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   710
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   711
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   712
  then show ?thesis by simp
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   713
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   714
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   715
  then have "a * 0 div a = 0"
64240
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
   716
    by (rule nonzero_mult_div_cancel_left)
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   717
  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
   718
qed
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   719
64240
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
   720
lemma div_by_1 [simp]: "a div 1 = a"
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
   721
  using nonzero_mult_div_cancel_left [of 1 a] by simp
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   722
64591
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   723
lemma dvd_div_eq_0_iff:
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   724
  assumes "b dvd a"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   725
  shows "a div b = 0 \<longleftrightarrow> a = 0"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   726
  using assms by (elim dvdE, cases "b = 0") simp_all  
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   727
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   728
lemma dvd_div_eq_cancel:
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   729
  "a div c = b div c \<Longrightarrow> c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> a = b"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   730
  by (elim dvdE, cases "c = 0") simp_all
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   731
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   732
lemma dvd_div_eq_iff:
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   733
  "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> a div c = b div c \<longleftrightarrow> a = b"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   734
  by (elim dvdE, cases "c = 0") simp_all
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   735
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   736
lemma inj_on_mult:
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   737
  "inj_on ((*) a) A" if "a \<noteq> 0"
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   738
proof (rule inj_onI)
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   739
  fix b c
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   740
  assume "a * b = a * c"
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   741
  then have "a * b div a = a * c div a"
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   742
    by (simp only:)
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   743
  with that show "b = c"
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   744
    by simp
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   745
qed
a03a63b81f44 tuned proofs
haftmann
parents: 69605
diff changeset
   746
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   747
end
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   748
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   749
class idom_divide = idom + semidom_divide
64591
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   750
begin
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   751
64592
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   752
lemma dvd_neg_div:
64591
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   753
  assumes "b dvd a"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   754
  shows "- a div b = - (a div b)"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   755
proof (cases "b = 0")
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   756
  case True
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   757
  then show ?thesis by simp
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   758
next
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   759
  case False
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   760
  from assms obtain c where "a = b * c" ..
64592
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   761
  then have "- a div b = (b * - c) div b"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   762
    by simp
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   763
  from False also have "\<dots> = - c"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   764
    by (rule nonzero_mult_div_cancel_left)  
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   765
  with False \<open>a = b * c\<close> show ?thesis
64591
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   766
    by simp
64592
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   767
qed
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   768
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   769
lemma dvd_div_neg:
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   770
  assumes "b dvd a"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   771
  shows "a div - b = - (a div b)"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   772
proof (cases "b = 0")
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   773
  case True
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   774
  then show ?thesis by simp
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   775
next
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   776
  case False
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   777
  then have "- b \<noteq> 0"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   778
    by simp
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   779
  from assms obtain c where "a = b * c" ..
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   780
  then have "a div - b = (- b * - c) div - b"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   781
    by simp
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   782
  from \<open>- b \<noteq> 0\<close> also have "\<dots> = - c"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   783
    by (rule nonzero_mult_div_cancel_left)  
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   784
  with False \<open>a = b * c\<close> show ?thesis
64591
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   785
    by simp
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   786
qed
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   787
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
   788
end
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   789
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   790
class algebraic_semidom = semidom_divide
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   791
begin
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   792
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   793
text \<open>
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 68253
diff changeset
   794
  Class \<^class>\<open>algebraic_semidom\<close> enriches a integral domain
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   795
  by notions from algebra, like units in a ring.
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   796
  It is a separate class to avoid spoiling fields with notions
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   797
  which are degenerated there.
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   798
\<close>
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   799
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   800
lemma dvd_times_left_cancel_iff [simp]:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   801
  assumes "a \<noteq> 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   802
  shows "a * b dvd a * c \<longleftrightarrow> b dvd c"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   803
    (is "?lhs \<longleftrightarrow> ?rhs")
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   804
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   805
  assume ?lhs
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   806
  then obtain d where "a * c = a * b * d" ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   807
  with assms have "c = b * d" by (simp add: ac_simps)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   808
  then show ?rhs ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   809
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   810
  assume ?rhs
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   811
  then obtain d where "c = b * d" ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   812
  then have "a * c = a * b * d" by (simp add: ac_simps)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   813
  then show ?lhs ..
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   814
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
   815
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   816
lemma dvd_times_right_cancel_iff [simp]:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   817
  assumes "a \<noteq> 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   818
  shows "b * a dvd c * a \<longleftrightarrow> b dvd c"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   819
  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
   820
60690
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   821
lemma div_dvd_iff_mult:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   822
  assumes "b \<noteq> 0" and "b dvd a"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   823
  shows "a div b dvd c \<longleftrightarrow> a dvd c * b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   824
proof -
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   825
  from \<open>b dvd a\<close> obtain d where "a = b * d" ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   826
  with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps)
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   827
qed
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   828
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   829
lemma dvd_div_iff_mult:
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   830
  assumes "c \<noteq> 0" and "c dvd b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   831
  shows "a dvd b div c \<longleftrightarrow> a * c dvd b"
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   832
proof -
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   833
  from \<open>c dvd b\<close> obtain d where "b = c * d" ..
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   834
  with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a])
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   835
qed
a9e45c9588c3 tuned facts
haftmann
parents: 60688
diff changeset
   836
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   837
lemma div_dvd_div [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   838
  assumes "a dvd b" and "a dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   839
  shows "b div a dvd c div a \<longleftrightarrow> b dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   840
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   841
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   842
  with assms show ?thesis by simp
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   843
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   844
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   845
  moreover from assms obtain k l where "b = a * k" and "c = a * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   846
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   847
  ultimately show ?thesis by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   848
qed
60353
838025c6e278 implicit partial divison operation in integral domains
haftmann
parents: 60352
diff changeset
   849
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   850
lemma div_add [simp]:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   851
  assumes "c dvd a" and "c dvd b"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   852
  shows "(a + b) div c = a div c + b div c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   853
proof (cases "c = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   854
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   855
  then show ?thesis by simp
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   856
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   857
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   858
  moreover from assms obtain k l where "a = c * k" and "b = c * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   859
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   860
  moreover have "c * k + c * l = c * (k + l)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   861
    by (simp add: algebra_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   862
  ultimately show ?thesis
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   863
    by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   864
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
   865
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   866
lemma div_mult_div_if_dvd:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   867
  assumes "b dvd a" and "d dvd c"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   868
  shows "(a div b) * (c div d) = (a * c) div (b * d)"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   869
proof (cases "b = 0 \<or> c = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   870
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   871
  with assms show ?thesis by auto
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   872
next
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   873
  case False
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   874
  moreover from assms obtain k l where "a = b * k" and "c = d * l"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   875
    by (auto elim!: dvdE)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   876
  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
   877
    by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   878
  ultimately show ?thesis by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   879
qed
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   880
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   881
lemma dvd_div_eq_mult:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   882
  assumes "a \<noteq> 0" and "a dvd b"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   883
  shows "b div a = c \<longleftrightarrow> b = c * a"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   884
    (is "?lhs \<longleftrightarrow> ?rhs")
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   885
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   886
  assume ?rhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   887
  then show ?lhs by (simp add: assms)
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   888
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   889
  assume ?lhs
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   890
  then have "b div a * a = c * a" by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   891
  moreover from assms have "b div a * a = b"
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   892
    by (auto elim!: dvdE simp add: ac_simps)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   893
  ultimately show ?rhs by simp
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   894
qed
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
   895
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   896
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
   897
  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
   898
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   899
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
   900
  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
   901
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   902
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
   903
  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
   904
  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
   905
proof (cases "c = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   906
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   907
  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
   908
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   909
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   910
  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
   911
  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
   912
    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
   913
  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
   914
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   915
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   916
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
   917
  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
   918
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   919
lemma dvd_div_mult2_eq:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   920
  assumes "b * c dvd a"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   921
  shows "a div (b * c) = a div b div c"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   922
proof -
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   923
  from assms obtain k where "a = b * c * k" ..
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   924
  then show ?thesis
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   925
    by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   926
qed
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
   927
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   928
lemma dvd_div_div_eq_mult:
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   929
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   930
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   931
    (is "?lhs \<longleftrightarrow> ?rhs")
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   932
proof -
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   933
  from assms have "a * c \<noteq> 0" by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   934
  then have "?lhs \<longleftrightarrow> b div a * (a * c) = d div c * (a * c)"
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   935
    by simp
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   936
  also have "\<dots> \<longleftrightarrow> (a * (b div a)) * c = (c * (d div c)) * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   937
    by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   938
  also have "\<dots> \<longleftrightarrow> (a * b div a) * c = (c * d div c) * a"
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   939
    using assms by (simp add: div_mult_swap)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   940
  also have "\<dots> \<longleftrightarrow> ?rhs"
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   941
    using assms by (simp add: ac_simps)
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   942
  finally show ?thesis .
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   943
qed
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60758
diff changeset
   944
63359
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   945
lemma dvd_mult_imp_div:
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   946
  assumes "a * c dvd b"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   947
  shows "a dvd b div c"
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   948
proof (cases "c = 0")
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   949
  case True then show ?thesis by simp
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   950
next
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   951
  case False
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   952
  from \<open>a * c dvd b\<close> obtain d where "b = a * c * d" ..
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   953
  with False show ?thesis
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
   954
    by (simp add: mult.commute [of a] mult.assoc)
63359
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   955
qed
99b51ba8da1c More lemmas on Gcd/Lcm
Manuel Eberl <eberlm@in.tum.de>
parents: 63325
diff changeset
   956
64592
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   957
lemma div_div_eq_right:
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   958
  assumes "c dvd b" "b dvd a"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   959
  shows   "a div (b div c) = a div b * c"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   960
proof (cases "c = 0 \<or> b = 0")
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   961
  case True
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   962
  then show ?thesis
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   963
    by auto
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   964
next
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   965
  case False
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   966
  from assms obtain r s where "b = c * r" and "a = c * r * s"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   967
    by (blast elim: dvdE)
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   968
  moreover with False have "r \<noteq> 0"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   969
    by auto
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   970
  ultimately show ?thesis using False
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   971
    by simp (simp add: mult.commute [of _ r] mult.assoc mult.commute [of c])
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   972
qed
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   973
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   974
lemma div_div_div_same:
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   975
  assumes "d dvd b" "b dvd a"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   976
  shows   "(a div d) div (b div d) = a div b"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   977
proof (cases "b = 0 \<or> d = 0")
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   978
  case True
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   979
  with assms show ?thesis
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   980
    by auto
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   981
next
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   982
  case False
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   983
  from assms obtain r s
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   984
    where "a = d * r * s" and "b = d * r"
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   985
    by (blast elim: dvdE)
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   986
  with False show ?thesis
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   987
    by simp (simp add: ac_simps)
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   988
qed
7759f1766189 more fine-grained type class hierarchy for div and mod
haftmann
parents: 64591
diff changeset
   989
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
   990
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   991
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
   992
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
   993
abbreviation is_unit :: "'a \<Rightarrow> bool"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   994
  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
   995
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   996
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
   997
  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
   998
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
   999
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
  1000
  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
  1001
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1002
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
  1003
  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
  1004
  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
  1005
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1006
  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
  1007
  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
  1008
  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
  1009
  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
  1010
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1011
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1012
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
  1013
  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
  1014
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1015
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
  1016
  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
  1017
  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
  1018
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1019
  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
  1020
  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
  1021
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1022
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1023
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
  1024
  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
  1025
  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
  1026
    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
  1027
    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
  1028
proof (rule that)
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62626
diff changeset
  1029
  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
  1030
  then show "1 div a = b" by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1031
  from assms b_def show "is_unit b" by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1032
  with assms show "a \<noteq> 0" and "b \<noteq> 0" by auto
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1033
  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
  1034
  then have "1 = a * b" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1035
  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
  1036
  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
  1037
  then obtain d where "c = a * d" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1038
  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
  1039
    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
  1040
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1041
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1042
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
  1043
  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
  1044
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1045
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
  1046
  by (auto dest: dvd_mult_left dvd_mult_right)
95c6cf433c91 more theorems
haftmann
parents: 62349
diff changeset
  1047
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1048
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
  1049
  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
  1050
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1051
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
  1052
  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
  1053
  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
  1054
proof
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1055
  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
  1056
  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
  1057
    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
  1058
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1059
  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
  1060
  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
  1061
  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
  1062
    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
  1063
  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
  1064
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1065
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
  1066
lemma mult_unit_dvd_iff': "is_unit a \<Longrightarrow> (a * b) dvd c \<longleftrightarrow> b dvd c"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
  1067
  using mult_unit_dvd_iff [of a b c] by (simp add: ac_simps)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
  1068
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1069
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
  1070
  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
  1071
  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
  1072
proof
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1073
  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
  1074
  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
  1075
    by (subst mult_assoc [symmetric]) simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1076
  also from assms have "b * (1 div b) = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1077
    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
  1078
  finally have "c * b dvd c" by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  1079
  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
  1080
next
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1081
  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
  1082
  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
  1083
qed
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1084
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
  1085
lemma dvd_mult_unit_iff': "is_unit b \<Longrightarrow> a dvd b * c \<longleftrightarrow> a dvd c"
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
  1086
  using dvd_mult_unit_iff [of b a c] by (simp add: ac_simps)
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
  1087
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1088
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
  1089
  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
  1090
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1091
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
  1092
  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
  1093
63924
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
  1094
lemmas unit_dvd_iff = mult_unit_dvd_iff mult_unit_dvd_iff'
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
  1095
  dvd_mult_unit_iff dvd_mult_unit_iff' 
f91766530e13 more generic algebraic lemmas
haftmann
parents: 63680
diff changeset
  1096
  div_unit_dvd_iff dvd_div_unit_iff (* FIXME consider named_theorems *)
60517
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1097
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1098
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
  1099
  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
  1100
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1101
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
  1102
  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
  1103
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1104
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
  1105
  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
  1106
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1107
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
  1108
  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
  1109
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1110
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
  1111
  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
  1112
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1113
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
  1114
  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
  1115
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1116
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
  1117
  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
  1118
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1119
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
  1120
  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
  1121
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1122
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
  1123
  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
  1124
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1125
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
  1126
  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
  1127
  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
  1128
proof -
f16e4fb20652 separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
haftmann
parents: 60516
diff changeset
  1129
  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
  1130
  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
  1131
    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
  1132
  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
  1133
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
  1134
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
  1135
lemma is_unit_div_mult2_eq:
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
  1136
  assumes "is_unit b" and "is_unit c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
  1137
  shows "a div (b * c) = a div b div c"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
  1138
proof -
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1139
  from assms have "is_unit (b * c)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1140
    by (simp add: unit_prod)
60570
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
  1141
  then have "b * c dvd a"
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
  1142
    by (rule unit_imp_dvd)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
  1143
  then show ?thesis
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
  1144
    by (rule dvd_div_mult2_eq)
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
  1145
qed
7ed2cde6806d more theorems
haftmann
parents: 60562
diff changeset
  1146
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
  1147
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
  1148
  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
  1149
  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
  1150
  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
  1151
64240
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
  1152
lemma is_unit_div_mult_cancel_left:
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1153
  assumes "a \<noteq> 0" and "is_unit b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1154
  shows "a div (a * b) = 1 div b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1155
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1156
  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
  1157
    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
  1158
  with assms show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1159
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1160
64240
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
  1161
lemma is_unit_div_mult_cancel_right:
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1162
  assumes "a \<noteq> 0" and "is_unit b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1163
  shows "a div (b * a) = 1 div b"
64240
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
  1164
  using assms is_unit_div_mult_cancel_left [of a b] by (simp add: ac_simps)
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1165
64591
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1166
lemma unit_div_eq_0_iff:
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1167
  assumes "is_unit b"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1168
  shows "a div b = 0 \<longleftrightarrow> a = 0"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1169
  by (rule dvd_div_eq_0_iff) (insert assms, auto)  
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1170
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1171
lemma div_mult_unit2:
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1172
  "is_unit c \<Longrightarrow> b dvd a \<Longrightarrow> a div (b * c) = a div b div c"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1173
  by (rule dvd_div_mult2_eq) (simp_all add: mult_unit_dvd_iff)
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1174
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1175
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1176
text \<open>Coprimality\<close>
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1177
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1178
definition coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1179
  where "coprime a b \<longleftrightarrow> (\<forall>c. c dvd a \<longrightarrow> c dvd b \<longrightarrow> is_unit c)"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1180
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1181
lemma coprimeI:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1182
  assumes "\<And>c. c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> is_unit c"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1183
  shows "coprime a b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1184
  using assms by (auto simp: coprime_def)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1185
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1186
lemma not_coprimeI:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1187
  assumes "c dvd a" and "c dvd b" and "\<not> is_unit c"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1188
  shows "\<not> coprime a b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1189
  using assms by (auto simp: coprime_def)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1190
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1191
lemma coprime_common_divisor:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1192
  "is_unit c" if "coprime a b" and "c dvd a" and "c dvd b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1193
  using that by (auto simp: coprime_def)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1194
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1195
lemma not_coprimeE:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1196
  assumes "\<not> coprime a b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1197
  obtains c where "c dvd a" and "c dvd b" and "\<not> is_unit c"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1198
  using assms by (auto simp: coprime_def)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1199
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1200
lemma coprime_imp_coprime:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1201
  "coprime a b" if "coprime c d"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1202
    and "\<And>e. \<not> is_unit e \<Longrightarrow> e dvd a \<Longrightarrow> e dvd b \<Longrightarrow> e dvd c"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1203
    and "\<And>e. \<not> is_unit e \<Longrightarrow> e dvd a \<Longrightarrow> e dvd b \<Longrightarrow> e dvd d"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1204
proof (rule coprimeI)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1205
  fix e
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1206
  assume "e dvd a" and "e dvd b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1207
  with that have "e dvd c" and "e dvd d"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1208
    by (auto intro: dvd_trans)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1209
  with \<open>coprime c d\<close> show "is_unit e"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1210
    by (rule coprime_common_divisor)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1211
qed
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1212
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1213
lemma coprime_divisors:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1214
  "coprime a b" if "a dvd c" "b dvd d" and "coprime c d"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1215
using \<open>coprime c d\<close> proof (rule coprime_imp_coprime)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1216
  fix e
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1217
  assume "e dvd a" then show "e dvd c"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1218
    using \<open>a dvd c\<close> by (rule dvd_trans)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1219
  assume "e dvd b" then show "e dvd d"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1220
    using \<open>b dvd d\<close> by (rule dvd_trans)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1221
qed
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1222
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1223
lemma coprime_self [simp]:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1224
  "coprime a a \<longleftrightarrow> is_unit a" (is "?P \<longleftrightarrow> ?Q")
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1225
proof
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1226
  assume ?P
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1227
  then show ?Q
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1228
    by (rule coprime_common_divisor) simp_all
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1229
next
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1230
  assume ?Q
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1231
  show ?P
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1232
    by (rule coprimeI) (erule dvd_unit_imp_unit, rule \<open>?Q\<close>)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1233
qed
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1234
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1235
lemma coprime_commute [ac_simps]:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1236
  "coprime b a \<longleftrightarrow> coprime a b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1237
  unfolding coprime_def by auto
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1238
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1239
lemma is_unit_left_imp_coprime:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1240
  "coprime a b" if "is_unit a"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1241
proof (rule coprimeI)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1242
  fix c
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1243
  assume "c dvd a"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1244
  with that show "is_unit c"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1245
    by (auto intro: dvd_unit_imp_unit)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1246
qed
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1247
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1248
lemma is_unit_right_imp_coprime:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1249
  "coprime a b" if "is_unit b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1250
  using that is_unit_left_imp_coprime [of b a] by (simp add: ac_simps)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1251
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1252
lemma coprime_1_left [simp]:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1253
  "coprime 1 a"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1254
  by (rule coprimeI)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1255
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1256
lemma coprime_1_right [simp]:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1257
  "coprime a 1"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1258
  by (rule coprimeI)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1259
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1260
lemma coprime_0_left_iff [simp]:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1261
  "coprime 0 a \<longleftrightarrow> is_unit a"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1262
  by (auto intro: coprimeI dvd_unit_imp_unit coprime_common_divisor [of 0 a a])
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1263
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1264
lemma coprime_0_right_iff [simp]:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1265
  "coprime a 0 \<longleftrightarrow> is_unit a"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1266
  using coprime_0_left_iff [of a] by (simp add: ac_simps)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1267
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1268
lemma coprime_mult_self_left_iff [simp]:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1269
  "coprime (c * a) (c * b) \<longleftrightarrow> is_unit c \<and> coprime a b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1270
  by (auto intro: coprime_common_divisor)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1271
    (rule coprimeI, auto intro: coprime_common_divisor simp add: dvd_mult_unit_iff')+
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1272
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1273
lemma coprime_mult_self_right_iff [simp]:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1274
  "coprime (a * c) (b * c) \<longleftrightarrow> is_unit c \<and> coprime a b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1275
  using coprime_mult_self_left_iff [of c a b] by (simp add: ac_simps)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1276
67234
ab10ea1d6fd0 Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents: 67226
diff changeset
  1277
lemma coprime_absorb_left:
ab10ea1d6fd0 Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents: 67226
diff changeset
  1278
  assumes "x dvd y"
ab10ea1d6fd0 Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents: 67226
diff changeset
  1279
  shows   "coprime x y \<longleftrightarrow> is_unit x"
ab10ea1d6fd0 Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents: 67226
diff changeset
  1280
  using assms coprime_common_divisor is_unit_left_imp_coprime by auto
ab10ea1d6fd0 Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents: 67226
diff changeset
  1281
ab10ea1d6fd0 Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents: 67226
diff changeset
  1282
lemma coprime_absorb_right:
ab10ea1d6fd0 Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents: 67226
diff changeset
  1283
  assumes "y dvd x"
ab10ea1d6fd0 Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents: 67226
diff changeset
  1284
  shows   "coprime x y \<longleftrightarrow> is_unit y"
ab10ea1d6fd0 Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents: 67226
diff changeset
  1285
  using assms coprime_common_divisor is_unit_right_imp_coprime by auto
ab10ea1d6fd0 Some lemmas on complex numbers and coprimality
eberlm <eberlm@in.tum.de>
parents: 67226
diff changeset
  1286
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1287
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1288
64848
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1289
class unit_factor =
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1290
  fixes unit_factor :: "'a \<Rightarrow> 'a"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1291
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1292
class semidom_divide_unit_factor = semidom_divide + unit_factor +
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1293
  assumes unit_factor_0 [simp]: "unit_factor 0 = 0"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1294
    and is_unit_unit_factor: "a dvd 1 \<Longrightarrow> unit_factor a = a"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1295
    and unit_factor_is_unit: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd 1"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1296
    and unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b"
67226
ec32cdaab97b isabelle update_cartouches -c -t;
wenzelm
parents: 67084
diff changeset
  1297
  \<comment> \<open>This fine-grained hierarchy will later on allow lean normalization of polynomials\<close>
64848
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1298
  
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1299
class normalization_semidom = algebraic_semidom + semidom_divide_unit_factor +
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1300
  fixes normalize :: "'a \<Rightarrow> 'a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1301
  assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1302
    and normalize_0 [simp]: "normalize 0 = 0"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1303
begin
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1304
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1305
text \<open>
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 68253
diff changeset
  1306
  Class \<^class>\<open>normalization_semidom\<close> cultivates the idea that each integral
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1307
  domain can be split into equivalence classes whose representants are
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 68253
diff changeset
  1308
  associated, i.e. divide each other. \<^const>\<open>normalize\<close> specifies a canonical
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1309
  representant for each equivalence class. The rationale behind this is that
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1310
  it is easier to reason about equality than equivalences, hence we prefer to
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1311
  think about equality of normalized values rather than associated elements.
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1312
\<close>
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1313
64848
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1314
declare unit_factor_is_unit [iff]
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1315
  
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1316
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
  1317
  by (rule unit_imp_dvd) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1318
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1319
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
  1320
  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
  1321
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1322
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
  1323
  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
  1324
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1325
lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1326
  (is "?lhs \<longleftrightarrow> ?rhs")
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1327
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1328
  assume ?lhs
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1329
  moreover have "unit_factor a * normalize a = a" by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1330
  ultimately show ?rhs by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1331
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1332
  assume ?rhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1333
  then show ?lhs by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1334
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1335
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1336
lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1337
  (is "?lhs \<longleftrightarrow> ?rhs")
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1338
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1339
  assume ?lhs
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1340
  moreover have "unit_factor a * normalize a = a" by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1341
  ultimately show ?rhs by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1342
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1343
  assume ?rhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1344
  then show ?lhs by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1345
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1346
64848
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1347
lemma div_unit_factor [simp]: "a div unit_factor a = normalize a"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1348
proof (cases "a = 0")
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1349
  case True
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1350
  then show ?thesis by simp
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1351
next
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1352
  case False
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1353
  then have "unit_factor a \<noteq> 0"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1354
    by simp
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1355
  with nonzero_mult_div_cancel_left
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1356
  have "unit_factor a * normalize a div unit_factor a = normalize a"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1357
    by blast
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1358
  then show ?thesis by simp
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1359
qed
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1360
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1361
lemma normalize_div [simp]: "normalize a div a = 1 div unit_factor a"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1362
proof (cases "a = 0")
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1363
  case True
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1364
  then show ?thesis by simp
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1365
next
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1366
  case False
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1367
  have "normalize a div a = normalize a div (unit_factor a * normalize a)"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1368
    by simp
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1369
  also have "\<dots> = 1 div unit_factor a"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1370
    using False by (subst is_unit_div_mult_cancel_right) simp_all
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1371
  finally show ?thesis .
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1372
qed
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1373
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1374
lemma is_unit_normalize:
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1375
  assumes "is_unit a"
64848
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1376
  shows "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
  1377
proof -
64848
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1378
  from assms have "unit_factor a = a"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1379
    by (rule is_unit_unit_factor)
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1380
  moreover from assms have "a \<noteq> 0"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1381
    by auto
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1382
  moreover have "normalize a = a div unit_factor a"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1383
    by simp
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1384
  ultimately show ?thesis
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  1385
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1386
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1387
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1388
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
  1389
  by (rule is_unit_unit_factor) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1390
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1391
lemma normalize_1 [simp]: "normalize 1 = 1"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1392
  by (rule is_unit_normalize) simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1393
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1394
lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1395
  (is "?lhs \<longleftrightarrow> ?rhs")
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1396
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1397
  assume ?rhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1398
  then show ?lhs by (rule is_unit_normalize)
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1399
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1400
  assume ?lhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1401
  then have "unit_factor a * normalize a = unit_factor a * 1"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1402
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1403
  then have "unit_factor a = a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1404
    by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1405
  moreover
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1406
  from \<open>?lhs\<close> have "a \<noteq> 0" by auto
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1407
  then have "is_unit (unit_factor a)" by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1408
  ultimately show ?rhs by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1409
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
  1410
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1411
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
  1412
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1413
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1414
  then show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1415
next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1416
  case False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1417
  then have "normalize a \<noteq> 0" by simp
64240
eabf80376aab more standardized names
haftmann
parents: 64239
diff changeset
  1418
  with nonzero_mult_div_cancel_right
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1419
  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
  1420
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1421
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1422
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1423
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
  1424
  by (cases "b = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1425
63947
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1426
lemma inv_unit_factor_eq_0_iff [simp]:
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1427
  "1 div unit_factor a = 0 \<longleftrightarrow> a = 0"
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1428
  (is "?lhs \<longleftrightarrow> ?rhs")
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1429
proof
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1430
  assume ?lhs
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1431
  then have "a * (1 div unit_factor a) = a * 0"
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1432
    by simp
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1433
  then show ?rhs
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1434
    by simp
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1435
next
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1436
  assume ?rhs
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1437
  then show ?lhs by simp
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1438
qed
559f0882d6a6 more lemmas
haftmann
parents: 63924
diff changeset
  1439
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1440
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
  1441
proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1442
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1443
  then show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1444
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1445
  case False
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1446
  have "unit_factor (a * b) * normalize (a * b) = a * b"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1447
    by (rule unit_factor_mult_normalize)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1448
  then have "normalize (a * b) = a * b div unit_factor (a * b)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1449
    by simp
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1450
  also have "\<dots> = a * b div unit_factor (b * a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1451
    by (simp add: ac_simps)
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1452
  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
  1453
    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
  1454
  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
  1455
    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
  1456
  also have "\<dots> = normalize a * normalize b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1457
    using False
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1458
    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
  1459
  finally show ?thesis .
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1460
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
  1461
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1462
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
  1463
  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
  1464
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1465
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
  1466
  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
  1467
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1468
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
  1469
proof (cases "a = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1470
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1471
  then show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1472
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1473
  case False
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1474
  have "normalize a = normalize (unit_factor a * normalize a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1475
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1476
  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
  1477
    by (simp only: normalize_mult)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1478
  finally show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1479
    using False by simp_all
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1480
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1481
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1482
lemma unit_factor_normalize [simp]:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1483
  assumes "a \<noteq> 0"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1484
  shows "unit_factor (normalize a) = 1"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1485
proof -
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1486
  from assms have *: "normalize a \<noteq> 0"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1487
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1488
  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
  1489
    by (simp only: unit_factor_mult_normalize)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1490
  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
  1491
    by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1492
  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
  1493
    by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1494
  with * show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1495
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1496
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1497
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1498
lemma dvd_unit_factor_div:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1499
  assumes "b dvd a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1500
  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
  1501
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1502
  from assms have "a = a div b * b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1503
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1504
  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
  1505
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1506
  then show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1507
    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
  1508
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1509
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1510
lemma dvd_normalize_div:
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1511
  assumes "b dvd a"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1512
  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
  1513
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1514
  from assms have "a = a div b * b"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1515
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1516
  then have "normalize a = normalize (a div b * b)"
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1517
    by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1518
  then show ?thesis
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1519
    by (cases "b = 0") (simp_all add: normalize_mult)
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1520
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1521
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1522
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
  1523
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1524
  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
  1525
    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
  1526
      by (cases "a = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1527
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1528
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1529
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1530
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
  1531
proof -
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1532
  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
  1533
    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
  1534
      by (cases "b = 0") simp_all
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1535
  then show ?thesis by simp
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1536
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1537
65811
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1538
lemma normalize_idem_imp_unit_factor_eq:
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1539
  assumes "normalize a = a"
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1540
  shows "unit_factor a = of_bool (a \<noteq> 0)"
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1541
proof (cases "a = 0")
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1542
  case True
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1543
  then show ?thesis
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1544
    by simp
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1545
next
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1546
  case False
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1547
  then show ?thesis
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1548
    using assms unit_factor_normalize [of a] by simp
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1549
qed
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1550
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1551
lemma normalize_idem_imp_is_unit_iff:
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1552
  assumes "normalize a = a"
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1553
  shows "is_unit a \<longleftrightarrow> a = 1"
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1554
  using assms by (cases "a = 0") (auto dest: is_unit_normalize)
2653f1cd8775 more lemmas
haftmann
parents: 64848
diff changeset
  1555
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1556
lemma coprime_normalize_left_iff [simp]:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1557
  "coprime (normalize a) b \<longleftrightarrow> coprime a b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1558
  by (rule; rule coprimeI) (auto intro: coprime_common_divisor)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1559
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1560
lemma coprime_normalize_right_iff [simp]:
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1561
  "coprime a (normalize b) \<longleftrightarrow> coprime a b"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1562
  using coprime_normalize_left_iff [of b a] by (simp add: ac_simps)
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  1563
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1564
text \<open>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1565
  We avoid an explicit definition of associated elements but prefer explicit
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 68253
diff changeset
  1566
  normalisation instead. In theory we could define an abbreviation like \<^prop>\<open>associated a b \<longleftrightarrow> normalize a = normalize b\<close> but this is counterproductive
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1567
  without suggestive infix syntax, which we do not want to sacrifice for this
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1568
  purpose here.
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1569
\<close>
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1570
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1571
lemma associatedI:
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1572
  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
  1573
  shows "normalize a = normalize b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1574
proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1575
  case True
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1576
  with assms show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1577
next
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1578
  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
  1579
  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
  1580
  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
  1581
  ultimately have "b * 1 = b * (c * d)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1582
    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
  1583
  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
  1584
    unfolding mult_cancel_left by simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1585
  then have "is_unit c" and "is_unit d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1586
    by auto
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1587
  with a b show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1588
    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
  1589
qed
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1590
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1591
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
  1592
  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
  1593
  by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1594
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1595
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
  1596
  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
  1597
  by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1598
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1599
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
  1600
  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
  1601
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1602
lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1603
  (is "?lhs \<longleftrightarrow> ?rhs")
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1604
proof
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1605
  assume ?rhs
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1606
  then show ?lhs by (auto intro!: associatedI)
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1607
next
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1608
  assume ?lhs
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1609
  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
  1610
    by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1611
  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
  1612
    by (simp add: ac_simps)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1613
  show ?rhs
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1614
  proof (cases "a = 0 \<or> b = 0")
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1615
    case True
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1616
    with \<open>?lhs\<close> show ?thesis by auto
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1617
  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
  1618
    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
  1619
    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
  1620
      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
  1621
    with * show ?thesis by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1622
  qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1623
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1624
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1625
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
  1626
  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
  1627
  assumes "normalize a = a" and "normalize b = b"
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1628
  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
  1629
proof -
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60685
diff changeset
  1630
  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
  1631
    unfolding associated_iff_dvd by simp
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1632
  with \<open>normalize a = a\<close> have "a = normalize b"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1633
    by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1634
  with \<open>normalize b = b\<close> show "a = b"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1635
    by simp
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1636
qed
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1637
64591
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1638
lemma normalize_unit_factor_eqI:
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1639
  assumes "normalize a = normalize b"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1640
    and "unit_factor a = unit_factor b"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1641
  shows "a = b"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1642
proof -
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1643
  from assms have "unit_factor a * normalize a = unit_factor b * normalize b"
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1644
    by simp
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1645
  then show ?thesis
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1646
    by simp
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1647
qed
240a39af9ec4 restructured matter on polynomials and normalized fractions
haftmann
parents: 64290
diff changeset
  1648
60685
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1649
end
cb21b7022b00 moved normalization and unit_factor into Main HOL corpus
haftmann
parents: 60615
diff changeset
  1650
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1651
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1652
text \<open>Syntactic division remainder operator\<close>
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1653
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1654
class modulo = dvd + divide +
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1655
  fixes modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "mod" 70)
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1656
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1657
text \<open>Arbitrary quotient and remainder partitions\<close>
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1658
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1659
class semiring_modulo = comm_semiring_1_cancel + divide + modulo +
64242
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1660
  assumes div_mult_mod_eq: "a div b * b + a mod b = a"
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1661
begin
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1662
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1663
lemma mod_div_decomp:
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1664
  fixes a b
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1665
  obtains q r where "q = a div b" and "r = a mod b"
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1666
    and "a = q * b + r"
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1667
proof -
64242
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1668
  from div_mult_mod_eq have "a = a div b * b + a mod b" by simp
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1669
  moreover have "a div b = a div b" ..
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1670
  moreover have "a mod b = a mod b" ..
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1671
  note that ultimately show thesis by blast
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1672
qed
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1673
64242
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1674
lemma mult_div_mod_eq: "b * (a div b) + a mod b = a"
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1675
  using div_mult_mod_eq [of a b] by (simp add: ac_simps)
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1676
64242
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1677
lemma mod_div_mult_eq: "a mod b + a div b * b = a"
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1678
  using div_mult_mod_eq [of a b] by (simp add: ac_simps)
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1679
64242
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1680
lemma mod_mult_div_eq: "a mod b + b * (a div b) = a"
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1681
  using div_mult_mod_eq [of a b] by (simp add: ac_simps)
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1682
64242
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1683
lemma minus_div_mult_eq_mod: "a - a div b * b = a mod b"
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1684
  by (rule add_implies_diff [symmetric]) (fact mod_div_mult_eq)
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1685
64242
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1686
lemma minus_mult_div_eq_mod: "a - b * (a div b) = a mod b"
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1687
  by (rule add_implies_diff [symmetric]) (fact mod_mult_div_eq)
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1688
64242
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1689
lemma minus_mod_eq_div_mult: "a - a mod b = a div b * b"
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1690
  by (rule add_implies_diff [symmetric]) (fact div_mult_mod_eq)
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1691
64242
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1692
lemma minus_mod_eq_mult_div: "a - a mod b = b * (a div b)"
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1693
  by (rule add_implies_diff [symmetric]) (fact mult_div_mod_eq)
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1694
68253
a8660a39e304 grouped material on numeral division
haftmann
parents: 68252
diff changeset
  1695
lemma [nitpick_unfold]:
a8660a39e304 grouped material on numeral division
haftmann
parents: 68252
diff changeset
  1696
  "a mod b = a - a div b * b"
a8660a39e304 grouped material on numeral division
haftmann
parents: 68252
diff changeset
  1697
  by (fact minus_div_mult_eq_mod [symmetric])
a8660a39e304 grouped material on numeral division
haftmann
parents: 68252
diff changeset
  1698
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1699
end
64242
93c6f0da5c70 more standardized theorem names for facts involving the div and mod identity
haftmann
parents: 64240
diff changeset
  1700
64164
38c407446400 separate type class for arbitrary quotient and remainder partitions
haftmann
parents: 63950
diff changeset
  1701
66807
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1702
text \<open>Quotient and remainder in integral domains\<close>
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1703
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1704
class semidom_modulo = algebraic_semidom + semiring_modulo
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1705
begin
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1706
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1707
lemma mod_0 [simp]: "0 mod a = 0"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1708
  using div_mult_mod_eq [of 0 a] by simp
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1709
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1710
lemma mod_by_0 [simp]: "a mod 0 = a"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1711
  using div_mult_mod_eq [of a 0] by simp
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1712
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1713
lemma mod_by_1 [simp]:
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1714
  "a mod 1 = 0"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1715
proof -
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1716
  from div_mult_mod_eq [of a one] div_by_1 have "a + a mod 1 = a" by simp
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1717
  then have "a + a mod 1 = a + 0" by simp
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1718
  then show ?thesis by (rule add_left_imp_eq)
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1719
qed
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1720
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1721
lemma mod_self [simp]:
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1722
  "a mod a = 0"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1723
  using div_mult_mod_eq [of a a] by simp
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1724
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1725
lemma dvd_imp_mod_0 [simp]:
67084
haftmann
parents: 67051
diff changeset
  1726
  "b mod a = 0" if "a dvd b"
haftmann
parents: 67051
diff changeset
  1727
  using that minus_div_mult_eq_mod [of b a] by simp
66807
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1728
68252
8b284d24f434 automatic classical rule to derive a dvd b from b mod a = 0
haftmann
parents: 68251
diff changeset
  1729
lemma mod_0_imp_dvd [dest!]: 
67084
haftmann
parents: 67051
diff changeset
  1730
  "b dvd a" if "a mod b = 0"
66807
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1731
proof -
67084
haftmann
parents: 67051
diff changeset
  1732
  have "b dvd (a div b) * b" by simp
66807
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1733
  also have "(a div b) * b = a"
67084
haftmann
parents: 67051
diff changeset
  1734
    using div_mult_mod_eq [of a b] by (simp add: that)
66807
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1735
  finally show ?thesis .
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1736
qed
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1737
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1738
lemma mod_eq_0_iff_dvd:
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1739
  "a mod b = 0 \<longleftrightarrow> b dvd a"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1740
  by (auto intro: mod_0_imp_dvd)
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1741
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1742
lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]:
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1743
  "a dvd b \<longleftrightarrow> b mod a = 0"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1744
  by (simp add: mod_eq_0_iff_dvd)
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1745
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1746
lemma dvd_mod_iff: 
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1747
  assumes "c dvd b"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1748
  shows "c dvd a mod b \<longleftrightarrow> c dvd a"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1749
proof -
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1750
  from assms have "(c dvd a mod b) \<longleftrightarrow> (c dvd ((a div b) * b + a mod b))" 
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1751
    by (simp add: dvd_add_right_iff)
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1752
  also have "(a div b) * b + a mod b = a"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1753
    using div_mult_mod_eq [of a b] by simp
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1754
  finally show ?thesis .
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1755
qed
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1756
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1757
lemma dvd_mod_imp_dvd:
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1758
  assumes "c dvd a mod b" and "c dvd b"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1759
  shows "c dvd a"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1760
  using assms dvd_mod_iff [of c b a] by simp
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1761
66808
1907167b6038 elementary definition of division on natural numbers
haftmann
parents: 66807
diff changeset
  1762
lemma dvd_minus_mod [simp]:
1907167b6038 elementary definition of division on natural numbers
haftmann
parents: 66807
diff changeset
  1763
  "b dvd a - a mod b"
1907167b6038 elementary definition of division on natural numbers
haftmann
parents: 66807
diff changeset
  1764
  by (simp add: minus_mod_eq_div_mult)
1907167b6038 elementary definition of division on natural numbers
haftmann
parents: 66807
diff changeset
  1765
66810
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1766
lemma cancel_div_mod_rules:
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1767
  "((a div b) * b + a mod b) + c = a + c"
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1768
  "(b * (a div b) + a mod b) + c = a + c"
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1769
  by (simp_all add: div_mult_mod_eq mult_div_mod_eq)
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1770
66807
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1771
end
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1772
66810
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1773
text \<open>Interlude: basic tool support for algebraic and arithmetic calculations\<close>
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1774
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1775
named_theorems arith "arith facts -- only ground formulas"
69605
a96320074298 isabelle update -u path_cartouches;
wenzelm
parents: 69593
diff changeset
  1776
ML_file \<open>Tools/arith_data.ML\<close>
a96320074298 isabelle update -u path_cartouches;
wenzelm
parents: 69593
diff changeset
  1777
a96320074298 isabelle update -u path_cartouches;
wenzelm
parents: 69593
diff changeset
  1778
ML_file \<open>~~/src/Provers/Arith/cancel_div_mod.ML\<close>
66810
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1779
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1780
ML \<open>
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1781
structure Cancel_Div_Mod_Ring = Cancel_Div_Mod
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1782
(
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 68253
diff changeset
  1783
  val div_name = \<^const_name>\<open>divide\<close>;
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 68253
diff changeset
  1784
  val mod_name = \<^const_name>\<open>modulo\<close>;
66810
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1785
  val mk_binop = HOLogic.mk_binop;
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1786
  val mk_sum = Arith_Data.mk_sum;
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1787
  val dest_sum = Arith_Data.dest_sum;
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1788
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1789
  val div_mod_eqs = map mk_meta_eq @{thms cancel_div_mod_rules};
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1790
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1791
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1792
    @{thms diff_conv_add_uminus add_0_left add_0_right ac_simps})
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1793
)
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1794
\<close>
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1795
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1796
simproc_setup cancel_div_mod_int ("(a::'a::semidom_modulo) + b") =
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1797
  \<open>K Cancel_Div_Mod_Ring.proc\<close>
cc2b490f9dc4 generalized simproc
haftmann
parents: 66808
diff changeset
  1798
66807
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1799
class idom_modulo = idom + semidom_modulo
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1800
begin
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1801
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1802
subclass idom_divide ..
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1803
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1804
lemma div_diff [simp]:
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1805
  "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> (a - b) div c = a div c - b div c"
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1806
  using div_add [of _  _ "- b"] by (simp add: dvd_neg_div)
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1807
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1808
end
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1809
c3631f32dfeb tuned structure
haftmann
parents: 66793
diff changeset
  1810
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
  1811
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
  1812
  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
  1813
  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
  1814
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1815
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1816
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
  1817
  apply (erule (1) mult_right_mono [THEN order_trans])
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1818
  apply (erule (1) mult_left_mono)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1819
  done
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1820
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1821
lemma mult_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1822
  by (rule mult_mono) (fast intro: order_trans)+
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1823
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1824
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
  1825
62377
ace69956d018 moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents: 62376
diff changeset
  1826
class ordered_semiring_0 = semiring_0 + ordered_semiring
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  1827
begin
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1828
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1829
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
  1830
  using mult_left_mono [of 0 b a] by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1831
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1832
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
  1833
  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
  1834
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  1835
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
  1836
  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
  1837
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1838
text \<open>Legacy -- use @{thm [source] mult_nonpos_nonneg}.\<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
  1839
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1840
  by (drule mult_right_mono [of b 0]) auto
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1841
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  1842
lemma split_mult_neg_le: "(0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1843
  by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1844
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1845
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1846
62377
ace69956d018 moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents: 62376
diff changeset
  1847
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
ace69956d018 moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents: 62376
diff changeset
  1848
begin
ace69956d018 moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents: 62376
diff changeset
  1849
ace69956d018 moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents: 62376
diff changeset
  1850
subclass semiring_0_cancel ..
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1851
62377
ace69956d018 moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents: 62376
diff changeset
  1852
subclass ordered_semiring_0 ..
ace69956d018 moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents: 62376
diff changeset
  1853
ace69956d018 moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents: 62376
diff changeset
  1854
end
ace69956d018 moved more proofs to ordered_comm_monoid_add; introduced strict_ordered_ab_semigroup/comm_monoid_add
hoelzl
parents: 62376
diff changeset
  1855
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  1856
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  1857
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1858
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1859
subclass ordered_cancel_semiring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1860
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
  1861
subclass ordered_cancel_comm_monoid_add ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1862
63456
3365c8ec67bd sharing simp rules between ordered monoids and rings
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63359
diff changeset
  1863
subclass ordered_ab_semigroup_monoid_add_imp_le ..
3365c8ec67bd sharing simp rules between ordered monoids and rings
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents: 63359
diff changeset
  1864
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1865
lemma mult_left_less_imp_less: "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1866
  by (force simp add: mult_left_mono not_le [symmetric])
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
  1867
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1868
lemma mult_right_less_imp_less: "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1869
  by (force simp add: mult_right_mono not_le [symmetric])
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
  1870
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
  1871
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
  1872
66937
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  1873
class zero_less_one = order + zero + one +
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  1874
  assumes zero_less_one [simp]: "0 < 1"
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  1875
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  1876
class linordered_semiring_1 = linordered_semiring + semiring_1 + zero_less_one
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1877
begin
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1878
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1879
lemma convex_bound_le:
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1880
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1881
  shows "u * x + v * y \<le> a"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1882
proof-
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1883
  from assms have "u * x + v * y \<le> u * a + v * a"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1884
    by (simp add: add_mono mult_left_mono)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1885
  with assms show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1886
    unfolding distrib_right[symmetric] by simp
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1887
qed
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1888
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1889
end
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
  1890
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
  1891
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1892
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1893
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  1894
begin
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
  1895
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
  1896
subclass semiring_0_cancel ..
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
  1897
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1898
subclass linordered_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
  1899
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
  1900
  fix a b c :: 'a
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1901
  assume *: "a \<le> b" "0 \<le> c"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1902
  then show "c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
  1903
    unfolding le_less
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
  1904
    using mult_strict_left_mono by (cases "c = 0") auto
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1905
  from * show "a * c \<le> b * c"
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
  1906
    unfolding le_less
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
  1907
    using mult_strict_right_mono by (cases "c = 0") auto
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
  1908
qed
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
  1909
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1910
lemma mult_left_le_imp_le: "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1911
  by (auto simp add: mult_strict_left_mono _not_less [symmetric])
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60529
diff changeset
  1912
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1913
lemma mult_right_le_imp_le: "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1914
  by (auto simp add: mult_strict_right_mono not_less [symmetric])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1915
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
  1916
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1917
  using mult_strict_left_mono [of 0 b 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
  1918
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  1919
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1920
  using mult_strict_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
  1921
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  1922
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1923
  using mult_strict_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
  1924
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1925
text \<open>Legacy -- use @{thm [source] mult_neg_pos}.\<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
  1926
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1927
  by (drule mult_strict_right_mono [of b 0]) auto
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1928
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1929
lemma zero_less_mult_pos: "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1930
  apply (cases "b \<le> 0")
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1931
   apply (auto simp add: le_less not_less)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1932
  apply (drule_tac mult_pos_neg [of a b])
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1933
   apply (auto dest: less_not_sym)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1934
  done
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1935
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1936
lemma zero_less_mult_pos2: "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1937
  apply (cases "b \<le> 0")
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1938
   apply (auto simp add: le_less not_less)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1939
  apply (drule_tac mult_pos_neg2 [of a b])
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1940
   apply (auto dest: less_not_sym)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1941
  done
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1942
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1943
text \<open>Strict monotonicity in both arguments\<close>
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1944
lemma mult_strict_mono:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1945
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1946
  shows "a * c < b * d"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1947
  using assms
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1948
  apply (cases "c = 0")
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1949
   apply simp
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1950
  apply (erule mult_strict_right_mono [THEN less_trans])
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1951
   apply (auto simp add: le_less)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1952
  apply (erule (1) mult_strict_left_mono)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1953
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1954
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1955
text \<open>This weaker variant has more natural premises\<close>
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1956
lemma mult_strict_mono':
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1957
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1958
  shows "a * c < b * d"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1959
  by (rule mult_strict_mono) (insert assms, auto)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1960
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1961
lemma mult_less_le_imp_less:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1962
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1963
  shows "a * c < b * d"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1964
  using assms
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1965
  apply (subgoal_tac "a * c < b * c")
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1966
   apply (erule less_le_trans)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1967
   apply (erule mult_left_mono)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1968
   apply simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1969
  apply (erule (1) mult_strict_right_mono)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1970
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1971
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1972
lemma mult_le_less_imp_less:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1973
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1974
  shows "a * c < b * d"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1975
  using assms
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1976
  apply (subgoal_tac "a * c \<le> b * c")
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1977
   apply (erule le_less_trans)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1978
   apply (erule mult_strict_left_mono)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  1979
   apply simp
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1980
  apply (erule (1) mult_right_mono)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1981
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1982
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1983
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1984
66937
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  1985
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1 + zero_less_one
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1986
begin
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1987
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1988
subclass linordered_semiring_1 ..
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1989
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1990
lemma convex_bound_lt:
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1991
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1992
  shows "u * x + v * y < a"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1993
proof -
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1994
  from assms have "u * x + v * y < u * a + v * a"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1995
    by (cases "u = 0") (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1996
  with assms show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  1997
    unfolding distrib_right[symmetric] by simp
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1998
qed
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  1999
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
  2000
end
33319
74f0dcc0b5fb moved algebraic classes to Ring_and_Field
haftmann
parents: 32960
diff changeset
  2001
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
  2002
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  2003
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
  2004
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
  2005
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2006
subclass ordered_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
  2007
proof
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
  2008
  fix a b c :: 'a
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
  2009
  assume "a \<le> b" "0 \<le> c"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2010
  then show "c * a \<le> c * b" by (rule comm_mult_left_mono)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2011
  then show "a * c \<le> b * c" by (simp only: mult.commute)
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
  2012
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  2013
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2014
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2015
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  2016
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2017
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  2018
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  2019
subclass comm_semiring_0_cancel ..
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2020
subclass ordered_comm_semiring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2021
subclass ordered_cancel_semiring ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2022
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2023
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2024
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2025
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
  2026
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2027
begin
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2028
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
  2029
subclass linordered_semiring_strict
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
  2030
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
  2031
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
  2032
  assume "a < b" "0 < c"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2033
  then show "c * a < c * b"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2034
    by (rule comm_mult_strict_left_mono)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2035
  then show "a * c < b * c"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2036
    by (simp only: mult.commute)
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
  2037
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
  2038
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2039
subclass ordered_cancel_comm_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
  2040
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
  2041
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
  2042
  assume "a \<le> b" "0 \<le> c"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2043
  then show "c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
  2044
    unfolding le_less
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2045
    using mult_strict_left_mono by (cases "c = 0") auto
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
  2046
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
  2047
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2048
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2049
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
  2050
class ordered_ring = ring + ordered_cancel_semiring
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2051
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2052
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2053
subclass ordered_ab_group_add ..
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  2054
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2055
lemma less_add_iff1: "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2056
  by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2057
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2058
lemma less_add_iff2: "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2059
  by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2060
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2061
lemma le_add_iff1: "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2062
  by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2063
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2064
lemma le_add_iff2: "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2065
  by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2066
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2067
lemma mult_left_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2068
  apply (drule mult_left_mono [of _ _ "- c"])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
  2069
  apply simp_all
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2070
  done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2071
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2072
lemma mult_right_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2073
  apply (drule mult_right_mono [of _ _ "- c"])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
  2074
  apply simp_all
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2075
  done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2076
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2077
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2078
  using mult_right_mono_neg [of a 0 b] by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2079
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2080
lemma split_mult_pos_le: "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2081
  by (auto simp add: mult_nonpos_nonpos)
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
  2082
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
  2083
end
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  2084
64290
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2085
class abs_if = minus + uminus + ord + zero + abs +
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2086
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2087
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2088
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2089
begin
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2090
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2091
subclass ordered_ring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2092
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2093
subclass ordered_ab_group_add_abs
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
  2094
proof
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2095
  fix a b
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2096
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2097
    by (auto simp add: abs_if not_le not_less algebra_simps
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2098
        simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2099
qed (auto simp: abs_if)
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2100
35631
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
  2101
lemma zero_le_square [simp]: "0 \<le> a * a"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2102
  using linear [of 0 a] by (auto simp add: mult_nonpos_nonpos)
35631
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
  2103
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
  2104
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
  2105
  by (simp add: not_less)
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
  2106
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
  2107
proposition abs_eq_iff: "\<bar>x\<bar> = \<bar>y\<bar> \<longleftrightarrow> x = y \<or> x = -y"
62390
842917225d56 more canonical names
nipkow
parents: 62378
diff changeset
  2108
  by (auto simp add: abs_if split: if_split_asm)
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  2109
64848
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  2110
lemma abs_eq_iff':
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  2111
  "\<bar>a\<bar> = b \<longleftrightarrow> b \<ge> 0 \<and> (a = b \<or> a = - b)"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  2112
  by (cases "a \<ge> 0") auto
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  2113
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  2114
lemma eq_abs_iff':
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  2115
  "a = \<bar>b\<bar> \<longleftrightarrow> a \<ge> 0 \<and> (b = a \<or> b = - a)"
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  2116
  using abs_eq_iff' [of b a] by auto
c50db2128048 slightly generalized type class hierarchy concerning unit factors, to allow for lean polynomial normalization
haftmann
parents: 64713
diff changeset
  2117
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2118
lemma sum_squares_ge_zero: "0 \<le> x * x + y * y"
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 61944
diff changeset
  2119
  by (intro add_nonneg_nonneg zero_le_square)
2230b7047376 generalized some lemmas;
haftmann
parents: 61944
diff changeset
  2120
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2121
lemma not_sum_squares_lt_zero: "\<not> x * x + y * y < 0"
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 61944
diff changeset
  2122
  by (simp add: not_less sum_squares_ge_zero)
2230b7047376 generalized some lemmas;
haftmann
parents: 61944
diff changeset
  2123
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2124
end
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
  2125
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
  2126
class linordered_ring_strict = ring + linordered_semiring_strict
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2127
  + ordered_ab_group_add + abs_if
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2128
begin
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
  2129
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2130
subclass linordered_ring ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2131
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2132
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2133
  using mult_strict_left_mono [of b a "- c"] by simp
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2134
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2135
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2136
  using mult_strict_right_mono [of b a "- c"] by simp
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2137
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2138
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2139
  using mult_strict_right_mono_neg [of a 0 b] by simp
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
  2140
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2141
subclass ring_no_zero_divisors
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
  2142
proof
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2143
  fix a b
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2144
  assume "a \<noteq> 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2145
  then have a: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2146
  assume "b \<noteq> 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2147
  then have b: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2148
  have "a * b < 0 \<or> 0 < a * b"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2149
  proof (cases "a < 0")
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2150
    case True
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2151
    show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2152
    proof (cases "b < 0")
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2153
      case True
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2154
      with \<open>a < 0\<close> show ?thesis by (auto dest: mult_neg_neg)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2155
    next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2156
      case False
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2157
      with b have "0 < b" by auto
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2158
      with \<open>a < 0\<close> show ?thesis by (auto dest: mult_strict_right_mono)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2159
    qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2160
  next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2161
    case False
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2162
    with a have "0 < a" by auto
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2163
    show ?thesis
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2164
    proof (cases "b < 0")
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2165
      case True
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2166
      with \<open>0 < a\<close> show ?thesis
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2167
        by (auto dest: mult_strict_right_mono_neg)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2168
    next
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2169
      case False
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2170
      with b have "0 < b" by auto
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2171
      with \<open>0 < a\<close> show ?thesis by auto
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2172
    qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2173
  qed
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2174
  then show "a * b \<noteq> 0"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2175
    by (simp add: neq_iff)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2176
qed
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2177
56480
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56217
diff changeset
  2178
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56217
diff changeset
  2179
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
  2180
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
  2181
56480
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56217
diff changeset
  2182
lemma zero_le_mult_iff: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56217
diff changeset
  2183
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  2184
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2185
lemma mult_less_0_iff: "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2186
  using zero_less_mult_iff [of "- a" b] by auto
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  2187
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2188
lemma mult_le_0_iff: "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2189
  using zero_le_mult_iff [of "- a" b] by auto
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2190
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2191
text \<open>
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 68253
diff changeset
  2192
  Cancellation laws for \<^term>\<open>c * a < c * b\<close> and \<^term>\<open>a * c < b * c\<close>,
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2193
  also with the relations \<open>\<le>\<close> and equality.
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2194
\<close>
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2195
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2196
text \<open>
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2197
  These ``disjunction'' versions produce two cases when the comparison is
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2198
  an assumption, but effectively four when the comparison is a goal.
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2199
\<close>
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2200
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2201
lemma mult_less_cancel_right_disj: "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2202
  apply (cases "c = 0")
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2203
   apply (auto simp add: neq_iff mult_strict_right_mono mult_strict_right_mono_neg)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2204
     apply (auto simp add: not_less not_le [symmetric, of "a*c"] not_le [symmetric, of a])
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2205
     apply (erule_tac [!] notE)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2206
     apply (auto simp add: less_imp_le mult_right_mono mult_right_mono_neg)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2207
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2208
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2209
lemma mult_less_cancel_left_disj: "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2210
  apply (cases "c = 0")
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2211
   apply (auto simp add: neq_iff mult_strict_left_mono mult_strict_left_mono_neg)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2212
     apply (auto simp add: not_less not_le [symmetric, of "c * a"] not_le [symmetric, of a])
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2213
     apply (erule_tac [!] notE)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2214
     apply (auto simp add: less_imp_le mult_left_mono mult_left_mono_neg)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2215
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2216
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2217
text \<open>
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2218
  The ``conjunction of implication'' lemmas produce two cases when the
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2219
  comparison is a goal, but give four when the comparison is an assumption.
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2220
\<close>
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2221
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2222
lemma mult_less_cancel_right: "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2223
  using mult_less_cancel_right_disj [of a c b] by auto
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2224
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2225
lemma mult_less_cancel_left: "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2226
  using mult_less_cancel_left_disj [of c a b] by auto
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2227
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2228
lemma mult_le_cancel_right: "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2229
  by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2230
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2231
lemma mult_le_cancel_left: "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2232
  by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2233
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2234
lemma mult_le_cancel_left_pos: "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2235
  by (auto simp: mult_le_cancel_left)
30649
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  2236
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2237
lemma mult_le_cancel_left_neg: "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2238
  by (auto simp: mult_le_cancel_left)
30649
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  2239
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2240
lemma mult_less_cancel_left_pos: "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2241
  by (auto simp: mult_less_cancel_left)
30649
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  2242
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2243
lemma mult_less_cancel_left_neg: "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2244
  by (auto simp: mult_less_cancel_left)
30649
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  2245
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2246
end
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  2247
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2248
lemmas mult_sign_intros =
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2249
  mult_nonneg_nonneg mult_nonneg_nonpos
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2250
  mult_nonpos_nonneg mult_nonpos_nonpos
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2251
  mult_pos_pos mult_pos_neg
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  2252
  mult_neg_pos mult_neg_neg
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2253
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2254
class ordered_comm_ring = comm_ring + ordered_comm_semiring
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2255
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2256
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2257
subclass ordered_ring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2258
subclass ordered_cancel_comm_semiring ..
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2259
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  2260
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2261
67689
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2262
class linordered_nonzero_semiring = ordered_comm_semiring + monoid_mult + linorder + zero_less_one +
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2263
  assumes add_mono1: "a < b \<Longrightarrow> a + 1 < b + 1"
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2264
begin
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2265
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2266
subclass zero_neq_one
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2267
  by standard (insert zero_less_one, blast)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2268
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2269
subclass comm_semiring_1
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2270
  by standard (rule mult_1_left)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2271
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2272
lemma zero_le_one [simp]: "0 \<le> 1"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2273
  by (rule zero_less_one [THEN less_imp_le])
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2274
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2275
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2276
  by (simp add: not_le)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2277
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2278
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2279
  by (simp add: not_less)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2280
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2281
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2282
  using mult_left_mono[of c 1 a] by simp
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2283
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2284
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2285
  using mult_mono[of a 1 b 1] by simp
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2286
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2287
lemma zero_less_two: "0 < 1 + 1"
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2288
  using add_pos_pos[OF zero_less_one zero_less_one] .
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2289
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2290
end
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2291
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2292
class linordered_semidom = semidom + linordered_comm_semiring_strict + zero_less_one +
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
  2293
  assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2294
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2295
67689
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2296
subclass linordered_nonzero_semiring 
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2297
proof
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2298
  show "a + 1 < b + 1" if "a < b" for a b
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2299
  proof (rule ccontr, simp add: not_less)
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2300
    assume "b \<le> a"
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2301
    with that show False
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2302
      by (simp add: )
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2303
  qed
2c38ffd6ec71 type class linordered_nonzero_semiring has new axiom to guarantee characteristic 0
paulson <lp15@cam.ac.uk>
parents: 67234
diff changeset
  2304
qed
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2305
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  2306
text \<open>Addition is the inverse of subtraction.\<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
  2307
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
  2308
lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"
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
  2309
  by (frule le_add_diff_inverse2) (simp add: add.commute)
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
  2310
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2311
lemma add_diff_inverse: "\<not> a < b \<Longrightarrow> b + (a - b) = a"
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
  2312
  by simp
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2313
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2314
lemma add_le_imp_le_diff: "i + k \<le> n \<Longrightarrow> i \<le> n - k"
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2315
  apply (subst add_le_cancel_right [where c=k, symmetric])
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2316
  apply (frule le_add_diff_inverse2)
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2317
  apply (simp only: add.assoc [symmetric])
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2318
  using add_implies_diff
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2319
  apply fastforce
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2320
  done
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2321
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
  2322
lemma add_le_add_imp_diff_le:
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2323
  assumes 1: "i + k \<le> n"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2324
    and 2: "n \<le> j + k"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2325
  shows "i + k \<le> n \<Longrightarrow> n \<le> j + k \<Longrightarrow> n - k \<le> j"
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2326
proof -
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2327
  have "n - (i + k) + (i + k) = n"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2328
    using 1 by simp
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2329
  moreover have "n - k = n - k - i + i"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2330
    using 1 by (simp add: add_le_imp_le_diff)
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2331
  ultimately show ?thesis
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2332
    using 2
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2333
    apply (simp add: add.assoc [symmetric])
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2334
    apply (rule add_le_imp_le_diff [of _ k "j + k", simplified add_diff_cancel_right'])
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2335
    apply (simp add: add.commute diff_diff_add)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2336
    done
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2337
qed
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60570
diff changeset
  2338
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2339
lemma less_1_mult: "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n"
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62377
diff changeset
  2340
  using mult_strict_mono [of 1 m 1 n] by (simp add: less_trans [OF zero_less_one])
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58952
diff changeset
  2341
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2342
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2343
66937
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2344
class linordered_idom = comm_ring_1 + linordered_comm_semiring_strict +
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2345
  ordered_ab_group_add + abs_if + sgn +
64290
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2346
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2347
begin
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2348
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
  2349
subclass linordered_ring_strict ..
66937
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2350
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2351
subclass linordered_semiring_1_strict
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2352
proof
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2353
  have "0 \<le> 1 * 1"
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2354
    by (fact zero_le_square)
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2355
  then show "0 < 1" 
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2356
    by (simp add: le_less)
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2357
qed
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2358
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2359
subclass ordered_comm_ring ..
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
  2360
subclass idom ..
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2361
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2362
subclass linordered_semidom
66937
a1a4a5e2933a rule out pathologic instances
haftmann
parents: 66816
diff changeset
  2363
  by standard simp
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2364
64290
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2365
subclass idom_abs_sgn
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2366
  by standard
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2367
    (auto simp add: sgn_if abs_if zero_less_mult_iff)
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2368
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2369
lemma linorder_neqE_linordered_idom:
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2370
  assumes "x \<noteq> y"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2371
  obtains "x < y" | "y < x"
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2372
  using assms by (rule neqE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  2373
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2374
text \<open>These cancellation simp rules also produce two cases when the comparison is a goal.\<close>
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  2375
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2376
lemma mult_le_cancel_right1: "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2377
  using mult_le_cancel_right [of 1 c b] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  2378
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2379
lemma mult_le_cancel_right2: "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2380
  using mult_le_cancel_right [of a c 1] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  2381
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2382
lemma mult_le_cancel_left1: "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2383
  using mult_le_cancel_left [of c 1 b] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  2384
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2385
lemma mult_le_cancel_left2: "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2386
  using mult_le_cancel_left [of c a 1] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  2387
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2388
lemma mult_less_cancel_right1: "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2389
  using mult_less_cancel_right [of 1 c b] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  2390
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2391
lemma mult_less_cancel_right2: "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2392
  using mult_less_cancel_right [of a c 1] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  2393
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2394
lemma mult_less_cancel_left1: "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2395
  using mult_less_cancel_left [of c 1 b] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  2396
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2397
lemma mult_less_cancel_left2: "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2398
  using mult_less_cancel_left [of c a 1] by simp
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  2399
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2400
lemma sgn_0_0: "sgn a = 0 \<longleftrightarrow> a = 0"
64290
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2401
  by (fact sgn_eq_0_iff)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  2402
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2403
lemma sgn_1_pos: "sgn a = 1 \<longleftrightarrow> a > 0"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2404
  unfolding sgn_if by simp
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  2405
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2406
lemma sgn_1_neg: "sgn a = - 1 \<longleftrightarrow> a < 0"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2407
  unfolding sgn_if by auto
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  2408
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2409
lemma sgn_pos [simp]: "0 < a \<Longrightarrow> sgn a = 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2410
  by (simp only: sgn_1_pos)
29940
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  2411
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2412
lemma sgn_neg [simp]: "a < 0 \<Longrightarrow> sgn a = - 1"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2413
  by (simp only: sgn_1_neg)
29940
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  2414
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2415
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2416
  unfolding sgn_if abs_if by auto
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  2417
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2418
lemma sgn_greater [simp]: "0 < sgn a \<longleftrightarrow> 0 < a"
29940
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  2419
  unfolding sgn_if by auto
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  2420
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2421
lemma sgn_less [simp]: "sgn a < 0 \<longleftrightarrow> a < 0"
29940
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  2422
  unfolding sgn_if by auto
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  2423
64239
de5cd9217d4c added lemma
haftmann
parents: 64164
diff changeset
  2424
lemma abs_sgn_eq_1 [simp]:
de5cd9217d4c added lemma
haftmann
parents: 64164
diff changeset
  2425
  "a \<noteq> 0 \<Longrightarrow> \<bar>sgn a\<bar> = 1"
64290
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2426
  by simp
64239
de5cd9217d4c added lemma
haftmann
parents: 64164
diff changeset
  2427
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2428
lemma abs_sgn_eq: "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 61944
diff changeset
  2429
  by (simp add: sgn_if)
2230b7047376 generalized some lemmas;
haftmann
parents: 61944
diff changeset
  2430
64713
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2431
lemma sgn_mult_self_eq [simp]:
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2432
  "sgn a * sgn a = of_bool (a \<noteq> 0)"
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2433
  by (cases "a > 0") simp_all
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2434
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2435
lemma abs_mult_self_eq [simp]:
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2436
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2437
  by (cases "a > 0") simp_all
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2438
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2439
lemma same_sgn_sgn_add:
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2440
  "sgn (a + b) = sgn a" if "sgn b = sgn a"
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2441
proof (cases a 0 rule: linorder_cases)
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2442
  case equal
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2443
  with that show ?thesis
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2444
    by simp
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2445
next
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2446
  case less
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2447
  with that have "b < 0"
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2448
    by (simp add: sgn_1_neg)
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2449
  with \<open>a < 0\<close> have "a + b < 0"
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2450
    by (rule add_neg_neg)
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2451
  with \<open>a < 0\<close> show ?thesis
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2452
    by simp
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2453
next
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2454
  case greater
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2455
  with that have "b > 0"
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2456
    by (simp add: sgn_1_pos)
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2457
  with \<open>a > 0\<close> have "a + b > 0"
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2458
    by (rule add_pos_pos)
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2459
  with \<open>a > 0\<close> show ?thesis
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2460
    by simp
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2461
qed
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2462
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2463
lemma same_sgn_abs_add:
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2464
  "\<bar>a + b\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" if "sgn b = sgn a"
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2465
proof -
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2466
  have "a + b = sgn a * \<bar>a\<bar> + sgn b * \<bar>b\<bar>"
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2467
    by (simp add: sgn_mult_abs)
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2468
  also have "\<dots> = sgn a * (\<bar>a\<bar> + \<bar>b\<bar>)"
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2469
    using that by (simp add: algebra_simps)
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2470
  finally show ?thesis
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2471
    by (auto simp add: abs_mult)
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2472
qed
9638c07283bc more facts on sgn, abs
haftmann
parents: 64592
diff changeset
  2473
66816
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66810
diff changeset
  2474
lemma sgn_not_eq_imp:
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66810
diff changeset
  2475
  "sgn a = - sgn b" if "sgn b \<noteq> sgn a" and "sgn a \<noteq> 0" and "sgn b \<noteq> 0"
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66810
diff changeset
  2476
  using that by (cases "a < 0") (auto simp add: sgn_0_0 sgn_1_pos sgn_1_neg)
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66810
diff changeset
  2477
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2478
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
29949
20a506b8256d lemmas abs_dvd_iff, dvd_abs_iff
huffman
parents: 29940
diff changeset
  2479
  by (simp add: abs_if)
20a506b8256d lemmas abs_dvd_iff, dvd_abs_iff
huffman
parents: 29940
diff changeset
  2480
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2481
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
29949
20a506b8256d lemmas abs_dvd_iff, dvd_abs_iff
huffman
parents: 29940
diff changeset
  2482
  by (simp add: abs_if)
29653
ece6a0e9f8af added lemma abs_sng
haftmann
parents: 29465
diff changeset
  2483
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2484
lemma dvd_if_abs_eq: "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2485
  by (subst abs_dvd_iff [symmetric]) simp
33676
802f5e233e48 moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents: 33364
diff changeset
  2486
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2487
text \<open>
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2488
  The following lemmas can be proven in more general structures, but
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2489
  are dangerous as simp rules in absence of @{thm neg_equal_zero},
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2490
  @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2491
\<close>
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2492
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2493
lemma equation_minus_iff_1 [simp, no_atp]: "1 = - a \<longleftrightarrow> a = - 1"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2494
  by (fact equation_minus_iff)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2495
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2496
lemma minus_equation_iff_1 [simp, no_atp]: "- a = 1 \<longleftrightarrow> a = - 1"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2497
  by (subst minus_equation_iff, auto)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2498
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2499
lemma le_minus_iff_1 [simp, no_atp]: "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2500
  by (fact le_minus_iff)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2501
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2502
lemma minus_le_iff_1 [simp, no_atp]: "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2503
  by (fact minus_le_iff)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2504
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2505
lemma less_minus_iff_1 [simp, no_atp]: "1 < - b \<longleftrightarrow> b < - 1"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2506
  by (fact less_minus_iff)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2507
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2508
lemma minus_less_iff_1 [simp, no_atp]: "- a < 1 \<longleftrightarrow> - 1 < a"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2509
  by (fact minus_less_iff)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  2510
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 65811
diff changeset
  2511
lemma add_less_zeroD:
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 65811
diff changeset
  2512
  shows "x+y < 0 \<Longrightarrow> x<0 \<or> y<0"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 65811
diff changeset
  2513
  by (auto simp: not_less intro: le_less_trans [of _ "x+y"])
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 65811
diff changeset
  2514
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  2515
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  2516
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  2517
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  2518
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
  2519
lemmas mult_compare_simps =
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2520
  mult_le_cancel_right mult_le_cancel_left
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2521
  mult_le_cancel_right1 mult_le_cancel_right2
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2522
  mult_le_cancel_left1 mult_le_cancel_left2
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2523
  mult_less_cancel_right mult_less_cancel_left
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2524
  mult_less_cancel_right1 mult_less_cancel_right2
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2525
  mult_less_cancel_left1 mult_less_cancel_left2
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2526
  mult_cancel_right mult_cancel_left
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2527
  mult_cancel_right1 mult_cancel_right2
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2528
  mult_cancel_left1 mult_cancel_left2
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2529
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  2530
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  2531
text \<open>Reasoning about inequalities with division\<close>
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  2532
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2533
context linordered_semidom
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  2534
begin
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  2535
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  2536
lemma less_add_one: "a < a + 1"
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  2537
proof -
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  2538
  have "a + 0 < a + 1"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  2539
    by (blast intro: zero_less_one add_strict_left_mono)
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2540
  then show ?thesis by simp
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  2541
qed
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  2542
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  2543
end
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  2544
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2545
context linordered_idom
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2546
begin
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  2547
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2548
lemma mult_right_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59832
diff changeset
  2549
  by (rule mult_left_le)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2550
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2551
lemma mult_left_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2552
  by (auto simp add: mult_le_cancel_right2)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2553
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2554
end
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2555
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60690
diff changeset
  2556
text \<open>Absolute Value\<close>
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  2557
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2558
context linordered_idom
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2559
begin
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2560
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2561
lemma mult_sgn_abs: "sgn x * \<bar>x\<bar> = x"
64290
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2562
  by (fact sgn_mult_abs)
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2563
64290
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2564
lemma abs_one: "\<bar>1\<bar> = 1"
fb5c74a58796 suitable logical type class for abs, sgn
haftmann
parents: 64242
diff changeset
  2565
  by (fact abs_1)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2566
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2567
end
24491
8d194c9198ae added constant sgn
nipkow
parents: 24427
diff changeset
  2568
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2569
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2570
  assumes abs_eq_mult:
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2571
    "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0) \<Longrightarrow> \<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  2572
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  2573
context linordered_idom
30961
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  2574
begin
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  2575
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2576
subclass ordered_ring_abs
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63456
diff changeset
  2577
  by standard (auto simp: abs_if not_less mult_less_0_iff)
30961
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  2578
67051
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  2579
lemma abs_mult_self: "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
e7e54a0b9197 dedicated definition for coprimality
haftmann
parents: 66937
diff changeset
  2580
  by (fact abs_mult_self_eq)
30961
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  2581
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  2582
lemma abs_mult_less:
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2583
  assumes ac: "\<bar>a\<bar> < c"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2584
    and bd: "\<bar>b\<bar> < d"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2585
  shows "\<bar>a\<bar> * \<bar>b\<bar> < c * d"
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  2586
proof -
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2587
  from ac have "0 < c"
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2588
    by (blast intro: le_less_trans abs_ge_zero)
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2589
  with bd show ?thesis by (simp add: ac mult_strict_mono)
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  2590
qed
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  2591
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2592
lemma abs_less_iff: "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2593
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
  2594
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2595
lemma abs_mult_pos: "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2596
  by (simp add: abs_mult)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2597
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2598
lemma abs_diff_less_iff: "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
51520
e9b361845809 move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents: 50420
diff changeset
  2599
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
e9b361845809 move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents: 50420
diff changeset
  2600
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2601
lemma abs_diff_le_iff: "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
59865
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59833
diff changeset
  2602
  by (auto simp add: diff_le_eq ac_simps abs_le_iff)
8a20dd967385 rationalised and generalised some theorems concerning abs and x^2.
paulson <lp15@cam.ac.uk>
parents: 59833
diff changeset
  2603
62626
de25474ce728 Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents: 62608
diff changeset
  2604
lemma abs_add_one_gt_zero: "0 < 1 + \<bar>x\<bar>"
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2605
  by (auto simp: abs_if not_less intro: zero_less_one add_strict_increasing less_trans)
62626
de25474ce728 Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents: 62608
diff changeset
  2606
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  2607
end
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  2608
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
  2609
subsection \<open>Dioids\<close>
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
  2610
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2611
text \<open>
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2612
  Dioids are the alternative extensions of semirings, a semiring can
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2613
  either be a ring or a dioid but never both.
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2614
\<close>
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
  2615
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
  2616
class dioid = semiring_1 + canonically_ordered_monoid_add
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
  2617
begin
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
  2618
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
  2619
subclass ordered_semiring
63325
1086d56cde86 misc tuning and modernization;
wenzelm
parents: 63040
diff changeset
  2620
  by standard (auto simp: le_iff_add distrib_left distrib_right)
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
  2621
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
  2622
end
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
  2623
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
  2624
59557
ebd8ecacfba6 establish unique preferred fact names
haftmann
parents: 59555
diff changeset
  2625
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
ebd8ecacfba6 establish unique preferred fact names
haftmann
parents: 59555
diff changeset
  2626
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51520
diff changeset
  2627
code_identifier
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51520
diff changeset
  2628
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
  2629
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  2630
end