src/HOL/Rings.thy
author haftmann
Mon, 26 Apr 2010 11:34:15 +0200
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dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
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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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header {* Rings *}
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theory Rings
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imports Groups
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begin
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes left_distrib[algebra_simps, field_simps]: "(a + b) * c = a * c + b * c"
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  assumes right_distrib[algebra_simps, field_simps]: "a * (b + c) = a * b + a * c"
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begin
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor:
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  "a * e + (b * e + c) = (a + b) * e + c"
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by (simp add: left_distrib add_ac)
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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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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0" by (simp add: left_distrib [symmetric])
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  thus "0 * a = 0" by (simp only: add_left_cancel)
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next
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  fix a :: 'a
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  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: right_distrib [symmetric])
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  thus "a * 0 = 0" by (simp only: add_left_cancel)
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qed
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end
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a + b) * c = a * c + b * c"
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begin
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subclass semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: mult_ac)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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end
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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begin
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subclass semiring_0 ..
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end
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class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0_cancel ..
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subclass comm_semiring_0 ..
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end
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "0 \<noteq> 1"
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begin
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lemma one_neq_zero [simp]: "1 \<noteq> 0"
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by (rule not_sym) (rule zero_neq_one)
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end
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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text {* Abstract divisibility *}
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class dvd = times
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begin
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definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where
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  [code del]: "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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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd
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  (*previously almost_semiring*)
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begin
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subclass semiring_1 ..
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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:
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  assumes "a dvd b" and "b dvd c"
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  shows "a dvd c"
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proof -
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  from assms obtain v where "b = a * v" by (auto elim!: dvdE)
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  moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
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  ultimately have "c = a * (v * w)" by (simp add: mult_assoc)
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  then show ?thesis ..
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qed
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lemma dvd_0_left_iff [no_atp, simp]: "0 dvd a \<longleftrightarrow> a = 0"
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by (auto intro: dvd_refl elim!: dvdE)
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lemma dvd_0_right [iff]: "a dvd 0"
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proof
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  show "0 = a * 0" by simp
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qed
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lemma one_dvd [simp]: "1 dvd a"
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by (auto intro!: dvdI)
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lemma dvd_mult[simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
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by (auto intro!: mult_left_commute dvdI elim!: dvdE)
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lemma dvd_mult2[simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
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  apply (subst mult_commute)
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  apply (erule dvd_mult)
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  done
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lemma dvd_triv_right [simp]: "a dvd b * a"
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by (rule dvd_mult) (rule dvd_refl)
27651
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parents: 27516
diff changeset
   151
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   152
lemma dvd_triv_left [simp]: "a dvd a * b"
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53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
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diff changeset
   153
by (rule dvd_mult2) (rule dvd_refl)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents: 27516
diff changeset
   154
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   155
lemma mult_dvd_mono:
30042
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   156
  assumes "a dvd b"
31039ee583fa Removed subsumed lemmas
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   157
    and "c dvd d"
27651
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   158
  shows "a * c dvd b * d"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents: 27516
diff changeset
   159
proof -
30042
31039ee583fa Removed subsumed lemmas
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   160
  from `a dvd b` obtain b' where "b = a * b'" ..
31039ee583fa Removed subsumed lemmas
nipkow
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diff changeset
   161
  moreover from `c dvd d` obtain d' where "d = c * d'" ..
27651
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parents: 27516
diff changeset
   162
  ultimately have "b * d = (a * c) * (b' * d')" by (simp add: mult_ac)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents: 27516
diff changeset
   163
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   164
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   165
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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diff changeset
   166
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
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53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
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parents: 29465
diff changeset
   167
by (simp add: dvd_def mult_assoc, blast)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents: 27516
diff changeset
   168
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents: 27516
diff changeset
   169
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents: 27516
diff changeset
   170
  unfolding mult_ac [of a] by (rule dvd_mult_left)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents: 27516
diff changeset
   171
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   172
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
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   173
by simp
27651
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parents: 27516
diff changeset
   174
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
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   175
lemma dvd_add[simp]:
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29915
diff changeset
   176
  assumes "a dvd b" and "a dvd c" 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
   177
proof -
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29915
diff changeset
   178
  from `a dvd b` obtain b' where "b = a * b'" ..
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29915
diff changeset
   179
  moreover from `a dvd c` obtain c' where "c = a * c'" ..
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents: 27516
diff changeset
   180
  ultimately have "b + c = a * (b' + c')" by (simp add: right_distrib)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   181
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   182
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   183
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   184
end
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   185
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29915
diff changeset
   186
22390
378f34b1e380 now using "class"
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   187
class no_zero_divisors = zero + times +
25062
af5ef0d4d655 global class syntax
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parents: 24748
diff changeset
   188
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   189
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   190
class semiring_1_cancel = semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   191
  + zero_neq_one + monoid_mult
25267
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haftmann
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diff changeset
   192
begin
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   193
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   194
subclass semiring_0_cancel ..
25512
4134f7c782e2 using intro_locales instead of unfold_locales if appropriate
haftmann
parents: 25450
diff changeset
   195
27516
9a5d4a8d4aac by intro_locales -> ..
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parents: 26274
diff changeset
   196
subclass semiring_1 ..
25267
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parents: 25238
diff changeset
   197
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   198
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   199
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
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diff changeset
   200
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   201
  + zero_neq_one + comm_monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
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parents: 25238
diff changeset
   202
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   203
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   204
subclass semiring_1_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   205
subclass comm_semiring_0_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   206
subclass comm_semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   207
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   208
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   209
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   210
class ring = semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   211
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   212
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   213
subclass semiring_0_cancel ..
25152
bfde2f8c0f63 partially localized
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parents: 25078
diff changeset
   214
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   215
text {* Distribution rules *}
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   216
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   217
lemma minus_mult_left: "- (a * b) = - a * b"
34146
14595e0c27e8 rename equals_zero_I to minus_unique (keep old name too)
huffman
parents: 33676
diff changeset
   218
by (rule minus_unique) (simp add: left_distrib [symmetric]) 
25152
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haftmann
parents: 25078
diff changeset
   219
bfde2f8c0f63 partially localized
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parents: 25078
diff changeset
   220
lemma minus_mult_right: "- (a * b) = a * - b"
34146
14595e0c27e8 rename equals_zero_I to minus_unique (keep old name too)
huffman
parents: 33676
diff changeset
   221
by (rule minus_unique) (simp add: right_distrib [symmetric]) 
25152
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haftmann
parents: 25078
diff changeset
   222
29407
5ef7e97fd9e4 move lemmas mult_minus{left,right} inside class ring
huffman
parents: 29406
diff changeset
   223
text{*Extract signs from products*}
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35631
diff changeset
   224
lemmas mult_minus_left [simp, no_atp] = minus_mult_left [symmetric]
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35631
diff changeset
   225
lemmas mult_minus_right [simp,no_atp] = minus_mult_right [symmetric]
29407
5ef7e97fd9e4 move lemmas mult_minus{left,right} inside class ring
huffman
parents: 29406
diff changeset
   226
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   227
lemma minus_mult_minus [simp]: "- a * - b = a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   228
by simp
25152
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parents: 25078
diff changeset
   229
bfde2f8c0f63 partially localized
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parents: 25078
diff changeset
   230
lemma minus_mult_commute: "- a * b = a * - b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   231
by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   232
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36304
diff changeset
   233
lemma right_diff_distrib[algebra_simps, field_simps]: "a * (b - c) = a * b - a * c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   234
by (simp add: right_distrib diff_minus)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   235
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36304
diff changeset
   236
lemma left_diff_distrib[algebra_simps, field_simps]: "(a - b) * c = a * c - b * c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   237
by (simp add: left_distrib diff_minus)
25152
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haftmann
parents: 25078
diff changeset
   238
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35631
diff changeset
   239
lemmas ring_distribs[no_atp] =
25152
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haftmann
parents: 25078
diff changeset
   240
  right_distrib left_distrib left_diff_distrib right_diff_distrib
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   241
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   242
lemma eq_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   243
  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   244
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   245
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   246
lemma eq_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   247
  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   248
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   249
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   250
end
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   251
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35631
diff changeset
   252
lemmas ring_distribs[no_atp] =
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   253
  right_distrib left_distrib left_diff_distrib right_diff_distrib
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   254
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   255
class comm_ring = comm_semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   256
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   257
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   258
subclass ring ..
28141
193c3ea0f63b instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents: 27651
diff changeset
   259
subclass comm_semiring_0_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   260
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   261
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   262
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   263
class ring_1 = ring + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   264
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   265
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   266
subclass semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   268
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   269
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   270
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   271
  (*previously ring*)
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   272
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   273
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   274
subclass ring_1 ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   275
subclass comm_semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   276
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
   277
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
   278
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   279
  assume "x dvd - y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   280
  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
   281
  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
   282
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   283
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   284
  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
   285
  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
   286
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   287
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
   288
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
   289
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   290
  assume "- x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   291
  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
   292
  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
   293
  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
   294
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   295
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   296
  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
   297
  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
   298
  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
   299
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   300
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   301
lemma dvd_diff[simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   302
by (simp only: diff_minus dvd_add dvd_minus_iff)
29409
f0a8fe83bc07 add lemma dvd_diff to class comm_ring_1
huffman
parents: 29408
diff changeset
   303
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   304
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   305
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   306
class ring_no_zero_divisors = ring + no_zero_divisors
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   307
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   308
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   309
lemma mult_eq_0_iff [simp]:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   310
  shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   311
proof (cases "a = 0 \<or> b = 0")
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   312
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   313
    then show ?thesis using no_zero_divisors by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   314
next
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   315
  case True then show ?thesis by auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   316
qed
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   317
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   318
text{*Cancellation of equalities with a common factor*}
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35631
diff changeset
   319
lemma mult_cancel_right [simp, no_atp]:
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   320
  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   321
proof -
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   322
  have "(a * c = b * c) = ((a - b) * c = 0)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   323
    by (simp add: algebra_simps)
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   324
  thus ?thesis by (simp add: disj_commute)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   325
qed
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   326
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35631
diff changeset
   327
lemma mult_cancel_left [simp, no_atp]:
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   328
  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   329
proof -
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   330
  have "(c * a = c * b) = (c * (a - b) = 0)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   331
    by (simp add: algebra_simps)
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   332
  thus ?thesis by simp
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   333
qed
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   334
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   335
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   336
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   337
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   338
begin
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   339
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   340
lemma mult_cancel_right1 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   341
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   342
by (insert mult_cancel_right [of 1 c b], force)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   343
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   344
lemma mult_cancel_right2 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   345
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   346
by (insert mult_cancel_right [of a c 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   347
 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   348
lemma mult_cancel_left1 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   349
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   350
by (insert mult_cancel_left [of c 1 b], force)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   351
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   352
lemma mult_cancel_left2 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   353
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   354
by (insert mult_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   355
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   356
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   357
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   358
class idom = comm_ring_1 + no_zero_divisors
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   359
begin
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   360
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   361
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
   362
29915
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   363
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   364
proof
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   365
  assume "a * a = b * b"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   366
  then have "(a - b) * (a + b) = 0"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   367
    by (simp add: algebra_simps)
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   368
  then show "a = b \<or> a = - b"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   369
    by (simp add: eq_neg_iff_add_eq_0)
29915
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   370
next
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   371
  assume "a = b \<or> a = - b"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   372
  then show "a * a = b * b" by auto
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   373
qed
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   374
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   375
lemma dvd_mult_cancel_right [simp]:
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   376
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   377
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   378
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   379
    unfolding dvd_def by (simp add: mult_ac)
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   380
  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
   381
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   382
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   383
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   384
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   385
lemma dvd_mult_cancel_left [simp]:
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   386
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   387
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   388
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   389
    unfolding dvd_def by (simp add: mult_ac)
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   390
  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
   391
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   392
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   393
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   394
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   395
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   396
35083
3246e66b0874 division ring assumes divide_inverse
haftmann
parents: 35050
diff changeset
   397
class inverse =
3246e66b0874 division ring assumes divide_inverse
haftmann
parents: 35050
diff changeset
   398
  fixes inverse :: "'a \<Rightarrow> 'a"
3246e66b0874 division ring assumes divide_inverse
haftmann
parents: 35050
diff changeset
   399
    and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)
3246e66b0874 division ring assumes divide_inverse
haftmann
parents: 35050
diff changeset
   400
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   401
class division_ring = ring_1 + inverse +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   402
  assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   403
  assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
35083
3246e66b0874 division ring assumes divide_inverse
haftmann
parents: 35050
diff changeset
   404
  assumes divide_inverse: "a / b = a * inverse b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   405
begin
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   406
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   407
subclass ring_1_no_zero_divisors
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   408
proof
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   409
  fix a b :: 'a
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   410
  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   411
  show "a * b \<noteq> 0"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   412
  proof
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   413
    assume ab: "a * b = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   414
    hence "0 = inverse a * (a * b) * inverse b" by simp
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   415
    also have "\<dots> = (inverse a * a) * (b * inverse b)"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   416
      by (simp only: mult_assoc)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   417
    also have "\<dots> = 1" using a b by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   418
    finally show False by simp
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   419
  qed
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   420
qed
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   421
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   422
lemma nonzero_imp_inverse_nonzero:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   423
  "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   424
proof
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   425
  assume ianz: "inverse a = 0"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   426
  assume "a \<noteq> 0"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   427
  hence "1 = a * inverse a" by simp
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   428
  also have "... = 0" by (simp add: ianz)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   429
  finally have "1 = 0" .
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   430
  thus False by (simp add: eq_commute)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   431
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   432
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   433
lemma inverse_zero_imp_zero:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   434
  "inverse a = 0 \<Longrightarrow> a = 0"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   435
apply (rule classical)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   436
apply (drule nonzero_imp_inverse_nonzero)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   437
apply auto
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   438
done
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   439
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   440
lemma inverse_unique: 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   441
  assumes ab: "a * b = 1"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   442
  shows "inverse a = b"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   443
proof -
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   444
  have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   445
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   446
  ultimately show ?thesis by (simp add: mult_assoc [symmetric])
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   447
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   448
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   449
lemma nonzero_inverse_minus_eq:
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   450
  "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   451
by (rule inverse_unique) simp
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   452
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   453
lemma nonzero_inverse_inverse_eq:
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   454
  "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   455
by (rule inverse_unique) simp
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   456
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   457
lemma nonzero_inverse_eq_imp_eq:
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   458
  assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   459
  shows "a = b"
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   460
proof -
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   461
  from `inverse a = inverse b`
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   462
  have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   463
  with `a \<noteq> 0` and `b \<noteq> 0` show "a = b"
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   464
    by (simp add: nonzero_inverse_inverse_eq)
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   465
qed
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   466
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   467
lemma inverse_1 [simp]: "inverse 1 = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   468
by (rule inverse_unique) simp
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   469
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   470
lemma nonzero_inverse_mult_distrib: 
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   471
  assumes "a \<noteq> 0" and "b \<noteq> 0"
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   472
  shows "inverse (a * b) = inverse b * inverse a"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   473
proof -
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   474
  have "a * (b * inverse b) * inverse a = 1" using assms by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   475
  hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   476
  thus ?thesis by (rule inverse_unique)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   477
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   478
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   479
lemma division_ring_inverse_add:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   480
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   481
by (simp add: algebra_simps)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   482
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   483
lemma division_ring_inverse_diff:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   484
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   485
by (simp add: algebra_simps)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   486
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   487
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   488
proof
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   489
  assume neq: "b \<noteq> 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   490
  {
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   491
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_assoc)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   492
    also assume "a / b = 1"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   493
    finally show "a = b" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   494
  next
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   495
    assume "a = b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   496
    with neq show "a / b = 1" by (simp add: divide_inverse)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   497
  }
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   498
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   499
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   500
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   501
by (simp add: divide_inverse)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   502
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   503
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   504
by (simp add: divide_inverse)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   505
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   506
lemma divide_zero_left [simp]: "0 / a = 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   507
by (simp add: divide_inverse)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   508
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   509
lemma inverse_eq_divide: "inverse a = 1 / a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   510
by (simp add: divide_inverse)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   511
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   512
lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   513
by (simp add: divide_inverse algebra_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   514
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   515
lemma divide_1 [simp]: "a / 1 = a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   516
  by (simp add: divide_inverse)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   517
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   518
lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   519
  by (simp add: divide_inverse mult_assoc)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   520
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   521
lemma minus_divide_left: "- (a / b) = (-a) / b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   522
  by (simp add: divide_inverse)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   523
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   524
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   525
  by (simp add: divide_inverse nonzero_inverse_minus_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   526
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   527
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   528
  by (simp add: divide_inverse nonzero_inverse_minus_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   529
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   530
lemma divide_minus_left [simp, no_atp]: "(-a) / b = - (a / b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   531
  by (simp add: divide_inverse)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   532
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   533
lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   534
  by (simp add: diff_minus add_divide_distrib)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   535
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36304
diff changeset
   536
lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   537
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   538
  assume [simp]: "c \<noteq> 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   539
  have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   540
  also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   541
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   542
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   543
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36304
diff changeset
   544
lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   545
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   546
  assume [simp]: "c \<noteq> 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   547
  have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   548
  also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   549
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   550
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   551
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   552
lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   553
  by (simp add: divide_inverse mult_assoc)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   554
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   555
lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   556
  by (drule sym) (simp add: divide_inverse mult_assoc)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   557
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   558
end
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   559
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36304
diff changeset
   560
class division_ring_inverse_zero = division_ring +
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   561
  assumes inverse_zero [simp]: "inverse 0 = 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   562
begin
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   563
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   564
lemma divide_zero [simp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   565
  "a / 0 = 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   566
  by (simp add: divide_inverse)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   567
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   568
lemma divide_self_if [simp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   569
  "a / a = (if a = 0 then 0 else 1)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   570
  by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   571
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   572
lemma inverse_nonzero_iff_nonzero [simp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   573
  "inverse a = 0 \<longleftrightarrow> a = 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   574
  by rule (fact inverse_zero_imp_zero, simp)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   575
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   576
lemma inverse_minus_eq [simp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   577
  "inverse (- a) = - inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   578
proof cases
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   579
  assume "a=0" thus ?thesis by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   580
next
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   581
  assume "a\<noteq>0" 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   582
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   583
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   584
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   585
lemma inverse_eq_imp_eq:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   586
  "inverse a = inverse b \<Longrightarrow> a = b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   587
apply (cases "a=0 | b=0") 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   588
 apply (force dest!: inverse_zero_imp_zero
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   589
              simp add: eq_commute [of "0::'a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   590
apply (force dest!: nonzero_inverse_eq_imp_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   591
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   592
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   593
lemma inverse_eq_iff_eq [simp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   594
  "inverse a = inverse b \<longleftrightarrow> a = b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   595
  by (force dest!: inverse_eq_imp_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   596
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   597
lemma inverse_inverse_eq [simp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   598
  "inverse (inverse a) = a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   599
proof cases
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   600
  assume "a=0" thus ?thesis by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   601
next
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   602
  assume "a\<noteq>0" 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   603
  thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   604
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   605
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   606
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   607
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   608
class mult_mono = times + zero + ord +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   609
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   610
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   611
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   612
text {*
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   613
  The theory of partially ordered rings is taken from the books:
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   614
  \begin{itemize}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   615
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   616
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   617
  \end{itemize}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   618
  Most of the used notions can also be looked up in 
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   619
  \begin{itemize}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   620
  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   621
  \item \emph{Algebra I} by van der Waerden, Springer.
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   622
  \end{itemize}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   623
*}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   624
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   625
class ordered_semiring = mult_mono + semiring_0 + ordered_ab_semigroup_add 
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   626
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   627
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   628
lemma mult_mono:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   629
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   630
     \<Longrightarrow> a * c \<le> b * d"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   631
apply (erule mult_right_mono [THEN order_trans], assumption)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   632
apply (erule mult_left_mono, assumption)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   633
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   634
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   635
lemma mult_mono':
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   636
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   637
     \<Longrightarrow> a * c \<le> b * d"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   638
apply (rule mult_mono)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   639
apply (fast intro: order_trans)+
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   640
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   641
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   642
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   643
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   644
class ordered_cancel_semiring = mult_mono + ordered_ab_semigroup_add
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   645
  + semiring + cancel_comm_monoid_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   646
begin
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   647
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   648
subclass semiring_0_cancel ..
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   649
subclass ordered_semiring ..
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   650
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   651
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   652
using mult_left_mono [of 0 b a] by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   653
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   654
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   655
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
   656
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   657
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   658
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
   659
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   660
text {* Legacy - use @{text mult_nonpos_nonneg} *}
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   661
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   662
by (drule mult_right_mono [of b 0], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   663
26234
1f6e28a88785 clarified proposition
haftmann
parents: 26193
diff changeset
   664
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   665
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   666
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   667
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   668
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   669
class linordered_semiring = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add + mult_mono
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   670
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   671
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   672
subclass ordered_cancel_semiring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   673
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   674
subclass ordered_comm_monoid_add ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   675
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   676
lemma mult_left_less_imp_less:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   677
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   678
by (force simp add: mult_left_mono not_le [symmetric])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   679
 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   680
lemma mult_right_less_imp_less:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   681
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   682
by (force simp add: mult_right_mono not_le [symmetric])
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   683
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   684
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   685
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   686
class linordered_semiring_1 = linordered_semiring + semiring_1
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   687
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   688
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   689
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   690
  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
   691
begin
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   692
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   693
subclass semiring_0_cancel ..
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   694
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   695
subclass linordered_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   696
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   697
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   698
  assume A: "a \<le> b" "0 \<le> c"
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   699
  from A show "c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   700
    unfolding le_less
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   701
    using mult_strict_left_mono by (cases "c = 0") auto
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   702
  from A show "a * c \<le> b * c"
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   703
    unfolding le_less
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   704
    using mult_strict_right_mono by (cases "c = 0") auto
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   705
qed
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   706
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   707
lemma mult_left_le_imp_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   708
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   709
by (force simp add: mult_strict_left_mono _not_less [symmetric])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   710
 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   711
lemma mult_right_le_imp_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   712
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   713
by (force simp add: mult_strict_right_mono not_less [symmetric])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   714
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   715
lemma mult_pos_pos: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   716
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
   717
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   718
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   719
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
   720
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   721
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   722
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
   723
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   724
text {* Legacy - use @{text mult_neg_pos} *}
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   725
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   726
by (drule mult_strict_right_mono [of b 0], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   727
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   728
lemma zero_less_mult_pos:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   729
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   730
apply (cases "b\<le>0")
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   731
 apply (auto simp add: le_less not_less)
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   732
apply (drule_tac mult_pos_neg [of a b])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   733
 apply (auto dest: less_not_sym)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   734
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   735
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   736
lemma zero_less_mult_pos2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   737
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   738
apply (cases "b\<le>0")
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   739
 apply (auto simp add: le_less not_less)
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   740
apply (drule_tac mult_pos_neg2 [of a b])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   741
 apply (auto dest: less_not_sym)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   742
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   743
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   744
text{*Strict monotonicity in both arguments*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   745
lemma mult_strict_mono:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   746
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   747
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   748
  using assms apply (cases "c=0")
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   749
  apply (simp add: mult_pos_pos)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   750
  apply (erule mult_strict_right_mono [THEN less_trans])
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   751
  apply (force simp add: le_less)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   752
  apply (erule mult_strict_left_mono, assumption)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   753
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   754
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   755
text{*This weaker variant has more natural premises*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   756
lemma mult_strict_mono':
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   757
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   758
  shows "a * c < b * d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   759
by (rule mult_strict_mono) (insert assms, auto)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   760
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   761
lemma mult_less_le_imp_less:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   762
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   763
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   764
  using assms apply (subgoal_tac "a * c < b * c")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   765
  apply (erule less_le_trans)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   766
  apply (erule mult_left_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   767
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   768
  apply (erule mult_strict_right_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   769
  apply assumption
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   770
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   771
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   772
lemma mult_le_less_imp_less:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   773
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   774
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   775
  using assms apply (subgoal_tac "a * c \<le> b * c")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   776
  apply (erule le_less_trans)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   777
  apply (erule mult_strict_left_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   778
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   779
  apply (erule mult_right_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   780
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   781
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   782
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   783
lemma mult_less_imp_less_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   784
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   785
  shows "a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   786
proof (rule ccontr)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   787
  assume "\<not>  a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   788
  hence "b \<le> a" by (simp add: linorder_not_less)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   789
  hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   790
  with this and less show False by (simp add: not_less [symmetric])
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   791
qed
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   792
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   793
lemma mult_less_imp_less_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   794
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   795
  shows "a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   796
proof (rule ccontr)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   797
  assume "\<not> a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   798
  hence "b \<le> a" by (simp add: linorder_not_less)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   799
  hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   800
  with this and less show False by (simp add: not_less [symmetric])
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   801
qed  
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   802
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   803
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   804
35097
4554bb2abfa3 dropped last occurence of the linlinordered accident
haftmann
parents: 35092
diff changeset
   805
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
33319
74f0dcc0b5fb moved algebraic classes to Ring_and_Field
haftmann
parents: 32960
diff changeset
   806
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   807
class mult_mono1 = times + zero + ord +
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   808
  assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   809
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   810
class ordered_comm_semiring = comm_semiring_0
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   811
  + ordered_ab_semigroup_add + mult_mono1
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   812
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   813
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   814
subclass ordered_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   815
proof
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   816
  fix a b c :: 'a
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   817
  assume "a \<le> b" "0 \<le> c"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   818
  thus "c * a \<le> c * b" by (rule mult_mono1)
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   819
  thus "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
   820
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   821
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   822
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   823
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   824
class ordered_cancel_comm_semiring = comm_semiring_0_cancel
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   825
  + ordered_ab_semigroup_add + mult_mono1
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   826
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   827
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   828
subclass ordered_comm_semiring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   829
subclass ordered_cancel_semiring ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   830
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   831
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   832
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   833
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   834
  assumes mult_strict_left_mono_comm: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   835
begin
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   836
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   837
subclass linordered_semiring_strict
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   838
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   839
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   840
  assume "a < b" "0 < c"
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   841
  thus "c * a < c * b" by (rule mult_strict_left_mono_comm)
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   842
  thus "a * c < b * c" by (simp only: mult_commute)
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   843
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   844
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   845
subclass ordered_cancel_comm_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   846
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   847
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   848
  assume "a \<le> b" "0 \<le> c"
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   849
  thus "c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   850
    unfolding le_less
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   851
    using mult_strict_left_mono by (cases "c = 0") auto
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   852
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   853
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   854
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   855
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   856
class ordered_ring = ring + ordered_cancel_semiring 
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   857
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   858
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   859
subclass ordered_ab_group_add ..
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   860
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   861
lemma less_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   862
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   863
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   864
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   865
lemma less_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   866
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   867
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   868
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   869
lemma le_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   870
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   871
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   872
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   873
lemma le_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   874
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   875
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   876
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   877
lemma mult_left_mono_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   878
  "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
   879
  apply (drule mult_left_mono [of _ _ "- c"])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   880
  apply simp_all
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   881
  done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   882
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   883
lemma mult_right_mono_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   884
  "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
   885
  apply (drule mult_right_mono [of _ _ "- c"])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   886
  apply simp_all
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   887
  done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   888
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   889
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   890
using mult_right_mono_neg [of a 0 b] by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   891
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   892
lemma split_mult_pos_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   893
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   894
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   895
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   896
end
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   897
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   898
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
   899
begin
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   900
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   901
subclass ordered_ring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   902
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   903
subclass ordered_ab_group_add_abs
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   904
proof
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   905
  fix a b
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   906
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   907
    by (auto simp add: abs_if not_less)
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   908
    (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric],
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   909
     auto intro: add_nonneg_nonneg, auto intro!: less_imp_le add_neg_neg)
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   910
qed (auto simp add: abs_if)
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   911
35631
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
   912
lemma zero_le_square [simp]: "0 \<le> a * a"
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
   913
  using linear [of 0 a]
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
   914
  by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
   915
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
   916
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
   917
  by (simp add: not_less)
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
   918
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   919
end
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   920
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   921
(* The "strict" suffix can be seen as describing the combination of linordered_ring and no_zero_divisors.
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   922
   Basically, linordered_ring + no_zero_divisors = linordered_ring_strict.
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   923
 *)
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   924
class linordered_ring_strict = ring + linordered_semiring_strict
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   925
  + ordered_ab_group_add + abs_if
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   926
begin
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   927
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   928
subclass linordered_ring ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   929
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   930
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   931
using mult_strict_left_mono [of b a "- c"] by simp
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   932
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   933
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   934
using mult_strict_right_mono [of b a "- c"] by simp
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   935
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   936
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   937
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
   938
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   939
subclass ring_no_zero_divisors
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   940
proof
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   941
  fix a b
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   942
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   943
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   944
  have "a * b < 0 \<or> 0 < a * b"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   945
  proof (cases "a < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   946
    case True note A' = this
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   947
    show ?thesis proof (cases "b < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   948
      case True with A'
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   949
      show ?thesis by (auto dest: mult_neg_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   950
    next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   951
      case False with B have "0 < b" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   952
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   953
    qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   954
  next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   955
    case False with A have A': "0 < a" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   956
    show ?thesis proof (cases "b < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   957
      case True with A'
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   958
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   959
    next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   960
      case False with B have "0 < b" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   961
      with A' show ?thesis by (auto dest: mult_pos_pos)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   962
    qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   963
  qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   964
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   965
qed
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   966
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   967
lemma zero_less_mult_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   968
  "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   969
  apply (auto simp add: mult_pos_pos mult_neg_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   970
  apply (simp_all add: not_less le_less)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   971
  apply (erule disjE) apply assumption defer
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   972
  apply (erule disjE) defer apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   973
  apply (erule disjE) defer apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   974
  apply (erule disjE) apply assumption apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   975
  apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   976
  apply (blast dest: zero_less_mult_pos)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   977
  apply (blast dest: zero_less_mult_pos2)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   978
  done
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   979
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   980
lemma zero_le_mult_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   981
  "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   982
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
   983
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   984
lemma mult_less_0_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   985
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   986
  apply (insert zero_less_mult_iff [of "-a" b])
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   987
  apply force
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   988
  done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   989
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   990
lemma mult_le_0_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   991
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   992
  apply (insert zero_le_mult_iff [of "-a" b]) 
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   993
  apply force
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   994
  done
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   995
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   996
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   997
   also with the relations @{text "\<le>"} and equality.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   998
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   999
text{*These ``disjunction'' versions produce two cases when the comparison is
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1000
 an assumption, but effectively four when the comparison is a goal.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1001
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1002
lemma mult_less_cancel_right_disj:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1003
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1004
  apply (cases "c = 0")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1005
  apply (auto simp add: neq_iff mult_strict_right_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1006
                      mult_strict_right_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1007
  apply (auto simp add: not_less 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1008
                      not_le [symmetric, of "a*c"]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1009
                      not_le [symmetric, of a])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1010
  apply (erule_tac [!] notE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1011
  apply (auto simp add: less_imp_le mult_right_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1012
                      mult_right_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1013
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1014
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1015
lemma mult_less_cancel_left_disj:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1016
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1017
  apply (cases "c = 0")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1018
  apply (auto simp add: neq_iff mult_strict_left_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1019
                      mult_strict_left_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1020
  apply (auto simp add: not_less 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1021
                      not_le [symmetric, of "c*a"]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1022
                      not_le [symmetric, of a])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1023
  apply (erule_tac [!] notE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1024
  apply (auto simp add: less_imp_le mult_left_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1025
                      mult_left_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1026
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1027
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1028
text{*The ``conjunction of implication'' lemmas produce two cases when the
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1029
comparison is a goal, but give four when the comparison is an assumption.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1030
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1031
lemma mult_less_cancel_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1032
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1033
  using mult_less_cancel_right_disj [of a c b] by auto
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1034
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1035
lemma mult_less_cancel_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1036
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1037
  using mult_less_cancel_left_disj [of c a b] by auto
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1038
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1039
lemma mult_le_cancel_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1040
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1041
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1042
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1043
lemma mult_le_cancel_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1044
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1045
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1046
30649
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1047
lemma mult_le_cancel_left_pos:
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1048
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1049
by (auto simp: mult_le_cancel_left)
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1050
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1051
lemma mult_le_cancel_left_neg:
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1052
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1053
by (auto simp: mult_le_cancel_left)
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1054
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1055
lemma mult_less_cancel_left_pos:
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1056
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1057
by (auto simp: mult_less_cancel_left)
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1058
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1059
lemma mult_less_cancel_left_neg:
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1060
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1061
by (auto simp: mult_less_cancel_left)
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
  1062
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1063
end
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  1064
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
  1065
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
  1066
  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
  1067
  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
  1068
  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
  1069
  mult_neg_pos mult_neg_neg
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1070
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1071
class ordered_comm_ring = comm_ring + ordered_comm_semiring
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  1072
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1073
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1074
subclass ordered_ring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1075
subclass ordered_cancel_comm_semiring ..
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1076
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  1077
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1078
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1079
class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1080
  (*previously linordered_semiring*)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1081
  assumes zero_less_one [simp]: "0 < 1"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1082
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1083
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1084
lemma pos_add_strict:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1085
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1086
  using add_strict_mono [of 0 a b c] by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1087
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1088
lemma zero_le_one [simp]: "0 \<le> 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1089
by (rule zero_less_one [THEN less_imp_le]) 
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1090
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1091
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1092
by (simp add: not_le) 
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1093
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1094
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1095
by (simp add: not_less) 
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1096
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1097
lemma less_1_mult:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1098
  assumes "1 < m" and "1 < n"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1099
  shows "1 < m * n"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1100
  using assms mult_strict_mono [of 1 m 1 n]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1101
    by (simp add:  less_trans [OF zero_less_one]) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1102
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1103
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1104
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1105
class linordered_idom = comm_ring_1 +
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1106
  linordered_comm_semiring_strict + ordered_ab_group_add +
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1107
  abs_if + sgn_if
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1108
  (*previously linordered_ring*)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1109
begin
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1110
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
  1111
subclass linordered_ring_strict ..
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1112
subclass ordered_comm_ring ..
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
  1113
subclass idom ..
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1114
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1115
subclass linordered_semidom
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
  1116
proof
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1117
  have "0 \<le> 1 * 1" by (rule zero_le_square)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1118
  thus "0 < 1" by (simp add: le_less)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1119
qed 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1120
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1121
lemma linorder_neqE_linordered_idom:
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1122
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1123
  using assms by (rule neqE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1124
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1125
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1126
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1127
lemma mult_le_cancel_right1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1128
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1129
by (insert mult_le_cancel_right [of 1 c b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1130
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1131
lemma mult_le_cancel_right2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1132
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1133
by (insert mult_le_cancel_right [of a c 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1134
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1135
lemma mult_le_cancel_left1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1136
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1137
by (insert mult_le_cancel_left [of c 1 b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1138
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1139
lemma mult_le_cancel_left2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1140
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1141
by (insert mult_le_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1142
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1143
lemma mult_less_cancel_right1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1144
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1145
by (insert mult_less_cancel_right [of 1 c b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1146
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1147
lemma mult_less_cancel_right2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1148
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1149
by (insert mult_less_cancel_right [of a c 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1150
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1151
lemma mult_less_cancel_left1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1152
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1153
by (insert mult_less_cancel_left [of c 1 b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1154
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1155
lemma mult_less_cancel_left2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1156
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1157
by (insert mult_less_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1158
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1159
lemma sgn_sgn [simp]:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1160
  "sgn (sgn a) = sgn a"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1161
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
  1162
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1163
lemma sgn_0_0:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1164
  "sgn a = 0 \<longleftrightarrow> a = 0"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1165
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
  1166
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1167
lemma sgn_1_pos:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1168
  "sgn a = 1 \<longleftrightarrow> a > 0"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
  1169
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
  1170
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1171
lemma sgn_1_neg:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1172
  "sgn a = - 1 \<longleftrightarrow> a < 0"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
  1173
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
  1174
29940
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1175
lemma sgn_pos [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1176
  "0 < a \<Longrightarrow> sgn a = 1"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1177
unfolding sgn_1_pos .
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1178
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1179
lemma sgn_neg [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1180
  "a < 0 \<Longrightarrow> sgn a = - 1"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1181
unfolding sgn_1_neg .
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1182
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1183
lemma sgn_times:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1184
  "sgn (a * b) = sgn a * sgn b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1185
by (auto simp add: sgn_if zero_less_mult_iff)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1186
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1187
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1188
unfolding sgn_if abs_if by auto
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1189
29940
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1190
lemma sgn_greater [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1191
  "0 < sgn a \<longleftrightarrow> 0 < a"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1192
  unfolding sgn_if by auto
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1193
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1194
lemma sgn_less [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1195
  "sgn a < 0 \<longleftrightarrow> a < 0"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1196
  unfolding sgn_if by auto
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1197
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1198
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
  1199
  by (simp add: abs_if)
20a506b8256d lemmas abs_dvd_iff, dvd_abs_iff
huffman
parents: 29940
diff changeset
  1200
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1201
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
  1202
  by (simp add: abs_if)
29653
ece6a0e9f8af added lemma abs_sng
haftmann
parents: 29465
diff changeset
  1203
33676
802f5e233e48 moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents: 33364
diff changeset
  1204
lemma dvd_if_abs_eq:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1205
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
33676
802f5e233e48 moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents: 33364
diff changeset
  1206
by(subst abs_dvd_iff[symmetric]) simp
802f5e233e48 moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents: 33364
diff changeset
  1207
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1208
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1209
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1210
text {* Simprules for comparisons where common factors can be cancelled. *}
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1211
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35631
diff changeset
  1212
lemmas mult_compare_simps[no_atp] =
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1213
    mult_le_cancel_right mult_le_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1214
    mult_le_cancel_right1 mult_le_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1215
    mult_le_cancel_left1 mult_le_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1216
    mult_less_cancel_right mult_less_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1217
    mult_less_cancel_right1 mult_less_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1218
    mult_less_cancel_left1 mult_less_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1219
    mult_cancel_right mult_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1220
    mult_cancel_right1 mult_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1221
    mult_cancel_left1 mult_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1222
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1223
text {* Reasoning about inequalities with division *}
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1224
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1225
context linordered_semidom
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1226
begin
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1227
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1228
lemma less_add_one: "a < a + 1"
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1229
proof -
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1230
  have "a + 0 < a + 1"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1231
    by (blast intro: zero_less_one add_strict_left_mono)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1232
  thus ?thesis by simp
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1233
qed
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1234
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1235
lemma zero_less_two: "0 < 1 + 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1236
by (blast intro: less_trans zero_less_one less_add_one)
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1237
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1238
end
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1239
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1240
context linordered_idom
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1241
begin
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1242
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1243
lemma mult_right_le_one_le:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1244
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1245
  by (auto simp add: mult_le_cancel_left2)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1246
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1247
lemma mult_left_le_one_le:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1248
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1249
  by (auto simp add: mult_le_cancel_right2)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1250
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1251
end
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1252
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1253
text {* Absolute Value *}
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1254
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1255
context linordered_idom
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1256
begin
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1257
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1258
lemma mult_sgn_abs:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1259
  "sgn x * \<bar>x\<bar> = x"
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1260
  unfolding abs_if sgn_if by auto
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1261
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1262
lemma abs_one [simp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1263
  "\<bar>1\<bar> = 1"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1264
  by (simp add: abs_if zero_less_one [THEN less_not_sym])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1265
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1266
end
24491
8d194c9198ae added constant sgn
nipkow
parents: 24427
diff changeset
  1267
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1268
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
  1269
  assumes abs_eq_mult:
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1270
    "(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
  1271
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1272
context linordered_idom
30961
541bfff659af more localisation
haftmann
parents: 30692
diff