src/HOL/Rings.thy
author haftmann
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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 distrib_right[algebra_simps, field_simps]: "(a + b) * c = a * c + b * c"
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  assumes distrib_left[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: distrib_right ac_simps)
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end
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "0 * a = 0"
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  assumes mult_zero_right [simp]: "a * 0 = 0"
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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begin
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lemma mult_not_zero:
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  "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
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  by auto
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end
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class semiring_0_cancel = semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
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  thus "0 * a = 0" by (simp only: add_left_cancel)
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next
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  fix a :: 'a
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  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
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  thus "a * 0 = 0" by (simp only: add_left_cancel)
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qed
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end
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a + b) * c = a * c + b * c"
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begin
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subclass semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: ac_simps)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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end
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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begin
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subclass semiring_0 ..
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end
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class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0_cancel ..
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subclass comm_semiring_0 ..
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end
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "0 \<noteq> 1"
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begin
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lemma one_neq_zero [simp]: "1 \<noteq> 0"
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by (rule not_sym) (rule zero_neq_one)
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definition of_bool :: "bool \<Rightarrow> 'a"
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where
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  "of_bool p = (if p then 1 else 0)" 
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lemma of_bool_eq [simp, code]:
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  "of_bool False = 0"
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  "of_bool True = 1"
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  by (simp_all add: of_bool_def)
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lemma of_bool_eq_iff:
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  "of_bool p = of_bool q \<longleftrightarrow> p = q"
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  by (simp add: of_bool_def)
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lemma split_of_bool [split]:
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  "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
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  by (cases p) simp_all
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lemma split_of_bool_asm:
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  "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
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  by (cases p) simp_all
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end  
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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text {* Abstract divisibility *}
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class dvd = times
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begin
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definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where
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  "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
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lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
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  unfolding dvd_def ..
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding dvd_def by blast 
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end
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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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16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   156
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   157
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   158
lemma dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   159
by (auto intro: dvd_refl elim!: dvdE)
28559
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   160
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   161
lemma dvd_0_right [iff]: "a dvd 0"
55c003a5600a tuned default rules of (dvd)
haftmann
parents: 28141
diff changeset
   162
proof
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   163
  show "0 = a * 0" by simp
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
haftmann
parents: 27516
diff changeset
   166
lemma one_dvd [simp]: "1 dvd a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   167
by (auto intro!: dvdI)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   168
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   169
lemma dvd_mult[simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56544
diff changeset
   170
by (auto intro!: mult.left_commute dvdI elim!: dvdE)
27651
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parents: 27516
diff changeset
   171
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   172
lemma dvd_mult2[simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
57512
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haftmann
parents: 56544
diff changeset
   173
  apply (subst mult.commute)
27651
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parents: 27516
diff changeset
   174
  apply (erule dvd_mult)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents: 27516
diff changeset
   175
  done
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   176
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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diff changeset
   177
lemma dvd_triv_right [simp]: "a dvd b * a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
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diff changeset
   178
by (rule dvd_mult) (rule dvd_refl)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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diff changeset
   179
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
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diff changeset
   180
lemma dvd_triv_left [simp]: "a dvd a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   181
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
   182
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   183
lemma mult_dvd_mono:
30042
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nipkow
parents: 29981
diff changeset
   184
  assumes "a dvd b"
31039ee583fa Removed subsumed lemmas
nipkow
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diff changeset
   185
    and "c dvd d"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   186
  shows "a * c dvd b * d"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   187
proof -
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   188
  from `a dvd b` obtain b' where "b = a * b'" ..
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   189
  moreover from `c dvd d` obtain d' where "d = c * d'" ..
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   190
  ultimately have "b * d = (a * c) * (b' * d')" by (simp add: ac_simps)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   191
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   192
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   193
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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diff changeset
   194
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
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diff changeset
   195
by (simp add: dvd_def mult.assoc, blast)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   196
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
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diff changeset
   197
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   198
  unfolding mult.commute [of a] by (rule dvd_mult_left)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   199
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
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diff changeset
   200
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
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diff changeset
   201
by simp
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   202
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29915
diff changeset
   203
lemma dvd_add[simp]:
17d1e32ef867 dvd and setprod lemmas
nipkow
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diff changeset
   204
  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
   205
proof -
29925
17d1e32ef867 dvd and setprod lemmas
nipkow
parents: 29915
diff changeset
   206
  from `a dvd b` obtain b' where "b = a * b'" ..
17d1e32ef867 dvd and setprod lemmas
nipkow
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diff changeset
   207
  moreover from `a dvd c` obtain c' where "c = a * c'" ..
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   208
  ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   209
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   210
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
   211
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   212
end
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   213
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   214
class no_zero_divisors = zero + times +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   215
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
36719
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   216
begin
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   217
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   218
lemma divisors_zero:
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   219
  assumes "a * b = 0"
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   220
  shows "a = 0 \<or> b = 0"
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   221
proof (rule classical)
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   222
  assume "\<not> (a = 0 \<or> b = 0)"
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   223
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   224
  with no_zero_divisors have "a * b \<noteq> 0" by blast
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   225
  with assms show ?thesis by simp
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   226
qed
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   227
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36622
diff changeset
   228
end
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   229
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   230
class semiring_1_cancel = semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   231
  + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   232
begin
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   233
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   234
subclass semiring_0_cancel ..
25512
4134f7c782e2 using intro_locales instead of unfold_locales if appropriate
haftmann
parents: 25450
diff changeset
   235
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   236
subclass semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   237
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   238
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   239
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   240
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   241
  + zero_neq_one + comm_monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   242
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   243
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   244
subclass semiring_1_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   245
subclass comm_semiring_0_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   246
subclass comm_semiring_1 ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   247
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   248
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   249
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   250
class ring = semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   251
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   252
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   253
subclass semiring_0_cancel ..
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   254
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   255
text {* Distribution rules *}
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   256
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   257
lemma minus_mult_left: "- (a * b) = - a * b"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   258
by (rule minus_unique) (simp add: distrib_right [symmetric]) 
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   259
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   260
lemma minus_mult_right: "- (a * b) = a * - b"
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   261
by (rule minus_unique) (simp add: distrib_left [symmetric]) 
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   262
29407
5ef7e97fd9e4 move lemmas mult_minus{left,right} inside class ring
huffman
parents: 29406
diff changeset
   263
text{*Extract signs from products*}
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   264
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   265
lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
29407
5ef7e97fd9e4 move lemmas mult_minus{left,right} inside class ring
huffman
parents: 29406
diff changeset
   266
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   267
lemma minus_mult_minus [simp]: "- a * - b = a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   268
by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   269
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   270
lemma minus_mult_commute: "- a * b = a * - b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   271
by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   272
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   273
lemma right_diff_distrib [algebra_simps, field_simps]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   274
  "a * (b - c) = a * b - a * c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   275
  using distrib_left [of a b "-c "] by simp
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   276
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   277
lemma left_diff_distrib [algebra_simps, field_simps]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   278
  "(a - b) * c = a * c - b * c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   279
  using distrib_right [of a "- b" c] by simp
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   280
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   281
lemmas ring_distribs =
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   282
  distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   283
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   284
lemma eq_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   285
  "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
   286
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   287
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   288
lemma eq_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   289
  "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
   290
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   291
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   292
end
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   293
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   294
lemmas ring_distribs =
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   295
  distrib_left distrib_right left_diff_distrib right_diff_distrib
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   296
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   297
class comm_ring = comm_semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   298
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   299
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   300
subclass ring ..
28141
193c3ea0f63b instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents: 27651
diff changeset
   301
subclass comm_semiring_0_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   302
44350
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   303
lemma square_diff_square_factored:
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   304
  "x * x - y * y = (x + y) * (x - y)"
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   305
  by (simp add: algebra_simps)
63cddfbc5a09 replace lemma realpow_two_diff with new lemma square_diff_square_factored
huffman
parents: 44346
diff changeset
   306
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   307
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   308
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   309
class ring_1 = ring + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   310
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   311
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   312
subclass semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   313
44346
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   314
lemma square_diff_one_factored:
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   315
  "x * x - 1 = (x + 1) * (x - 1)"
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   316
  by (simp add: algebra_simps)
00dd3c4dabe0 rename real_squared_diff_one_factored to square_diff_one_factored and move to Rings.thy
huffman
parents: 44064
diff changeset
   317
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   318
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   319
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   320
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   321
  (*previously ring*)
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   322
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   323
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   324
subclass ring_1 ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   325
subclass comm_semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   326
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
   327
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
   328
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   329
  assume "x dvd - y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   330
  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
   331
  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
   332
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   333
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   334
  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
   335
  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
   336
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   337
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
   338
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
   339
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   340
  assume "- x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   341
  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
   342
  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
   343
  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
   344
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   345
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   346
  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
   347
  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
   348
  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
   349
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   350
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   351
lemma dvd_diff [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   352
  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   353
  using dvd_add [of x y "- z"] by simp
29409
f0a8fe83bc07 add lemma dvd_diff to class comm_ring_1
huffman
parents: 29408
diff changeset
   354
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   355
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   356
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   357
class ring_no_zero_divisors = ring + no_zero_divisors
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   358
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   359
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   360
lemma mult_eq_0_iff [simp]:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   361
  shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   362
proof (cases "a = 0 \<or> b = 0")
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   363
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   364
    then show ?thesis using no_zero_divisors by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   365
next
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   366
  case True then show ?thesis by auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   367
qed
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   368
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   369
text{*Cancellation of equalities with a common factor*}
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   370
lemma mult_cancel_right [simp]:
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   371
  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   372
proof -
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   373
  have "(a * c = b * c) = ((a - b) * c = 0)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   374
    by (simp add: algebra_simps)
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   375
  thus ?thesis by (simp add: disj_commute)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   376
qed
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   377
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   378
lemma mult_cancel_left [simp]:
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   379
  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   380
proof -
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   381
  have "(c * a = c * b) = (c * (a - b) = 0)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   382
    by (simp add: algebra_simps)
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   383
  thus ?thesis by simp
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   384
qed
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   385
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   386
lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> (c*a=c*b) = (a=b)"
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   387
by simp 
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   388
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   389
lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> (a*c=b*c) = (a=b)"
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   390
by simp 
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 55912
diff changeset
   391
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   392
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   393
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   394
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   395
begin
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   396
36970
fb3fdb4b585e remove simp attribute from square_eq_1_iff
huffman
parents: 36821
diff changeset
   397
lemma square_eq_1_iff:
36821
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   398
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   399
proof -
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   400
  have "(x - 1) * (x + 1) = x * x - 1"
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   401
    by (simp add: algebra_simps)
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   402
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   403
    by simp
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   404
  thus ?thesis
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   405
    by (simp add: eq_neg_iff_add_eq_0)
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   406
qed
9207505d1ee5 move lemma real_mult_is_one to Rings.thy, renamed to square_eq_1_iff
huffman
parents: 36719
diff changeset
   407
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   408
lemma mult_cancel_right1 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   409
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   410
by (insert mult_cancel_right [of 1 c b], force)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   411
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   412
lemma mult_cancel_right2 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   413
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   414
by (insert mult_cancel_right [of a c 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   415
 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   416
lemma mult_cancel_left1 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   417
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   418
by (insert mult_cancel_left [of c 1 b], force)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   419
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   420
lemma mult_cancel_left2 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   421
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   422
by (insert mult_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   423
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   424
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   425
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   426
class idom = comm_ring_1 + no_zero_divisors
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   427
begin
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   428
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   429
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
   430
29915
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   431
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
   432
proof
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   433
  assume "a * a = b * b"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   434
  then have "(a - b) * (a + b) = 0"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   435
    by (simp add: algebra_simps)
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   436
  then show "a = b \<or> a = - b"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   437
    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
   438
next
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   439
  assume "a = b \<or> a = - b"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   440
  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
   441
qed
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   442
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   443
lemma dvd_mult_cancel_right [simp]:
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   444
  "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
   445
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   446
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   447
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   448
  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
   449
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   450
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   451
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   452
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   453
lemma dvd_mult_cancel_left [simp]:
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   454
  "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
   455
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   456
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   457
    unfolding dvd_def by (simp add: ac_simps)
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   458
  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
   459
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   460
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   461
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   462
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   463
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   464
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   465
text {*
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   466
  The theory of partially ordered rings is taken from the books:
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   467
  \begin{itemize}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   468
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   469
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   470
  \end{itemize}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   471
  Most of the used notions can also be looked up in 
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   472
  \begin{itemize}
54703
499f92dc6e45 more antiquotations;
wenzelm
parents: 54489
diff changeset
   473
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
35302
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   474
  \item \emph{Algebra I} by van der Waerden, Springer.
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   475
  \end{itemize}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   476
*}
4bc6b4d70e08 tuned text
haftmann
parents: 35216
diff changeset
   477
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   478
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   479
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   480
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   481
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   482
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   483
lemma mult_mono:
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   484
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   485
apply (erule mult_right_mono [THEN order_trans], assumption)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   486
apply (erule mult_left_mono, assumption)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   487
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   488
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   489
lemma mult_mono':
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   490
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   491
apply (rule mult_mono)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   492
apply (fast intro: order_trans)+
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   493
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   494
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   495
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   496
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   497
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   498
begin
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   499
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   500
subclass semiring_0_cancel ..
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   501
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56480
diff changeset
   502
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   503
using mult_left_mono [of 0 b a] by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   504
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   505
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
   506
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
   507
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   508
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
   509
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
   510
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   511
text {* Legacy - use @{text mult_nonpos_nonneg} *}
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   512
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
   513
by (drule mult_right_mono [of b 0], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   514
26234
1f6e28a88785 clarified proposition
haftmann
parents: 26193
diff changeset
   515
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
   516
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   517
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   518
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   519
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   520
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   521
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   522
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   523
subclass ordered_cancel_semiring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   524
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   525
subclass ordered_comm_monoid_add ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   526
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   527
lemma mult_left_less_imp_less:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   528
  "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
   529
by (force simp add: mult_left_mono not_le [symmetric])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   530
 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   531
lemma mult_right_less_imp_less:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   532
  "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
   533
by (force simp add: mult_right_mono not_le [symmetric])
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   534
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   535
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   536
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   537
class linordered_semiring_1 = linordered_semiring + semiring_1
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   538
begin
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   539
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   540
lemma convex_bound_le:
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   541
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   542
  shows "u * x + v * y \<le> a"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   543
proof-
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   544
  from assms have "u * x + v * y \<le> u * a + v * a"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   545
    by (simp add: add_mono mult_left_mono)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   546
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   547
qed
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   548
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   549
end
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   550
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   551
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   552
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   553
  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
   554
begin
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   555
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   556
subclass semiring_0_cancel ..
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   557
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   558
subclass linordered_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   559
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   560
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   561
  assume A: "a \<le> b" "0 \<le> c"
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   562
  from A show "c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   563
    unfolding le_less
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   564
    using mult_strict_left_mono by (cases "c = 0") auto
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   565
  from A show "a * c \<le> b * c"
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   566
    unfolding le_less
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   567
    using mult_strict_right_mono by (cases "c = 0") auto
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   568
qed
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   569
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   570
lemma mult_left_le_imp_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   571
  "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
   572
by (force simp add: mult_strict_left_mono _not_less [symmetric])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   573
 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   574
lemma mult_right_le_imp_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   575
  "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
   576
by (force simp add: mult_strict_right_mono not_less [symmetric])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   577
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   578
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   579
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
   580
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   581
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
   582
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
   583
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   584
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
   585
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
   586
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   587
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
   588
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
   589
by (drule mult_strict_right_mono [of b 0], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   590
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   591
lemma zero_less_mult_pos:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   592
  "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
   593
apply (cases "b\<le>0")
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   594
 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
   595
apply (drule_tac mult_pos_neg [of a b])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   596
 apply (auto dest: less_not_sym)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   597
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   598
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   599
lemma zero_less_mult_pos2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   600
  "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
   601
apply (cases "b\<le>0")
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   602
 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
   603
apply (drule_tac mult_pos_neg2 [of a b])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   604
 apply (auto dest: less_not_sym)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   605
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   606
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   607
text{*Strict monotonicity in both arguments*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   608
lemma mult_strict_mono:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   609
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   610
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   611
  using assms apply (cases "c=0")
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   612
  apply (simp)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   613
  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
   614
  apply (force simp add: le_less)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   615
  apply (erule mult_strict_left_mono, assumption)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   616
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   617
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   618
text{*This weaker variant has more natural premises*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   619
lemma mult_strict_mono':
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   620
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   621
  shows "a * c < b * d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   622
by (rule mult_strict_mono) (insert assms, auto)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   623
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   624
lemma mult_less_le_imp_less:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   625
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   626
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   627
  using assms apply (subgoal_tac "a * c < b * c")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   628
  apply (erule less_le_trans)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   629
  apply (erule mult_left_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   630
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   631
  apply (erule mult_strict_right_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   632
  apply assumption
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   633
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   634
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   635
lemma mult_le_less_imp_less:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   636
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   637
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   638
  using assms apply (subgoal_tac "a * c \<le> b * c")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   639
  apply (erule le_less_trans)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   640
  apply (erule mult_strict_left_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   641
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   642
  apply (erule mult_right_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   643
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   644
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   645
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   646
lemma mult_less_imp_less_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   647
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   648
  shows "a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   649
proof (rule ccontr)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   650
  assume "\<not>  a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   651
  hence "b \<le> a" by (simp add: linorder_not_less)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   652
  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
   653
  with this and less show False by (simp add: not_less [symmetric])
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   654
qed
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   655
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   656
lemma mult_less_imp_less_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   657
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   658
  shows "a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   659
proof (rule ccontr)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   660
  assume "\<not> a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   661
  hence "b \<le> a" by (simp add: linorder_not_less)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   662
  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
   663
  with this and less show False by (simp add: not_less [symmetric])
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   664
qed  
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   665
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   666
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   667
35097
4554bb2abfa3 dropped last occurence of the linlinordered accident
haftmann
parents: 35092
diff changeset
   668
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   669
begin
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   670
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   671
subclass linordered_semiring_1 ..
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   672
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   673
lemma convex_bound_lt:
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   674
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   675
  shows "u * x + v * y < a"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   676
proof -
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   677
  from assms have "u * x + v * y < u * a + v * a"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   678
    by (cases "u = 0")
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   679
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 44921
diff changeset
   680
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   681
qed
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   682
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   683
end
33319
74f0dcc0b5fb moved algebraic classes to Ring_and_Field
haftmann
parents: 32960
diff changeset
   684
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   685
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add + 
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   686
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   687
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   688
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   689
subclass ordered_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   690
proof
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   691
  fix a b c :: 'a
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   692
  assume "a \<le> b" "0 \<le> c"
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   693
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56544
diff changeset
   694
  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
   695
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   696
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   697
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   698
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   699
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   700
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   701
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   702
subclass comm_semiring_0_cancel ..
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   703
subclass ordered_comm_semiring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   704
subclass ordered_cancel_semiring ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   705
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   706
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   707
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   708
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   709
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   710
begin
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   711
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   712
subclass linordered_semiring_strict
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   713
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   714
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   715
  assume "a < b" "0 < c"
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37767
diff changeset
   716
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56544
diff changeset
   717
  thus "a * c < b * c" by (simp only: mult.commute)
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   718
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   719
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   720
subclass ordered_cancel_comm_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   721
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   722
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   723
  assume "a \<le> b" "0 \<le> c"
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   724
  thus "c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   725
    unfolding le_less
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   726
    using mult_strict_left_mono by (cases "c = 0") auto
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   727
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   728
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   729
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   730
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   731
class ordered_ring = ring + ordered_cancel_semiring 
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   732
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   733
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   734
subclass ordered_ab_group_add ..
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   735
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   736
lemma less_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   737
  "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
   738
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   739
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   740
lemma less_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   741
  "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
   742
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   743
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   744
lemma le_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   745
  "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
   746
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   747
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   748
lemma le_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   749
  "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
   750
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   751
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   752
lemma mult_left_mono_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   753
  "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
   754
  apply (drule mult_left_mono [of _ _ "- c"])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   755
  apply simp_all
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   756
  done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   757
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   758
lemma mult_right_mono_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   759
  "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
   760
  apply (drule mult_right_mono [of _ _ "- c"])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   761
  apply simp_all
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   762
  done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   763
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   764
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
   765
using mult_right_mono_neg [of a 0 b] by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   766
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   767
lemma split_mult_pos_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   768
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56480
diff changeset
   769
by (auto simp add: mult_nonpos_nonpos)
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   770
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   771
end
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   772
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   773
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
   774
begin
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   775
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   776
subclass ordered_ring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   777
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   778
subclass ordered_ab_group_add_abs
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   779
proof
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   780
  fix a b
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   781
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54225
diff changeset
   782
    by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   783
qed (auto simp add: abs_if)
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   784
35631
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
   785
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
   786
  using linear [of 0 a]
56536
aefb4a8da31f made mult_nonneg_nonneg a simp rule
nipkow
parents: 56480
diff changeset
   787
  by (auto simp add: mult_nonpos_nonpos)
35631
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
   788
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
   789
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
   790
  by (simp add: not_less)
0b8a5fd339ab generalize some lemmas from class linordered_ring_strict to linordered_ring
huffman
parents: 35302
diff changeset
   791
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   792
end
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   793
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   794
(* 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
   795
   Basically, linordered_ring + no_zero_divisors = linordered_ring_strict.
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   796
 *)
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   797
class linordered_ring_strict = ring + linordered_semiring_strict
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   798
  + ordered_ab_group_add + abs_if
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   799
begin
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   800
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   801
subclass linordered_ring ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   802
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   803
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
   804
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
   805
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   806
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
   807
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
   808
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   809
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
   810
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
   811
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   812
subclass ring_no_zero_divisors
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   813
proof
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   814
  fix a b
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   815
  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
   816
  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
   817
  have "a * b < 0 \<or> 0 < a * b"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   818
  proof (cases "a < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   819
    case True note A' = this
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   820
    show ?thesis proof (cases "b < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   821
      case True with A'
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   822
      show ?thesis by (auto dest: mult_neg_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   823
    next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   824
      case False with B have "0 < b" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   825
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   826
    qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   827
  next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   828
    case False with A have A': "0 < a" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   829
    show ?thesis proof (cases "b < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   830
      case True with A'
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   831
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   832
    next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   833
      case False with B have "0 < b" by auto
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   834
      with A' show ?thesis by auto
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   835
    qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   836
  qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   837
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   838
qed
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   839
56480
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56217
diff changeset
   840
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56217
diff changeset
   841
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
56544
b60d5d119489 made mult_pos_pos a simp rule
nipkow
parents: 56536
diff changeset
   842
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   843
56480
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56217
diff changeset
   844
lemma zero_le_mult_iff: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
093ea91498e6 field_simps: better support for negation and division, and power
hoelzl
parents: 56217
diff changeset
   845
  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
   846
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   847
lemma mult_less_0_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   848
  "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
   849
  apply (insert zero_less_mult_iff [of "-a" b])
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   850
  apply force
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   851
  done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   852
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   853
lemma mult_le_0_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   854
  "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
   855
  apply (insert zero_le_mult_iff [of "-a" b]) 
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
   856
  apply force
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   857
  done
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   858
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   859
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   860
   also with the relations @{text "\<le>"} and equality.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   861
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   862
text{*These ``disjunction'' versions produce two cases when the comparison is
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   863
 an assumption, but effectively four when the comparison is a goal.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   864
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   865
lemma mult_less_cancel_right_disj:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   866
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   867
  apply (cases "c = 0")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   868
  apply (auto simp add: neq_iff mult_strict_right_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   869
                      mult_strict_right_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   870
  apply (auto simp add: not_less 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   871
                      not_le [symmetric, of "a*c"]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   872
                      not_le [symmetric, of a])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   873
  apply (erule_tac [!] notE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   874
  apply (auto simp add: less_imp_le mult_right_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   875
                      mult_right_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   876
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   877
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   878
lemma mult_less_cancel_left_disj:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   879
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   880
  apply (cases "c = 0")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   881
  apply (auto simp add: neq_iff mult_strict_left_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   882
                      mult_strict_left_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   883
  apply (auto simp add: not_less 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   884
                      not_le [symmetric, of "c*a"]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   885
                      not_le [symmetric, of a])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   886
  apply (erule_tac [!] notE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   887
  apply (auto simp add: less_imp_le mult_left_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   888
                      mult_left_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   889
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   890
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   891
text{*The ``conjunction of implication'' lemmas produce two cases when the
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   892
comparison is a goal, but give four when the comparison is an assumption.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   893
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   894
lemma mult_less_cancel_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   895
  "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
   896
  using mult_less_cancel_right_disj [of a c b] by auto
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   897
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   898
lemma mult_less_cancel_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   899
  "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
   900
  using mult_less_cancel_left_disj [of c a b] by auto
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   901
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   902
lemma mult_le_cancel_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   903
   "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
   904
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   905
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   906
lemma mult_le_cancel_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   907
  "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
   908
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   909
30649
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   910
lemma mult_le_cancel_left_pos:
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   911
  "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
   912
by (auto simp: mult_le_cancel_left)
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   913
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   914
lemma mult_le_cancel_left_neg:
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   915
  "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
   916
by (auto simp: mult_le_cancel_left)
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   917
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   918
lemma mult_less_cancel_left_pos:
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   919
  "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
   920
by (auto simp: mult_less_cancel_left)
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   921
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   922
lemma mult_less_cancel_left_neg:
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   923
  "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
   924
by (auto simp: mult_less_cancel_left)
57753e0ec1d4 1. New cancellation simprocs for common factors in inequations
nipkow
parents: 30242
diff changeset
   925
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   926
end
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   927
30692
44ea10bc07a7 clean up proofs of sign rules for multiplication; add list of lemmas mult_sign_intros
huffman
parents: 30650
diff changeset
   928
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
   929
  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
   930
  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
   931
  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
   932
  mult_neg_pos mult_neg_neg
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   933
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   934
class ordered_comm_ring = comm_ring + ordered_comm_semiring
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   935
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   936
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   937
subclass ordered_ring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   938
subclass ordered_cancel_comm_semiring ..
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   939
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   940
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   941
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   942
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
   943
  (*previously linordered_semiring*)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   944
  assumes zero_less_one [simp]: "0 < 1"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   945
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   946
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   947
lemma pos_add_strict:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   948
  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
   949
  using add_strict_mono [of 0 a b c] by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   950
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   951
lemma zero_le_one [simp]: "0 \<le> 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   952
by (rule zero_less_one [THEN less_imp_le]) 
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   953
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   954
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
   955
by (simp add: not_le) 
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   956
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   957
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   958
by (simp add: not_less) 
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   959
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   960
lemma less_1_mult:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   961
  assumes "1 < m" and "1 < n"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   962
  shows "1 < m * n"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   963
  using assms mult_strict_mono [of 1 m 1 n]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   964
    by (simp add:  less_trans [OF zero_less_one]) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   965
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   966
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   967
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   968
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
   969
  linordered_comm_semiring_strict + ordered_ab_group_add +
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   970
  abs_if + sgn_if
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   971
  (*previously linordered_ring*)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   972
begin
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   973
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36348
diff changeset
   974
subclass linordered_semiring_1_strict ..
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   975
subclass linordered_ring_strict ..
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   976
subclass ordered_comm_ring ..
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   977
subclass idom ..
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   978
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   979
subclass linordered_semidom
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   980
proof
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   981
  have "0 \<le> 1 * 1" by (rule zero_le_square)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   982
  thus "0 < 1" by (simp add: le_less)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   983
qed 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   984
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   985
lemma linorder_neqE_linordered_idom:
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   986
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   987
  using assms by (rule neqE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   988
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   989
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   990
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   991
lemma mult_le_cancel_right1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   992
  "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
   993
by (insert mult_le_cancel_right [of 1 c b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   994
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   995
lemma mult_le_cancel_right2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   996
  "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
   997
by (insert mult_le_cancel_right [of a c 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   998
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   999
lemma mult_le_cancel_left1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1000
  "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
  1001
by (insert mult_le_cancel_left [of c 1 b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1002
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1003
lemma mult_le_cancel_left2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1004
  "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
  1005
by (insert mult_le_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1006
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1007
lemma mult_less_cancel_right1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1008
  "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
  1009
by (insert mult_less_cancel_right [of 1 c b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1010
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1011
lemma mult_less_cancel_right2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1012
  "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
  1013
by (insert mult_less_cancel_right [of a c 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1014
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1015
lemma mult_less_cancel_left1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1016
  "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
  1017
by (insert mult_less_cancel_left [of c 1 b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1018
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1019
lemma mult_less_cancel_left2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1020
  "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
  1021
by (insert mult_less_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1022
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1023
lemma sgn_sgn [simp]:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1024
  "sgn (sgn a) = sgn a"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1025
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
  1026
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1027
lemma sgn_0_0:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1028
  "sgn a = 0 \<longleftrightarrow> a = 0"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1029
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
  1030
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1031
lemma sgn_1_pos:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1032
  "sgn a = 1 \<longleftrightarrow> a > 0"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
  1033
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
  1034
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1035
lemma sgn_1_neg:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1036
  "sgn a = - 1 \<longleftrightarrow> a < 0"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
  1037
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
  1038
29940
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1039
lemma sgn_pos [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1040
  "0 < a \<Longrightarrow> sgn a = 1"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1041
unfolding sgn_1_pos .
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1042
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1043
lemma sgn_neg [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1044
  "a < 0 \<Longrightarrow> sgn a = - 1"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1045
unfolding sgn_1_neg .
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1046
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1047
lemma sgn_times:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1048
  "sgn (a * b) = sgn a * sgn b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1049
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
  1050
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1051
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1052
unfolding sgn_if abs_if by auto
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1053
29940
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1054
lemma sgn_greater [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1055
  "0 < sgn a \<longleftrightarrow> 0 < a"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1056
  unfolding sgn_if by auto
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1057
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1058
lemma sgn_less [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1059
  "sgn a < 0 \<longleftrightarrow> a < 0"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1060
  unfolding sgn_if by auto
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1061
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1062
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
  1063
  by (simp add: abs_if)
20a506b8256d lemmas abs_dvd_iff, dvd_abs_iff
huffman
parents: 29940
diff changeset
  1064
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1065
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
  1066
  by (simp add: abs_if)
29653
ece6a0e9f8af added lemma abs_sng
haftmann
parents: 29465
diff changeset
  1067
33676
802f5e233e48 moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents: 33364
diff changeset
  1068
lemma dvd_if_abs_eq:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1069
  "\<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
  1070
by(subst abs_dvd_iff[symmetric]) simp
802f5e233e48 moved lemma from Algebra/IntRing to Ring_and_Field
nipkow
parents: 33364
diff changeset
  1071
55912
e12a0ab9917c fix typo
huffman
parents: 55187
diff changeset
  1072
text {* The following lemmas can be proven in more general structures, but
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1073
are dangerous as simp rules in absence of @{thm neg_equal_zero}, 
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1074
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1075
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1076
lemma equation_minus_iff_1 [simp, no_atp]:
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1077
  "1 = - a \<longleftrightarrow> a = - 1"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1078
  by (fact equation_minus_iff)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1079
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1080
lemma minus_equation_iff_1 [simp, no_atp]:
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1081
  "- a = 1 \<longleftrightarrow> a = - 1"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1082
  by (subst minus_equation_iff, auto)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1083
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1084
lemma le_minus_iff_1 [simp, no_atp]:
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1085
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1086
  by (fact le_minus_iff)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1087
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1088
lemma minus_le_iff_1 [simp, no_atp]:
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1089
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1090
  by (fact minus_le_iff)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1091
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1092
lemma less_minus_iff_1 [simp, no_atp]:
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1093
  "1 < - b \<longleftrightarrow> b < - 1"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1094
  by (fact less_minus_iff)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1095
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1096
lemma minus_less_iff_1 [simp, no_atp]:
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1097
  "- a < 1 \<longleftrightarrow> - 1 < a"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1098
  by (fact minus_less_iff)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54250
diff changeset
  1099
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1100
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1101
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1102
text {* Simprules for comparisons where common factors can be cancelled. *}
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1103
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
  1104
lemmas mult_compare_simps =
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1105
    mult_le_cancel_right mult_le_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1106
    mult_le_cancel_right1 mult_le_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1107
    mult_le_cancel_left1 mult_le_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1108
    mult_less_cancel_right mult_less_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1109
    mult_less_cancel_right1 mult_less_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1110
    mult_less_cancel_left1 mult_less_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1111
    mult_cancel_right mult_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1112
    mult_cancel_right1 mult_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1113
    mult_cancel_left1 mult_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1114
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1115
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
  1116
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1117
context linordered_semidom
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1118
begin
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1119
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1120
lemma less_add_one: "a < a + 1"
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1121
proof -
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1122
  have "a + 0 < a + 1"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1123
    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
  1124
  thus ?thesis by simp
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1125
qed
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1126
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1127
lemma zero_less_two: "0 < 1 + 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1128
by (blast intro: less_trans zero_less_one less_add_one)
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1129
e2e1a4b00de3 various localizations
haftmann
parents: 25186
diff changeset
  1130
end
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1131
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1132
context linordered_idom
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1133
begin
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1134
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1135
lemma mult_right_le_one_le:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1136
  "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
  1137
  by (auto simp add: mult_le_cancel_left2)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1138
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1139
lemma mult_left_le_one_le:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1140
  "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
  1141
  by (auto simp add: mult_le_cancel_right2)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1142
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1143
end
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1144
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1145
text {* Absolute Value *}
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1146
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1147
context linordered_idom
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1148
begin
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1149
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1150
lemma mult_sgn_abs:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1151
  "sgn x * \<bar>x\<bar> = x"
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1152
  unfolding abs_if sgn_if by auto
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1153
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1154
lemma abs_one [simp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1155
  "\<bar>1\<bar> = 1"
44921
58eef4843641 tuned proofs
huffman
parents: 44350
diff changeset
  1156
  by (simp add: abs_if)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1157
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1158
end
24491
8d194c9198ae added constant sgn
nipkow
parents: 24427
diff changeset
  1159
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1160
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
  1161
  assumes abs_eq_mult:
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1162
    "(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
  1163
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1164
context linordered_idom
30961
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  1165
begin
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  1166
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1167
subclass ordered_ring_abs proof
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35097
diff changeset
  1168
qed (auto simp add: abs_if not_less mult_less_0_iff)
30961
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  1169
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  1170
lemma abs_mult:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1171
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" 
30961
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  1172
  by (rule abs_eq_mult) auto
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  1173
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  1174
lemma abs_mult_self:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1175
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
30961
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  1176
  by (simp add: abs_if) 
541bfff659af more localisation
haftmann
parents: 30692
diff changeset
  1177
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1178
lemma abs_mult_less:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1179
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1180
proof -
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1181
  assume ac: "\<bar>a\<bar> < c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1182
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1183
  assume "\<bar>b\<bar> < d"
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1184
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1185
qed
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1186
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1187
lemma abs_less_iff:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1188
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1189
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
  1190
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1191
lemma abs_mult_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1192
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1193
  by (simp add: abs_mult)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1194
51520
e9b361845809 move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents: 50420
diff changeset
  1195
lemma abs_diff_less_iff:
e9b361845809 move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents: 50420
diff changeset
  1196
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
e9b361845809 move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents: 50420
diff changeset
  1197
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
e9b361845809 move real_isLub_unique to isLub_unique in Lubs; real_sum_of_halves to RealDef; abs_diff_less_iff to Rings
hoelzl
parents: 50420
diff changeset
  1198
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1199
end
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1200
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51520
diff changeset
  1201
code_identifier
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51520
diff changeset
  1202
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
  1203
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  1204
end
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51520
diff changeset
  1205