src/HOL/Ring_and_Field.thy
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(*  Title:   HOL/Ring_and_Field.thy
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    Author:  Gertrud Bauer, Steven Obua, Tobias Nipkow, Lawrence C Paulson, and Markus Wenzel,
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             with contributions by Jeremy Avigad
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*)
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header {* (Ordered) Rings and Fields *}
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theory Ring_and_Field
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imports OrderedGroup
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begin
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text {*
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  The theory of partially ordered rings is taken from the books:
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  \begin{itemize}
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  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
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  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
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  \end{itemize}
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  Most of the used notions can also be looked up in 
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  \begin{itemize}
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  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
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  \item \emph{Algebra I} by van der Waerden, Springer.
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  \end{itemize}
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*}
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes left_distrib[algebra_simps]: "(a + b) * c = a * c + b * c"
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  assumes right_distrib[algebra_simps]: "a * (b + c) = a * b + a * c"
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begin
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor:
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  "a * e + (b * e + c) = (a + b) * e + c"
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by (simp add: left_distrib add_ac)
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end
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "0 * a = 0"
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  assumes mult_zero_right [simp]: "a * 0 = 0"
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0" by (simp add: left_distrib [symmetric])
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  thus "0 * a = 0" by (simp only: add_left_cancel)
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next
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  fix a :: 'a
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  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: right_distrib [symmetric])
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  thus "a * 0 = 0" by (simp only: add_left_cancel)
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qed
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end
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a + b) * c = a * c + b * c"
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begin
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subclass semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: mult_ac)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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end
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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begin
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subclass semiring_0 ..
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end
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class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0_cancel ..
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subclass comm_semiring_0 ..
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end
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "0 \<noteq> 1"
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begin
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lemma one_neq_zero [simp]: "1 \<noteq> 0"
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by (rule not_sym) (rule zero_neq_one)
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end
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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text {* Abstract divisibility *}
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class dvd = times
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begin
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definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where
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  [code del]: "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
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lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
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  unfolding dvd_def ..
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding dvd_def by blast 
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end
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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd
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  (*previously almost_semiring*)
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begin
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subclass semiring_1 ..
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lemma dvd_refl[simp]: "a dvd a"
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proof
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  show "a = a * 1" by simp
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qed
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lemma dvd_trans:
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  assumes "a dvd b" and "b dvd c"
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  shows "a dvd c"
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proof -
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  from assms obtain v where "b = a * v" by (auto elim!: dvdE)
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  moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
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  ultimately have "c = a * (v * w)" by (simp add: mult_assoc)
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  then show ?thesis ..
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qed
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lemma dvd_0_left_iff [noatp, simp]: "0 dvd a \<longleftrightarrow> a = 0"
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by (auto intro: dvd_refl elim!: dvdE)
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lemma dvd_0_right [iff]: "a dvd 0"
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proof
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  show "0 = a * 0" by simp
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qed
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lemma one_dvd [simp]: "1 dvd a"
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by (auto intro!: dvdI)
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lemma dvd_mult[simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
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by (auto intro!: mult_left_commute dvdI elim!: dvdE)
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   153
lemma dvd_mult2[simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
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  apply (subst mult_commute)
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   155
  apply (erule dvd_mult)
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   156
  done
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   157
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   158
lemma dvd_triv_right [simp]: "a dvd b * a"
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   159
by (rule dvd_mult) (rule dvd_refl)
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   160
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   161
lemma dvd_triv_left [simp]: "a dvd a * b"
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diff changeset
   162
by (rule dvd_mult2) (rule dvd_refl)
27651
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   163
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   164
lemma mult_dvd_mono:
30042
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   165
  assumes "a dvd b"
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   166
    and "c dvd d"
27651
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   167
  shows "a * c dvd b * d"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   168
proof -
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   169
  from `a dvd b` obtain b' where "b = a * b'" ..
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   170
  moreover from `c dvd d` obtain d' where "d = c * d'" ..
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   171
  ultimately have "b * d = (a * c) * (b' * d')" by (simp add: mult_ac)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   172
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   173
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   174
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   175
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
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   176
by (simp add: dvd_def mult_assoc, blast)
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   177
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   178
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   179
  unfolding mult_ac [of a] by (rule dvd_mult_left)
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   180
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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   181
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
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   182
by simp
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   183
29925
17d1e32ef867 dvd and setprod lemmas
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   184
lemma dvd_add[simp]:
17d1e32ef867 dvd and setprod lemmas
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   185
  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
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diff changeset
   186
proof -
29925
17d1e32ef867 dvd and setprod lemmas
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diff changeset
   187
  from `a dvd b` obtain b' where "b = a * b'" ..
17d1e32ef867 dvd and setprod lemmas
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diff changeset
   188
  moreover from `a dvd c` obtain c' where "c = a * c'" ..
27651
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diff changeset
   189
  ultimately have "b + c = a * (b' + c')" by (simp add: right_distrib)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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diff changeset
   190
  then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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parents: 27516
diff changeset
   191
qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
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diff changeset
   192
25152
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diff changeset
   193
end
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   194
29925
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diff changeset
   195
22390
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   196
class no_zero_divisors = zero + times +
25062
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   197
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   198
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
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   199
class semiring_1_cancel = semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
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   200
  + zero_neq_one + monoid_mult
25267
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   201
begin
14940
b9ab8babd8b3 Further development of matrix theory
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   202
27516
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   203
subclass semiring_0_cancel ..
25512
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   204
27516
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   205
subclass semiring_1 ..
25267
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diff changeset
   206
1f745c599b5c proper reinitialisation after subclass
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diff changeset
   207
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   208
29904
856f16a3b436 add class cancel_comm_monoid_add
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diff changeset
   209
class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
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diff changeset
   210
  + zero_neq_one + comm_monoid_mult
25267
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diff changeset
   211
begin
14738
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diff changeset
   212
27516
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diff changeset
   213
subclass semiring_1_cancel ..
9a5d4a8d4aac by intro_locales -> ..
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diff changeset
   214
subclass comm_semiring_0_cancel ..
9a5d4a8d4aac by intro_locales -> ..
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diff changeset
   215
subclass comm_semiring_1 ..
25267
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diff changeset
   216
1f745c599b5c proper reinitialisation after subclass
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diff changeset
   217
end
25152
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diff changeset
   218
22390
378f34b1e380 now using "class"
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diff changeset
   219
class ring = semiring + ab_group_add
25267
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diff changeset
   220
begin
25152
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diff changeset
   221
27516
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diff changeset
   222
subclass semiring_0_cancel ..
25152
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diff changeset
   223
bfde2f8c0f63 partially localized
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diff changeset
   224
text {* Distribution rules *}
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diff changeset
   225
bfde2f8c0f63 partially localized
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diff changeset
   226
lemma minus_mult_left: "- (a * b) = - a * b"
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53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
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diff changeset
   227
by (rule equals_zero_I) (simp add: left_distrib [symmetric]) 
25152
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diff changeset
   228
bfde2f8c0f63 partially localized
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diff changeset
   229
lemma minus_mult_right: "- (a * b) = a * - b"
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nipkow
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diff changeset
   230
by (rule equals_zero_I) (simp add: right_distrib [symmetric]) 
25152
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haftmann
parents: 25078
diff changeset
   231
29407
5ef7e97fd9e4 move lemmas mult_minus{left,right} inside class ring
huffman
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diff changeset
   232
text{*Extract signs from products*}
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
   233
lemmas mult_minus_left [simp, noatp] = minus_mult_left [symmetric]
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
   234
lemmas mult_minus_right [simp,noatp] = minus_mult_right [symmetric]
29407
5ef7e97fd9e4 move lemmas mult_minus{left,right} inside class ring
huffman
parents: 29406
diff changeset
   235
25152
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parents: 25078
diff changeset
   236
lemma minus_mult_minus [simp]: "- a * - b = a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   237
by simp
25152
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diff changeset
   238
bfde2f8c0f63 partially localized
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diff changeset
   239
lemma minus_mult_commute: "- a * b = a * - b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
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diff changeset
   240
by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   241
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   242
lemma right_diff_distrib[algebra_simps]: "a * (b - c) = a * b - a * c"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   243
by (simp add: right_distrib diff_minus)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   244
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   245
lemma left_diff_distrib[algebra_simps]: "(a - b) * c = a * c - b * c"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   246
by (simp add: left_distrib diff_minus)
25152
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parents: 25078
diff changeset
   247
29833
409138c4de12 added noatps
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parents: 29700
diff changeset
   248
lemmas ring_distribs[noatp] =
25152
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parents: 25078
diff changeset
   249
  right_distrib left_distrib left_diff_distrib right_diff_distrib
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   250
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
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diff changeset
   251
text{*Legacy - use @{text algebra_simps} *}
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
   252
lemmas ring_simps[noatp] = algebra_simps
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   253
022029099a83 continued localization
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parents: 25193
diff changeset
   254
lemma eq_add_iff1:
022029099a83 continued localization
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parents: 25193
diff changeset
   255
  "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
   256
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   257
022029099a83 continued localization
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parents: 25193
diff changeset
   258
lemma eq_add_iff2:
022029099a83 continued localization
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parents: 25193
diff changeset
   259
  "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
   260
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   261
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   262
end
bfde2f8c0f63 partially localized
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parents: 25078
diff changeset
   263
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
   264
lemmas ring_distribs[noatp] =
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   265
  right_distrib left_distrib left_diff_distrib right_diff_distrib
bfde2f8c0f63 partially localized
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parents: 25078
diff changeset
   266
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   267
class comm_ring = comm_semiring + ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   268
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   269
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   270
subclass ring ..
28141
193c3ea0f63b instances comm_semiring_0_cancel < comm_semiring_0, comm_ring < comm_semiring_0_cancel
huffman
parents: 27651
diff changeset
   271
subclass comm_semiring_0_cancel ..
25267
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haftmann
parents: 25238
diff changeset
   272
1f745c599b5c proper reinitialisation after subclass
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parents: 25238
diff changeset
   273
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   274
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   275
class ring_1 = ring + zero_neq_one + monoid_mult
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
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diff changeset
   276
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   277
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   278
subclass semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
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diff changeset
   279
1f745c599b5c proper reinitialisation after subclass
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parents: 25238
diff changeset
   280
end
25152
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parents: 25078
diff changeset
   281
22390
378f34b1e380 now using "class"
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parents: 21328
diff changeset
   282
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   283
  (*previously ring*)
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   284
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   285
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   286
subclass ring_1 ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   287
subclass comm_semiring_1_cancel ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   288
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
   289
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
   290
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   291
  assume "x dvd - y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   292
  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
   293
  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
   294
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   295
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   296
  then 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
   297
  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
   298
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   299
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
   300
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
   301
proof
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   302
  assume "- x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   303
  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
   304
  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
   305
  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
   306
next
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   307
  assume "x dvd y"
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   308
  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
   309
  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
   310
  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
   311
qed
6d10cf26b5dc add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
huffman
parents: 29407
diff changeset
   312
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   313
lemma dvd_diff[simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
31039ee583fa Removed subsumed lemmas
nipkow
parents: 29981
diff changeset
   314
by (simp add: diff_minus dvd_minus_iff)
29409
f0a8fe83bc07 add lemma dvd_diff to class comm_ring_1
huffman
parents: 29408
diff changeset
   315
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   316
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   317
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   318
class ring_no_zero_divisors = ring + no_zero_divisors
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   319
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   320
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   321
lemma mult_eq_0_iff [simp]:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   322
  shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   323
proof (cases "a = 0 \<or> b = 0")
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   324
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   325
    then show ?thesis using no_zero_divisors by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   326
next
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   327
  case True then show ?thesis by auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   328
qed
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   329
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   330
text{*Cancellation of equalities with a common factor*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   331
lemma mult_cancel_right [simp, noatp]:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   332
  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   333
proof -
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   334
  have "(a * c = b * c) = ((a - b) * c = 0)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   335
    by (simp add: algebra_simps right_minus_eq)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   336
  thus ?thesis by (simp add: disj_commute right_minus_eq)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   337
qed
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   338
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   339
lemma mult_cancel_left [simp, noatp]:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   340
  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   341
proof -
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   342
  have "(c * a = c * b) = (c * (a - b) = 0)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   343
    by (simp add: algebra_simps right_minus_eq)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   344
  thus ?thesis by (simp add: right_minus_eq)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   345
qed
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   346
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   347
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   348
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   349
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   350
begin
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   351
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   352
lemma mult_cancel_right1 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   353
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   354
by (insert mult_cancel_right [of 1 c b], force)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   355
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   356
lemma mult_cancel_right2 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   357
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   358
by (insert mult_cancel_right [of a c 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   359
 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   360
lemma mult_cancel_left1 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   361
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   362
by (insert mult_cancel_left [of c 1 b], force)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   363
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   364
lemma mult_cancel_left2 [simp]:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   365
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   366
by (insert mult_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   367
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   368
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   369
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   370
class idom = comm_ring_1 + no_zero_divisors
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   371
begin
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   372
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   373
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
   374
29915
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   375
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
   376
proof
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   377
  assume "a * a = b * b"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   378
  then have "(a - b) * (a + b) = 0"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   379
    by (simp add: algebra_simps)
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   380
  then show "a = b \<or> a = - b"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   381
    by (simp add: right_minus_eq eq_neg_iff_add_eq_0)
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   382
next
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   383
  assume "a = b \<or> a = - b"
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   384
  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
   385
qed
2146e512cec9 generalize lemma fps_square_eq_iff, move to Ring_and_Field
huffman
parents: 29904
diff changeset
   386
29981
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   387
lemma dvd_mult_cancel_right [simp]:
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   388
  "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
   389
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   390
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   391
    unfolding dvd_def by (simp add: mult_ac)
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   392
  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
   393
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   394
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   395
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   396
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   397
lemma dvd_mult_cancel_left [simp]:
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   398
  "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
   399
proof -
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   400
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   401
    unfolding dvd_def by (simp add: mult_ac)
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   402
  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
   403
    unfolding dvd_def by simp
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   404
  finally show ?thesis .
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   405
qed
7d0ed261b712 generalize int_dvd_cancel_factor simproc to idom class
huffman
parents: 29949
diff changeset
   406
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   407
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   408
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   409
class division_ring = ring_1 + inverse +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   410
  assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   411
  assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   412
begin
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   413
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   414
subclass ring_1_no_zero_divisors
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   415
proof
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   416
  fix a b :: 'a
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   417
  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   418
  show "a * b \<noteq> 0"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   419
  proof
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   420
    assume ab: "a * b = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   421
    hence "0 = inverse a * (a * b) * inverse b" by simp
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   422
    also have "\<dots> = (inverse a * a) * (b * inverse b)"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   423
      by (simp only: mult_assoc)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   424
    also have "\<dots> = 1" using a b by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   425
    finally show False by simp
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   426
  qed
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   427
qed
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   428
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   429
lemma nonzero_imp_inverse_nonzero:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   430
  "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   431
proof
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   432
  assume ianz: "inverse a = 0"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   433
  assume "a \<noteq> 0"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   434
  hence "1 = a * inverse a" by simp
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   435
  also have "... = 0" by (simp add: ianz)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   436
  finally have "1 = 0" .
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   437
  thus False by (simp add: eq_commute)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   438
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   439
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   440
lemma inverse_zero_imp_zero:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   441
  "inverse a = 0 \<Longrightarrow> a = 0"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   442
apply (rule classical)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   443
apply (drule nonzero_imp_inverse_nonzero)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   444
apply auto
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   445
done
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   446
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   447
lemma inverse_unique: 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   448
  assumes ab: "a * b = 1"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   449
  shows "inverse a = b"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   450
proof -
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   451
  have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   452
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   453
  ultimately show ?thesis by (simp add: mult_assoc [symmetric])
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   454
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   455
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   456
lemma nonzero_inverse_minus_eq:
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   457
  "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   458
by (rule inverse_unique) simp
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   459
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   460
lemma nonzero_inverse_inverse_eq:
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   461
  "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   462
by (rule inverse_unique) simp
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   463
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   464
lemma nonzero_inverse_eq_imp_eq:
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   465
  assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   466
  shows "a = b"
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   467
proof -
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   468
  from `inverse a = inverse b`
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   469
  have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   470
  with `a \<noteq> 0` and `b \<noteq> 0` show "a = b"
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   471
    by (simp add: nonzero_inverse_inverse_eq)
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   472
qed
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   473
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   474
lemma inverse_1 [simp]: "inverse 1 = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   475
by (rule inverse_unique) simp
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   476
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   477
lemma nonzero_inverse_mult_distrib: 
29406
54bac26089bd clean up division_ring proofs
huffman
parents: 28823
diff changeset
   478
  assumes "a \<noteq> 0" and "b \<noteq> 0"
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   479
  shows "inverse (a * b) = inverse b * inverse a"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   480
proof -
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   481
  have "a * (b * inverse b) * inverse a = 1" using assms by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   482
  hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   483
  thus ?thesis by (rule inverse_unique)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   484
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   485
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   486
lemma division_ring_inverse_add:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   487
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   488
by (simp add: algebra_simps)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   489
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   490
lemma division_ring_inverse_diff:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   491
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   492
by (simp add: algebra_simps)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   493
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   494
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   495
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   496
class field = comm_ring_1 + inverse +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   497
  assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   498
  assumes divide_inverse: "a / b = a * inverse b"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   499
begin
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   500
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   501
subclass division_ring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   502
proof
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   503
  fix a :: 'a
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   504
  assume "a \<noteq> 0"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   505
  thus "inverse a * a = 1" by (rule field_inverse)
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   506
  thus "a * inverse a = 1" by (simp only: mult_commute)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   507
qed
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   508
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   509
subclass idom ..
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   510
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   511
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   512
proof
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   513
  assume neq: "b \<noteq> 0"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   514
  {
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   515
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   516
    also assume "a / b = 1"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   517
    finally show "a = b" by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   518
  next
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   519
    assume "a = b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   520
    with neq show "a / b = 1" by (simp add: divide_inverse)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   521
  }
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   522
qed
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   523
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   524
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   525
by (simp add: divide_inverse)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   526
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   527
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   528
by (simp add: divide_inverse)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   529
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   530
lemma divide_zero_left [simp]: "0 / a = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   531
by (simp add: divide_inverse)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   532
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   533
lemma inverse_eq_divide: "inverse a = 1 / a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   534
by (simp add: divide_inverse)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   535
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   536
lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   537
by (simp add: divide_inverse algebra_simps) 
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   538
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   539
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   540
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   541
class division_by_zero = zero + inverse +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   542
  assumes inverse_zero [simp]: "inverse 0 = 0"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   543
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   544
lemma divide_zero [simp]:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   545
  "a / 0 = (0::'a::{field,division_by_zero})"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   546
by (simp add: divide_inverse)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   547
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   548
lemma divide_self_if [simp]:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   549
  "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   550
by simp
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   551
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   552
class mult_mono = times + zero + ord +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   553
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   554
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   555
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   556
class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   557
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   558
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   559
lemma mult_mono:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   560
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   561
     \<Longrightarrow> a * c \<le> b * d"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   562
apply (erule mult_right_mono [THEN order_trans], assumption)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   563
apply (erule mult_left_mono, assumption)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   564
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   565
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   566
lemma mult_mono':
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   567
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   568
     \<Longrightarrow> a * c \<le> b * d"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   569
apply (rule mult_mono)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   570
apply (fast intro: order_trans)+
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   571
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   572
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   573
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   574
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   575
class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29833
diff changeset
   576
  + semiring + cancel_comm_monoid_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   577
begin
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   578
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   579
subclass semiring_0_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   580
subclass pordered_semiring ..
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   581
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   582
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   583
by (drule mult_left_mono [of zero b], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   584
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   585
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   586
by (drule mult_left_mono [of b zero], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   587
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   588
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   589
by (drule mult_right_mono [of b zero], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   590
26234
1f6e28a88785 clarified proposition
haftmann
parents: 26193
diff changeset
   591
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
   592
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   593
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   594
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   595
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   596
class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   597
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   598
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   599
subclass pordered_cancel_semiring ..
25512
4134f7c782e2 using intro_locales instead of unfold_locales if appropriate
haftmann
parents: 25450
diff changeset
   600
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   601
subclass pordered_comm_monoid_add ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   602
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   603
lemma mult_left_less_imp_less:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   604
  "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
   605
by (force simp add: mult_left_mono not_le [symmetric])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   606
 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   607
lemma mult_right_less_imp_less:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   608
  "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
   609
by (force simp add: mult_right_mono not_le [symmetric])
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   610
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   611
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   612
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   613
class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   614
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   615
  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
   616
begin
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   617
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   618
subclass semiring_0_cancel ..
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   619
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   620
subclass ordered_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   621
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   622
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   623
  assume A: "a \<le> b" "0 \<le> c"
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   624
  from A show "c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   625
    unfolding le_less
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   626
    using mult_strict_left_mono by (cases "c = 0") auto
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   627
  from A show "a * c \<le> b * c"
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   628
    unfolding le_less
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   629
    using mult_strict_right_mono by (cases "c = 0") auto
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   630
qed
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   631
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   632
lemma mult_left_le_imp_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   633
  "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
   634
by (force simp add: mult_strict_left_mono _not_less [symmetric])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   635
 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   636
lemma mult_right_le_imp_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   637
  "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
   638
by (force simp add: mult_strict_right_mono not_less [symmetric])
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   639
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   640
lemma mult_pos_pos:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   641
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   642
by (drule mult_strict_left_mono [of zero b], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   643
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   644
lemma mult_pos_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   645
  "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   646
by (drule mult_strict_left_mono [of b zero], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   647
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   648
lemma mult_pos_neg2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   649
  "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   650
by (drule mult_strict_right_mono [of b zero], auto)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   651
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   652
lemma zero_less_mult_pos:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   653
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   654
apply (cases "b\<le>0") 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   655
 apply (auto simp add: le_less not_less)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   656
apply (drule_tac mult_pos_neg [of a b]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   657
 apply (auto dest: less_not_sym)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   658
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   659
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   660
lemma zero_less_mult_pos2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   661
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   662
apply (cases "b\<le>0") 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   663
 apply (auto simp add: le_less not_less)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   664
apply (drule_tac mult_pos_neg2 [of a b]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   665
 apply (auto dest: less_not_sym)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   666
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   667
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   668
text{*Strict monotonicity in both arguments*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   669
lemma mult_strict_mono:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   670
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   671
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   672
  using assms apply (cases "c=0")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   673
  apply (simp add: mult_pos_pos) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   674
  apply (erule mult_strict_right_mono [THEN less_trans])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   675
  apply (force simp add: le_less) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   676
  apply (erule mult_strict_left_mono, assumption)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   677
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   678
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   679
text{*This weaker variant has more natural premises*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   680
lemma mult_strict_mono':
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   681
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   682
  shows "a * c < b * d"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   683
by (rule mult_strict_mono) (insert assms, auto)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   684
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   685
lemma mult_less_le_imp_less:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   686
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   687
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   688
  using assms apply (subgoal_tac "a * c < b * c")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   689
  apply (erule less_le_trans)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   690
  apply (erule mult_left_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   691
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   692
  apply (erule mult_strict_right_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   693
  apply assumption
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   694
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   695
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   696
lemma mult_le_less_imp_less:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   697
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   698
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   699
  using assms apply (subgoal_tac "a * c \<le> b * c")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   700
  apply (erule le_less_trans)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   701
  apply (erule mult_strict_left_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   702
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   703
  apply (erule mult_right_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   704
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   705
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   706
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   707
lemma mult_less_imp_less_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   708
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   709
  shows "a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   710
proof (rule ccontr)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   711
  assume "\<not>  a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   712
  hence "b \<le> a" by (simp add: linorder_not_less)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   713
  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
   714
  with this and less show False by (simp add: not_less [symmetric])
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   715
qed
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   716
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   717
lemma mult_less_imp_less_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   718
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   719
  shows "a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   720
proof (rule ccontr)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   721
  assume "\<not> a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   722
  hence "b \<le> a" by (simp add: linorder_not_less)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   723
  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
   724
  with this and less show False by (simp add: not_less [symmetric])
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   725
qed  
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   726
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   727
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   728
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   729
class mult_mono1 = times + zero + ord +
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   730
  assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   731
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   732
class pordered_comm_semiring = comm_semiring_0
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   733
  + pordered_ab_semigroup_add + mult_mono1
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   734
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   735
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   736
subclass pordered_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   737
proof
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   738
  fix a b c :: 'a
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   739
  assume "a \<le> b" "0 \<le> c"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   740
  thus "c * a \<le> c * b" by (rule mult_mono1)
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   741
  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
   742
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   743
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   744
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   745
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   746
class pordered_cancel_comm_semiring = comm_semiring_0_cancel
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   747
  + pordered_ab_semigroup_add + mult_mono1
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   748
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   749
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   750
subclass pordered_comm_semiring ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   751
subclass pordered_cancel_semiring ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   752
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   753
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   754
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   755
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   756
  assumes mult_strict_left_mono_comm: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   757
begin
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   758
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   759
subclass ordered_semiring_strict
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   760
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   761
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   762
  assume "a < b" "0 < c"
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   763
  thus "c * a < c * b" by (rule mult_strict_left_mono_comm)
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   764
  thus "a * c < b * c" by (simp only: mult_commute)
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   765
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   766
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   767
subclass pordered_cancel_comm_semiring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   768
proof
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   769
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   770
  assume "a \<le> b" "0 \<le> c"
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   771
  thus "c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   772
    unfolding le_less
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   773
    using mult_strict_left_mono by (cases "c = 0") auto
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   774
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   775
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   776
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   777
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   778
class pordered_ring = ring + pordered_cancel_semiring 
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   779
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   780
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   781
subclass pordered_ab_group_add ..
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   782
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   783
text{*Legacy - use @{text algebra_simps} *}
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
   784
lemmas ring_simps[noatp] = algebra_simps
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   785
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   786
lemma less_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   787
  "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
   788
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   789
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   790
lemma less_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   791
  "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
   792
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   793
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   794
lemma le_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   795
  "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
   796
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   797
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   798
lemma le_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   799
  "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
   800
by (simp add: algebra_simps)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   801
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   802
lemma mult_left_mono_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   803
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   804
  apply (drule mult_left_mono [of _ _ "uminus c"])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   805
  apply (simp_all add: minus_mult_left [symmetric]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   806
  done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   807
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   808
lemma mult_right_mono_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   809
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   810
  apply (drule mult_right_mono [of _ _ "uminus c"])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   811
  apply (simp_all add: minus_mult_right [symmetric]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   812
  done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   813
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   814
lemma mult_nonpos_nonpos:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   815
  "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   816
by (drule mult_right_mono_neg [of a zero b]) auto
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   817
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   818
lemma split_mult_pos_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   819
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   820
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   821
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   822
end
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   823
25762
c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25564
diff changeset
   824
class abs_if = minus + uminus + ord + zero + abs +
c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25564
diff changeset
   825
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25564
diff changeset
   826
c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25564
diff changeset
   827
class sgn_if = minus + uminus + zero + one + ord + sgn +
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   828
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 24491
diff changeset
   829
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25512
diff changeset
   830
lemma (in sgn_if) sgn0[simp]: "sgn 0 = 0"
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25512
diff changeset
   831
by(simp add:sgn_if)
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25512
diff changeset
   832
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   833
class ordered_ring = ring + ordered_semiring
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   834
  + ordered_ab_group_add + abs_if
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   835
begin
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   836
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   837
subclass pordered_ring ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   838
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   839
subclass pordered_ab_group_add_abs
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   840
proof
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   841
  fix a b
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   842
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   843
by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   844
   (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   845
     neg_less_eq_nonneg less_eq_neg_nonpos, auto intro: add_nonneg_nonneg,
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   846
      auto intro!: less_imp_le add_neg_neg)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   847
qed (auto simp add: abs_if less_eq_neg_nonpos neg_equal_zero)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   848
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   849
end
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   850
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   851
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   852
   Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   853
 *)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   854
class ordered_ring_strict = ring + ordered_semiring_strict
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   855
  + ordered_ab_group_add + abs_if
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   856
begin
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   857
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   858
subclass ordered_ring ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   859
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   860
lemma mult_strict_left_mono_neg:
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   861
  "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   862
  apply (drule mult_strict_left_mono [of _ _ "uminus c"])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   863
  apply (simp_all add: minus_mult_left [symmetric]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   864
  done
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   865
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   866
lemma mult_strict_right_mono_neg:
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   867
  "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   868
  apply (drule mult_strict_right_mono [of _ _ "uminus c"])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   869
  apply (simp_all add: minus_mult_right [symmetric]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   870
  done
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   871
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   872
lemma mult_neg_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   873
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   874
by (drule mult_strict_right_mono_neg, auto)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   875
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   876
subclass ring_no_zero_divisors
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   877
proof
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   878
  fix a b
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   879
  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
   880
  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
   881
  have "a * b < 0 \<or> 0 < a * b"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   882
  proof (cases "a < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   883
    case True note A' = this
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   884
    show ?thesis proof (cases "b < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   885
      case True with A'
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   886
      show ?thesis by (auto dest: mult_neg_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   887
    next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   888
      case False with B have "0 < b" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   889
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   890
    qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   891
  next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   892
    case False with A have A': "0 < a" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   893
    show ?thesis proof (cases "b < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   894
      case True with A'
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   895
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   896
    next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   897
      case False with B have "0 < b" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   898
      with A' show ?thesis by (auto dest: mult_pos_pos)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   899
    qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   900
  qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   901
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   902
qed
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   903
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   904
lemma zero_less_mult_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   905
  "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   906
  apply (auto simp add: mult_pos_pos mult_neg_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   907
  apply (simp_all add: not_less le_less)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   908
  apply (erule disjE) apply assumption defer
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   909
  apply (erule disjE) defer apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   910
  apply (erule disjE) defer apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   911
  apply (erule disjE) apply assumption apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   912
  apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   913
  apply (blast dest: zero_less_mult_pos)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   914
  apply (blast dest: zero_less_mult_pos2)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   915
  done
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   916
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   917
lemma zero_le_mult_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   918
  "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   919
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
   920
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   921
lemma mult_less_0_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   922
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   923
  apply (insert zero_less_mult_iff [of "-a" b]) 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   924
  apply (force simp add: minus_mult_left[symmetric]) 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   925
  done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   926
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   927
lemma mult_le_0_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   928
  "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
   929
  apply (insert zero_le_mult_iff [of "-a" b]) 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   930
  apply (force simp add: minus_mult_left[symmetric]) 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   931
  done
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   932
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   933
lemma zero_le_square [simp]: "0 \<le> a * a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   934
by (simp add: zero_le_mult_iff linear)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   935
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   936
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   937
by (simp add: not_less)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   938
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   939
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   940
   also with the relations @{text "\<le>"} and equality.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   941
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   942
text{*These ``disjunction'' versions produce two cases when the comparison is
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   943
 an assumption, but effectively four when the comparison is a goal.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   944
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   945
lemma mult_less_cancel_right_disj:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   946
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   947
  apply (cases "c = 0")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   948
  apply (auto simp add: neq_iff mult_strict_right_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   949
                      mult_strict_right_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   950
  apply (auto simp add: not_less 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   951
                      not_le [symmetric, of "a*c"]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   952
                      not_le [symmetric, of a])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   953
  apply (erule_tac [!] notE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   954
  apply (auto simp add: less_imp_le mult_right_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   955
                      mult_right_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   956
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   957
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   958
lemma mult_less_cancel_left_disj:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   959
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   960
  apply (cases "c = 0")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   961
  apply (auto simp add: neq_iff mult_strict_left_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   962
                      mult_strict_left_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   963
  apply (auto simp add: not_less 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   964
                      not_le [symmetric, of "c*a"]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   965
                      not_le [symmetric, of a])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   966
  apply (erule_tac [!] notE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   967
  apply (auto simp add: less_imp_le mult_left_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   968
                      mult_left_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   969
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   970
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   971
text{*The ``conjunction of implication'' lemmas produce two cases when the
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   972
comparison is a goal, but give four when the comparison is an assumption.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   973
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   974
lemma mult_less_cancel_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   975
  "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
   976
  using mult_less_cancel_right_disj [of a c b] by auto
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   977
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   978
lemma mult_less_cancel_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   979
  "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
   980
  using mult_less_cancel_left_disj [of c a b] by auto
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   981
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   982
lemma mult_le_cancel_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   983
   "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
   984
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   985
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   986
lemma mult_le_cancel_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   987
  "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
   988
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   989
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   990
end
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   991
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   992
text{*Legacy - use @{text algebra_simps} *}
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
   993
lemmas ring_simps[noatp] = algebra_simps
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   994
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   995
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   996
class pordered_comm_ring = comm_ring + pordered_comm_semiring
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   997
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   998
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   999
subclass pordered_ring ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
  1000
subclass pordered_cancel_comm_semiring ..
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1001
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
  1002
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1003
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1004
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1005
  (*previously ordered_semiring*)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1006
  assumes zero_less_one [simp]: "0 < 1"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1007
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1008
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1009
lemma pos_add_strict:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1010
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1011
  using add_strict_mono [of zero a b c] by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1012
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1013
lemma zero_le_one [simp]: "0 \<le> 1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1014
by (rule zero_less_one [THEN less_imp_le]) 
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1015
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1016
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
  1017
by (simp add: not_le) 
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1018
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1019
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1020
by (simp add: not_less) 
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1021
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1022
lemma less_1_mult:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1023
  assumes "1 < m" and "1 < n"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1024
  shows "1 < m * n"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1025
  using assms mult_strict_mono [of 1 m 1 n]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1026
    by (simp add:  less_trans [OF zero_less_one]) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1027
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1028
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1029
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1030
class ordered_idom = comm_ring_1 +
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1031
  ordered_comm_semiring_strict + ordered_ab_group_add +
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1032
  abs_if + sgn_if
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1033
  (*previously ordered_ring*)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1034
begin
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1035
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
  1036
subclass ordered_ring_strict ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
  1037
subclass pordered_comm_ring ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
  1038
subclass idom ..
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1039
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1040
subclass ordered_semidom
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
  1041
proof
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1042
  have "0 \<le> 1 * 1" by (rule zero_le_square)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1043
  thus "0 < 1" by (simp add: le_less)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1044
qed 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1045
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1046
lemma linorder_neqE_ordered_idom:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1047
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1048
  using assms by (rule neqE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
  1049
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1050
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1051
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1052
lemma mult_le_cancel_right1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1053
  "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
  1054
by (insert mult_le_cancel_right [of 1 c b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1055
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1056
lemma mult_le_cancel_right2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1057
  "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
  1058
by (insert mult_le_cancel_right [of a c 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1059
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1060
lemma mult_le_cancel_left1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1061
  "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
  1062
by (insert mult_le_cancel_left [of c 1 b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1063
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1064
lemma mult_le_cancel_left2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1065
  "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
  1066
by (insert mult_le_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1067
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1068
lemma mult_less_cancel_right1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1069
  "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
  1070
by (insert mult_less_cancel_right [of 1 c b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1071
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1072
lemma mult_less_cancel_right2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1073
  "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
  1074
by (insert mult_less_cancel_right [of a c 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1075
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1076
lemma mult_less_cancel_left1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1077
  "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
  1078
by (insert mult_less_cancel_left [of c 1 b], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1079
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1080
lemma mult_less_cancel_left2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1081
  "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
  1082
by (insert mult_less_cancel_left [of c a 1], simp)
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1083
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1084
lemma sgn_sgn [simp]:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1085
  "sgn (sgn a) = sgn a"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1086
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
  1087
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1088
lemma sgn_0_0:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1089
  "sgn a = 0 \<longleftrightarrow> a = 0"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1090
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
  1091
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1092
lemma sgn_1_pos:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1093
  "sgn a = 1 \<longleftrightarrow> a > 0"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1094
unfolding sgn_if by (simp add: neg_equal_zero)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1095
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1096
lemma sgn_1_neg:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1097
  "sgn a = - 1 \<longleftrightarrow> a < 0"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1098
unfolding sgn_if by (auto simp add: equal_neg_zero)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1099
29940
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1100
lemma sgn_pos [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1101
  "0 < a \<Longrightarrow> sgn a = 1"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1102
unfolding sgn_1_pos .
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1103
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1104
lemma sgn_neg [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1105
  "a < 0 \<Longrightarrow> sgn a = - 1"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1106
unfolding sgn_1_neg .
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1107
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1108
lemma sgn_times:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27516
diff changeset
  1109
  "sgn (a * b) = sgn a * sgn b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1110
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
  1111
29653
ece6a0e9f8af added lemma abs_sng
haftmann
parents: 29465
diff changeset
  1112
lemma abs_sgn: "abs k = k * sgn k"
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1113
unfolding sgn_if abs_if by auto
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
  1114
29940
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1115
lemma sgn_greater [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1116
  "0 < sgn a \<longleftrightarrow> 0 < a"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1117
  unfolding sgn_if by auto
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1118
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1119
lemma sgn_less [simp]:
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1120
  "sgn a < 0 \<longleftrightarrow> a < 0"
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1121
  unfolding sgn_if by auto
83b373f61d41 more default simp rules for sgn
haftmann
parents: 29925
diff changeset
  1122
29949
20a506b8256d lemmas abs_dvd_iff, dvd_abs_iff
huffman
parents: 29940
diff changeset
  1123
lemma abs_dvd_iff [simp]: "(abs m) dvd k \<longleftrightarrow> m dvd k"
20a506b8256d lemmas abs_dvd_iff, dvd_abs_iff
huffman
parents: 29940
diff changeset
  1124
  by (simp add: abs_if)
20a506b8256d lemmas abs_dvd_iff, dvd_abs_iff
huffman
parents: 29940
diff changeset
  1125
20a506b8256d lemmas abs_dvd_iff, dvd_abs_iff
huffman
parents: 29940
diff changeset
  1126
lemma dvd_abs_iff [simp]: "m dvd (abs k) \<longleftrightarrow> m dvd k"
20a506b8256d lemmas abs_dvd_iff, dvd_abs_iff
huffman
parents: 29940
diff changeset
  1127
  by (simp add: abs_if)
29653
ece6a0e9f8af added lemma abs_sng
haftmann
parents: 29465
diff changeset
  1128
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
  1129
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1130
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1131
class ordered_field = field + ordered_idom
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
  1132
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1133
text {* Simprules for comparisons where common factors can be cancelled. *}
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1134
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
  1135
lemmas mult_compare_simps[noatp] =
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1136
    mult_le_cancel_right mult_le_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1137
    mult_le_cancel_right1 mult_le_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1138
    mult_le_cancel_left1 mult_le_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1139
    mult_less_cancel_right mult_less_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1140
    mult_less_cancel_right1 mult_less_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1141
    mult_less_cancel_left1 mult_less_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1142
    mult_cancel_right mult_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1143
    mult_cancel_right1 mult_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1144
    mult_cancel_left1 mult_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1145
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1146
-- {* FIXME continue localization here *}
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1147
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1148
lemma inverse_nonzero_iff_nonzero [simp]:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
  1149
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
  1150
by (force dest: inverse_zero_imp_zero) 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1151
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1152
lemma inverse_minus_eq [simp]:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
  1153
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1154
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1155
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1156
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1157
  assume "a\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1158
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1159
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1160
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1161
lemma inverse_eq_imp_eq:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
  1162
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1163
apply (cases "a=0 | b=0") 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1164
 apply (force dest!: inverse_zero_imp_zero
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1165
              simp add: eq_commute [of "0::'a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1166
apply (force dest!: nonzero_inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1167
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1168
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1169
lemma inverse_eq_iff_eq [simp]:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
  1170
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
  1171
by (force dest!: inverse_eq_imp_eq)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1172
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1173
lemma inverse_inverse_eq [simp]:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
  1174
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1175
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1176
    assume "a=0" thus ?thesis by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1177
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1178
    assume "a\<noteq>0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1179
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1180
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1181
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1182
text{*This version builds in division by zero while also re-orienting
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1183
      the right-hand side.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1184
lemma inverse_mult_distrib [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1185
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1186
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1187
    assume "a \<noteq> 0 & b \<noteq> 0" 
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1188
    thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1189
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1190
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1191
    thus ?thesis by force
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1192
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1193
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1194
text{*There is no slick version using division by zero.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1195
lemma inverse_add:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1196
  "[|a \<noteq> 0;  b \<noteq> 0|]
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1197
   ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
  1198
by (simp add: division_ring_inverse_add mult_ac)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1199
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1200
lemma inverse_divide [simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1201
  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1202
by (simp add: divide_inverse mult_commute)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1203
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1204
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1205
subsection {* Calculations with fractions *}
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1206
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1207
text{* There is a whole bunch of simp-rules just for class @{text
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1208
field} but none for class @{text field} and @{text nonzero_divides}
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1209
because the latter are covered by a simproc. *}
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1210
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1211
lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1212
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1213
proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1214
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1215
    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1216
  also have "... =  a * inverse b * (inverse c * c)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1217
    by (simp only: mult_ac)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1218
  also have "... =  a * inverse b" by simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
  1219
    finally show ?thesis by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1220
qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1221
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1222
lemma mult_divide_mult_cancel_left:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1223
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1224
apply (cases "b = 0")
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1225
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1226
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1227
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1228
lemma nonzero_mult_divide_mult_cancel_right [noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1229
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1230
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1231
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1232
lemma mult_divide_mult_cancel_right:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1233
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1234
apply (cases "b = 0")
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1235
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1236
done
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1237
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
  1238
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1239
by (simp add: divide_inverse)
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
  1240
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1241
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1242
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1243
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1244
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1245
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1246
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
  1247
lemmas times_divide_eq[noatp] = times_divide_eq_right times_divide_eq_left
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1248
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
  1249
lemma divide_divide_eq_right [simp,noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1250
  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1251
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1252
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
  1253
lemma divide_divide_eq_left [simp,noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1254
  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1255
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1256
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1257
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1258
    x / y + w / z = (x * z + w * y) / (y * z)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1259
apply (subgoal_tac "x / y = (x * z) / (y * z)")
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1260
apply (erule ssubst)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1261
apply (subgoal_tac "w / z = (w * y) / (y * z)")
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1262
apply (erule ssubst)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1263
apply (rule add_divide_distrib [THEN sym])
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1264
apply (subst mult_commute)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1265
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1266
apply assumption
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1267
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1268
apply assumption
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1269
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1270
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1271
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1272
subsubsection{*Special Cancellation Simprules for Division*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1273
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: