src/HOL/Ring_and_Field.thy
author haftmann
Mon Nov 17 17:00:55 2008 +0100 (2008-11-17)
changeset 28823 dcbef866c9e2
parent 28559 55c003a5600a
child 29406 54bac26089bd
permissions -rw-r--r--
tuned unfold_locales invocation
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(*  Title:   HOL/Ring_and_Field.thy
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    ID:      $Id$
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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: "(a + b) * c = a * c + b * c"
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  assumes right_distrib: "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 + comm_monoid_add + cancel_ab_semigroup_add
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begin
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subclass semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0"
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    by (simp add: left_distrib [symmetric])
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  thus "0 * a = 0"
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    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"
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    by (simp add: right_distrib [symmetric])
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  thus "a * 0 = 0"
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    by (simp only: add_left_cancel)
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qed
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end
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a + b) * c = a * c + b * c"
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begin
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subclass semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" 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 + comm_monoid_add + cancel_ab_semigroup_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: "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: "a dvd c \<Longrightarrow> a dvd (b * c)"
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  by (auto intro!: mult_left_commute dvdI elim!: dvdE)
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lemma dvd_mult2: "a dvd b \<Longrightarrow> a dvd (b * c)"
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  apply (subst mult_commute)
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  apply (erule dvd_mult)
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  done
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lemma dvd_triv_right [simp]: "a dvd b * a"
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  by (rule dvd_mult) (rule dvd_refl)
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lemma dvd_triv_left [simp]: "a dvd a * b"
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  by (rule dvd_mult2) (rule dvd_refl)
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lemma mult_dvd_mono:
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  assumes ab: "a dvd b"
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    and "cd": "c dvd d"
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  shows "a * c dvd b * d"
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proof -
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  from ab obtain b' where "b = a * b'" ..
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  moreover from "cd" obtain d' where "d = c * d'" ..
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  ultimately have "b * d = (a * c) * (b' * d')" by (simp add: mult_ac)
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  then show ?thesis ..
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qed
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lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
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  by (simp add: dvd_def mult_assoc, blast)
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lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
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  unfolding mult_ac [of a] by (rule dvd_mult_left)
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lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
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  by simp
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lemma dvd_add:
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  assumes ab: "a dvd b"
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    and ac: "a dvd c"
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    shows "a dvd (b + c)"
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proof -
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  from ab obtain b' where "b = a * b'" ..
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  moreover from ac obtain c' where "c = a * c'" ..
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  ultimately have "b + c = a * (b' + c')" by (simp add: right_distrib)
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  then show ?thesis ..
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qed
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end
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class no_zero_divisors = zero + times +
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  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
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class semiring_1_cancel = semiring + comm_monoid_add + zero_neq_one
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  + cancel_ab_semigroup_add + monoid_mult
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begin
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subclass semiring_0_cancel ..
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subclass semiring_1 ..
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end
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class comm_semiring_1_cancel = comm_semiring + comm_monoid_add + comm_monoid_mult
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  + zero_neq_one + cancel_ab_semigroup_add
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begin
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subclass semiring_1_cancel ..
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subclass comm_semiring_0_cancel ..
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subclass comm_semiring_1 ..
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end
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class ring = semiring + ab_group_add
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begin
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subclass semiring_0_cancel ..
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text {* Distribution rules *}
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lemma minus_mult_left: "- (a * b) = - a * b"
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  by (rule equals_zero_I) (simp add: left_distrib [symmetric]) 
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lemma minus_mult_right: "- (a * b) = a * - b"
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  by (rule equals_zero_I) (simp add: right_distrib [symmetric]) 
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lemma minus_mult_minus [simp]: "- a * - b = a * b"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma minus_mult_commute: "- a * b = a * - b"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma right_diff_distrib: "a * (b - c) = a * b - a * c"
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  by (simp add: right_distrib diff_minus 
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    minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma left_diff_distrib: "(a - b) * c = a * c - b * c"
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  by (simp add: left_distrib diff_minus 
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    minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemmas ring_distribs =
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  right_distrib left_distrib left_diff_distrib right_diff_distrib
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lemmas ring_simps =
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  add_ac
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  add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
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  diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
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  ring_distribs
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lemma eq_add_iff1:
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  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
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  by (simp add: ring_simps)
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lemma eq_add_iff2:
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  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
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  by (simp add: ring_simps)
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end
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lemmas ring_distribs =
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  right_distrib left_distrib left_diff_distrib right_diff_distrib
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class comm_ring = comm_semiring + ab_group_add
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begin
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subclass ring ..
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subclass comm_semiring_0_cancel ..
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end
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class ring_1 = ring + zero_neq_one + monoid_mult
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begin
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subclass semiring_1_cancel ..
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end
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class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
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  (*previously ring*)
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begin
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subclass ring_1 ..
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subclass comm_semiring_1_cancel ..
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end
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class ring_no_zero_divisors = ring + no_zero_divisors
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begin
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lemma mult_eq_0_iff [simp]:
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  shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)"
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proof (cases "a = 0 \<or> b = 0")
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  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
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    then show ?thesis using no_zero_divisors by simp
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next
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  case True then show ?thesis by auto
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qed
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text{*Cancellation of equalities with a common factor*}
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lemma mult_cancel_right [simp, noatp]:
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  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
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proof -
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  have "(a * c = b * c) = ((a - b) * c = 0)"
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    by (simp add: ring_distribs right_minus_eq)
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  thus ?thesis
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    by (simp add: disj_commute right_minus_eq)
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qed
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lemma mult_cancel_left [simp, noatp]:
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  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
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proof -
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  have "(c * a = c * b) = (c * (a - b) = 0)"
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    by (simp add: ring_distribs right_minus_eq)
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  thus ?thesis
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    by (simp add: right_minus_eq)
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qed
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end
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class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
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begin
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lemma mult_cancel_right1 [simp]:
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  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
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  by (insert mult_cancel_right [of 1 c b], force)
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lemma mult_cancel_right2 [simp]:
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  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
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  by (insert mult_cancel_right [of a c 1], simp)
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lemma mult_cancel_left1 [simp]:
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  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
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  by (insert mult_cancel_left [of c 1 b], force)
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lemma mult_cancel_left2 [simp]:
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  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
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  by (insert mult_cancel_left [of c a 1], simp)
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end
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class idom = comm_ring_1 + no_zero_divisors
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begin
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subclass ring_1_no_zero_divisors ..
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   357
end
haftmann@25152
   358
haftmann@22390
   359
class division_ring = ring_1 + inverse +
haftmann@25062
   360
  assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
haftmann@25062
   361
  assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
haftmann@25186
   362
begin
huffman@20496
   363
haftmann@25186
   364
subclass ring_1_no_zero_divisors
haftmann@28823
   365
proof
huffman@22987
   366
  fix a b :: 'a
huffman@22987
   367
  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
huffman@22987
   368
  show "a * b \<noteq> 0"
huffman@22987
   369
  proof
huffman@22987
   370
    assume ab: "a * b = 0"
huffman@22987
   371
    hence "0 = inverse a * (a * b) * inverse b"
huffman@22987
   372
      by simp
huffman@22987
   373
    also have "\<dots> = (inverse a * a) * (b * inverse b)"
huffman@22987
   374
      by (simp only: mult_assoc)
huffman@22987
   375
    also have "\<dots> = 1"
huffman@22987
   376
      using a b by simp
huffman@22987
   377
    finally show False
huffman@22987
   378
      by simp
huffman@22987
   379
  qed
huffman@22987
   380
qed
huffman@20496
   381
haftmann@26274
   382
lemma nonzero_imp_inverse_nonzero:
haftmann@26274
   383
  "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
haftmann@26274
   384
proof
haftmann@26274
   385
  assume ianz: "inverse a = 0"
haftmann@26274
   386
  assume "a \<noteq> 0"
haftmann@26274
   387
  hence "1 = a * inverse a" by simp
haftmann@26274
   388
  also have "... = 0" by (simp add: ianz)
haftmann@26274
   389
  finally have "1 = 0" .
haftmann@26274
   390
  thus False by (simp add: eq_commute)
haftmann@26274
   391
qed
haftmann@26274
   392
haftmann@26274
   393
lemma inverse_zero_imp_zero:
haftmann@26274
   394
  "inverse a = 0 \<Longrightarrow> a = 0"
haftmann@26274
   395
apply (rule classical)
haftmann@26274
   396
apply (drule nonzero_imp_inverse_nonzero)
haftmann@26274
   397
apply auto
haftmann@26274
   398
done
haftmann@26274
   399
haftmann@26274
   400
lemma nonzero_inverse_minus_eq:
haftmann@26274
   401
  assumes "a \<noteq> 0"
haftmann@26274
   402
  shows "inverse (- a) = - inverse a"
haftmann@26274
   403
proof -
haftmann@26274
   404
  have "- a * inverse (- a) = - a * - inverse a"
haftmann@26274
   405
    using assms by simp
haftmann@26274
   406
  then show ?thesis unfolding mult_cancel_left using assms by simp 
haftmann@26274
   407
qed
haftmann@26274
   408
haftmann@26274
   409
lemma nonzero_inverse_inverse_eq:
haftmann@26274
   410
  assumes "a \<noteq> 0"
haftmann@26274
   411
  shows "inverse (inverse a) = a"
haftmann@26274
   412
proof -
haftmann@26274
   413
  have "(inverse (inverse a) * inverse a) * a = a" 
haftmann@26274
   414
    using assms by (simp add: nonzero_imp_inverse_nonzero)
haftmann@26274
   415
  then show ?thesis using assms by (simp add: mult_assoc)
haftmann@26274
   416
qed
haftmann@26274
   417
haftmann@26274
   418
lemma nonzero_inverse_eq_imp_eq:
haftmann@26274
   419
  assumes inveq: "inverse a = inverse b"
haftmann@26274
   420
    and anz:  "a \<noteq> 0"
haftmann@26274
   421
    and bnz:  "b \<noteq> 0"
haftmann@26274
   422
  shows "a = b"
haftmann@26274
   423
proof -
haftmann@26274
   424
  have "a * inverse b = a * inverse a"
haftmann@26274
   425
    by (simp add: inveq)
haftmann@26274
   426
  hence "(a * inverse b) * b = (a * inverse a) * b"
haftmann@26274
   427
    by simp
haftmann@26274
   428
  then show "a = b"
haftmann@26274
   429
    by (simp add: mult_assoc anz bnz)
haftmann@26274
   430
qed
haftmann@26274
   431
haftmann@26274
   432
lemma inverse_1 [simp]: "inverse 1 = 1"
haftmann@26274
   433
proof -
haftmann@26274
   434
  have "inverse 1 * 1 = 1" 
haftmann@26274
   435
    by (rule left_inverse) (rule one_neq_zero)
haftmann@26274
   436
  then show ?thesis by simp
haftmann@26274
   437
qed
haftmann@26274
   438
haftmann@26274
   439
lemma inverse_unique: 
haftmann@26274
   440
  assumes ab: "a * b = 1"
haftmann@26274
   441
  shows "inverse a = b"
haftmann@26274
   442
proof -
haftmann@26274
   443
  have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
haftmann@26274
   444
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
haftmann@26274
   445
  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
haftmann@26274
   446
qed
haftmann@26274
   447
haftmann@26274
   448
lemma nonzero_inverse_mult_distrib: 
haftmann@26274
   449
  assumes anz: "a \<noteq> 0"
haftmann@26274
   450
    and bnz: "b \<noteq> 0"
haftmann@26274
   451
  shows "inverse (a * b) = inverse b * inverse a"
haftmann@26274
   452
proof -
haftmann@26274
   453
  have "inverse (a * b) * (a * b) * inverse b = inverse b" 
haftmann@26274
   454
    by (simp add: anz bnz)
haftmann@26274
   455
  hence "inverse (a * b) * a = inverse b" 
haftmann@26274
   456
    by (simp add: mult_assoc bnz)
haftmann@26274
   457
  hence "inverse (a * b) * a * inverse a = inverse b * inverse a" 
haftmann@26274
   458
    by simp
haftmann@26274
   459
  thus ?thesis
haftmann@26274
   460
    by (simp add: mult_assoc anz)
haftmann@26274
   461
qed
haftmann@26274
   462
haftmann@26274
   463
lemma division_ring_inverse_add:
haftmann@26274
   464
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
haftmann@26274
   465
  by (simp add: ring_simps mult_assoc)
haftmann@26274
   466
haftmann@26274
   467
lemma division_ring_inverse_diff:
haftmann@26274
   468
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
haftmann@26274
   469
  by (simp add: ring_simps mult_assoc)
haftmann@26274
   470
haftmann@25186
   471
end
haftmann@25152
   472
huffman@22987
   473
class field = comm_ring_1 + inverse +
haftmann@25062
   474
  assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
haftmann@25062
   475
  assumes divide_inverse: "a / b = a * inverse b"
haftmann@25267
   476
begin
huffman@20496
   477
haftmann@25267
   478
subclass division_ring
haftmann@28823
   479
proof
huffman@22987
   480
  fix a :: 'a
huffman@22987
   481
  assume "a \<noteq> 0"
huffman@22987
   482
  thus "inverse a * a = 1" by (rule field_inverse)
huffman@22987
   483
  thus "a * inverse a = 1" by (simp only: mult_commute)
obua@14738
   484
qed
haftmann@25230
   485
huffman@27516
   486
subclass idom ..
haftmann@25230
   487
haftmann@25230
   488
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
haftmann@25230
   489
proof
haftmann@25230
   490
  assume neq: "b \<noteq> 0"
haftmann@25230
   491
  {
haftmann@25230
   492
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
haftmann@25230
   493
    also assume "a / b = 1"
haftmann@25230
   494
    finally show "a = b" by simp
haftmann@25230
   495
  next
haftmann@25230
   496
    assume "a = b"
haftmann@25230
   497
    with neq show "a / b = 1" by (simp add: divide_inverse)
haftmann@25230
   498
  }
haftmann@25230
   499
qed
haftmann@25230
   500
haftmann@25230
   501
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
haftmann@25230
   502
  by (simp add: divide_inverse)
haftmann@25230
   503
haftmann@25230
   504
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
haftmann@25230
   505
  by (simp add: divide_inverse)
haftmann@25230
   506
haftmann@25230
   507
lemma divide_zero_left [simp]: "0 / a = 0"
haftmann@25230
   508
  by (simp add: divide_inverse)
haftmann@25230
   509
haftmann@25230
   510
lemma inverse_eq_divide: "inverse a = 1 / a"
haftmann@25230
   511
  by (simp add: divide_inverse)
haftmann@25230
   512
haftmann@25230
   513
lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
haftmann@25230
   514
  by (simp add: divide_inverse ring_distribs) 
haftmann@25230
   515
haftmann@25230
   516
end
haftmann@25230
   517
haftmann@22390
   518
class division_by_zero = zero + inverse +
haftmann@25062
   519
  assumes inverse_zero [simp]: "inverse 0 = 0"
paulson@14265
   520
haftmann@25230
   521
lemma divide_zero [simp]:
haftmann@25230
   522
  "a / 0 = (0::'a::{field,division_by_zero})"
haftmann@25230
   523
  by (simp add: divide_inverse)
haftmann@25230
   524
haftmann@25230
   525
lemma divide_self_if [simp]:
haftmann@25230
   526
  "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
haftmann@28559
   527
  by simp
haftmann@25230
   528
haftmann@22390
   529
class mult_mono = times + zero + ord +
haftmann@25062
   530
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25062
   531
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
paulson@14267
   532
haftmann@22390
   533
class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
haftmann@25230
   534
begin
haftmann@25230
   535
haftmann@25230
   536
lemma mult_mono:
haftmann@25230
   537
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c
haftmann@25230
   538
     \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   539
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   540
apply (erule mult_left_mono, assumption)
haftmann@25230
   541
done
haftmann@25230
   542
haftmann@25230
   543
lemma mult_mono':
haftmann@25230
   544
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c
haftmann@25230
   545
     \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   546
apply (rule mult_mono)
haftmann@25230
   547
apply (fast intro: order_trans)+
haftmann@25230
   548
done
haftmann@25230
   549
haftmann@25230
   550
end
krauss@21199
   551
haftmann@22390
   552
class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
huffman@22987
   553
  + semiring + comm_monoid_add + cancel_ab_semigroup_add
haftmann@25267
   554
begin
paulson@14268
   555
huffman@27516
   556
subclass semiring_0_cancel ..
huffman@27516
   557
subclass pordered_semiring ..
obua@23521
   558
haftmann@25230
   559
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@25230
   560
  by (drule mult_left_mono [of zero b], auto)
haftmann@25230
   561
haftmann@25230
   562
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@25230
   563
  by (drule mult_left_mono [of b zero], auto)
haftmann@25230
   564
haftmann@25230
   565
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
haftmann@25230
   566
  by (drule mult_right_mono [of b zero], auto)
haftmann@25230
   567
haftmann@26234
   568
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
haftmann@25230
   569
  by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   570
haftmann@25230
   571
end
haftmann@25230
   572
haftmann@25230
   573
class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
haftmann@25267
   574
begin
haftmann@25230
   575
huffman@27516
   576
subclass pordered_cancel_semiring ..
haftmann@25512
   577
huffman@27516
   578
subclass pordered_comm_monoid_add ..
haftmann@25304
   579
haftmann@25230
   580
lemma mult_left_less_imp_less:
haftmann@25230
   581
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
haftmann@25230
   582
  by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   583
 
haftmann@25230
   584
lemma mult_right_less_imp_less:
haftmann@25230
   585
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
haftmann@25230
   586
  by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   587
haftmann@25186
   588
end
haftmann@25152
   589
haftmann@22390
   590
class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
haftmann@25062
   591
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   592
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   593
begin
paulson@14341
   594
huffman@27516
   595
subclass semiring_0_cancel ..
obua@14940
   596
haftmann@25267
   597
subclass ordered_semiring
haftmann@28823
   598
proof
huffman@23550
   599
  fix a b c :: 'a
huffman@23550
   600
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   601
  from A show "c * a \<le> c * b"
haftmann@25186
   602
    unfolding le_less
haftmann@25186
   603
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   604
  from A show "a * c \<le> b * c"
haftmann@25152
   605
    unfolding le_less
haftmann@25186
   606
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   607
qed
haftmann@25152
   608
haftmann@25230
   609
lemma mult_left_le_imp_le:
haftmann@25230
   610
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
haftmann@25230
   611
  by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   612
 
haftmann@25230
   613
lemma mult_right_le_imp_le:
haftmann@25230
   614
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
haftmann@25230
   615
  by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   616
haftmann@25230
   617
lemma mult_pos_pos:
haftmann@25230
   618
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@25230
   619
  by (drule mult_strict_left_mono [of zero b], auto)
haftmann@25230
   620
haftmann@25230
   621
lemma mult_pos_neg:
haftmann@25230
   622
  "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@25230
   623
  by (drule mult_strict_left_mono [of b zero], auto)
haftmann@25230
   624
haftmann@25230
   625
lemma mult_pos_neg2:
haftmann@25230
   626
  "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
haftmann@25230
   627
  by (drule mult_strict_right_mono [of b zero], auto)
haftmann@25230
   628
haftmann@25230
   629
lemma zero_less_mult_pos:
haftmann@25230
   630
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
haftmann@25230
   631
apply (cases "b\<le>0") 
haftmann@25230
   632
 apply (auto simp add: le_less not_less)
haftmann@25230
   633
apply (drule_tac mult_pos_neg [of a b]) 
haftmann@25230
   634
 apply (auto dest: less_not_sym)
haftmann@25230
   635
done
haftmann@25230
   636
haftmann@25230
   637
lemma zero_less_mult_pos2:
haftmann@25230
   638
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
haftmann@25230
   639
apply (cases "b\<le>0") 
haftmann@25230
   640
 apply (auto simp add: le_less not_less)
haftmann@25230
   641
apply (drule_tac mult_pos_neg2 [of a b]) 
haftmann@25230
   642
 apply (auto dest: less_not_sym)
haftmann@25230
   643
done
haftmann@25230
   644
haftmann@26193
   645
text{*Strict monotonicity in both arguments*}
haftmann@26193
   646
lemma mult_strict_mono:
haftmann@26193
   647
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
   648
  shows "a * c < b * d"
haftmann@26193
   649
  using assms apply (cases "c=0")
haftmann@26193
   650
  apply (simp add: mult_pos_pos) 
haftmann@26193
   651
  apply (erule mult_strict_right_mono [THEN less_trans])
haftmann@26193
   652
  apply (force simp add: le_less) 
haftmann@26193
   653
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
   654
  done
haftmann@26193
   655
haftmann@26193
   656
text{*This weaker variant has more natural premises*}
haftmann@26193
   657
lemma mult_strict_mono':
haftmann@26193
   658
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
   659
  shows "a * c < b * d"
haftmann@26193
   660
  by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
   661
haftmann@26193
   662
lemma mult_less_le_imp_less:
haftmann@26193
   663
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
   664
  shows "a * c < b * d"
haftmann@26193
   665
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
   666
  apply (erule less_le_trans)
haftmann@26193
   667
  apply (erule mult_left_mono)
haftmann@26193
   668
  apply simp
haftmann@26193
   669
  apply (erule mult_strict_right_mono)
haftmann@26193
   670
  apply assumption
haftmann@26193
   671
  done
haftmann@26193
   672
haftmann@26193
   673
lemma mult_le_less_imp_less:
haftmann@26193
   674
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
   675
  shows "a * c < b * d"
haftmann@26193
   676
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
   677
  apply (erule le_less_trans)
haftmann@26193
   678
  apply (erule mult_strict_left_mono)
haftmann@26193
   679
  apply simp
haftmann@26193
   680
  apply (erule mult_right_mono)
haftmann@26193
   681
  apply simp
haftmann@26193
   682
  done
haftmann@26193
   683
haftmann@26193
   684
lemma mult_less_imp_less_left:
haftmann@26193
   685
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
haftmann@26193
   686
  shows "a < b"
haftmann@26193
   687
proof (rule ccontr)
haftmann@26193
   688
  assume "\<not>  a < b"
haftmann@26193
   689
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   690
  hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)
haftmann@26193
   691
  with this and less show False 
haftmann@26193
   692
    by (simp add: not_less [symmetric])
haftmann@26193
   693
qed
haftmann@26193
   694
haftmann@26193
   695
lemma mult_less_imp_less_right:
haftmann@26193
   696
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
haftmann@26193
   697
  shows "a < b"
haftmann@26193
   698
proof (rule ccontr)
haftmann@26193
   699
  assume "\<not> a < b"
haftmann@26193
   700
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   701
  hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)
haftmann@26193
   702
  with this and less show False 
haftmann@26193
   703
    by (simp add: not_less [symmetric])
haftmann@26193
   704
qed  
haftmann@26193
   705
haftmann@25230
   706
end
haftmann@25230
   707
haftmann@22390
   708
class mult_mono1 = times + zero + ord +
haftmann@25230
   709
  assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
paulson@14270
   710
haftmann@22390
   711
class pordered_comm_semiring = comm_semiring_0
haftmann@22390
   712
  + pordered_ab_semigroup_add + mult_mono1
haftmann@25186
   713
begin
haftmann@25152
   714
haftmann@25267
   715
subclass pordered_semiring
haftmann@28823
   716
proof
krauss@21199
   717
  fix a b c :: 'a
huffman@23550
   718
  assume "a \<le> b" "0 \<le> c"
haftmann@25230
   719
  thus "c * a \<le> c * b" by (rule mult_mono1)
huffman@23550
   720
  thus "a * c \<le> b * c" by (simp only: mult_commute)
krauss@21199
   721
qed
paulson@14265
   722
haftmann@25267
   723
end
haftmann@25267
   724
haftmann@25267
   725
class pordered_cancel_comm_semiring = comm_semiring_0_cancel
haftmann@25267
   726
  + pordered_ab_semigroup_add + mult_mono1
haftmann@25267
   727
begin
paulson@14265
   728
huffman@27516
   729
subclass pordered_comm_semiring ..
huffman@27516
   730
subclass pordered_cancel_semiring ..
haftmann@25267
   731
haftmann@25267
   732
end
haftmann@25267
   733
haftmann@25267
   734
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
haftmann@26193
   735
  assumes mult_strict_left_mono_comm: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
   736
begin
haftmann@25267
   737
haftmann@25267
   738
subclass ordered_semiring_strict
haftmann@28823
   739
proof
huffman@23550
   740
  fix a b c :: 'a
huffman@23550
   741
  assume "a < b" "0 < c"
haftmann@26193
   742
  thus "c * a < c * b" by (rule mult_strict_left_mono_comm)
huffman@23550
   743
  thus "a * c < b * c" by (simp only: mult_commute)
huffman@23550
   744
qed
paulson@14272
   745
haftmann@25267
   746
subclass pordered_cancel_comm_semiring
haftmann@28823
   747
proof
huffman@23550
   748
  fix a b c :: 'a
huffman@23550
   749
  assume "a \<le> b" "0 \<le> c"
huffman@23550
   750
  thus "c * a \<le> c * b"
haftmann@25186
   751
    unfolding le_less
haftmann@26193
   752
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   753
qed
paulson@14272
   754
haftmann@25267
   755
end
haftmann@25230
   756
haftmann@25267
   757
class pordered_ring = ring + pordered_cancel_semiring 
haftmann@25267
   758
begin
haftmann@25230
   759
huffman@27516
   760
subclass pordered_ab_group_add ..
paulson@14270
   761
haftmann@25230
   762
lemmas ring_simps = ring_simps group_simps
haftmann@25230
   763
haftmann@25230
   764
lemma less_add_iff1:
haftmann@25230
   765
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
haftmann@25230
   766
  by (simp add: ring_simps)
haftmann@25230
   767
haftmann@25230
   768
lemma less_add_iff2:
haftmann@25230
   769
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
haftmann@25230
   770
  by (simp add: ring_simps)
haftmann@25230
   771
haftmann@25230
   772
lemma le_add_iff1:
haftmann@25230
   773
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
haftmann@25230
   774
  by (simp add: ring_simps)
haftmann@25230
   775
haftmann@25230
   776
lemma le_add_iff2:
haftmann@25230
   777
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
haftmann@25230
   778
  by (simp add: ring_simps)
haftmann@25230
   779
haftmann@25230
   780
lemma mult_left_mono_neg:
haftmann@25230
   781
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@25230
   782
  apply (drule mult_left_mono [of _ _ "uminus c"])
haftmann@25230
   783
  apply (simp_all add: minus_mult_left [symmetric]) 
haftmann@25230
   784
  done
haftmann@25230
   785
haftmann@25230
   786
lemma mult_right_mono_neg:
haftmann@25230
   787
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   788
  apply (drule mult_right_mono [of _ _ "uminus c"])
haftmann@25230
   789
  apply (simp_all add: minus_mult_right [symmetric]) 
haftmann@25230
   790
  done
haftmann@25230
   791
haftmann@25230
   792
lemma mult_nonpos_nonpos:
haftmann@25230
   793
  "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@25230
   794
  by (drule mult_right_mono_neg [of a zero b]) auto
haftmann@25230
   795
haftmann@25230
   796
lemma split_mult_pos_le:
haftmann@25230
   797
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
haftmann@25230
   798
  by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
haftmann@25186
   799
haftmann@25186
   800
end
paulson@14270
   801
haftmann@25762
   802
class abs_if = minus + uminus + ord + zero + abs +
haftmann@25762
   803
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@25762
   804
haftmann@25762
   805
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@25186
   806
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
nipkow@24506
   807
nipkow@25564
   808
lemma (in sgn_if) sgn0[simp]: "sgn 0 = 0"
nipkow@25564
   809
by(simp add:sgn_if)
nipkow@25564
   810
haftmann@25230
   811
class ordered_ring = ring + ordered_semiring
haftmann@25304
   812
  + ordered_ab_group_add + abs_if
haftmann@25304
   813
begin
haftmann@25304
   814
huffman@27516
   815
subclass pordered_ring ..
haftmann@25304
   816
haftmann@25304
   817
subclass pordered_ab_group_add_abs
haftmann@28823
   818
proof
haftmann@25304
   819
  fix a b
haftmann@25304
   820
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25304
   821
  by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
haftmann@25304
   822
   (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]
haftmann@25304
   823
     neg_less_eq_nonneg less_eq_neg_nonpos, auto intro: add_nonneg_nonneg,
haftmann@25304
   824
      auto intro!: less_imp_le add_neg_neg)
haftmann@25304
   825
qed (auto simp add: abs_if less_eq_neg_nonpos neg_equal_zero)
haftmann@25304
   826
haftmann@25304
   827
end
obua@23521
   828
haftmann@25230
   829
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
haftmann@25230
   830
   Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
haftmann@25230
   831
 *)
haftmann@25230
   832
class ordered_ring_strict = ring + ordered_semiring_strict
haftmann@25304
   833
  + ordered_ab_group_add + abs_if
haftmann@25230
   834
begin
paulson@14348
   835
huffman@27516
   836
subclass ordered_ring ..
haftmann@25304
   837
paulson@14265
   838
lemma mult_strict_left_mono_neg:
haftmann@25230
   839
  "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
haftmann@25230
   840
  apply (drule mult_strict_left_mono [of _ _ "uminus c"])
haftmann@25230
   841
  apply (simp_all add: minus_mult_left [symmetric]) 
haftmann@25230
   842
  done
obua@14738
   843
paulson@14265
   844
lemma mult_strict_right_mono_neg:
haftmann@25230
   845
  "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
haftmann@25230
   846
  apply (drule mult_strict_right_mono [of _ _ "uminus c"])
haftmann@25230
   847
  apply (simp_all add: minus_mult_right [symmetric]) 
haftmann@25230
   848
  done
obua@14738
   849
haftmann@25230
   850
lemma mult_neg_neg:
haftmann@25230
   851
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@25230
   852
  by (drule mult_strict_right_mono_neg, auto)
obua@14738
   853
haftmann@25917
   854
subclass ring_no_zero_divisors
haftmann@28823
   855
proof
haftmann@25917
   856
  fix a b
haftmann@25917
   857
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
   858
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
   859
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
   860
  proof (cases "a < 0")
haftmann@25917
   861
    case True note A' = this
haftmann@25917
   862
    show ?thesis proof (cases "b < 0")
haftmann@25917
   863
      case True with A'
haftmann@25917
   864
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
   865
    next
haftmann@25917
   866
      case False with B have "0 < b" by auto
haftmann@25917
   867
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
   868
    qed
haftmann@25917
   869
  next
haftmann@25917
   870
    case False with A have A': "0 < a" by auto
haftmann@25917
   871
    show ?thesis proof (cases "b < 0")
haftmann@25917
   872
      case True with A'
haftmann@25917
   873
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
   874
    next
haftmann@25917
   875
      case False with B have "0 < b" by auto
haftmann@25917
   876
      with A' show ?thesis by (auto dest: mult_pos_pos)
haftmann@25917
   877
    qed
haftmann@25917
   878
  qed
haftmann@25917
   879
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
   880
qed
haftmann@25304
   881
paulson@14265
   882
lemma zero_less_mult_iff:
haftmann@25917
   883
  "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
haftmann@25917
   884
  apply (auto simp add: mult_pos_pos mult_neg_neg)
haftmann@25917
   885
  apply (simp_all add: not_less le_less)
haftmann@25917
   886
  apply (erule disjE) apply assumption defer
haftmann@25917
   887
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
   888
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
   889
  apply (erule disjE) apply assumption apply (drule sym) apply simp
haftmann@25917
   890
  apply (drule sym) apply simp
haftmann@25917
   891
  apply (blast dest: zero_less_mult_pos)
haftmann@25230
   892
  apply (blast dest: zero_less_mult_pos2)
haftmann@25230
   893
  done
huffman@22990
   894
paulson@14265
   895
lemma zero_le_mult_iff:
haftmann@25917
   896
  "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
haftmann@25917
   897
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
   898
paulson@14265
   899
lemma mult_less_0_iff:
haftmann@25917
   900
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
haftmann@25917
   901
  apply (insert zero_less_mult_iff [of "-a" b]) 
haftmann@25917
   902
  apply (force simp add: minus_mult_left[symmetric]) 
haftmann@25917
   903
  done
paulson@14265
   904
paulson@14265
   905
lemma mult_le_0_iff:
haftmann@25917
   906
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
   907
  apply (insert zero_le_mult_iff [of "-a" b]) 
haftmann@25917
   908
  apply (force simp add: minus_mult_left[symmetric]) 
haftmann@25917
   909
  done
haftmann@25917
   910
haftmann@25917
   911
lemma zero_le_square [simp]: "0 \<le> a * a"
haftmann@25917
   912
  by (simp add: zero_le_mult_iff linear)
haftmann@25917
   913
haftmann@25917
   914
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
haftmann@25917
   915
  by (simp add: not_less)
haftmann@25917
   916
haftmann@26193
   917
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
   918
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
   919
haftmann@26193
   920
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
   921
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
   922
haftmann@26193
   923
lemma mult_less_cancel_right_disj:
haftmann@26193
   924
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   925
  apply (cases "c = 0")
haftmann@26193
   926
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
   927
                      mult_strict_right_mono_neg)
haftmann@26193
   928
  apply (auto simp add: not_less 
haftmann@26193
   929
                      not_le [symmetric, of "a*c"]
haftmann@26193
   930
                      not_le [symmetric, of a])
haftmann@26193
   931
  apply (erule_tac [!] notE)
haftmann@26193
   932
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
   933
                      mult_right_mono_neg)
haftmann@26193
   934
  done
haftmann@26193
   935
haftmann@26193
   936
lemma mult_less_cancel_left_disj:
haftmann@26193
   937
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   938
  apply (cases "c = 0")
haftmann@26193
   939
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
   940
                      mult_strict_left_mono_neg)
haftmann@26193
   941
  apply (auto simp add: not_less 
haftmann@26193
   942
                      not_le [symmetric, of "c*a"]
haftmann@26193
   943
                      not_le [symmetric, of a])
haftmann@26193
   944
  apply (erule_tac [!] notE)
haftmann@26193
   945
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
   946
                      mult_left_mono_neg)
haftmann@26193
   947
  done
haftmann@26193
   948
haftmann@26193
   949
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
   950
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
   951
haftmann@26193
   952
lemma mult_less_cancel_right:
haftmann@26193
   953
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   954
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
   955
haftmann@26193
   956
lemma mult_less_cancel_left:
haftmann@26193
   957
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   958
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
   959
haftmann@26193
   960
lemma mult_le_cancel_right:
haftmann@26193
   961
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
haftmann@26193
   962
  by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
   963
haftmann@26193
   964
lemma mult_le_cancel_left:
haftmann@26193
   965
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
haftmann@26193
   966
  by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
   967
haftmann@25917
   968
end
paulson@14265
   969
haftmann@25230
   970
text{*This list of rewrites simplifies ring terms by multiplying
haftmann@25230
   971
everything out and bringing sums and products into a canonical form
haftmann@25230
   972
(by ordered rewriting). As a result it decides ring equalities but
haftmann@25230
   973
also helps with inequalities. *}
haftmann@25230
   974
lemmas ring_simps = group_simps ring_distribs
haftmann@25230
   975
haftmann@25230
   976
haftmann@25230
   977
class pordered_comm_ring = comm_ring + pordered_comm_semiring
haftmann@25267
   978
begin
haftmann@25230
   979
huffman@27516
   980
subclass pordered_ring ..
huffman@27516
   981
subclass pordered_cancel_comm_semiring ..
haftmann@25230
   982
haftmann@25267
   983
end
haftmann@25230
   984
haftmann@25230
   985
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
haftmann@25230
   986
  (*previously ordered_semiring*)
haftmann@25230
   987
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
   988
begin
haftmann@25230
   989
haftmann@25230
   990
lemma pos_add_strict:
haftmann@25230
   991
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@25230
   992
  using add_strict_mono [of zero a b c] by simp
haftmann@25230
   993
haftmann@26193
   994
lemma zero_le_one [simp]: "0 \<le> 1"
haftmann@26193
   995
  by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
   996
haftmann@26193
   997
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
haftmann@26193
   998
  by (simp add: not_le) 
haftmann@26193
   999
haftmann@26193
  1000
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
haftmann@26193
  1001
  by (simp add: not_less) 
haftmann@26193
  1002
haftmann@26193
  1003
lemma less_1_mult:
haftmann@26193
  1004
  assumes "1 < m" and "1 < n"
haftmann@26193
  1005
  shows "1 < m * n"
haftmann@26193
  1006
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
  1007
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
  1008
haftmann@25230
  1009
end
haftmann@25230
  1010
haftmann@26193
  1011
class ordered_idom = comm_ring_1 +
haftmann@26193
  1012
  ordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1013
  abs_if + sgn_if
haftmann@25230
  1014
  (*previously ordered_ring*)
haftmann@25917
  1015
begin
haftmann@25917
  1016
huffman@27516
  1017
subclass ordered_ring_strict ..
huffman@27516
  1018
subclass pordered_comm_ring ..
huffman@27516
  1019
subclass idom ..
haftmann@25917
  1020
haftmann@25917
  1021
subclass ordered_semidom
haftmann@28823
  1022
proof
haftmann@26193
  1023
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1024
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
  1025
qed 
haftmann@25917
  1026
haftmann@26193
  1027
lemma linorder_neqE_ordered_idom:
haftmann@26193
  1028
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1029
  using assms by (rule neqE)
haftmann@26193
  1030
haftmann@26274
  1031
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1032
haftmann@26274
  1033
lemma mult_le_cancel_right1:
haftmann@26274
  1034
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
haftmann@26274
  1035
  by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1036
haftmann@26274
  1037
lemma mult_le_cancel_right2:
haftmann@26274
  1038
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
haftmann@26274
  1039
  by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1040
haftmann@26274
  1041
lemma mult_le_cancel_left1:
haftmann@26274
  1042
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
haftmann@26274
  1043
  by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1044
haftmann@26274
  1045
lemma mult_le_cancel_left2:
haftmann@26274
  1046
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
haftmann@26274
  1047
  by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1048
haftmann@26274
  1049
lemma mult_less_cancel_right1:
haftmann@26274
  1050
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
haftmann@26274
  1051
  by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1052
haftmann@26274
  1053
lemma mult_less_cancel_right2:
haftmann@26274
  1054
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
haftmann@26274
  1055
  by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1056
haftmann@26274
  1057
lemma mult_less_cancel_left1:
haftmann@26274
  1058
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
haftmann@26274
  1059
  by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1060
haftmann@26274
  1061
lemma mult_less_cancel_left2:
haftmann@26274
  1062
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
haftmann@26274
  1063
  by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1064
haftmann@27651
  1065
lemma sgn_sgn [simp]:
haftmann@27651
  1066
  "sgn (sgn a) = sgn a"
haftmann@27651
  1067
  unfolding sgn_if by simp
haftmann@27651
  1068
haftmann@27651
  1069
lemma sgn_0_0:
haftmann@27651
  1070
  "sgn a = 0 \<longleftrightarrow> a = 0"
haftmann@27651
  1071
  unfolding sgn_if by simp
haftmann@27651
  1072
haftmann@27651
  1073
lemma sgn_1_pos:
haftmann@27651
  1074
  "sgn a = 1 \<longleftrightarrow> a > 0"
haftmann@27651
  1075
  unfolding sgn_if by (simp add: neg_equal_zero)
haftmann@27651
  1076
haftmann@27651
  1077
lemma sgn_1_neg:
haftmann@27651
  1078
  "sgn a = - 1 \<longleftrightarrow> a < 0"
haftmann@27651
  1079
  unfolding sgn_if by (auto simp add: equal_neg_zero)
haftmann@27651
  1080
haftmann@27651
  1081
lemma sgn_times:
haftmann@27651
  1082
  "sgn (a * b) = sgn a * sgn b"
haftmann@27651
  1083
  by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1084
haftmann@25917
  1085
end
haftmann@25230
  1086
haftmann@25230
  1087
class ordered_field = field + ordered_idom
haftmann@25230
  1088
haftmann@26274
  1089
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1090
paulson@15234
  1091
lemmas mult_compare_simps =
paulson@15234
  1092
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1093
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1094
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1095
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1096
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1097
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1098
    mult_cancel_right mult_cancel_left
paulson@15234
  1099
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1100
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1101
haftmann@26274
  1102
-- {* FIXME continue localization here *}
paulson@14268
  1103
paulson@14268
  1104
lemma inverse_nonzero_iff_nonzero [simp]:
huffman@20496
  1105
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
haftmann@26274
  1106
by (force dest: inverse_zero_imp_zero) 
paulson@14268
  1107
paulson@14268
  1108
lemma inverse_minus_eq [simp]:
huffman@20496
  1109
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
paulson@14377
  1110
proof cases
paulson@14377
  1111
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
  1112
next
paulson@14377
  1113
  assume "a\<noteq>0" 
paulson@14377
  1114
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
  1115
qed
paulson@14268
  1116
paulson@14268
  1117
lemma inverse_eq_imp_eq:
huffman@20496
  1118
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
haftmann@21328
  1119
apply (cases "a=0 | b=0") 
paulson@14268
  1120
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
  1121
              simp add: eq_commute [of "0::'a"])
paulson@14268
  1122
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
  1123
done
paulson@14268
  1124
paulson@14268
  1125
lemma inverse_eq_iff_eq [simp]:
huffman@20496
  1126
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
huffman@20496
  1127
by (force dest!: inverse_eq_imp_eq)
paulson@14268
  1128
paulson@14270
  1129
lemma inverse_inverse_eq [simp]:
huffman@20496
  1130
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
paulson@14270
  1131
  proof cases
paulson@14270
  1132
    assume "a=0" thus ?thesis by simp
paulson@14270
  1133
  next
paulson@14270
  1134
    assume "a\<noteq>0" 
paulson@14270
  1135
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
  1136
  qed
paulson@14270
  1137
paulson@14270
  1138
text{*This version builds in division by zero while also re-orienting
paulson@14270
  1139
      the right-hand side.*}
paulson@14270
  1140
lemma inverse_mult_distrib [simp]:
paulson@14270
  1141
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
  1142
  proof cases
paulson@14270
  1143
    assume "a \<noteq> 0 & b \<noteq> 0" 
haftmann@22993
  1144
    thus ?thesis
haftmann@22993
  1145
      by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
  1146
  next
paulson@14270
  1147
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
haftmann@22993
  1148
    thus ?thesis
haftmann@22993
  1149
      by force
paulson@14270
  1150
  qed
paulson@14270
  1151
paulson@14270
  1152
text{*There is no slick version using division by zero.*}
paulson@14270
  1153
lemma inverse_add:
nipkow@23477
  1154
  "[|a \<noteq> 0;  b \<noteq> 0|]
nipkow@23477
  1155
   ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
huffman@20496
  1156
by (simp add: division_ring_inverse_add mult_ac)
paulson@14270
  1157
paulson@14365
  1158
lemma inverse_divide [simp]:
nipkow@23477
  1159
  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
nipkow@23477
  1160
by (simp add: divide_inverse mult_commute)
paulson@14365
  1161
wenzelm@23389
  1162
avigad@16775
  1163
subsection {* Calculations with fractions *}
avigad@16775
  1164
nipkow@23413
  1165
text{* There is a whole bunch of simp-rules just for class @{text
nipkow@23413
  1166
field} but none for class @{text field} and @{text nonzero_divides}
nipkow@23413
  1167
because the latter are covered by a simproc. *}
nipkow@23413
  1168
paulson@24427
  1169
lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]:
nipkow@23477
  1170
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
  1171
proof -
paulson@14277
  1172
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
nipkow@23482
  1173
    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
paulson@14277
  1174
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
  1175
    by (simp only: mult_ac)
paulson@14277
  1176
  also have "... =  a * inverse b"
paulson@14277
  1177
    by simp
paulson@14277
  1178
    finally show ?thesis 
paulson@14277
  1179
    by (simp add: divide_inverse)
paulson@14277
  1180
qed
paulson@14277
  1181
nipkow@23413
  1182
lemma mult_divide_mult_cancel_left:
nipkow@23477
  1183
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1184
apply (cases "b = 0")
nipkow@23413
  1185
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
paulson@14277
  1186
done
paulson@14277
  1187
paulson@24427
  1188
lemma nonzero_mult_divide_mult_cancel_right [noatp]:
nipkow@23477
  1189
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1190
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
paulson@14321
  1191
nipkow@23413
  1192
lemma mult_divide_mult_cancel_right:
nipkow@23477
  1193
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1194
apply (cases "b = 0")
nipkow@23413
  1195
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
paulson@14321
  1196
done
nipkow@23413
  1197
paulson@14284
  1198
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
nipkow@23477
  1199
by (simp add: divide_inverse)
paulson@14284
  1200
paulson@15234
  1201
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
  1202
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1203
paulson@14430
  1204
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
  1205
by (simp add: divide_inverse mult_ac)
paulson@14288
  1206
nipkow@23482
  1207
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
nipkow@23482
  1208
paulson@24286
  1209
lemma divide_divide_eq_right [simp,noatp]:
nipkow@23477
  1210
  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
  1211
by (simp add: divide_inverse mult_ac)
paulson@14288
  1212
paulson@24286
  1213
lemma divide_divide_eq_left [simp,noatp]:
nipkow@23477
  1214
  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
  1215
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1216
avigad@16775
  1217
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1218
    x / y + w / z = (x * z + w * y) / (y * z)"
nipkow@23477
  1219
apply (subgoal_tac "x / y = (x * z) / (y * z)")
nipkow@23477
  1220
apply (erule ssubst)
nipkow@23477
  1221
apply (subgoal_tac "w / z = (w * y) / (y * z)")
nipkow@23477
  1222
apply (erule ssubst)
nipkow@23477
  1223
apply (rule add_divide_distrib [THEN sym])
nipkow@23477
  1224
apply (subst mult_commute)
nipkow@23477
  1225
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
nipkow@23477
  1226
apply assumption
nipkow@23477
  1227
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
nipkow@23477
  1228
apply assumption
avigad@16775
  1229
done
paulson@14268
  1230
wenzelm@23389
  1231
paulson@15234
  1232
subsubsection{*Special Cancellation Simprules for Division*}
paulson@15234
  1233
paulson@24427
  1234
lemma mult_divide_mult_cancel_left_if[simp,noatp]:
nipkow@23477
  1235
fixes c :: "'a :: {field,division_by_zero}"
nipkow@23477
  1236
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
nipkow@23413
  1237
by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
  1238
paulson@24427
  1239
lemma nonzero_mult_divide_cancel_right[simp,noatp]:
nipkow@23413
  1240
  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
nipkow@23413
  1241
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
nipkow@23413
  1242
paulson@24427
  1243
lemma nonzero_mult_divide_cancel_left[simp,noatp]:
nipkow@23413
  1244
  "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
nipkow@23413
  1245
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
nipkow@23413
  1246
nipkow@23413
  1247
paulson@24427
  1248
lemma nonzero_divide_mult_cancel_right[simp,noatp]:
nipkow@23413
  1249
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
nipkow@23413
  1250
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
nipkow@23413
  1251
paulson@24427
  1252
lemma nonzero_divide_mult_cancel_left[simp,noatp]:
nipkow@23413
  1253
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
nipkow@23413
  1254
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
nipkow@23413
  1255
nipkow@23413
  1256
paulson@24427
  1257
lemma nonzero_mult_divide_mult_cancel_left2[simp,noatp]:
nipkow@23477
  1258
  "[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1259
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
nipkow@23413
  1260
paulson@24427
  1261
lemma nonzero_mult_divide_mult_cancel_right2[simp,noatp]:
nipkow@23477
  1262
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
nipkow@23413
  1263
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
nipkow@23413
  1264
paulson@15234
  1265
paulson@14293
  1266
subsection {* Division and Unary Minus *}
paulson@14293
  1267
paulson@14293
  1268
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
paulson@14293
  1269
by (simp add: divide_inverse minus_mult_left)
paulson@14293
  1270
paulson@14293
  1271
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
paulson@14293
  1272
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
paulson@14293
  1273
paulson@14293
  1274
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
  1275
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
  1276
paulson@14430
  1277
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
paulson@14430
  1278
by (simp add: divide_inverse minus_mult_left [symmetric])
paulson@14293
  1279
paulson@14293
  1280
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
paulson@14430
  1281
by (simp add: divide_inverse minus_mult_right [symmetric])
paulson@14430
  1282
paulson@14293
  1283
paulson@14293
  1284
text{*The effect is to extract signs from divisions*}
paulson@17085
  1285
lemmas divide_minus_left = minus_divide_left [symmetric]
paulson@17085
  1286
lemmas divide_minus_right = minus_divide_right [symmetric]
paulson@17085
  1287
declare divide_minus_left [simp]   divide_minus_right [simp]
paulson@14293
  1288
paulson@14387
  1289
text{*Also, extract signs from products*}
paulson@17085
  1290
lemmas mult_minus_left = minus_mult_left [symmetric]
paulson@17085
  1291
lemmas mult_minus_right = minus_mult_right [symmetric]
paulson@17085
  1292
declare mult_minus_left [simp]   mult_minus_right [simp]
paulson@14387
  1293
paulson@14293
  1294
lemma minus_divide_divide [simp]:
nipkow@23477
  1295
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1296
apply (cases "b=0", simp) 
paulson@14293
  1297
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
  1298
done
paulson@14293
  1299
paulson@14430
  1300
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
  1301
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
  1302
nipkow@23482
  1303
lemma add_divide_eq_iff:
nipkow@23482
  1304
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
nipkow@23482
  1305
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1306
nipkow@23482
  1307
lemma divide_add_eq_iff:
nipkow@23482
  1308
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
nipkow@23482
  1309
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1310
nipkow@23482
  1311
lemma diff_divide_eq_iff:
nipkow@23482
  1312
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
nipkow@23482
  1313
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1314
nipkow@23482
  1315
lemma divide_diff_eq_iff:
nipkow@23482
  1316
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
nipkow@23482
  1317
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1318
nipkow@23482
  1319
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
nipkow@23482
  1320
proof -
nipkow@23482
  1321
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1322
  have "(a = b/c) = (a*c = (b/c)*c)" by simp
nipkow@23496
  1323
  also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
nipkow@23482
  1324
  finally show ?thesis .
nipkow@23482
  1325
qed
nipkow@23482
  1326
nipkow@23482
  1327
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
nipkow@23482
  1328
proof -
nipkow@23482
  1329
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1330
  have "(b/c = a) = ((b/c)*c = a*c)"  by simp
nipkow@23496
  1331
  also have "... = (b = a*c)"  by (simp add: divide_inverse mult_assoc) 
nipkow@23482
  1332
  finally show ?thesis .
nipkow@23482
  1333
qed
nipkow@23482
  1334
nipkow@23482
  1335
lemma eq_divide_eq:
nipkow@23482
  1336
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
nipkow@23482
  1337
by (simp add: nonzero_eq_divide_eq) 
nipkow@23482
  1338
nipkow@23482
  1339
lemma divide_eq_eq:
nipkow@23482
  1340
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
nipkow@23482
  1341
by (force simp add: nonzero_divide_eq_eq) 
nipkow@23482
  1342
nipkow@23482
  1343
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1344
    b = a * c ==> b / c = a"
nipkow@23482
  1345
  by (subst divide_eq_eq, simp)
nipkow@23482
  1346
nipkow@23482
  1347
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1348
    a * c = b ==> a = b / c"
nipkow@23482
  1349
  by (subst eq_divide_eq, simp)
nipkow@23482
  1350
nipkow@23482
  1351
nipkow@23482
  1352
lemmas field_eq_simps = ring_simps
nipkow@23482
  1353
  (* pull / out*)
nipkow@23482
  1354
  add_divide_eq_iff divide_add_eq_iff
nipkow@23482
  1355
  diff_divide_eq_iff divide_diff_eq_iff
nipkow@23482
  1356
  (* multiply eqn *)
nipkow@23482
  1357
  nonzero_eq_divide_eq nonzero_divide_eq_eq
nipkow@23482
  1358
(* is added later:
nipkow@23482
  1359
  times_divide_eq_left times_divide_eq_right
nipkow@23482
  1360
*)
nipkow@23482
  1361
nipkow@23482
  1362
text{*An example:*}
nipkow@23482
  1363
lemma fixes a b c d e f :: "'a::field"
nipkow@23482
  1364
shows "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f \<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1"
nipkow@23482
  1365
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
nipkow@23482
  1366
 apply(simp add:field_eq_simps)
nipkow@23482
  1367
apply(simp)
nipkow@23482
  1368
done
nipkow@23482
  1369
nipkow@23482
  1370
avigad@16775
  1371
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1372
    x / y - w / z = (x * z - w * y) / (y * z)"
nipkow@23482
  1373
by (simp add:field_eq_simps times_divide_eq)
nipkow@23482
  1374
nipkow@23482
  1375
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
nipkow@23482
  1376
    (x / y = w / z) = (x * z = w * y)"
nipkow@23482
  1377
by (simp add:field_eq_simps times_divide_eq)
paulson@14293
  1378
wenzelm@23389
  1379
paulson@14268
  1380
subsection {* Ordered Fields *}
paulson@14268
  1381
paulson@14277
  1382
lemma positive_imp_inverse_positive: 
nipkow@23482
  1383
assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
nipkow@23482
  1384
proof -
paulson@14268
  1385
  have "0 < a * inverse a" 
paulson@14268
  1386
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
  1387
  thus "0 < inverse a" 
paulson@14268
  1388
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
nipkow@23482
  1389
qed
paulson@14268
  1390
paulson@14277
  1391
lemma negative_imp_inverse_negative:
nipkow@23482
  1392
  "a < 0 ==> inverse a < (0::'a::ordered_field)"
nipkow@23482
  1393
by (insert positive_imp_inverse_positive [of "-a"], 
nipkow@23482
  1394
    simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
paulson@14268
  1395
paulson@14268
  1396
lemma inverse_le_imp_le:
nipkow@23482
  1397
assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
nipkow@23482
  1398
shows "b \<le> (a::'a::ordered_field)"
nipkow@23482
  1399
proof (rule classical)
paulson@14268
  1400
  assume "~ b \<le> a"
nipkow@23482
  1401
  hence "a < b"  by (simp add: linorder_not_le)
nipkow@23482
  1402
  hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
paulson@14268
  1403
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
  1404
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
  1405
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
  1406
    by (simp add: bpos order_less_imp_le mult_right_mono)
nipkow@23482
  1407
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
nipkow@23482
  1408
qed
paulson@14268
  1409
paulson@14277
  1410
lemma inverse_positive_imp_positive:
nipkow@23482
  1411
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
nipkow@23482
  1412
shows "0 < (a::'a::ordered_field)"
wenzelm@23389
  1413
proof -
paulson@14277
  1414
  have "0 < inverse (inverse a)"
wenzelm@23389
  1415
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
  1416
  thus "0 < a"
wenzelm@23389
  1417
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1418
qed
paulson@14277
  1419
paulson@14277
  1420
lemma inverse_positive_iff_positive [simp]:
nipkow@23482
  1421
  "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1422
apply (cases "a = 0", simp)
paulson@14277
  1423
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
  1424
done
paulson@14277
  1425
paulson@14277
  1426
lemma inverse_negative_imp_negative:
nipkow@23482
  1427
assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
nipkow@23482
  1428
shows "a < (0::'a::ordered_field)"
wenzelm@23389
  1429
proof -
paulson@14277
  1430
  have "inverse (inverse a) < 0"
wenzelm@23389
  1431
    using inv_less_0 by (rule negative_imp_inverse_negative)
nipkow@23482
  1432
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1433
qed
paulson@14277
  1434
paulson@14277
  1435
lemma inverse_negative_iff_negative [simp]:
nipkow@23482
  1436
  "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1437
apply (cases "a = 0", simp)
paulson@14277
  1438
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
  1439
done
paulson@14277
  1440
paulson@14277
  1441
lemma inverse_nonnegative_iff_nonnegative [simp]:
nipkow@23482
  1442
  "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1443
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1444
paulson@14277
  1445
lemma inverse_nonpositive_iff_nonpositive [simp]:
nipkow@23482
  1446
  "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1447
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1448
chaieb@23406
  1449
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
chaieb@23406
  1450
proof
chaieb@23406
  1451
  fix x::'a
chaieb@23406
  1452
  have m1: "- (1::'a) < 0" by simp
chaieb@23406
  1453
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1454
  have "(- 1) + x < x" by simp
chaieb@23406
  1455
  thus "\<exists>y. y < x" by blast
chaieb@23406
  1456
qed
chaieb@23406
  1457
chaieb@23406
  1458
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
chaieb@23406
  1459
proof
chaieb@23406
  1460
  fix x::'a
chaieb@23406
  1461
  have m1: " (1::'a) > 0" by simp
chaieb@23406
  1462
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1463
  have "1 + x > x" by simp
chaieb@23406
  1464
  thus "\<exists>y. y > x" by blast
chaieb@23406
  1465
qed
paulson@14277
  1466
paulson@14277
  1467
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
  1468
paulson@14268
  1469
lemma less_imp_inverse_less:
nipkow@23482
  1470
assumes less: "a < b" and apos:  "0 < a"
nipkow@23482
  1471
shows "inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1472
proof (rule ccontr)
paulson@14268
  1473
  assume "~ inverse b < inverse a"
paulson@14268
  1474
  hence "inverse a \<le> inverse b"
paulson@14268
  1475
    by (simp add: linorder_not_less)
paulson@14268
  1476
  hence "~ (a < b)"
paulson@14268
  1477
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
  1478
  thus False
paulson@14268
  1479
    by (rule notE [OF _ less])
nipkow@23482
  1480
qed
paulson@14268
  1481
paulson@14268
  1482
lemma inverse_less_imp_less:
nipkow@23482
  1483
  "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1484
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
  1485
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
  1486
done
paulson@14268
  1487
paulson@14268
  1488
text{*Both premises are essential. Consider -1 and 1.*}
paulson@24286
  1489
lemma inverse_less_iff_less [simp,noatp]:
nipkow@23482
  1490
  "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1491
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
  1492
paulson@14268
  1493
lemma le_imp_inverse_le:
nipkow@23482
  1494
  "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1495
by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
  1496
paulson@24286
  1497
lemma inverse_le_iff_le [simp,noatp]:
nipkow@23482
  1498
 "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1499
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
  1500
paulson@14268
  1501
paulson@14268
  1502
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
  1503
case is trivial, since inverse preserves signs.*}
paulson@14268
  1504
lemma inverse_le_imp_le_neg:
nipkow@23482
  1505
  "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
nipkow@23482
  1506
apply (rule classical) 
nipkow@23482
  1507
apply (subgoal_tac "a < 0") 
nipkow@23482
  1508
 prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
nipkow@23482
  1509
apply (insert inverse_le_imp_le [of "-b" "-a"])
nipkow@23482
  1510
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1511
done
paulson@14268
  1512
paulson@14268
  1513
lemma less_imp_inverse_less_neg:
paulson@14268
  1514
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1515
apply (subgoal_tac "a < 0") 
nipkow@23482
  1516
 prefer 2 apply (blast intro: order_less_trans) 
nipkow@23482
  1517
apply (insert less_imp_inverse_less [of "-b" "-a"])
nipkow@23482
  1518
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1519
done
paulson@14268
  1520
paulson@14268
  1521
lemma inverse_less_imp_less_neg:
paulson@14268
  1522
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
nipkow@23482
  1523
apply (rule classical) 
nipkow@23482
  1524
apply (subgoal_tac "a < 0") 
nipkow@23482
  1525
 prefer 2
nipkow@23482
  1526
 apply (force simp add: linorder_not_less intro: order_le_less_trans) 
nipkow@23482
  1527
apply (insert inverse_less_imp_less [of "-b" "-a"])
nipkow@23482
  1528
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1529
done
paulson@14268
  1530
paulson@24286
  1531
lemma inverse_less_iff_less_neg [simp,noatp]:
nipkow@23482
  1532
  "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
nipkow@23482
  1533
apply (insert inverse_less_iff_less [of "-b" "-a"])
nipkow@23482
  1534
apply (simp del: inverse_less_iff_less 
nipkow@23482
  1535
            add: order_less_imp_not_eq nonzero_inverse_minus_eq)
nipkow@23482
  1536
done
paulson@14268
  1537
paulson@14268
  1538
lemma le_imp_inverse_le_neg:
nipkow@23482
  1539
  "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1540
by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1541
paulson@24286
  1542
lemma inverse_le_iff_le_neg [simp,noatp]:
nipkow@23482
  1543
 "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1544
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1545
paulson@14277
  1546
paulson@14365
  1547
subsection{*Inverses and the Number One*}
paulson@14365
  1548
paulson@14365
  1549
lemma one_less_inverse_iff:
nipkow@23482
  1550
  "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
nipkow@23482
  1551
proof cases
paulson@14365
  1552
  assume "0 < x"
paulson@14365
  1553
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1554
    show ?thesis by simp
paulson@14365
  1555
next
paulson@14365
  1556
  assume notless: "~ (0 < x)"
paulson@14365
  1557
  have "~ (1 < inverse x)"
paulson@14365
  1558
  proof
paulson@14365
  1559
    assume "1 < inverse x"
paulson@14365
  1560
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1561
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1562
    finally show False by auto
paulson@14365
  1563
  qed
paulson@14365
  1564
  with notless show ?thesis by simp
paulson@14365
  1565
qed
paulson@14365
  1566
paulson@14365
  1567
lemma inverse_eq_1_iff [simp]:
nipkow@23482
  1568
  "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1569
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1570
paulson@14365
  1571
lemma one_le_inverse_iff:
nipkow@23482
  1572
  "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1573
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1574
                    eq_commute [of 1]) 
paulson@14365
  1575
paulson@14365
  1576
lemma inverse_less_1_iff:
nipkow@23482
  1577
  "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1578
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1579
paulson@14365
  1580
lemma inverse_le_1_iff:
nipkow@23482
  1581
  "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1582
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1583
wenzelm@23389
  1584
paulson@14288
  1585
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1586
paulson@14288
  1587
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1588
proof -
paulson@14288
  1589
  assume less: "0<c"
paulson@14288
  1590
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1591
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1592
  also have "... = (a*c \<le> b)"
paulson@14288
  1593
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1594
  finally show ?thesis .
paulson@14288
  1595
qed
paulson@14288
  1596
paulson@14288
  1597
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1598
proof -
paulson@14288
  1599
  assume less: "c<0"
paulson@14288
  1600
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1601
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1602
  also have "... = (b \<le> a*c)"
paulson@14288
  1603
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1604
  finally show ?thesis .
paulson@14288
  1605
qed
paulson@14288
  1606
paulson@14288
  1607
lemma le_divide_eq:
paulson@14288
  1608
  "(a \<le> b/c) = 
paulson@14288
  1609
   (if 0 < c then a*c \<le> b
paulson@14288
  1610
             else if c < 0 then b \<le> a*c
paulson@14288
  1611
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1612
apply (cases "c=0", simp) 
paulson@14288
  1613
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1614
done
paulson@14288
  1615
paulson@14288
  1616
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1617
proof -
paulson@14288
  1618
  assume less: "0<c"
paulson@14288
  1619
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1620
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1621
  also have "... = (b \<le> a*c)"
paulson@14288
  1622
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1623
  finally show ?thesis .
paulson@14288
  1624
qed
paulson@14288
  1625
paulson@14288
  1626
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1627
proof -
paulson@14288
  1628
  assume less: "c<0"
paulson@14288
  1629
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1630
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1631
  also have "... = (a*c \<le> b)"
paulson@14288
  1632
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1633
  finally show ?thesis .
paulson@14288
  1634
qed
paulson@14288
  1635
paulson@14288
  1636
lemma divide_le_eq:
paulson@14288
  1637
  "(b/c \<le> a) = 
paulson@14288
  1638
   (if 0 < c then b \<le> a*c
paulson@14288
  1639
             else if c < 0 then a*c \<le> b
paulson@14288
  1640
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1641
apply (cases "c=0", simp) 
paulson@14288
  1642
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1643
done
paulson@14288
  1644
paulson@14288
  1645
lemma pos_less_divide_eq:
paulson@14288
  1646
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1647
proof -
paulson@14288
  1648
  assume less: "0<c"
paulson@14288
  1649
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@15234
  1650
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1651
  also have "... = (a*c < b)"
paulson@14288
  1652
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1653
  finally show ?thesis .
paulson@14288
  1654
qed
paulson@14288
  1655
paulson@14288
  1656
lemma neg_less_divide_eq:
paulson@14288
  1657
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1658
proof -
paulson@14288
  1659
  assume less: "c<0"
paulson@14288
  1660
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@15234
  1661
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1662
  also have "... = (b < a*c)"
paulson@14288
  1663
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1664
  finally show ?thesis .
paulson@14288
  1665
qed
paulson@14288
  1666
paulson@14288
  1667
lemma less_divide_eq:
paulson@14288
  1668
  "(a < b/c) = 
paulson@14288
  1669
   (if 0 < c then a*c < b
paulson@14288
  1670
             else if c < 0 then b < a*c
paulson@14288
  1671
             else  a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1672
apply (cases "c=0", simp) 
paulson@14288
  1673
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1674
done
paulson@14288
  1675
paulson@14288
  1676
lemma pos_divide_less_eq:
paulson@14288
  1677
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1678
proof -
paulson@14288
  1679
  assume less: "0<c"
paulson@14288
  1680
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@15234
  1681
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1682
  also have "... = (b < a*c)"
paulson@14288
  1683
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1684
  finally show ?thesis .
paulson@14288
  1685
qed
paulson@14288
  1686
paulson@14288
  1687
lemma neg_divide_less_eq:
paulson@14288
  1688
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1689
proof -
paulson@14288
  1690
  assume less: "c<0"
paulson@14288
  1691
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@15234
  1692
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1693
  also have "... = (a*c < b)"
paulson@14288
  1694
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1695
  finally show ?thesis .
paulson@14288
  1696
qed
paulson@14288
  1697
paulson@14288
  1698
lemma divide_less_eq:
paulson@14288
  1699
  "(b/c < a) = 
paulson@14288
  1700
   (if 0 < c then b < a*c
paulson@14288
  1701
             else if c < 0 then a*c < b
paulson@14288
  1702
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1703
apply (cases "c=0", simp) 
paulson@14288
  1704
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1705
done
paulson@14288
  1706
nipkow@23482
  1707
nipkow@23482
  1708
subsection{*Field simplification*}
nipkow@23482
  1709
nipkow@23482
  1710
text{* Lemmas @{text field_simps} multiply with denominators in
nipkow@23482
  1711
in(equations) if they can be proved to be non-zero (for equations) or
nipkow@23482
  1712
positive/negative (for inequations). *}
paulson@14288
  1713
nipkow@23482
  1714
lemmas field_simps = field_eq_simps
nipkow@23482
  1715
  (* multiply ineqn *)
nipkow@23482
  1716
  pos_divide_less_eq neg_divide_less_eq
nipkow@23482
  1717
  pos_less_divide_eq neg_less_divide_eq
nipkow@23482
  1718
  pos_divide_le_eq neg_divide_le_eq
nipkow@23482
  1719
  pos_le_divide_eq neg_le_divide_eq
paulson@14288
  1720
nipkow@23482
  1721
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
nipkow@23483
  1722
of positivity/negativity needed for @{text field_simps}. Have not added @{text
nipkow@23482
  1723
sign_simps} to @{text field_simps} because the former can lead to case
nipkow@23482
  1724
explosions. *}
paulson@14288
  1725
nipkow@23482
  1726
lemmas sign_simps = group_simps
nipkow@23482
  1727
  zero_less_mult_iff  mult_less_0_iff
paulson@14288
  1728
nipkow@23482
  1729
(* Only works once linear arithmetic is installed:
nipkow@23482
  1730
text{*An example:*}
nipkow@23482
  1731
lemma fixes a b c d e f :: "'a::ordered_field"
nipkow@23482
  1732
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
nipkow@23482
  1733
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
nipkow@23482
  1734
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
nipkow@23482
  1735
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
nipkow@23482
  1736
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1737
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
nipkow@23482
  1738
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1739
apply(simp add:field_simps)
avigad@16775
  1740
done
nipkow@23482
  1741
*)
avigad@16775
  1742
wenzelm@23389
  1743
avigad@16775
  1744
subsection{*Division and Signs*}
avigad@16775
  1745
avigad@16775
  1746
lemma zero_less_divide_iff:
avigad@16775
  1747
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
  1748
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
  1749
avigad@16775
  1750
lemma divide_less_0_iff:
avigad@16775
  1751
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
avigad@16775
  1752
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
  1753
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
  1754
avigad@16775
  1755
lemma zero_le_divide_iff:
avigad@16775
  1756
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
avigad@16775
  1757
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
  1758
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
  1759
avigad@16775
  1760
lemma divide_le_0_iff:
avigad@16775
  1761
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
avigad@16775
  1762
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
  1763
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
  1764
paulson@24286
  1765
lemma divide_eq_0_iff [simp,noatp]:
avigad@16775
  1766
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
nipkow@23482
  1767
by (simp add: divide_inverse)
avigad@16775
  1768
nipkow@23482
  1769
lemma divide_pos_pos:
nipkow@23482
  1770
  "0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
nipkow@23482
  1771
by(simp add:field_simps)
nipkow@23482
  1772
avigad@16775
  1773
nipkow@23482
  1774
lemma divide_nonneg_pos:
nipkow@23482
  1775
  "0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
nipkow@23482
  1776
by(simp add:field_simps)
avigad@16775
  1777
nipkow@23482
  1778
lemma divide_neg_pos:
nipkow@23482
  1779
  "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
nipkow@23482
  1780
by(simp add:field_simps)
avigad@16775
  1781
nipkow@23482
  1782
lemma divide_nonpos_pos:
nipkow@23482
  1783
  "(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
nipkow@23482
  1784
by(simp add:field_simps)
avigad@16775
  1785
nipkow@23482
  1786
lemma divide_pos_neg:
nipkow@23482
  1787
  "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
nipkow@23482
  1788
by(simp add:field_simps)
avigad@16775
  1789
nipkow@23482
  1790
lemma divide_nonneg_neg:
nipkow@23482
  1791
  "0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" 
nipkow@23482
  1792
by(simp add:field_simps)
avigad@16775
  1793
nipkow@23482
  1794
lemma divide_neg_neg:
nipkow@23482
  1795
  "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
nipkow@23482
  1796
by(simp add:field_simps)
avigad@16775
  1797
nipkow@23482
  1798
lemma divide_nonpos_neg:
nipkow@23482
  1799
  "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
nipkow@23482
  1800
by(simp add:field_simps)
paulson@15234
  1801
wenzelm@23389
  1802
paulson@14288
  1803
subsection{*Cancellation Laws for Division*}
paulson@14288
  1804
paulson@24286
  1805
lemma divide_cancel_right [simp,noatp]:
paulson@14288
  1806
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1807
apply (cases "c=0", simp)
nipkow@23496
  1808
apply (simp add: divide_inverse)
paulson@14288
  1809
done
paulson@14288
  1810
paulson@24286
  1811
lemma divide_cancel_left [simp,noatp]:
paulson@14288
  1812
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
nipkow@23482
  1813
apply (cases "c=0", simp)
nipkow@23496
  1814
apply (simp add: divide_inverse)
paulson@14288
  1815
done
paulson@14288
  1816
wenzelm@23389
  1817
paulson@14353
  1818
subsection {* Division and the Number One *}
paulson@14353
  1819
paulson@14353
  1820
text{*Simplify expressions equated with 1*}
paulson@24286
  1821
lemma divide_eq_1_iff [simp,noatp]:
paulson@14353
  1822
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1823
apply (cases "b=0", simp)
nipkow@23482
  1824
apply (simp add: right_inverse_eq)
paulson@14353
  1825
done
paulson@14353
  1826
paulson@24286
  1827
lemma one_eq_divide_iff [simp,noatp]:
paulson@14353
  1828
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1829
by (simp add: eq_commute [of 1])
paulson@14353
  1830
paulson@24286
  1831
lemma zero_eq_1_divide_iff [simp,noatp]:
paulson@14353
  1832
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
nipkow@23482
  1833
apply (cases "a=0", simp)
nipkow@23482
  1834
apply (auto simp add: nonzero_eq_divide_eq)
paulson@14353
  1835
done
paulson@14353
  1836
paulson@24286
  1837
lemma one_divide_eq_0_iff [simp,noatp]:
paulson@14353
  1838
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
nipkow@23482
  1839
apply (cases "a=0", simp)
nipkow@23482
  1840
apply (insert zero_neq_one [THEN not_sym])
nipkow@23482
  1841
apply (auto simp add: nonzero_divide_eq_eq)
paulson@14353
  1842
done
paulson@14353
  1843
paulson@14353
  1844
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@18623
  1845
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
paulson@18623
  1846
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
paulson@18623
  1847
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
paulson@18623
  1848
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
paulson@17085
  1849
paulson@17085
  1850
declare zero_less_divide_1_iff [simp]
paulson@24286
  1851
declare divide_less_0_1_iff [simp,noatp]
paulson@17085
  1852
declare zero_le_divide_1_iff [simp]
paulson@24286
  1853
declare divide_le_0_1_iff [simp,noatp]
paulson@14353
  1854
wenzelm@23389
  1855
paulson@14293
  1856
subsection {* Ordering Rules for Division *}
paulson@14293
  1857
paulson@14293
  1858
lemma divide_strict_right_mono:
paulson@14293
  1859
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1860
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
nipkow@23482
  1861
              positive_imp_inverse_positive)
paulson@14293
  1862
paulson@14293
  1863
lemma divide_right_mono:
paulson@14293
  1864
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
nipkow@23482
  1865
by (force simp add: divide_strict_right_mono order_le_less)
paulson@14293
  1866
avigad@16775
  1867
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1868
    ==> c <= 0 ==> b / c <= a / c"
nipkow@23482
  1869
apply (drule divide_right_mono [of _ _ "- c"])
nipkow@23482
  1870
apply auto
avigad@16775
  1871
done
avigad@16775
  1872
avigad@16775
  1873
lemma divide_strict_right_mono_neg:
avigad@16775
  1874
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
nipkow@23482
  1875
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
nipkow@23482
  1876
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
avigad@16775
  1877
done
paulson@14293
  1878
paulson@14293
  1879
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1880
      have the same sign*}
paulson@14293
  1881
lemma divide_strict_left_mono:
nipkow@23482
  1882
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1883
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
paulson@14293
  1884
paulson@14293
  1885
lemma divide_left_mono:
nipkow@23482
  1886
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
nipkow@23482
  1887
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
paulson@14293
  1888
avigad@16775
  1889
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1890
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1891
  apply (drule divide_left_mono [of _ _ "- c"])
avigad@16775
  1892
  apply (auto simp add: mult_commute)
avigad@16775
  1893
done
avigad@16775
  1894
paulson@14293
  1895
lemma divide_strict_left_mono_neg:
nipkow@23482
  1896
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1897
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
nipkow@23482
  1898
paulson@14293
  1899
avigad@16775
  1900
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1901
paulson@24286
  1902
lemma le_divide_eq_1 [noatp]:
avigad@16775
  1903
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1904
  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1905
by (auto simp add: le_divide_eq)
avigad@16775
  1906
paulson@24286
  1907
lemma divide_le_eq_1 [noatp]:
avigad@16775
  1908
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1909
  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1910
by (auto simp add: divide_le_eq)
avigad@16775
  1911
paulson@24286
  1912
lemma less_divide_eq_1 [noatp]:
avigad@16775
  1913
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1914
  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1915
by (auto simp add: less_divide_eq)
avigad@16775
  1916
paulson@24286
  1917
lemma divide_less_eq_1 [noatp]:
avigad@16775
  1918
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1919
  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1920
by (auto simp add: divide_less_eq)
avigad@16775
  1921
wenzelm@23389
  1922
avigad@16775
  1923
subsection{*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1924
paulson@24286
  1925
lemma le_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1926
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1927
  shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1928
by (auto simp add: le_divide_eq)
avigad@16775
  1929
paulson@24286
  1930
lemma le_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1931
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1932
  shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1933
by (auto simp add: le_divide_eq)
avigad@16775
  1934
paulson@24286
  1935
lemma divide_le_eq_1_pos [simp,noatp]:
avigad@16775
  1936
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1937
  shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1938
by (auto simp add: divide_le_eq)
avigad@16775
  1939
paulson@24286
  1940
lemma divide_le_eq_1_neg [simp,noatp]:
avigad@16775
  1941
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1942
  shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1943
by (auto simp add: divide_le_eq)
avigad@16775
  1944
paulson@24286
  1945
lemma less_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1946
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1947
  shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1948
by (auto simp add: less_divide_eq)
avigad@16775
  1949
paulson@24286
  1950
lemma less_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1951
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1952
  shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1953
by (auto simp add: less_divide_eq)
avigad@16775
  1954
paulson@24286
  1955
lemma divide_less_eq_1_pos [simp,noatp]:
avigad@16775
  1956
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1957
  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1958
by (auto simp add: divide_less_eq)
paulson@18649
  1959
paulson@24286
  1960
lemma divide_less_eq_1_neg [simp,noatp]:
paulson@18649
  1961
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1962
  shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1963
by (auto simp add: divide_less_eq)
avigad@16775
  1964
paulson@24286
  1965
lemma eq_divide_eq_1 [simp,noatp]:
avigad@16775
  1966
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1967
  shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1968
by (auto simp add: eq_divide_eq)
avigad@16775
  1969
paulson@24286
  1970
lemma divide_eq_eq_1 [simp,noatp]:
avigad@16775
  1971
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1972
  shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1973
by (auto simp add: divide_eq_eq)
avigad@16775
  1974
wenzelm@23389
  1975
avigad@16775
  1976
subsection {* Reasoning about inequalities with division *}
avigad@16775
  1977
avigad@16775
  1978
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1979
    ==> x * y <= x"
avigad@16775
  1980
  by (auto simp add: mult_compare_simps);
avigad@16775
  1981
avigad@16775
  1982
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1983
    ==> y * x <= x"
avigad@16775
  1984
  by (auto simp add: mult_compare_simps);
avigad@16775
  1985
avigad@16775
  1986
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
avigad@16775
  1987
    x / y <= z";
avigad@16775
  1988
  by (subst pos_divide_le_eq, assumption+);
avigad@16775
  1989
avigad@16775
  1990
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
nipkow@23482
  1991
    z <= x / y"
nipkow@23482
  1992
by(simp add:field_simps)
avigad@16775
  1993
avigad@16775
  1994
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
avigad@16775
  1995
    x / y < z"
nipkow@23482
  1996
by(simp add:field_simps)
avigad@16775
  1997
avigad@16775
  1998
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
avigad@16775
  1999
    z < x / y"
nipkow@23482
  2000
by(simp add:field_simps)
avigad@16775
  2001
avigad@16775
  2002
lemma frac_le: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  2003
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
avigad@16775
  2004
  apply (rule mult_imp_div_pos_le)
haftmann@25230
  2005
  apply simp
haftmann@25230
  2006
  apply (subst times_divide_eq_left)
avigad@16775
  2007
  apply (rule mult_imp_le_div_pos, assumption)
avigad@16775
  2008
  apply (rule mult_mono)
avigad@16775
  2009
  apply simp_all
paulson@14293
  2010
done
paulson@14293
  2011
avigad@16775
  2012
lemma frac_less: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  2013
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
avigad@16775
  2014
  apply (rule mult_imp_div_pos_less)
avigad@16775
  2015
  apply simp;
avigad@16775
  2016
  apply (subst times_divide_eq_left);
avigad@16775
  2017
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  2018
  apply (erule mult_less_le_imp_less)
avigad@16775
  2019
  apply simp_all
avigad@16775
  2020
done
avigad@16775
  2021
avigad@16775
  2022
lemma frac_less2: "(0::'a::ordered_field) < x ==> 
avigad@16775
  2023
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
avigad@16775
  2024
  apply (rule mult_imp_div_pos_less)
avigad@16775
  2025
  apply simp_all
avigad@16775
  2026
  apply (subst times_divide_eq_left);
avigad@16775
  2027
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  2028
  apply (erule mult_le_less_imp_less)
avigad@16775
  2029
  apply simp_all
avigad@16775
  2030
done
avigad@16775
  2031
avigad@16775
  2032
text{*It's not obvious whether these should be simprules or not. 
avigad@16775
  2033
  Their effect is to gather terms into one big fraction, like
avigad@16775
  2034
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
avigad@16775
  2035
  seem to need them.*}
avigad@16775
  2036
avigad@16775
  2037
declare times_divide_eq [simp]
paulson@14293
  2038
wenzelm@23389
  2039
paulson@14293
  2040
subsection {* Ordered Fields are Dense *}
paulson@14293
  2041
haftmann@25193
  2042
context ordered_semidom
haftmann@25193
  2043
begin
haftmann@25193
  2044
haftmann@25193
  2045
lemma less_add_one: "a < a + 1"
paulson@14293
  2046
proof -
haftmann@25193
  2047
  have "a + 0 < a + 1"
nipkow@23482
  2048
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  2049
  thus ?thesis by simp
paulson@14293
  2050
qed
paulson@14293
  2051
haftmann@25193
  2052
lemma zero_less_two: "0 < 1 + 1"
haftmann@25193
  2053
  by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  2054
haftmann@25193
  2055
end
paulson@14365
  2056
paulson@14293
  2057
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
nipkow@23482
  2058
by (simp add: field_simps zero_less_two)
paulson@14293
  2059
paulson@14293
  2060
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
nipkow@23482
  2061
by (simp add: field_simps zero_less_two)
paulson@14293
  2062
haftmann@24422
  2063
instance ordered_field < dense_linear_order
haftmann@24422
  2064
proof
haftmann@24422
  2065
  fix x y :: 'a
haftmann@24422
  2066
  have "x < x + 1" by simp
haftmann@24422
  2067
  then show "\<exists>y. x < y" .. 
haftmann@24422
  2068
  have "x - 1 < x" by simp
haftmann@24422
  2069
  then show "\<exists>y. y < x" ..
haftmann@24422
  2070
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
haftmann@24422
  2071
qed
paulson@14293
  2072
paulson@15234
  2073
paulson@14293
  2074
subsection {* Absolute Value *}
paulson@14293
  2075
haftmann@25304
  2076
context ordered_idom
haftmann@25304
  2077
begin
haftmann@25304
  2078
haftmann@25304
  2079
lemma mult_sgn_abs: "sgn x * abs x = x"
haftmann@25304
  2080
  unfolding abs_if sgn_if by auto
haftmann@25304
  2081
haftmann@25304
  2082
end
nipkow@24491
  2083
obua@14738
  2084
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
haftmann@25304
  2085
  by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
haftmann@25304
  2086
haftmann@25304
  2087
class pordered_ring_abs = pordered_ring + pordered_ab_group_add_abs +
haftmann@25304
  2088
  assumes abs_eq_mult:
haftmann@25304
  2089
    "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0) \<Longrightarrow> \<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
haftmann@25304
  2090
haftmann@25304
  2091
haftmann@25304
  2092
class lordered_ring = pordered_ring + lordered_ab_group_add_abs
haftmann@25304
  2093
begin
haftmann@25304
  2094
huffman@27516
  2095
subclass lordered_ab_group_add_meet ..
huffman@27516
  2096
subclass lordered_ab_group_add_join ..
haftmann@25304
  2097
haftmann@25304
  2098
end
paulson@14294
  2099
obua@14738
  2100
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  2101
proof -
obua@14738
  2102
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  2103
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  2104
  have a: "(abs a) * (abs b) = ?x"
nipkow@23477
  2105
    by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
obua@14738
  2106
  {
obua@14738
  2107
    fix u v :: 'a
paulson@15481
  2108
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
paulson@15481
  2109
              u * v = pprt a * pprt b + pprt a * nprt b + 
paulson@15481
  2110
                      nprt a * pprt b + nprt a * nprt b"
obua@14738
  2111
      apply (subst prts[of u], subst prts[of v])
nipkow@23477
  2112
      apply (simp add: ring_simps) 
obua@14738
  2113
      done
obua@14738
  2114
  }
obua@14738
  2115
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  2116
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  2117
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  2118
  have xy: "- ?x <= ?y"
obua@14754
  2119
    apply (simp)
obua@14754
  2120
    apply (rule_tac y="0::'a" in order_trans)
nipkow@16568
  2121
    apply (rule addm2)
avigad@16775
  2122
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
nipkow@16568
  2123
    apply (rule addm)
avigad@16775
  2124
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14754
  2125
    done
obua@14738
  2126
  have yx: "?y <= ?x"
nipkow@16568
  2127
    apply (simp add:diff_def)
obua@14754
  2128
    apply (rule_tac y=0 in order_trans)
avigad@16775
  2129
    apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
avigad@16775
  2130
    apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
obua@14738
  2131
    done
obua@14738
  2132
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  2133
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  2134
  show ?thesis
obua@14738
  2135
    apply (rule abs_leI)
obua@14738
  2136
    apply (simp add: i1)
obua@14738
  2137
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  2138
    done
obua@14738
  2139
qed
paulson@14294
  2140
haftmann@25304
  2141
instance lordered_ring \<subseteq> pordered_ring_abs
haftmann@25304
  2142
proof
haftmann@25304
  2143
  fix a b :: "'a\<Colon> lordered_ring"
haftmann@25304
  2144
  assume "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
haftmann@25304
  2145
  show "abs (a*b) = abs a * abs b"
obua@14738
  2146
proof -
obua@14738
  2147
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  2148
    apply (auto)    
obua@14738
  2149
    apply (rule_tac split_mult_pos_le)
obua@14738
  2150
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  2151
    apply (simp)
obua@14738
  2152
    apply (rule_tac split_mult_neg_le)
obua@14738
  2153
    apply (insert prems)
obua@14738
  2154
    apply (blast)
obua@14738
  2155
    done
obua@14738
  2156
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  2157
    by (simp add: prts[symmetric])
obua@14738
  2158
  show ?thesis
obua@14738
  2159
  proof cases
obua@14738
  2160
    assume "0 <= a * b"
obua@14738
  2161
    then show ?thesis
obua@14738
  2162
      apply (simp_all add: mulprts abs_prts)
obua@14738
  2163
      apply (insert prems)
obua@14754
  2164
      apply (auto simp add: 
nipkow@23477
  2165
	ring_simps 
haftmann@25078
  2166
	iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
haftmann@25078
  2167
	iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
avigad@16775
  2168
	apply(drule (1) mult_nonneg_nonpos[of a b], simp)
avigad@16775
  2169
	apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
obua@14738
  2170
      done
obua@14738
  2171
  next
obua@14738
  2172
    assume "~(0 <= a*b)"
obua@14738
  2173
    with s have "a*b <= 0" by simp
obua@14738
  2174
    then show ?thesis
obua@14738
  2175
      apply (simp_all add: mulprts abs_prts)
obua@14738
  2176
      apply (insert prems)
nipkow@23477
  2177
      apply (auto simp add: ring_simps)
avigad@16775
  2178
      apply(drule (1) mult_nonneg_nonneg[of a b],simp)
avigad@16775
  2179
      apply(drule (1) mult_nonpos_nonpos[of a b],simp)
obua@14738
  2180
      done
obua@14738
  2181
  qed
obua@14738
  2182
qed
haftmann@25304
  2183
qed
haftmann@25304
  2184
haftmann@25304
  2185
instance ordered_idom \<subseteq> pordered_ring_abs
haftmann@25304
  2186
by default (auto simp add: abs_if not_less
haftmann@25304
  2187
  equal_neg_zero neg_equal_zero mult_less_0_iff)
paulson@14294
  2188
obua@14738
  2189
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
haftmann@25304
  2190
  by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  2191
obua@14738
  2192
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
haftmann@25304
  2193
  by (simp add: abs_if) 
paulson@14294
  2194
paulson@14294
  2195
lemma nonzero_abs_inverse:
paulson@14294
  2196
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  2197
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  2198
                      negative_imp_inverse_negative)
paulson@14294
  2199
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  2200
done
paulson@14294
  2201
paulson@14294
  2202
lemma abs_inverse [simp]:
paulson@14294
  2203
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  2204
      inverse (abs a)"
haftmann@21328
  2205
apply (cases "a=0", simp) 
paulson@14294
  2206
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  2207
done
paulson@14294
  2208
paulson@14294
  2209
lemma nonzero_abs_divide:
paulson@14294
  2210
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  2211
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  2212
paulson@15234
  2213
lemma abs_divide [simp]:
paulson@14294
  2214
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
haftmann@21328
  2215
apply (cases "b=0", simp) 
paulson@14294
  2216
apply (simp add: nonzero_abs_divide) 
paulson@14294
  2217
done
paulson@14294
  2218
paulson@14294
  2219
lemma abs_mult_less:
obua@14738
  2220
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  2221
proof -
paulson@14294
  2222
  assume ac: "abs a < c"
paulson@14294
  2223
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  2224
  assume "abs b < d"
paulson@14294
  2225
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  2226
qed
paulson@14293
  2227
haftmann@25304
  2228
lemmas eq_minus_self_iff = equal_neg_zero
obua@14738
  2229
obua@14738
  2230
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
haftmann@25304
  2231
  unfolding order_less_le less_eq_neg_nonpos equal_neg_zero ..
obua@14738
  2232
obua@14738
  2233
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  2234
apply (simp add: order_less_le abs_le_iff)  
haftmann@25304
  2235
apply (auto simp add: abs_if neg_less_eq_nonneg less_eq_neg_nonpos)
obua@14738
  2236
done
obua@14738
  2237
avigad@16775
  2238
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
haftmann@25304
  2239
    (abs y) * x = abs (y * x)"
haftmann@25304
  2240
  apply (subst abs_mult)
haftmann@25304
  2241
  apply simp
haftmann@25304
  2242
done
avigad@16775
  2243
avigad@16775
  2244
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
haftmann@25304
  2245
    abs x / y = abs (x / y)"
haftmann@25304
  2246
  apply (subst abs_divide)
haftmann@25304
  2247
  apply (simp add: order_less_imp_le)
haftmann@25304
  2248
done
avigad@16775
  2249
wenzelm@23389
  2250
obua@19404
  2251
subsection {* Bounds of products via negative and positive Part *}
obua@15178
  2252
obua@15580
  2253
lemma mult_le_prts:
obua@15580
  2254
  assumes
obua@15580
  2255
  "a1 <= (a::'a::lordered_ring)"
obua@15580
  2256
  "a <= a2"
obua@15580
  2257
  "b1 <= b"
obua@15580
  2258
  "b <= b2"
obua@15580
  2259
  shows
obua@15580
  2260
  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
obua@15580
  2261
proof - 
obua@15580
  2262
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
obua@15580
  2263
    apply (subst prts[symmetric])+
obua@15580
  2264
    apply simp
obua@15580
  2265
    done
obua@15580
  2266
  then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
nipkow@23477
  2267
    by (simp add: ring_simps)
obua@15580
  2268
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
obua@15580
  2269
    by (simp_all add: prems mult_mono)
obua@15580
  2270
  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
obua@15580
  2271
  proof -
obua@15580
  2272
    have "pprt a * nprt b <= pprt a * nprt b2"
obua@15580
  2273
      by (simp add: mult_left_mono prems)
obua@15580
  2274
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
obua@15580
  2275
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2276
    ultimately show ?thesis
obua@15580
  2277
      by simp
obua@15580
  2278
  qed
obua@15580
  2279
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
obua@15580
  2280
  proof - 
obua@15580
  2281
    have "nprt a * pprt b <= nprt a2 * pprt b"
obua@15580
  2282
      by (simp add: mult_right_mono prems)
obua@15580
  2283
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
obua@15580
  2284
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2285
    ultimately show ?thesis
obua@15580
  2286
      by simp
obua@15580
  2287
  qed
obua@15580
  2288
  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
obua@15580
  2289
  proof -
obua@15580
  2290
    have "nprt a * nprt b <= nprt a * nprt b1"
obua@15580
  2291
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2292
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
obua@15580
  2293
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2294
    ultimately show ?thesis
obua@15580
  2295
      by simp
obua@15580
  2296
  qed
obua@15580
  2297
  ultimately show ?thesis
obua@15580
  2298
    by - (rule add_mono | simp)+
obua@15580
  2299
qed
obua@19404
  2300
obua@19404
  2301
lemma mult_ge_prts:
obua@15178
  2302
  assumes
obua@19404
  2303
  "a1 <= (a::'a::lordered_ring)"
obua@19404
  2304
  "a <= a2"
obua@19404
  2305
  "b1 <= b"
obua@19404
  2306
  "b <= b2"
obua@15178
  2307
  shows
obua@19404
  2308
  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
obua@19404
  2309
proof - 
obua@19404
  2310
  from prems have a1:"- a2 <= -a" by auto
obua@19404
  2311
  from prems have a2: "-a <= -a1" by auto
obua@19404
  2312
  from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 prems(3) prems(4), simplified nprt_neg pprt_neg] 
obua@19404
  2313
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
obua@19404
  2314
  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
obua@19404
  2315
    by (simp only: minus_le_iff)
obua@19404
  2316
  then show ?thesis by simp
obua@15178
  2317
qed
obua@15178
  2318
paulson@14265
  2319
end