src/HOL/Ring_and_Field.thy
author huffman
Thu Jan 08 10:26:50 2009 -0800 (2009-01-08)
changeset 29408 6d10cf26b5dc
parent 29407 5ef7e97fd9e4
child 29409 f0a8fe83bc07
permissions -rw-r--r--
add lemmas dvd_minus_iff and minus_dvd_iff in class comm_ring_1
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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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text{*Extract signs from products*}
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lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
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lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
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lemma minus_mult_minus [simp]: "- a * - b = a * b"
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  by simp
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lemma minus_mult_commute: "- a * b = a * - b"
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  by simp
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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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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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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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lemma dvd_minus_iff: "x dvd - y \<longleftrightarrow> x dvd y"
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proof
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  assume "x dvd - y"
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  then have "x dvd - 1 * - y" by (rule dvd_mult)
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  then show "x dvd y" by simp
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next
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  assume "x dvd y"
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  then have "x dvd - 1 * y" by (rule dvd_mult)
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  then show "x dvd - y" by simp
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qed
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lemma minus_dvd_iff: "- x dvd y \<longleftrightarrow> x dvd y"
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proof
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  assume "- x dvd y"
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  then obtain k where "y = - x * k" ..
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  then have "y = x * - k" by simp
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  then show "x dvd y" ..
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next
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  assume "x dvd y"
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  then obtain k where "y = x * k" ..
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  then have "y = - x * - k" by simp
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  then show "- x dvd y" ..
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qed
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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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huffman@23544
   357
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   358
begin
haftmann@26274
   359
haftmann@26274
   360
lemma mult_cancel_right1 [simp]:
haftmann@26274
   361
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
haftmann@26274
   362
  by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   363
haftmann@26274
   364
lemma mult_cancel_right2 [simp]:
haftmann@26274
   365
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
haftmann@26274
   366
  by (insert mult_cancel_right [of a c 1], simp)
haftmann@26274
   367
 
haftmann@26274
   368
lemma mult_cancel_left1 [simp]:
haftmann@26274
   369
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
haftmann@26274
   370
  by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   371
haftmann@26274
   372
lemma mult_cancel_left2 [simp]:
haftmann@26274
   373
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
haftmann@26274
   374
  by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   375
haftmann@26274
   376
end
huffman@22990
   377
haftmann@22390
   378
class idom = comm_ring_1 + no_zero_divisors
haftmann@25186
   379
begin
paulson@14421
   380
huffman@27516
   381
subclass ring_1_no_zero_divisors ..
huffman@22990
   382
haftmann@25186
   383
end
haftmann@25152
   384
haftmann@22390
   385
class division_ring = ring_1 + inverse +
haftmann@25062
   386
  assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
haftmann@25062
   387
  assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
haftmann@25186
   388
begin
huffman@20496
   389
haftmann@25186
   390
subclass ring_1_no_zero_divisors
haftmann@28823
   391
proof
huffman@22987
   392
  fix a b :: 'a
huffman@22987
   393
  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
huffman@22987
   394
  show "a * b \<noteq> 0"
huffman@22987
   395
  proof
huffman@22987
   396
    assume ab: "a * b = 0"
huffman@22987
   397
    hence "0 = inverse a * (a * b) * inverse b"
huffman@22987
   398
      by simp
huffman@22987
   399
    also have "\<dots> = (inverse a * a) * (b * inverse b)"
huffman@22987
   400
      by (simp only: mult_assoc)
huffman@22987
   401
    also have "\<dots> = 1"
huffman@22987
   402
      using a b by simp
huffman@22987
   403
    finally show False
huffman@22987
   404
      by simp
huffman@22987
   405
  qed
huffman@22987
   406
qed
huffman@20496
   407
haftmann@26274
   408
lemma nonzero_imp_inverse_nonzero:
haftmann@26274
   409
  "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
haftmann@26274
   410
proof
haftmann@26274
   411
  assume ianz: "inverse a = 0"
haftmann@26274
   412
  assume "a \<noteq> 0"
haftmann@26274
   413
  hence "1 = a * inverse a" by simp
haftmann@26274
   414
  also have "... = 0" by (simp add: ianz)
haftmann@26274
   415
  finally have "1 = 0" .
haftmann@26274
   416
  thus False by (simp add: eq_commute)
haftmann@26274
   417
qed
haftmann@26274
   418
haftmann@26274
   419
lemma inverse_zero_imp_zero:
haftmann@26274
   420
  "inverse a = 0 \<Longrightarrow> a = 0"
haftmann@26274
   421
apply (rule classical)
haftmann@26274
   422
apply (drule nonzero_imp_inverse_nonzero)
haftmann@26274
   423
apply auto
haftmann@26274
   424
done
haftmann@26274
   425
haftmann@26274
   426
lemma inverse_unique: 
haftmann@26274
   427
  assumes ab: "a * b = 1"
haftmann@26274
   428
  shows "inverse a = b"
haftmann@26274
   429
proof -
haftmann@26274
   430
  have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
huffman@29406
   431
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
huffman@29406
   432
  ultimately show ?thesis by (simp add: mult_assoc [symmetric])
haftmann@26274
   433
qed
haftmann@26274
   434
huffman@29406
   435
lemma nonzero_inverse_minus_eq:
huffman@29406
   436
  "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
huffman@29406
   437
  by (rule inverse_unique) simp
huffman@29406
   438
huffman@29406
   439
lemma nonzero_inverse_inverse_eq:
huffman@29406
   440
  "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
huffman@29406
   441
  by (rule inverse_unique) simp
huffman@29406
   442
huffman@29406
   443
lemma nonzero_inverse_eq_imp_eq:
huffman@29406
   444
  assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
huffman@29406
   445
  shows "a = b"
huffman@29406
   446
proof -
huffman@29406
   447
  from `inverse a = inverse b`
huffman@29406
   448
  have "inverse (inverse a) = inverse (inverse b)"
huffman@29406
   449
    by (rule arg_cong)
huffman@29406
   450
  with `a \<noteq> 0` and `b \<noteq> 0` show "a = b"
huffman@29406
   451
    by (simp add: nonzero_inverse_inverse_eq)
huffman@29406
   452
qed
huffman@29406
   453
huffman@29406
   454
lemma inverse_1 [simp]: "inverse 1 = 1"
huffman@29406
   455
  by (rule inverse_unique) simp
huffman@29406
   456
haftmann@26274
   457
lemma nonzero_inverse_mult_distrib: 
huffman@29406
   458
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@26274
   459
  shows "inverse (a * b) = inverse b * inverse a"
haftmann@26274
   460
proof -
huffman@29406
   461
  have "a * (b * inverse b) * inverse a = 1"
huffman@29406
   462
    using assms by simp
huffman@29406
   463
  hence "a * b * (inverse b * inverse a) = 1"
huffman@29406
   464
    by (simp only: mult_assoc)
haftmann@26274
   465
  thus ?thesis
huffman@29406
   466
    by (rule inverse_unique)
haftmann@26274
   467
qed
haftmann@26274
   468
haftmann@26274
   469
lemma division_ring_inverse_add:
haftmann@26274
   470
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
haftmann@26274
   471
  by (simp add: ring_simps mult_assoc)
haftmann@26274
   472
haftmann@26274
   473
lemma division_ring_inverse_diff:
haftmann@26274
   474
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
haftmann@26274
   475
  by (simp add: ring_simps mult_assoc)
haftmann@26274
   476
haftmann@25186
   477
end
haftmann@25152
   478
huffman@22987
   479
class field = comm_ring_1 + inverse +
haftmann@25062
   480
  assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
haftmann@25062
   481
  assumes divide_inverse: "a / b = a * inverse b"
haftmann@25267
   482
begin
huffman@20496
   483
haftmann@25267
   484
subclass division_ring
haftmann@28823
   485
proof
huffman@22987
   486
  fix a :: 'a
huffman@22987
   487
  assume "a \<noteq> 0"
huffman@22987
   488
  thus "inverse a * a = 1" by (rule field_inverse)
huffman@22987
   489
  thus "a * inverse a = 1" by (simp only: mult_commute)
obua@14738
   490
qed
haftmann@25230
   491
huffman@27516
   492
subclass idom ..
haftmann@25230
   493
haftmann@25230
   494
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
haftmann@25230
   495
proof
haftmann@25230
   496
  assume neq: "b \<noteq> 0"
haftmann@25230
   497
  {
haftmann@25230
   498
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
haftmann@25230
   499
    also assume "a / b = 1"
haftmann@25230
   500
    finally show "a = b" by simp
haftmann@25230
   501
  next
haftmann@25230
   502
    assume "a = b"
haftmann@25230
   503
    with neq show "a / b = 1" by (simp add: divide_inverse)
haftmann@25230
   504
  }
haftmann@25230
   505
qed
haftmann@25230
   506
haftmann@25230
   507
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
haftmann@25230
   508
  by (simp add: divide_inverse)
haftmann@25230
   509
haftmann@25230
   510
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
haftmann@25230
   511
  by (simp add: divide_inverse)
haftmann@25230
   512
haftmann@25230
   513
lemma divide_zero_left [simp]: "0 / a = 0"
haftmann@25230
   514
  by (simp add: divide_inverse)
haftmann@25230
   515
haftmann@25230
   516
lemma inverse_eq_divide: "inverse a = 1 / a"
haftmann@25230
   517
  by (simp add: divide_inverse)
haftmann@25230
   518
haftmann@25230
   519
lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
haftmann@25230
   520
  by (simp add: divide_inverse ring_distribs) 
haftmann@25230
   521
haftmann@25230
   522
end
haftmann@25230
   523
haftmann@22390
   524
class division_by_zero = zero + inverse +
haftmann@25062
   525
  assumes inverse_zero [simp]: "inverse 0 = 0"
paulson@14265
   526
haftmann@25230
   527
lemma divide_zero [simp]:
haftmann@25230
   528
  "a / 0 = (0::'a::{field,division_by_zero})"
haftmann@25230
   529
  by (simp add: divide_inverse)
haftmann@25230
   530
haftmann@25230
   531
lemma divide_self_if [simp]:
haftmann@25230
   532
  "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
haftmann@28559
   533
  by simp
haftmann@25230
   534
haftmann@22390
   535
class mult_mono = times + zero + ord +
haftmann@25062
   536
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25062
   537
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
paulson@14267
   538
haftmann@22390
   539
class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
haftmann@25230
   540
begin
haftmann@25230
   541
haftmann@25230
   542
lemma mult_mono:
haftmann@25230
   543
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c
haftmann@25230
   544
     \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   545
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   546
apply (erule mult_left_mono, assumption)
haftmann@25230
   547
done
haftmann@25230
   548
haftmann@25230
   549
lemma mult_mono':
haftmann@25230
   550
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c
haftmann@25230
   551
     \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   552
apply (rule mult_mono)
haftmann@25230
   553
apply (fast intro: order_trans)+
haftmann@25230
   554
done
haftmann@25230
   555
haftmann@25230
   556
end
krauss@21199
   557
haftmann@22390
   558
class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
huffman@22987
   559
  + semiring + comm_monoid_add + cancel_ab_semigroup_add
haftmann@25267
   560
begin
paulson@14268
   561
huffman@27516
   562
subclass semiring_0_cancel ..
huffman@27516
   563
subclass pordered_semiring ..
obua@23521
   564
haftmann@25230
   565
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@25230
   566
  by (drule mult_left_mono [of zero b], auto)
haftmann@25230
   567
haftmann@25230
   568
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@25230
   569
  by (drule mult_left_mono [of b zero], auto)
haftmann@25230
   570
haftmann@25230
   571
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
haftmann@25230
   572
  by (drule mult_right_mono [of b zero], auto)
haftmann@25230
   573
haftmann@26234
   574
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
haftmann@25230
   575
  by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   576
haftmann@25230
   577
end
haftmann@25230
   578
haftmann@25230
   579
class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
haftmann@25267
   580
begin
haftmann@25230
   581
huffman@27516
   582
subclass pordered_cancel_semiring ..
haftmann@25512
   583
huffman@27516
   584
subclass pordered_comm_monoid_add ..
haftmann@25304
   585
haftmann@25230
   586
lemma mult_left_less_imp_less:
haftmann@25230
   587
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
haftmann@25230
   588
  by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   589
 
haftmann@25230
   590
lemma mult_right_less_imp_less:
haftmann@25230
   591
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
haftmann@25230
   592
  by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   593
haftmann@25186
   594
end
haftmann@25152
   595
haftmann@22390
   596
class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
haftmann@25062
   597
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   598
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   599
begin
paulson@14341
   600
huffman@27516
   601
subclass semiring_0_cancel ..
obua@14940
   602
haftmann@25267
   603
subclass ordered_semiring
haftmann@28823
   604
proof
huffman@23550
   605
  fix a b c :: 'a
huffman@23550
   606
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   607
  from A show "c * a \<le> c * b"
haftmann@25186
   608
    unfolding le_less
haftmann@25186
   609
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   610
  from A show "a * c \<le> b * c"
haftmann@25152
   611
    unfolding le_less
haftmann@25186
   612
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   613
qed
haftmann@25152
   614
haftmann@25230
   615
lemma mult_left_le_imp_le:
haftmann@25230
   616
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
haftmann@25230
   617
  by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   618
 
haftmann@25230
   619
lemma mult_right_le_imp_le:
haftmann@25230
   620
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
haftmann@25230
   621
  by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   622
haftmann@25230
   623
lemma mult_pos_pos:
haftmann@25230
   624
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@25230
   625
  by (drule mult_strict_left_mono [of zero b], auto)
haftmann@25230
   626
haftmann@25230
   627
lemma mult_pos_neg:
haftmann@25230
   628
  "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@25230
   629
  by (drule mult_strict_left_mono [of b zero], auto)
haftmann@25230
   630
haftmann@25230
   631
lemma mult_pos_neg2:
haftmann@25230
   632
  "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
haftmann@25230
   633
  by (drule mult_strict_right_mono [of b zero], auto)
haftmann@25230
   634
haftmann@25230
   635
lemma zero_less_mult_pos:
haftmann@25230
   636
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
haftmann@25230
   637
apply (cases "b\<le>0") 
haftmann@25230
   638
 apply (auto simp add: le_less not_less)
haftmann@25230
   639
apply (drule_tac mult_pos_neg [of a b]) 
haftmann@25230
   640
 apply (auto dest: less_not_sym)
haftmann@25230
   641
done
haftmann@25230
   642
haftmann@25230
   643
lemma zero_less_mult_pos2:
haftmann@25230
   644
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
haftmann@25230
   645
apply (cases "b\<le>0") 
haftmann@25230
   646
 apply (auto simp add: le_less not_less)
haftmann@25230
   647
apply (drule_tac mult_pos_neg2 [of a b]) 
haftmann@25230
   648
 apply (auto dest: less_not_sym)
haftmann@25230
   649
done
haftmann@25230
   650
haftmann@26193
   651
text{*Strict monotonicity in both arguments*}
haftmann@26193
   652
lemma mult_strict_mono:
haftmann@26193
   653
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
   654
  shows "a * c < b * d"
haftmann@26193
   655
  using assms apply (cases "c=0")
haftmann@26193
   656
  apply (simp add: mult_pos_pos) 
haftmann@26193
   657
  apply (erule mult_strict_right_mono [THEN less_trans])
haftmann@26193
   658
  apply (force simp add: le_less) 
haftmann@26193
   659
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
   660
  done
haftmann@26193
   661
haftmann@26193
   662
text{*This weaker variant has more natural premises*}
haftmann@26193
   663
lemma mult_strict_mono':
haftmann@26193
   664
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
   665
  shows "a * c < b * d"
haftmann@26193
   666
  by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
   667
haftmann@26193
   668
lemma mult_less_le_imp_less:
haftmann@26193
   669
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
   670
  shows "a * c < b * d"
haftmann@26193
   671
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
   672
  apply (erule less_le_trans)
haftmann@26193
   673
  apply (erule mult_left_mono)
haftmann@26193
   674
  apply simp
haftmann@26193
   675
  apply (erule mult_strict_right_mono)
haftmann@26193
   676
  apply assumption
haftmann@26193
   677
  done
haftmann@26193
   678
haftmann@26193
   679
lemma mult_le_less_imp_less:
haftmann@26193
   680
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
   681
  shows "a * c < b * d"
haftmann@26193
   682
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
   683
  apply (erule le_less_trans)
haftmann@26193
   684
  apply (erule mult_strict_left_mono)
haftmann@26193
   685
  apply simp
haftmann@26193
   686
  apply (erule mult_right_mono)
haftmann@26193
   687
  apply simp
haftmann@26193
   688
  done
haftmann@26193
   689
haftmann@26193
   690
lemma mult_less_imp_less_left:
haftmann@26193
   691
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
haftmann@26193
   692
  shows "a < b"
haftmann@26193
   693
proof (rule ccontr)
haftmann@26193
   694
  assume "\<not>  a < b"
haftmann@26193
   695
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   696
  hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)
haftmann@26193
   697
  with this and less show False 
haftmann@26193
   698
    by (simp add: not_less [symmetric])
haftmann@26193
   699
qed
haftmann@26193
   700
haftmann@26193
   701
lemma mult_less_imp_less_right:
haftmann@26193
   702
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
haftmann@26193
   703
  shows "a < b"
haftmann@26193
   704
proof (rule ccontr)
haftmann@26193
   705
  assume "\<not> a < b"
haftmann@26193
   706
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   707
  hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)
haftmann@26193
   708
  with this and less show False 
haftmann@26193
   709
    by (simp add: not_less [symmetric])
haftmann@26193
   710
qed  
haftmann@26193
   711
haftmann@25230
   712
end
haftmann@25230
   713
haftmann@22390
   714
class mult_mono1 = times + zero + ord +
haftmann@25230
   715
  assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
paulson@14270
   716
haftmann@22390
   717
class pordered_comm_semiring = comm_semiring_0
haftmann@22390
   718
  + pordered_ab_semigroup_add + mult_mono1
haftmann@25186
   719
begin
haftmann@25152
   720
haftmann@25267
   721
subclass pordered_semiring
haftmann@28823
   722
proof
krauss@21199
   723
  fix a b c :: 'a
huffman@23550
   724
  assume "a \<le> b" "0 \<le> c"
haftmann@25230
   725
  thus "c * a \<le> c * b" by (rule mult_mono1)
huffman@23550
   726
  thus "a * c \<le> b * c" by (simp only: mult_commute)
krauss@21199
   727
qed
paulson@14265
   728
haftmann@25267
   729
end
haftmann@25267
   730
haftmann@25267
   731
class pordered_cancel_comm_semiring = comm_semiring_0_cancel
haftmann@25267
   732
  + pordered_ab_semigroup_add + mult_mono1
haftmann@25267
   733
begin
paulson@14265
   734
huffman@27516
   735
subclass pordered_comm_semiring ..
huffman@27516
   736
subclass pordered_cancel_semiring ..
haftmann@25267
   737
haftmann@25267
   738
end
haftmann@25267
   739
haftmann@25267
   740
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
haftmann@26193
   741
  assumes mult_strict_left_mono_comm: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
   742
begin
haftmann@25267
   743
haftmann@25267
   744
subclass ordered_semiring_strict
haftmann@28823
   745
proof
huffman@23550
   746
  fix a b c :: 'a
huffman@23550
   747
  assume "a < b" "0 < c"
haftmann@26193
   748
  thus "c * a < c * b" by (rule mult_strict_left_mono_comm)
huffman@23550
   749
  thus "a * c < b * c" by (simp only: mult_commute)
huffman@23550
   750
qed
paulson@14272
   751
haftmann@25267
   752
subclass pordered_cancel_comm_semiring
haftmann@28823
   753
proof
huffman@23550
   754
  fix a b c :: 'a
huffman@23550
   755
  assume "a \<le> b" "0 \<le> c"
huffman@23550
   756
  thus "c * a \<le> c * b"
haftmann@25186
   757
    unfolding le_less
haftmann@26193
   758
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   759
qed
paulson@14272
   760
haftmann@25267
   761
end
haftmann@25230
   762
haftmann@25267
   763
class pordered_ring = ring + pordered_cancel_semiring 
haftmann@25267
   764
begin
haftmann@25230
   765
huffman@27516
   766
subclass pordered_ab_group_add ..
paulson@14270
   767
haftmann@25230
   768
lemmas ring_simps = ring_simps group_simps
haftmann@25230
   769
haftmann@25230
   770
lemma less_add_iff1:
haftmann@25230
   771
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
haftmann@25230
   772
  by (simp add: ring_simps)
haftmann@25230
   773
haftmann@25230
   774
lemma less_add_iff2:
haftmann@25230
   775
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
haftmann@25230
   776
  by (simp add: ring_simps)
haftmann@25230
   777
haftmann@25230
   778
lemma le_add_iff1:
haftmann@25230
   779
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
haftmann@25230
   780
  by (simp add: ring_simps)
haftmann@25230
   781
haftmann@25230
   782
lemma le_add_iff2:
haftmann@25230
   783
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
haftmann@25230
   784
  by (simp add: ring_simps)
haftmann@25230
   785
haftmann@25230
   786
lemma mult_left_mono_neg:
haftmann@25230
   787
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@25230
   788
  apply (drule mult_left_mono [of _ _ "uminus c"])
haftmann@25230
   789
  apply (simp_all add: minus_mult_left [symmetric]) 
haftmann@25230
   790
  done
haftmann@25230
   791
haftmann@25230
   792
lemma mult_right_mono_neg:
haftmann@25230
   793
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   794
  apply (drule mult_right_mono [of _ _ "uminus c"])
haftmann@25230
   795
  apply (simp_all add: minus_mult_right [symmetric]) 
haftmann@25230
   796
  done
haftmann@25230
   797
haftmann@25230
   798
lemma mult_nonpos_nonpos:
haftmann@25230
   799
  "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@25230
   800
  by (drule mult_right_mono_neg [of a zero b]) auto
haftmann@25230
   801
haftmann@25230
   802
lemma split_mult_pos_le:
haftmann@25230
   803
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
haftmann@25230
   804
  by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
haftmann@25186
   805
haftmann@25186
   806
end
paulson@14270
   807
haftmann@25762
   808
class abs_if = minus + uminus + ord + zero + abs +
haftmann@25762
   809
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@25762
   810
haftmann@25762
   811
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@25186
   812
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
nipkow@24506
   813
nipkow@25564
   814
lemma (in sgn_if) sgn0[simp]: "sgn 0 = 0"
nipkow@25564
   815
by(simp add:sgn_if)
nipkow@25564
   816
haftmann@25230
   817
class ordered_ring = ring + ordered_semiring
haftmann@25304
   818
  + ordered_ab_group_add + abs_if
haftmann@25304
   819
begin
haftmann@25304
   820
huffman@27516
   821
subclass pordered_ring ..
haftmann@25304
   822
haftmann@25304
   823
subclass pordered_ab_group_add_abs
haftmann@28823
   824
proof
haftmann@25304
   825
  fix a b
haftmann@25304
   826
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25304
   827
  by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
haftmann@25304
   828
   (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]
haftmann@25304
   829
     neg_less_eq_nonneg less_eq_neg_nonpos, auto intro: add_nonneg_nonneg,
haftmann@25304
   830
      auto intro!: less_imp_le add_neg_neg)
haftmann@25304
   831
qed (auto simp add: abs_if less_eq_neg_nonpos neg_equal_zero)
haftmann@25304
   832
haftmann@25304
   833
end
obua@23521
   834
haftmann@25230
   835
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
haftmann@25230
   836
   Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
haftmann@25230
   837
 *)
haftmann@25230
   838
class ordered_ring_strict = ring + ordered_semiring_strict
haftmann@25304
   839
  + ordered_ab_group_add + abs_if
haftmann@25230
   840
begin
paulson@14348
   841
huffman@27516
   842
subclass ordered_ring ..
haftmann@25304
   843
paulson@14265
   844
lemma mult_strict_left_mono_neg:
haftmann@25230
   845
  "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
haftmann@25230
   846
  apply (drule mult_strict_left_mono [of _ _ "uminus c"])
haftmann@25230
   847
  apply (simp_all add: minus_mult_left [symmetric]) 
haftmann@25230
   848
  done
obua@14738
   849
paulson@14265
   850
lemma mult_strict_right_mono_neg:
haftmann@25230
   851
  "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
haftmann@25230
   852
  apply (drule mult_strict_right_mono [of _ _ "uminus c"])
haftmann@25230
   853
  apply (simp_all add: minus_mult_right [symmetric]) 
haftmann@25230
   854
  done
obua@14738
   855
haftmann@25230
   856
lemma mult_neg_neg:
haftmann@25230
   857
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@25230
   858
  by (drule mult_strict_right_mono_neg, auto)
obua@14738
   859
haftmann@25917
   860
subclass ring_no_zero_divisors
haftmann@28823
   861
proof
haftmann@25917
   862
  fix a b
haftmann@25917
   863
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
   864
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
   865
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
   866
  proof (cases "a < 0")
haftmann@25917
   867
    case True note A' = this
haftmann@25917
   868
    show ?thesis proof (cases "b < 0")
haftmann@25917
   869
      case True with A'
haftmann@25917
   870
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
   871
    next
haftmann@25917
   872
      case False with B have "0 < b" by auto
haftmann@25917
   873
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
   874
    qed
haftmann@25917
   875
  next
haftmann@25917
   876
    case False with A have A': "0 < a" by auto
haftmann@25917
   877
    show ?thesis proof (cases "b < 0")
haftmann@25917
   878
      case True with A'
haftmann@25917
   879
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
   880
    next
haftmann@25917
   881
      case False with B have "0 < b" by auto
haftmann@25917
   882
      with A' show ?thesis by (auto dest: mult_pos_pos)
haftmann@25917
   883
    qed
haftmann@25917
   884
  qed
haftmann@25917
   885
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
   886
qed
haftmann@25304
   887
paulson@14265
   888
lemma zero_less_mult_iff:
haftmann@25917
   889
  "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
haftmann@25917
   890
  apply (auto simp add: mult_pos_pos mult_neg_neg)
haftmann@25917
   891
  apply (simp_all add: not_less le_less)
haftmann@25917
   892
  apply (erule disjE) apply assumption defer
haftmann@25917
   893
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
   894
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
   895
  apply (erule disjE) apply assumption apply (drule sym) apply simp
haftmann@25917
   896
  apply (drule sym) apply simp
haftmann@25917
   897
  apply (blast dest: zero_less_mult_pos)
haftmann@25230
   898
  apply (blast dest: zero_less_mult_pos2)
haftmann@25230
   899
  done
huffman@22990
   900
paulson@14265
   901
lemma zero_le_mult_iff:
haftmann@25917
   902
  "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
haftmann@25917
   903
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
   904
paulson@14265
   905
lemma mult_less_0_iff:
haftmann@25917
   906
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
haftmann@25917
   907
  apply (insert zero_less_mult_iff [of "-a" b]) 
haftmann@25917
   908
  apply (force simp add: minus_mult_left[symmetric]) 
haftmann@25917
   909
  done
paulson@14265
   910
paulson@14265
   911
lemma mult_le_0_iff:
haftmann@25917
   912
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
   913
  apply (insert zero_le_mult_iff [of "-a" b]) 
haftmann@25917
   914
  apply (force simp add: minus_mult_left[symmetric]) 
haftmann@25917
   915
  done
haftmann@25917
   916
haftmann@25917
   917
lemma zero_le_square [simp]: "0 \<le> a * a"
haftmann@25917
   918
  by (simp add: zero_le_mult_iff linear)
haftmann@25917
   919
haftmann@25917
   920
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
haftmann@25917
   921
  by (simp add: not_less)
haftmann@25917
   922
haftmann@26193
   923
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
   924
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
   925
haftmann@26193
   926
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
   927
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
   928
haftmann@26193
   929
lemma mult_less_cancel_right_disj:
haftmann@26193
   930
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   931
  apply (cases "c = 0")
haftmann@26193
   932
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
   933
                      mult_strict_right_mono_neg)
haftmann@26193
   934
  apply (auto simp add: not_less 
haftmann@26193
   935
                      not_le [symmetric, of "a*c"]
haftmann@26193
   936
                      not_le [symmetric, of a])
haftmann@26193
   937
  apply (erule_tac [!] notE)
haftmann@26193
   938
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
   939
                      mult_right_mono_neg)
haftmann@26193
   940
  done
haftmann@26193
   941
haftmann@26193
   942
lemma mult_less_cancel_left_disj:
haftmann@26193
   943
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   944
  apply (cases "c = 0")
haftmann@26193
   945
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
   946
                      mult_strict_left_mono_neg)
haftmann@26193
   947
  apply (auto simp add: not_less 
haftmann@26193
   948
                      not_le [symmetric, of "c*a"]
haftmann@26193
   949
                      not_le [symmetric, of a])
haftmann@26193
   950
  apply (erule_tac [!] notE)
haftmann@26193
   951
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
   952
                      mult_left_mono_neg)
haftmann@26193
   953
  done
haftmann@26193
   954
haftmann@26193
   955
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
   956
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
   957
haftmann@26193
   958
lemma mult_less_cancel_right:
haftmann@26193
   959
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   960
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
   961
haftmann@26193
   962
lemma mult_less_cancel_left:
haftmann@26193
   963
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   964
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
   965
haftmann@26193
   966
lemma mult_le_cancel_right:
haftmann@26193
   967
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
haftmann@26193
   968
  by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
   969
haftmann@26193
   970
lemma mult_le_cancel_left:
haftmann@26193
   971
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
haftmann@26193
   972
  by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
   973
haftmann@25917
   974
end
paulson@14265
   975
haftmann@25230
   976
text{*This list of rewrites simplifies ring terms by multiplying
haftmann@25230
   977
everything out and bringing sums and products into a canonical form
haftmann@25230
   978
(by ordered rewriting). As a result it decides ring equalities but
haftmann@25230
   979
also helps with inequalities. *}
haftmann@25230
   980
lemmas ring_simps = group_simps ring_distribs
haftmann@25230
   981
haftmann@25230
   982
haftmann@25230
   983
class pordered_comm_ring = comm_ring + pordered_comm_semiring
haftmann@25267
   984
begin
haftmann@25230
   985
huffman@27516
   986
subclass pordered_ring ..
huffman@27516
   987
subclass pordered_cancel_comm_semiring ..
haftmann@25230
   988
haftmann@25267
   989
end
haftmann@25230
   990
haftmann@25230
   991
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
haftmann@25230
   992
  (*previously ordered_semiring*)
haftmann@25230
   993
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
   994
begin
haftmann@25230
   995
haftmann@25230
   996
lemma pos_add_strict:
haftmann@25230
   997
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@25230
   998
  using add_strict_mono [of zero a b c] by simp
haftmann@25230
   999
haftmann@26193
  1000
lemma zero_le_one [simp]: "0 \<le> 1"
haftmann@26193
  1001
  by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
  1002
haftmann@26193
  1003
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
haftmann@26193
  1004
  by (simp add: not_le) 
haftmann@26193
  1005
haftmann@26193
  1006
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
haftmann@26193
  1007
  by (simp add: not_less) 
haftmann@26193
  1008
haftmann@26193
  1009
lemma less_1_mult:
haftmann@26193
  1010
  assumes "1 < m" and "1 < n"
haftmann@26193
  1011
  shows "1 < m * n"
haftmann@26193
  1012
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
  1013
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
  1014
haftmann@25230
  1015
end
haftmann@25230
  1016
haftmann@26193
  1017
class ordered_idom = comm_ring_1 +
haftmann@26193
  1018
  ordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1019
  abs_if + sgn_if
haftmann@25230
  1020
  (*previously ordered_ring*)
haftmann@25917
  1021
begin
haftmann@25917
  1022
huffman@27516
  1023
subclass ordered_ring_strict ..
huffman@27516
  1024
subclass pordered_comm_ring ..
huffman@27516
  1025
subclass idom ..
haftmann@25917
  1026
haftmann@25917
  1027
subclass ordered_semidom
haftmann@28823
  1028
proof
haftmann@26193
  1029
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1030
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
  1031
qed 
haftmann@25917
  1032
haftmann@26193
  1033
lemma linorder_neqE_ordered_idom:
haftmann@26193
  1034
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1035
  using assms by (rule neqE)
haftmann@26193
  1036
haftmann@26274
  1037
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1038
haftmann@26274
  1039
lemma mult_le_cancel_right1:
haftmann@26274
  1040
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
haftmann@26274
  1041
  by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1042
haftmann@26274
  1043
lemma mult_le_cancel_right2:
haftmann@26274
  1044
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
haftmann@26274
  1045
  by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1046
haftmann@26274
  1047
lemma mult_le_cancel_left1:
haftmann@26274
  1048
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
haftmann@26274
  1049
  by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1050
haftmann@26274
  1051
lemma mult_le_cancel_left2:
haftmann@26274
  1052
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
haftmann@26274
  1053
  by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1054
haftmann@26274
  1055
lemma mult_less_cancel_right1:
haftmann@26274
  1056
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
haftmann@26274
  1057
  by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1058
haftmann@26274
  1059
lemma mult_less_cancel_right2:
haftmann@26274
  1060
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
haftmann@26274
  1061
  by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1062
haftmann@26274
  1063
lemma mult_less_cancel_left1:
haftmann@26274
  1064
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
haftmann@26274
  1065
  by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1066
haftmann@26274
  1067
lemma mult_less_cancel_left2:
haftmann@26274
  1068
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
haftmann@26274
  1069
  by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1070
haftmann@27651
  1071
lemma sgn_sgn [simp]:
haftmann@27651
  1072
  "sgn (sgn a) = sgn a"
haftmann@27651
  1073
  unfolding sgn_if by simp
haftmann@27651
  1074
haftmann@27651
  1075
lemma sgn_0_0:
haftmann@27651
  1076
  "sgn a = 0 \<longleftrightarrow> a = 0"
haftmann@27651
  1077
  unfolding sgn_if by simp
haftmann@27651
  1078
haftmann@27651
  1079
lemma sgn_1_pos:
haftmann@27651
  1080
  "sgn a = 1 \<longleftrightarrow> a > 0"
haftmann@27651
  1081
  unfolding sgn_if by (simp add: neg_equal_zero)
haftmann@27651
  1082
haftmann@27651
  1083
lemma sgn_1_neg:
haftmann@27651
  1084
  "sgn a = - 1 \<longleftrightarrow> a < 0"
haftmann@27651
  1085
  unfolding sgn_if by (auto simp add: equal_neg_zero)
haftmann@27651
  1086
haftmann@27651
  1087
lemma sgn_times:
haftmann@27651
  1088
  "sgn (a * b) = sgn a * sgn b"
haftmann@27651
  1089
  by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1090
haftmann@25917
  1091
end
haftmann@25230
  1092
haftmann@25230
  1093
class ordered_field = field + ordered_idom
haftmann@25230
  1094
haftmann@26274
  1095
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1096
paulson@15234
  1097
lemmas mult_compare_simps =
paulson@15234
  1098
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1099
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1100
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1101
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1102
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1103
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1104
    mult_cancel_right mult_cancel_left
paulson@15234
  1105
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1106
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1107
haftmann@26274
  1108
-- {* FIXME continue localization here *}
paulson@14268
  1109
paulson@14268
  1110
lemma inverse_nonzero_iff_nonzero [simp]:
huffman@20496
  1111
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
haftmann@26274
  1112
by (force dest: inverse_zero_imp_zero) 
paulson@14268
  1113
paulson@14268
  1114
lemma inverse_minus_eq [simp]:
huffman@20496
  1115
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
paulson@14377
  1116
proof cases
paulson@14377
  1117
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
  1118
next
paulson@14377
  1119
  assume "a\<noteq>0" 
paulson@14377
  1120
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
  1121
qed
paulson@14268
  1122
paulson@14268
  1123
lemma inverse_eq_imp_eq:
huffman@20496
  1124
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
haftmann@21328
  1125
apply (cases "a=0 | b=0") 
paulson@14268
  1126
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
  1127
              simp add: eq_commute [of "0::'a"])
paulson@14268
  1128
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
  1129
done
paulson@14268
  1130
paulson@14268
  1131
lemma inverse_eq_iff_eq [simp]:
huffman@20496
  1132
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
huffman@20496
  1133
by (force dest!: inverse_eq_imp_eq)
paulson@14268
  1134
paulson@14270
  1135
lemma inverse_inverse_eq [simp]:
huffman@20496
  1136
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
paulson@14270
  1137
  proof cases
paulson@14270
  1138
    assume "a=0" thus ?thesis by simp
paulson@14270
  1139
  next
paulson@14270
  1140
    assume "a\<noteq>0" 
paulson@14270
  1141
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
  1142
  qed
paulson@14270
  1143
paulson@14270
  1144
text{*This version builds in division by zero while also re-orienting
paulson@14270
  1145
      the right-hand side.*}
paulson@14270
  1146
lemma inverse_mult_distrib [simp]:
paulson@14270
  1147
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
  1148
  proof cases
paulson@14270
  1149
    assume "a \<noteq> 0 & b \<noteq> 0" 
haftmann@22993
  1150
    thus ?thesis
haftmann@22993
  1151
      by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
  1152
  next
paulson@14270
  1153
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
haftmann@22993
  1154
    thus ?thesis
haftmann@22993
  1155
      by force
paulson@14270
  1156
  qed
paulson@14270
  1157
paulson@14270
  1158
text{*There is no slick version using division by zero.*}
paulson@14270
  1159
lemma inverse_add:
nipkow@23477
  1160
  "[|a \<noteq> 0;  b \<noteq> 0|]
nipkow@23477
  1161
   ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
huffman@20496
  1162
by (simp add: division_ring_inverse_add mult_ac)
paulson@14270
  1163
paulson@14365
  1164
lemma inverse_divide [simp]:
nipkow@23477
  1165
  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
nipkow@23477
  1166
by (simp add: divide_inverse mult_commute)
paulson@14365
  1167
wenzelm@23389
  1168
avigad@16775
  1169
subsection {* Calculations with fractions *}
avigad@16775
  1170
nipkow@23413
  1171
text{* There is a whole bunch of simp-rules just for class @{text
nipkow@23413
  1172
field} but none for class @{text field} and @{text nonzero_divides}
nipkow@23413
  1173
because the latter are covered by a simproc. *}
nipkow@23413
  1174
paulson@24427
  1175
lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]:
nipkow@23477
  1176
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
  1177
proof -
paulson@14277
  1178
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
nipkow@23482
  1179
    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
paulson@14277
  1180
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
  1181
    by (simp only: mult_ac)
paulson@14277
  1182
  also have "... =  a * inverse b"
paulson@14277
  1183
    by simp
paulson@14277
  1184
    finally show ?thesis 
paulson@14277
  1185
    by (simp add: divide_inverse)
paulson@14277
  1186
qed
paulson@14277
  1187
nipkow@23413
  1188
lemma mult_divide_mult_cancel_left:
nipkow@23477
  1189
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1190
apply (cases "b = 0")
nipkow@23413
  1191
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
paulson@14277
  1192
done
paulson@14277
  1193
paulson@24427
  1194
lemma nonzero_mult_divide_mult_cancel_right [noatp]:
nipkow@23477
  1195
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1196
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
paulson@14321
  1197
nipkow@23413
  1198
lemma mult_divide_mult_cancel_right:
nipkow@23477
  1199
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1200
apply (cases "b = 0")
nipkow@23413
  1201
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
paulson@14321
  1202
done
nipkow@23413
  1203
paulson@14284
  1204
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
nipkow@23477
  1205
by (simp add: divide_inverse)
paulson@14284
  1206
paulson@15234
  1207
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
  1208
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1209
paulson@14430
  1210
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
  1211
by (simp add: divide_inverse mult_ac)
paulson@14288
  1212
nipkow@23482
  1213
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
nipkow@23482
  1214
paulson@24286
  1215
lemma divide_divide_eq_right [simp,noatp]:
nipkow@23477
  1216
  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
  1217
by (simp add: divide_inverse mult_ac)
paulson@14288
  1218
paulson@24286
  1219
lemma divide_divide_eq_left [simp,noatp]:
nipkow@23477
  1220
  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
  1221
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1222
avigad@16775
  1223
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1224
    x / y + w / z = (x * z + w * y) / (y * z)"
nipkow@23477
  1225
apply (subgoal_tac "x / y = (x * z) / (y * z)")
nipkow@23477
  1226
apply (erule ssubst)
nipkow@23477
  1227
apply (subgoal_tac "w / z = (w * y) / (y * z)")
nipkow@23477
  1228
apply (erule ssubst)
nipkow@23477
  1229
apply (rule add_divide_distrib [THEN sym])
nipkow@23477
  1230
apply (subst mult_commute)
nipkow@23477
  1231
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
nipkow@23477
  1232
apply assumption
nipkow@23477
  1233
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
nipkow@23477
  1234
apply assumption
avigad@16775
  1235
done
paulson@14268
  1236
wenzelm@23389
  1237
paulson@15234
  1238
subsubsection{*Special Cancellation Simprules for Division*}
paulson@15234
  1239
paulson@24427
  1240
lemma mult_divide_mult_cancel_left_if[simp,noatp]:
nipkow@23477
  1241
fixes c :: "'a :: {field,division_by_zero}"
nipkow@23477
  1242
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
nipkow@23413
  1243
by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
  1244
paulson@24427
  1245
lemma nonzero_mult_divide_cancel_right[simp,noatp]:
nipkow@23413
  1246
  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
nipkow@23413
  1247
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
nipkow@23413
  1248
paulson@24427
  1249
lemma nonzero_mult_divide_cancel_left[simp,noatp]:
nipkow@23413
  1250
  "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
nipkow@23413
  1251
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
nipkow@23413
  1252
nipkow@23413
  1253
paulson@24427
  1254
lemma nonzero_divide_mult_cancel_right[simp,noatp]:
nipkow@23413
  1255
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
nipkow@23413
  1256
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
nipkow@23413
  1257
paulson@24427
  1258
lemma nonzero_divide_mult_cancel_left[simp,noatp]:
nipkow@23413
  1259
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
nipkow@23413
  1260
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
nipkow@23413
  1261
nipkow@23413
  1262
paulson@24427
  1263
lemma nonzero_mult_divide_mult_cancel_left2[simp,noatp]:
nipkow@23477
  1264
  "[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1265
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
nipkow@23413
  1266
paulson@24427
  1267
lemma nonzero_mult_divide_mult_cancel_right2[simp,noatp]:
nipkow@23477
  1268
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
nipkow@23413
  1269
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
nipkow@23413
  1270
paulson@15234
  1271
paulson@14293
  1272
subsection {* Division and Unary Minus *}
paulson@14293
  1273
paulson@14293
  1274
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
huffman@29407
  1275
by (simp add: divide_inverse)
paulson@14293
  1276
paulson@14293
  1277
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
huffman@29407
  1278
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
  1279
paulson@14293
  1280
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
  1281
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
  1282
paulson@14430
  1283
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
huffman@29407
  1284
by (simp add: divide_inverse)
paulson@14293
  1285
paulson@14293
  1286
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
huffman@29407
  1287
by (simp add: divide_inverse)
paulson@14430
  1288
paulson@14293
  1289
paulson@14293
  1290
text{*The effect is to extract signs from divisions*}
paulson@17085
  1291
lemmas divide_minus_left = minus_divide_left [symmetric]
paulson@17085
  1292
lemmas divide_minus_right = minus_divide_right [symmetric]
paulson@17085
  1293
declare divide_minus_left [simp]   divide_minus_right [simp]
paulson@14293
  1294
paulson@14293
  1295
lemma minus_divide_divide [simp]:
nipkow@23477
  1296
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1297
apply (cases "b=0", simp) 
paulson@14293
  1298
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
  1299
done
paulson@14293
  1300
paulson@14430
  1301
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
  1302
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
  1303
nipkow@23482
  1304
lemma add_divide_eq_iff:
nipkow@23482
  1305
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
nipkow@23482
  1306
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1307
nipkow@23482
  1308
lemma divide_add_eq_iff:
nipkow@23482
  1309
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
nipkow@23482
  1310
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1311
nipkow@23482
  1312
lemma diff_divide_eq_iff:
nipkow@23482
  1313
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
nipkow@23482
  1314
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1315
nipkow@23482
  1316
lemma divide_diff_eq_iff:
nipkow@23482
  1317
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
nipkow@23482
  1318
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1319
nipkow@23482
  1320
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
nipkow@23482
  1321
proof -
nipkow@23482
  1322
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1323
  have "(a = b/c) = (a*c = (b/c)*c)" by simp
nipkow@23496
  1324
  also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
nipkow@23482
  1325
  finally show ?thesis .
nipkow@23482
  1326
qed
nipkow@23482
  1327
nipkow@23482
  1328
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
nipkow@23482
  1329
proof -
nipkow@23482
  1330
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1331
  have "(b/c = a) = ((b/c)*c = a*c)"  by simp
nipkow@23496
  1332
  also have "... = (b = a*c)"  by (simp add: divide_inverse mult_assoc) 
nipkow@23482
  1333
  finally show ?thesis .
nipkow@23482
  1334
qed
nipkow@23482
  1335
nipkow@23482
  1336
lemma eq_divide_eq:
nipkow@23482
  1337
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
nipkow@23482
  1338
by (simp add: nonzero_eq_divide_eq) 
nipkow@23482
  1339
nipkow@23482
  1340
lemma divide_eq_eq:
nipkow@23482
  1341
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
nipkow@23482
  1342
by (force simp add: nonzero_divide_eq_eq) 
nipkow@23482
  1343
nipkow@23482
  1344
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1345
    b = a * c ==> b / c = a"
nipkow@23482
  1346
  by (subst divide_eq_eq, simp)
nipkow@23482
  1347
nipkow@23482
  1348
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1349
    a * c = b ==> a = b / c"
nipkow@23482
  1350
  by (subst eq_divide_eq, simp)
nipkow@23482
  1351
nipkow@23482
  1352
nipkow@23482
  1353
lemmas field_eq_simps = ring_simps
nipkow@23482
  1354
  (* pull / out*)
nipkow@23482
  1355
  add_divide_eq_iff divide_add_eq_iff
nipkow@23482
  1356
  diff_divide_eq_iff divide_diff_eq_iff
nipkow@23482
  1357
  (* multiply eqn *)
nipkow@23482
  1358
  nonzero_eq_divide_eq nonzero_divide_eq_eq
nipkow@23482
  1359
(* is added later:
nipkow@23482
  1360
  times_divide_eq_left times_divide_eq_right
nipkow@23482
  1361
*)
nipkow@23482
  1362
nipkow@23482
  1363
text{*An example:*}
nipkow@23482
  1364
lemma fixes a b c d e f :: "'a::field"
nipkow@23482
  1365
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
  1366
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
nipkow@23482
  1367
 apply(simp add:field_eq_simps)
nipkow@23482
  1368
apply(simp)
nipkow@23482
  1369
done
nipkow@23482
  1370
nipkow@23482
  1371
avigad@16775
  1372
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1373
    x / y - w / z = (x * z - w * y) / (y * z)"
nipkow@23482
  1374
by (simp add:field_eq_simps times_divide_eq)
nipkow@23482
  1375
nipkow@23482
  1376
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
nipkow@23482
  1377
    (x / y = w / z) = (x * z = w * y)"
nipkow@23482
  1378
by (simp add:field_eq_simps times_divide_eq)
paulson@14293
  1379
wenzelm@23389
  1380
paulson@14268
  1381
subsection {* Ordered Fields *}
paulson@14268
  1382
paulson@14277
  1383
lemma positive_imp_inverse_positive: 
nipkow@23482
  1384
assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
nipkow@23482
  1385
proof -
paulson@14268
  1386
  have "0 < a * inverse a" 
paulson@14268
  1387
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
  1388
  thus "0 < inverse a" 
paulson@14268
  1389
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
nipkow@23482
  1390
qed
paulson@14268
  1391
paulson@14277
  1392
lemma negative_imp_inverse_negative:
nipkow@23482
  1393
  "a < 0 ==> inverse a < (0::'a::ordered_field)"
nipkow@23482
  1394
by (insert positive_imp_inverse_positive [of "-a"], 
nipkow@23482
  1395
    simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
paulson@14268
  1396
paulson@14268
  1397
lemma inverse_le_imp_le:
nipkow@23482
  1398
assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
nipkow@23482
  1399
shows "b \<le> (a::'a::ordered_field)"
nipkow@23482
  1400
proof (rule classical)
paulson@14268
  1401
  assume "~ b \<le> a"
nipkow@23482
  1402
  hence "a < b"  by (simp add: linorder_not_le)
nipkow@23482
  1403
  hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
paulson@14268
  1404
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
  1405
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
  1406
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
  1407
    by (simp add: bpos order_less_imp_le mult_right_mono)
nipkow@23482
  1408
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
nipkow@23482
  1409
qed
paulson@14268
  1410
paulson@14277
  1411
lemma inverse_positive_imp_positive:
nipkow@23482
  1412
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
nipkow@23482
  1413
shows "0 < (a::'a::ordered_field)"
wenzelm@23389
  1414
proof -
paulson@14277
  1415
  have "0 < inverse (inverse a)"
wenzelm@23389
  1416
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
  1417
  thus "0 < a"
wenzelm@23389
  1418
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1419
qed
paulson@14277
  1420
paulson@14277
  1421
lemma inverse_positive_iff_positive [simp]:
nipkow@23482
  1422
  "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1423
apply (cases "a = 0", simp)
paulson@14277
  1424
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
  1425
done
paulson@14277
  1426
paulson@14277
  1427
lemma inverse_negative_imp_negative:
nipkow@23482
  1428
assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
nipkow@23482
  1429
shows "a < (0::'a::ordered_field)"
wenzelm@23389
  1430
proof -
paulson@14277
  1431
  have "inverse (inverse a) < 0"
wenzelm@23389
  1432
    using inv_less_0 by (rule negative_imp_inverse_negative)
nipkow@23482
  1433
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1434
qed
paulson@14277
  1435
paulson@14277
  1436
lemma inverse_negative_iff_negative [simp]:
nipkow@23482
  1437
  "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1438
apply (cases "a = 0", simp)
paulson@14277
  1439
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
  1440
done
paulson@14277
  1441
paulson@14277
  1442
lemma inverse_nonnegative_iff_nonnegative [simp]:
nipkow@23482
  1443
  "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1444
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1445
paulson@14277
  1446
lemma inverse_nonpositive_iff_nonpositive [simp]:
nipkow@23482
  1447
  "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1448
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1449
chaieb@23406
  1450
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
chaieb@23406
  1451
proof
chaieb@23406
  1452
  fix x::'a
chaieb@23406
  1453
  have m1: "- (1::'a) < 0" by simp
chaieb@23406
  1454
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1455
  have "(- 1) + x < x" by simp
chaieb@23406
  1456
  thus "\<exists>y. y < x" by blast
chaieb@23406
  1457
qed
chaieb@23406
  1458
chaieb@23406
  1459
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
chaieb@23406
  1460
proof
chaieb@23406
  1461
  fix x::'a
chaieb@23406
  1462
  have m1: " (1::'a) > 0" by simp
chaieb@23406
  1463
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1464
  have "1 + x > x" by simp
chaieb@23406
  1465
  thus "\<exists>y. y > x" by blast
chaieb@23406
  1466
qed
paulson@14277
  1467
paulson@14277
  1468
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
  1469
paulson@14268
  1470
lemma less_imp_inverse_less:
nipkow@23482
  1471
assumes less: "a < b" and apos:  "0 < a"
nipkow@23482
  1472
shows "inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1473
proof (rule ccontr)
paulson@14268
  1474
  assume "~ inverse b < inverse a"
paulson@14268
  1475
  hence "inverse a \<le> inverse b"
paulson@14268
  1476
    by (simp add: linorder_not_less)
paulson@14268
  1477
  hence "~ (a < b)"
paulson@14268
  1478
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
  1479
  thus False
paulson@14268
  1480
    by (rule notE [OF _ less])
nipkow@23482
  1481
qed
paulson@14268
  1482
paulson@14268
  1483
lemma inverse_less_imp_less:
nipkow@23482
  1484
  "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1485
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
  1486
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
  1487
done
paulson@14268
  1488
paulson@14268
  1489
text{*Both premises are essential. Consider -1 and 1.*}
paulson@24286
  1490
lemma inverse_less_iff_less [simp,noatp]:
nipkow@23482
  1491
  "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1492
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
  1493
paulson@14268
  1494
lemma le_imp_inverse_le:
nipkow@23482
  1495
  "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1496
by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
  1497
paulson@24286
  1498
lemma inverse_le_iff_le [simp,noatp]:
nipkow@23482
  1499
 "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1500
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
  1501
paulson@14268
  1502
paulson@14268
  1503
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
  1504
case is trivial, since inverse preserves signs.*}
paulson@14268
  1505
lemma inverse_le_imp_le_neg:
nipkow@23482
  1506
  "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
nipkow@23482
  1507
apply (rule classical) 
nipkow@23482
  1508
apply (subgoal_tac "a < 0") 
nipkow@23482
  1509
 prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
nipkow@23482
  1510
apply (insert inverse_le_imp_le [of "-b" "-a"])
nipkow@23482
  1511
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1512
done
paulson@14268
  1513
paulson@14268
  1514
lemma less_imp_inverse_less_neg:
paulson@14268
  1515
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1516
apply (subgoal_tac "a < 0") 
nipkow@23482
  1517
 prefer 2 apply (blast intro: order_less_trans) 
nipkow@23482
  1518
apply (insert less_imp_inverse_less [of "-b" "-a"])
nipkow@23482
  1519
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1520
done
paulson@14268
  1521
paulson@14268
  1522
lemma inverse_less_imp_less_neg:
paulson@14268
  1523
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
nipkow@23482
  1524
apply (rule classical) 
nipkow@23482
  1525
apply (subgoal_tac "a < 0") 
nipkow@23482
  1526
 prefer 2
nipkow@23482
  1527
 apply (force simp add: linorder_not_less intro: order_le_less_trans) 
nipkow@23482
  1528
apply (insert inverse_less_imp_less [of "-b" "-a"])
nipkow@23482
  1529
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1530
done
paulson@14268
  1531
paulson@24286
  1532
lemma inverse_less_iff_less_neg [simp,noatp]:
nipkow@23482
  1533
  "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
nipkow@23482
  1534
apply (insert inverse_less_iff_less [of "-b" "-a"])
nipkow@23482
  1535
apply (simp del: inverse_less_iff_less 
nipkow@23482
  1536
            add: order_less_imp_not_eq nonzero_inverse_minus_eq)
nipkow@23482
  1537
done
paulson@14268
  1538
paulson@14268
  1539
lemma le_imp_inverse_le_neg:
nipkow@23482
  1540
  "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1541
by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1542
paulson@24286
  1543
lemma inverse_le_iff_le_neg [simp,noatp]:
nipkow@23482
  1544
 "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1545
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1546
paulson@14277
  1547
paulson@14365
  1548
subsection{*Inverses and the Number One*}
paulson@14365
  1549
paulson@14365
  1550
lemma one_less_inverse_iff:
nipkow@23482
  1551
  "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
nipkow@23482
  1552
proof cases
paulson@14365
  1553
  assume "0 < x"
paulson@14365
  1554
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1555
    show ?thesis by simp
paulson@14365
  1556
next
paulson@14365
  1557
  assume notless: "~ (0 < x)"
paulson@14365
  1558
  have "~ (1 < inverse x)"
paulson@14365
  1559
  proof
paulson@14365
  1560
    assume "1 < inverse x"
paulson@14365
  1561
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1562
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1563
    finally show False by auto
paulson@14365
  1564
  qed
paulson@14365
  1565
  with notless show ?thesis by simp
paulson@14365
  1566
qed
paulson@14365
  1567
paulson@14365
  1568
lemma inverse_eq_1_iff [simp]:
nipkow@23482
  1569
  "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1570
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1571
paulson@14365
  1572
lemma one_le_inverse_iff:
nipkow@23482
  1573
  "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1574
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1575
                    eq_commute [of 1]) 
paulson@14365
  1576
paulson@14365
  1577
lemma inverse_less_1_iff:
nipkow@23482
  1578
  "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1579
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1580
paulson@14365
  1581
lemma inverse_le_1_iff:
nipkow@23482
  1582
  "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1583
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1584
wenzelm@23389
  1585
paulson@14288
  1586
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1587
paulson@14288
  1588
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1589
proof -
paulson@14288
  1590
  assume less: "0<c"
paulson@14288
  1591
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1592
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1593
  also have "... = (a*c \<le> b)"
paulson@14288
  1594
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1595
  finally show ?thesis .
paulson@14288
  1596
qed
paulson@14288
  1597
paulson@14288
  1598
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1599
proof -
paulson@14288
  1600
  assume less: "c<0"
paulson@14288
  1601
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1602
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1603
  also have "... = (b \<le> a*c)"
paulson@14288
  1604
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1605
  finally show ?thesis .
paulson@14288
  1606
qed
paulson@14288
  1607
paulson@14288
  1608
lemma le_divide_eq:
paulson@14288
  1609
  "(a \<le> b/c) = 
paulson@14288
  1610
   (if 0 < c then a*c \<le> b
paulson@14288
  1611
             else if c < 0 then b \<le> a*c
paulson@14288
  1612
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1613
apply (cases "c=0", simp) 
paulson@14288
  1614
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1615
done
paulson@14288
  1616
paulson@14288
  1617
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1618
proof -
paulson@14288
  1619
  assume less: "0<c"
paulson@14288
  1620
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1621
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1622
  also have "... = (b \<le> a*c)"
paulson@14288
  1623
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1624
  finally show ?thesis .
paulson@14288
  1625
qed
paulson@14288
  1626
paulson@14288
  1627
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1628
proof -
paulson@14288
  1629
  assume less: "c<0"
paulson@14288
  1630
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1631
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1632
  also have "... = (a*c \<le> b)"
paulson@14288
  1633
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1634
  finally show ?thesis .
paulson@14288
  1635
qed
paulson@14288
  1636
paulson@14288
  1637
lemma divide_le_eq:
paulson@14288
  1638
  "(b/c \<le> a) = 
paulson@14288
  1639
   (if 0 < c then b \<le> a*c
paulson@14288
  1640
             else if c < 0 then a*c \<le> b
paulson@14288
  1641
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1642
apply (cases "c=0", simp) 
paulson@14288
  1643
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1644
done
paulson@14288
  1645
paulson@14288
  1646
lemma pos_less_divide_eq:
paulson@14288
  1647
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1648
proof -
paulson@14288
  1649
  assume less: "0<c"
paulson@14288
  1650
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@15234
  1651
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1652
  also have "... = (a*c < b)"
paulson@14288
  1653
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1654
  finally show ?thesis .
paulson@14288
  1655
qed
paulson@14288
  1656
paulson@14288
  1657
lemma neg_less_divide_eq:
paulson@14288
  1658
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1659
proof -
paulson@14288
  1660
  assume less: "c<0"
paulson@14288
  1661
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@15234
  1662
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1663
  also have "... = (b < a*c)"
paulson@14288
  1664
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1665
  finally show ?thesis .
paulson@14288
  1666
qed
paulson@14288
  1667
paulson@14288
  1668
lemma less_divide_eq:
paulson@14288
  1669
  "(a < b/c) = 
paulson@14288
  1670
   (if 0 < c then a*c < b
paulson@14288
  1671
             else if c < 0 then b < a*c
paulson@14288
  1672
             else  a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1673
apply (cases "c=0", simp) 
paulson@14288
  1674
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1675
done
paulson@14288
  1676
paulson@14288
  1677
lemma pos_divide_less_eq:
paulson@14288
  1678
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1679
proof -
paulson@14288
  1680
  assume less: "0<c"
paulson@14288
  1681
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@15234
  1682
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1683
  also have "... = (b < a*c)"
paulson@14288
  1684
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1685
  finally show ?thesis .
paulson@14288
  1686
qed
paulson@14288
  1687
paulson@14288
  1688
lemma neg_divide_less_eq:
paulson@14288
  1689
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1690
proof -
paulson@14288
  1691
  assume less: "c<0"
paulson@14288
  1692
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@15234
  1693
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1694
  also have "... = (a*c < b)"
paulson@14288
  1695
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1696
  finally show ?thesis .
paulson@14288
  1697
qed
paulson@14288
  1698
paulson@14288
  1699
lemma divide_less_eq:
paulson@14288
  1700
  "(b/c < a) = 
paulson@14288
  1701
   (if 0 < c then b < a*c
paulson@14288
  1702
             else if c < 0 then a*c < b
paulson@14288
  1703
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1704
apply (cases "c=0", simp) 
paulson@14288
  1705
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1706
done
paulson@14288
  1707
nipkow@23482
  1708
nipkow@23482
  1709
subsection{*Field simplification*}
nipkow@23482
  1710
nipkow@23482
  1711
text{* Lemmas @{text field_simps} multiply with denominators in
nipkow@23482
  1712
in(equations) if they can be proved to be non-zero (for equations) or
nipkow@23482
  1713
positive/negative (for inequations). *}
paulson@14288
  1714
nipkow@23482
  1715
lemmas field_simps = field_eq_simps
nipkow@23482
  1716
  (* multiply ineqn *)
nipkow@23482
  1717
  pos_divide_less_eq neg_divide_less_eq
nipkow@23482
  1718
  pos_less_divide_eq neg_less_divide_eq
nipkow@23482
  1719
  pos_divide_le_eq neg_divide_le_eq
nipkow@23482
  1720
  pos_le_divide_eq neg_le_divide_eq
paulson@14288
  1721
nipkow@23482
  1722
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
nipkow@23483
  1723
of positivity/negativity needed for @{text field_simps}. Have not added @{text
nipkow@23482
  1724
sign_simps} to @{text field_simps} because the former can lead to case
nipkow@23482
  1725
explosions. *}
paulson@14288
  1726
nipkow@23482
  1727
lemmas sign_simps = group_simps
nipkow@23482
  1728
  zero_less_mult_iff  mult_less_0_iff
paulson@14288
  1729
nipkow@23482
  1730
(* Only works once linear arithmetic is installed:
nipkow@23482
  1731
text{*An example:*}
nipkow@23482
  1732
lemma fixes a b c d e f :: "'a::ordered_field"
nipkow@23482
  1733
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
nipkow@23482
  1734
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
nipkow@23482
  1735
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
nipkow@23482
  1736
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
nipkow@23482
  1737
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1738
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
nipkow@23482
  1739
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1740
apply(simp add:field_simps)
avigad@16775
  1741
done
nipkow@23482
  1742
*)
avigad@16775
  1743
wenzelm@23389
  1744
avigad@16775
  1745
subsection{*Division and Signs*}
avigad@16775
  1746
avigad@16775
  1747
lemma zero_less_divide_iff:
avigad@16775
  1748
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
  1749
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
  1750
avigad@16775
  1751
lemma divide_less_0_iff:
avigad@16775
  1752
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
avigad@16775
  1753
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
  1754
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
  1755
avigad@16775
  1756
lemma zero_le_divide_iff:
avigad@16775
  1757
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
avigad@16775
  1758
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
  1759
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
  1760
avigad@16775
  1761
lemma divide_le_0_iff:
avigad@16775
  1762
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
avigad@16775
  1763
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
  1764
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
  1765
paulson@24286
  1766
lemma divide_eq_0_iff [simp,noatp]:
avigad@16775
  1767
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
nipkow@23482
  1768
by (simp add: divide_inverse)
avigad@16775
  1769
nipkow@23482
  1770
lemma divide_pos_pos:
nipkow@23482
  1771
  "0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
nipkow@23482
  1772
by(simp add:field_simps)
nipkow@23482
  1773
avigad@16775
  1774
nipkow@23482
  1775
lemma divide_nonneg_pos:
nipkow@23482
  1776
  "0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
nipkow@23482
  1777
by(simp add:field_simps)
avigad@16775
  1778
nipkow@23482
  1779
lemma divide_neg_pos:
nipkow@23482
  1780
  "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
nipkow@23482
  1781
by(simp add:field_simps)
avigad@16775
  1782
nipkow@23482
  1783
lemma divide_nonpos_pos:
nipkow@23482
  1784
  "(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
nipkow@23482
  1785
by(simp add:field_simps)
avigad@16775
  1786
nipkow@23482
  1787
lemma divide_pos_neg:
nipkow@23482
  1788
  "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
nipkow@23482
  1789
by(simp add:field_simps)
avigad@16775
  1790
nipkow@23482
  1791
lemma divide_nonneg_neg:
nipkow@23482
  1792
  "0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" 
nipkow@23482
  1793
by(simp add:field_simps)
avigad@16775
  1794
nipkow@23482
  1795
lemma divide_neg_neg:
nipkow@23482
  1796
  "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
nipkow@23482
  1797
by(simp add:field_simps)
avigad@16775
  1798
nipkow@23482
  1799
lemma divide_nonpos_neg:
nipkow@23482
  1800
  "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
nipkow@23482
  1801
by(simp add:field_simps)
paulson@15234
  1802
wenzelm@23389
  1803
paulson@14288
  1804
subsection{*Cancellation Laws for Division*}
paulson@14288
  1805
paulson@24286
  1806
lemma divide_cancel_right [simp,noatp]:
paulson@14288
  1807
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1808
apply (cases "c=0", simp)
nipkow@23496
  1809
apply (simp add: divide_inverse)
paulson@14288
  1810
done
paulson@14288
  1811
paulson@24286
  1812
lemma divide_cancel_left [simp,noatp]:
paulson@14288
  1813
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
nipkow@23482
  1814
apply (cases "c=0", simp)
nipkow@23496
  1815
apply (simp add: divide_inverse)
paulson@14288
  1816
done
paulson@14288
  1817
wenzelm@23389
  1818
paulson@14353
  1819
subsection {* Division and the Number One *}
paulson@14353
  1820
paulson@14353
  1821
text{*Simplify expressions equated with 1*}
paulson@24286
  1822
lemma divide_eq_1_iff [simp,noatp]:
paulson@14353
  1823
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1824
apply (cases "b=0", simp)
nipkow@23482
  1825
apply (simp add: right_inverse_eq)
paulson@14353
  1826
done
paulson@14353
  1827
paulson@24286
  1828
lemma one_eq_divide_iff [simp,noatp]:
paulson@14353
  1829
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1830
by (simp add: eq_commute [of 1])
paulson@14353
  1831
paulson@24286
  1832
lemma zero_eq_1_divide_iff [simp,noatp]:
paulson@14353
  1833
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
nipkow@23482
  1834
apply (cases "a=0", simp)
nipkow@23482
  1835
apply (auto simp add: nonzero_eq_divide_eq)
paulson@14353
  1836
done
paulson@14353
  1837
paulson@24286
  1838
lemma one_divide_eq_0_iff [simp,noatp]:
paulson@14353
  1839
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
nipkow@23482
  1840
apply (cases "a=0", simp)
nipkow@23482
  1841
apply (insert zero_neq_one [THEN not_sym])
nipkow@23482
  1842
apply (auto simp add: nonzero_divide_eq_eq)
paulson@14353
  1843
done
paulson@14353
  1844
paulson@14353
  1845
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@18623
  1846
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
paulson@18623
  1847
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
paulson@18623
  1848
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
paulson@18623
  1849
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
paulson@17085
  1850
paulson@17085
  1851
declare zero_less_divide_1_iff [simp]
paulson@24286
  1852
declare divide_less_0_1_iff [simp,noatp]
paulson@17085
  1853
declare zero_le_divide_1_iff [simp]
paulson@24286
  1854
declare divide_le_0_1_iff [simp,noatp]
paulson@14353
  1855
wenzelm@23389
  1856
paulson@14293
  1857
subsection {* Ordering Rules for Division *}
paulson@14293
  1858
paulson@14293
  1859
lemma divide_strict_right_mono:
paulson@14293
  1860
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1861
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
nipkow@23482
  1862
              positive_imp_inverse_positive)
paulson@14293
  1863
paulson@14293
  1864
lemma divide_right_mono:
paulson@14293
  1865
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
nipkow@23482
  1866
by (force simp add: divide_strict_right_mono order_le_less)
paulson@14293
  1867
avigad@16775
  1868
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1869
    ==> c <= 0 ==> b / c <= a / c"
nipkow@23482
  1870
apply (drule divide_right_mono [of _ _ "- c"])
nipkow@23482
  1871
apply auto
avigad@16775
  1872
done
avigad@16775
  1873
avigad@16775
  1874
lemma divide_strict_right_mono_neg:
avigad@16775
  1875
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
nipkow@23482
  1876
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
nipkow@23482
  1877
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
avigad@16775
  1878
done
paulson@14293
  1879
paulson@14293
  1880
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1881
      have the same sign*}
paulson@14293
  1882
lemma divide_strict_left_mono:
nipkow@23482
  1883
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1884
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
paulson@14293
  1885
paulson@14293
  1886
lemma divide_left_mono:
nipkow@23482
  1887
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
nipkow@23482
  1888
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
paulson@14293
  1889
avigad@16775
  1890
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1891
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1892
  apply (drule divide_left_mono [of _ _ "- c"])
avigad@16775
  1893
  apply (auto simp add: mult_commute)
avigad@16775
  1894
done
avigad@16775
  1895
paulson@14293
  1896
lemma divide_strict_left_mono_neg:
nipkow@23482
  1897
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1898
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
nipkow@23482
  1899
paulson@14293
  1900
avigad@16775
  1901
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1902
paulson@24286
  1903
lemma le_divide_eq_1 [noatp]:
avigad@16775
  1904
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1905
  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1906
by (auto simp add: le_divide_eq)
avigad@16775
  1907
paulson@24286
  1908
lemma divide_le_eq_1 [noatp]:
avigad@16775
  1909
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1910
  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1911
by (auto simp add: divide_le_eq)
avigad@16775
  1912
paulson@24286
  1913
lemma less_divide_eq_1 [noatp]:
avigad@16775
  1914
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1915
  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1916
by (auto simp add: less_divide_eq)
avigad@16775
  1917
paulson@24286
  1918
lemma divide_less_eq_1 [noatp]:
avigad@16775
  1919
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1920
  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1921
by (auto simp add: divide_less_eq)
avigad@16775
  1922
wenzelm@23389
  1923
avigad@16775
  1924
subsection{*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1925
paulson@24286
  1926
lemma le_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1927
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1928
  shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1929
by (auto simp add: le_divide_eq)
avigad@16775
  1930
paulson@24286
  1931
lemma le_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1932
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1933
  shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1934
by (auto simp add: le_divide_eq)
avigad@16775
  1935
paulson@24286
  1936
lemma divide_le_eq_1_pos [simp,noatp]:
avigad@16775
  1937
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1938
  shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1939
by (auto simp add: divide_le_eq)
avigad@16775
  1940
paulson@24286
  1941
lemma divide_le_eq_1_neg [simp,noatp]:
avigad@16775
  1942
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1943
  shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1944
by (auto simp add: divide_le_eq)
avigad@16775
  1945
paulson@24286
  1946
lemma less_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1947
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1948
  shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1949
by (auto simp add: less_divide_eq)
avigad@16775
  1950
paulson@24286
  1951
lemma less_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1952
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1953
  shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1954
by (auto simp add: less_divide_eq)
avigad@16775
  1955
paulson@24286
  1956
lemma divide_less_eq_1_pos [simp,noatp]:
avigad@16775
  1957
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1958
  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1959
by (auto simp add: divide_less_eq)
paulson@18649
  1960
paulson@24286
  1961
lemma divide_less_eq_1_neg [simp,noatp]:
paulson@18649
  1962
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1963
  shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1964
by (auto simp add: divide_less_eq)
avigad@16775
  1965
paulson@24286
  1966
lemma eq_divide_eq_1 [simp,noatp]:
avigad@16775
  1967
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1968
  shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1969
by (auto simp add: eq_divide_eq)
avigad@16775
  1970
paulson@24286
  1971
lemma divide_eq_eq_1 [simp,noatp]:
avigad@16775
  1972
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1973
  shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1974
by (auto simp add: divide_eq_eq)
avigad@16775
  1975
wenzelm@23389
  1976
avigad@16775
  1977
subsection {* Reasoning about inequalities with division *}
avigad@16775
  1978
avigad@16775
  1979
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1980
    ==> x * y <= x"
avigad@16775
  1981
  by (auto simp add: mult_compare_simps);
avigad@16775
  1982
avigad@16775
  1983
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1984
    ==> y * x <= x"
avigad@16775
  1985
  by (auto simp add: mult_compare_simps);
avigad@16775
  1986
avigad@16775
  1987
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
avigad@16775
  1988
    x / y <= z";
avigad@16775
  1989
  by (subst pos_divide_le_eq, assumption+);
avigad@16775
  1990
avigad@16775
  1991
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
nipkow@23482
  1992
    z <= x / y"
nipkow@23482
  1993
by(simp add:field_simps)
avigad@16775
  1994
avigad@16775
  1995
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
avigad@16775
  1996
    x / y < z"
nipkow@23482
  1997
by(simp add:field_simps)
avigad@16775
  1998
avigad@16775
  1999
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
avigad@16775
  2000
    z < x / y"
nipkow@23482
  2001
by(simp add:field_simps)
avigad@16775
  2002
avigad@16775
  2003
lemma frac_le: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  2004
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
avigad@16775
  2005
  apply (rule mult_imp_div_pos_le)
haftmann@25230
  2006
  apply simp
haftmann@25230
  2007
  apply (subst times_divide_eq_left)
avigad@16775
  2008
  apply (rule mult_imp_le_div_pos, assumption)
avigad@16775
  2009
  apply (rule mult_mono)
avigad@16775
  2010
  apply simp_all
paulson@14293
  2011
done
paulson@14293
  2012
avigad@16775
  2013
lemma frac_less: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  2014
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
avigad@16775
  2015
  apply (rule mult_imp_div_pos_less)
avigad@16775
  2016
  apply simp;
avigad@16775
  2017
  apply (subst times_divide_eq_left);
avigad@16775
  2018
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  2019
  apply (erule mult_less_le_imp_less)
avigad@16775
  2020
  apply simp_all
avigad@16775
  2021
done
avigad@16775
  2022
avigad@16775
  2023
lemma frac_less2: "(0::'a::ordered_field) < x ==> 
avigad@16775
  2024
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
avigad@16775
  2025
  apply (rule mult_imp_div_pos_less)
avigad@16775
  2026
  apply simp_all
avigad@16775
  2027
  apply (subst times_divide_eq_left);
avigad@16775
  2028
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  2029
  apply (erule mult_le_less_imp_less)
avigad@16775
  2030
  apply simp_all
avigad@16775
  2031
done
avigad@16775
  2032
avigad@16775
  2033
text{*It's not obvious whether these should be simprules or not. 
avigad@16775
  2034
  Their effect is to gather terms into one big fraction, like
avigad@16775
  2035
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
avigad@16775
  2036
  seem to need them.*}
avigad@16775
  2037
avigad@16775
  2038
declare times_divide_eq [simp]
paulson@14293
  2039
wenzelm@23389
  2040
paulson@14293
  2041
subsection {* Ordered Fields are Dense *}
paulson@14293
  2042
haftmann@25193
  2043
context ordered_semidom
haftmann@25193
  2044
begin
haftmann@25193
  2045
haftmann@25193
  2046
lemma less_add_one: "a < a + 1"
paulson@14293
  2047
proof -
haftmann@25193
  2048
  have "a + 0 < a + 1"
nipkow@23482
  2049
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  2050
  thus ?thesis by simp
paulson@14293
  2051
qed
paulson@14293
  2052
haftmann@25193
  2053
lemma zero_less_two: "0 < 1 + 1"
haftmann@25193
  2054
  by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  2055
haftmann@25193
  2056
end
paulson@14365
  2057
paulson@14293
  2058
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
nipkow@23482
  2059
by (simp add: field_simps zero_less_two)
paulson@14293
  2060
paulson@14293
  2061
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
nipkow@23482
  2062
by (simp add: field_simps zero_less_two)
paulson@14293
  2063
haftmann@24422
  2064
instance ordered_field < dense_linear_order
haftmann@24422
  2065
proof
haftmann@24422
  2066
  fix x y :: 'a
haftmann@24422
  2067
  have "x < x + 1" by simp
haftmann@24422
  2068
  then show "\<exists>y. x < y" .. 
haftmann@24422
  2069
  have "x - 1 < x" by simp
haftmann@24422
  2070
  then show "\<exists>y. y < x" ..
haftmann@24422
  2071
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
haftmann@24422
  2072
qed
paulson@14293
  2073
paulson@15234
  2074
paulson@14293
  2075
subsection {* Absolute Value *}
paulson@14293
  2076
haftmann@25304
  2077
context ordered_idom
haftmann@25304
  2078
begin
haftmann@25304
  2079
haftmann@25304
  2080
lemma mult_sgn_abs: "sgn x * abs x = x"
haftmann@25304
  2081
  unfolding abs_if sgn_if by auto
haftmann@25304
  2082
haftmann@25304
  2083
end
nipkow@24491
  2084
obua@14738
  2085
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
haftmann@25304
  2086
  by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
haftmann@25304
  2087
haftmann@25304
  2088
class pordered_ring_abs = pordered_ring + pordered_ab_group_add_abs +
haftmann@25304
  2089
  assumes abs_eq_mult:
haftmann@25304
  2090
    "(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
  2091
haftmann@25304
  2092
haftmann@25304
  2093
class lordered_ring = pordered_ring + lordered_ab_group_add_abs
haftmann@25304
  2094
begin
haftmann@25304
  2095
huffman@27516
  2096
subclass lordered_ab_group_add_meet ..
huffman@27516
  2097
subclass lordered_ab_group_add_join ..
haftmann@25304
  2098
haftmann@25304
  2099
end
paulson@14294
  2100
obua@14738
  2101
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  2102
proof -
obua@14738
  2103
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  2104
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  2105
  have a: "(abs a) * (abs b) = ?x"
nipkow@23477
  2106
    by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
obua@14738
  2107
  {
obua@14738
  2108
    fix u v :: 'a
paulson@15481
  2109
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
paulson@15481
  2110
              u * v = pprt a * pprt b + pprt a * nprt b + 
paulson@15481
  2111
                      nprt a * pprt b + nprt a * nprt b"
obua@14738
  2112
      apply (subst prts[of u], subst prts[of v])
nipkow@23477
  2113
      apply (simp add: ring_simps) 
obua@14738
  2114
      done
obua@14738
  2115
  }
obua@14738
  2116
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  2117
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  2118
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  2119
  have xy: "- ?x <= ?y"
obua@14754
  2120
    apply (simp)
obua@14754
  2121
    apply (rule_tac y="0::'a" in order_trans)
nipkow@16568
  2122
    apply (rule addm2)
avigad@16775
  2123
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
nipkow@16568
  2124
    apply (rule addm)
avigad@16775
  2125
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14754
  2126
    done
obua@14738
  2127
  have yx: "?y <= ?x"
nipkow@16568
  2128
    apply (simp add:diff_def)
obua@14754
  2129
    apply (rule_tac y=0 in order_trans)
avigad@16775
  2130
    apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
avigad@16775
  2131
    apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
obua@14738
  2132
    done
obua@14738
  2133
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  2134
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  2135
  show ?thesis
obua@14738
  2136
    apply (rule abs_leI)
obua@14738
  2137
    apply (simp add: i1)
obua@14738
  2138
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  2139
    done
obua@14738
  2140
qed
paulson@14294
  2141
haftmann@25304
  2142
instance lordered_ring \<subseteq> pordered_ring_abs
haftmann@25304
  2143
proof
haftmann@25304
  2144
  fix a b :: "'a\<Colon> lordered_ring"
haftmann@25304
  2145
  assume "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
haftmann@25304
  2146
  show "abs (a*b) = abs a * abs b"
obua@14738
  2147
proof -
obua@14738
  2148
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  2149
    apply (auto)    
obua@14738
  2150
    apply (rule_tac split_mult_pos_le)
obua@14738
  2151
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  2152
    apply (simp)
obua@14738
  2153
    apply (rule_tac split_mult_neg_le)
obua@14738
  2154
    apply (insert prems)
obua@14738
  2155
    apply (blast)
obua@14738
  2156
    done
obua@14738
  2157
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  2158
    by (simp add: prts[symmetric])
obua@14738
  2159
  show ?thesis
obua@14738
  2160
  proof cases
obua@14738
  2161
    assume "0 <= a * b"
obua@14738
  2162
    then show ?thesis
obua@14738
  2163
      apply (simp_all add: mulprts abs_prts)
obua@14738
  2164
      apply (insert prems)
obua@14754
  2165
      apply (auto simp add: 
nipkow@23477
  2166
	ring_simps 
haftmann@25078
  2167
	iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
haftmann@25078
  2168
	iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
avigad@16775
  2169
	apply(drule (1) mult_nonneg_nonpos[of a b], simp)
avigad@16775
  2170
	apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
obua@14738
  2171
      done
obua@14738
  2172
  next
obua@14738
  2173
    assume "~(0 <= a*b)"
obua@14738
  2174
    with s have "a*b <= 0" by simp
obua@14738
  2175
    then show ?thesis
obua@14738
  2176
      apply (simp_all add: mulprts abs_prts)
obua@14738
  2177
      apply (insert prems)
nipkow@23477
  2178
      apply (auto simp add: ring_simps)
avigad@16775
  2179
      apply(drule (1) mult_nonneg_nonneg[of a b],simp)
avigad@16775
  2180
      apply(drule (1) mult_nonpos_nonpos[of a b],simp)
obua@14738
  2181
      done
obua@14738
  2182
  qed
obua@14738
  2183
qed
haftmann@25304
  2184
qed
haftmann@25304
  2185
haftmann@25304
  2186
instance ordered_idom \<subseteq> pordered_ring_abs
haftmann@25304
  2187
by default (auto simp add: abs_if not_less
haftmann@25304
  2188
  equal_neg_zero neg_equal_zero mult_less_0_iff)
paulson@14294
  2189
obua@14738
  2190
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
haftmann@25304
  2191
  by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  2192
obua@14738
  2193
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
haftmann@25304
  2194
  by (simp add: abs_if) 
paulson@14294
  2195
paulson@14294
  2196
lemma nonzero_abs_inverse:
paulson@14294
  2197
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  2198
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  2199
                      negative_imp_inverse_negative)
paulson@14294
  2200
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  2201
done
paulson@14294
  2202
paulson@14294
  2203
lemma abs_inverse [simp]:
paulson@14294
  2204
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  2205
      inverse (abs a)"
haftmann@21328
  2206
apply (cases "a=0", simp) 
paulson@14294
  2207
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  2208
done
paulson@14294
  2209
paulson@14294
  2210
lemma nonzero_abs_divide:
paulson@14294
  2211
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  2212
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  2213
paulson@15234
  2214
lemma abs_divide [simp]:
paulson@14294
  2215
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
haftmann@21328
  2216
apply (cases "b=0", simp) 
paulson@14294
  2217
apply (simp add: nonzero_abs_divide) 
paulson@14294
  2218
done
paulson@14294
  2219
paulson@14294
  2220
lemma abs_mult_less:
obua@14738
  2221
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  2222
proof -
paulson@14294
  2223
  assume ac: "abs a < c"
paulson@14294
  2224
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  2225
  assume "abs b < d"
paulson@14294
  2226
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  2227
qed
paulson@14293
  2228
haftmann@25304
  2229
lemmas eq_minus_self_iff = equal_neg_zero
obua@14738
  2230
obua@14738
  2231
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
haftmann@25304
  2232
  unfolding order_less_le less_eq_neg_nonpos equal_neg_zero ..
obua@14738
  2233
obua@14738
  2234
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  2235
apply (simp add: order_less_le abs_le_iff)  
haftmann@25304
  2236
apply (auto simp add: abs_if neg_less_eq_nonneg less_eq_neg_nonpos)
obua@14738
  2237
done
obua@14738
  2238
avigad@16775
  2239
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
haftmann@25304
  2240
    (abs y) * x = abs (y * x)"
haftmann@25304
  2241
  apply (subst abs_mult)
haftmann@25304
  2242
  apply simp
haftmann@25304
  2243
done
avigad@16775
  2244
avigad@16775
  2245
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
haftmann@25304
  2246
    abs x / y = abs (x / y)"
haftmann@25304
  2247
  apply (subst abs_divide)
haftmann@25304
  2248
  apply (simp add: order_less_imp_le)
haftmann@25304
  2249
done
avigad@16775
  2250
wenzelm@23389
  2251
obua@19404
  2252
subsection {* Bounds of products via negative and positive Part *}
obua@15178
  2253
obua@15580
  2254
lemma mult_le_prts:
obua@15580
  2255
  assumes
obua@15580
  2256
  "a1 <= (a::'a::lordered_ring)"
obua@15580
  2257
  "a <= a2"
obua@15580
  2258
  "b1 <= b"
obua@15580
  2259
  "b <= b2"
obua@15580
  2260
  shows
obua@15580
  2261
  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
obua@15580
  2262
proof - 
obua@15580
  2263
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
obua@15580
  2264
    apply (subst prts[symmetric])+
obua@15580
  2265
    apply simp
obua@15580
  2266
    done
obua@15580
  2267
  then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
nipkow@23477
  2268
    by (simp add: ring_simps)
obua@15580
  2269
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
obua@15580
  2270
    by (simp_all add: prems mult_mono)
obua@15580
  2271
  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
obua@15580
  2272
  proof -
obua@15580
  2273
    have "pprt a * nprt b <= pprt a * nprt b2"
obua@15580
  2274
      by (simp add: mult_left_mono prems)
obua@15580
  2275
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
obua@15580
  2276
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2277
    ultimately show ?thesis
obua@15580
  2278
      by simp
obua@15580
  2279
  qed
obua@15580
  2280
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
obua@15580
  2281
  proof - 
obua@15580
  2282
    have "nprt a * pprt b <= nprt a2 * pprt b"
obua@15580
  2283
      by (simp add: mult_right_mono prems)
obua@15580
  2284
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
obua@15580
  2285
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2286
    ultimately show ?thesis
obua@15580
  2287
      by simp
obua@15580
  2288
  qed
obua@15580
  2289
  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
obua@15580
  2290
  proof -
obua@15580
  2291
    have "nprt a * nprt b <= nprt a * nprt b1"
obua@15580
  2292
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2293
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
obua@15580
  2294
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2295
    ultimately show ?thesis
obua@15580
  2296
      by simp
obua@15580
  2297
  qed
obua@15580
  2298
  ultimately show ?thesis
obua@15580
  2299
    by - (rule add_mono | simp)+
obua@15580
  2300
qed
obua@19404
  2301
obua@19404
  2302
lemma mult_ge_prts:
obua@15178
  2303
  assumes
obua@19404
  2304
  "a1 <= (a::'a::lordered_ring)"
obua@19404
  2305
  "a <= a2"
obua@19404
  2306
  "b1 <= b"
obua@19404
  2307
  "b <= b2"
obua@15178
  2308
  shows
obua@19404
  2309
  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
obua@19404
  2310
proof - 
obua@19404
  2311
  from prems have a1:"- a2 <= -a" by auto
obua@19404
  2312
  from prems have a2: "-a <= -a1" by auto
obua@19404
  2313
  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
  2314
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
obua@19404
  2315
  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
obua@19404
  2316
    by (simp only: minus_le_iff)
obua@19404
  2317
  then show ?thesis by simp
obua@15178
  2318
qed
obua@15178
  2319
paulson@14265
  2320
end