src/HOL/Ring_and_Field.thy
author huffman
Sat Feb 14 19:01:31 2009 -0800 (2009-02-14)
changeset 29915 2146e512cec9
parent 29904 856f16a3b436
child 29925 17d1e32ef867
permissions -rw-r--r--
generalize lemma fps_square_eq_iff, move to Ring_and_Field
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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[algebra_simps]: "(a + b) * c = a * c + b * c"
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  assumes right_distrib[algebra_simps]: "a * (b + c) = a * b + a * c"
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begin
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor:
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  "a * e + (b * e + c) = (a + b) * e + c"
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by (simp add: left_distrib add_ac)
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end
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "0 * a = 0"
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  assumes mult_zero_right [simp]: "a * 0 = 0"
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0" by (simp add: left_distrib [symmetric])
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  thus "0 * a = 0" by (simp only: add_left_cancel)
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next
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  fix a :: 'a
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  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: right_distrib [symmetric])
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  thus "a * 0 = 0" by (simp only: add_left_cancel)
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qed
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end
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a + b) * c = a * c + b * c"
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begin
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subclass semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: mult_ac)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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end
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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begin
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subclass semiring_0 ..
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end
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class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0_cancel ..
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subclass comm_semiring_0 ..
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end
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "0 \<noteq> 1"
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begin
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lemma one_neq_zero [simp]: "1 \<noteq> 0"
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by (rule not_sym) (rule zero_neq_one)
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end
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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text {* Abstract divisibility *}
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class dvd = times
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begin
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definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where
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  [code del]: "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
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lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
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  unfolding dvd_def ..
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding dvd_def by blast 
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end
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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd
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  (*previously almost_semiring*)
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begin
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subclass semiring_1 ..
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lemma dvd_refl: "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 + cancel_comm_monoid_add
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  + zero_neq_one + 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 + cancel_comm_monoid_add
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  + zero_neq_one + comm_monoid_mult
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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, noatp] = minus_mult_left [symmetric]
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lemmas mult_minus_right [simp,noatp] = 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[algebra_simps]: "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[algebra_simps]: "(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[noatp] =
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  right_distrib left_distrib left_diff_distrib right_diff_distrib
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text{*Legacy - use @{text algebra_simps} *}
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lemmas ring_simps[noatp] = algebra_simps
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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: algebra_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: algebra_simps)
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end
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lemmas ring_distribs[noatp] =
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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 [simp]: "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 [simp]: "- 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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lemma dvd_diff: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
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by (simp add: diff_minus dvd_add dvd_minus_iff)
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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: algebra_simps right_minus_eq)
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  thus ?thesis 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: algebra_simps right_minus_eq)
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  thus ?thesis by (simp add: right_minus_eq)
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qed
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end
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class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
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begin
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lemma mult_cancel_right1 [simp]:
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  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   356
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   357
haftmann@26274
   358
lemma mult_cancel_right2 [simp]:
haftmann@26274
   359
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   360
by (insert mult_cancel_right [of a c 1], simp)
haftmann@26274
   361
 
haftmann@26274
   362
lemma mult_cancel_left1 [simp]:
haftmann@26274
   363
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   364
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   365
haftmann@26274
   366
lemma mult_cancel_left2 [simp]:
haftmann@26274
   367
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   368
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   369
haftmann@26274
   370
end
huffman@22990
   371
haftmann@22390
   372
class idom = comm_ring_1 + no_zero_divisors
haftmann@25186
   373
begin
paulson@14421
   374
huffman@27516
   375
subclass ring_1_no_zero_divisors ..
huffman@22990
   376
huffman@29915
   377
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)"
huffman@29915
   378
proof
huffman@29915
   379
  assume "a * a = b * b"
huffman@29915
   380
  then have "(a - b) * (a + b) = 0"
huffman@29915
   381
    by (simp add: algebra_simps)
huffman@29915
   382
  then show "a = b \<or> a = - b"
huffman@29915
   383
    by (simp add: right_minus_eq eq_neg_iff_add_eq_0)
huffman@29915
   384
next
huffman@29915
   385
  assume "a = b \<or> a = - b"
huffman@29915
   386
  then show "a * a = b * b" by auto
huffman@29915
   387
qed
huffman@29915
   388
haftmann@25186
   389
end
haftmann@25152
   390
haftmann@22390
   391
class division_ring = ring_1 + inverse +
haftmann@25062
   392
  assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
haftmann@25062
   393
  assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
haftmann@25186
   394
begin
huffman@20496
   395
haftmann@25186
   396
subclass ring_1_no_zero_divisors
haftmann@28823
   397
proof
huffman@22987
   398
  fix a b :: 'a
huffman@22987
   399
  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
huffman@22987
   400
  show "a * b \<noteq> 0"
huffman@22987
   401
  proof
huffman@22987
   402
    assume ab: "a * b = 0"
nipkow@29667
   403
    hence "0 = inverse a * (a * b) * inverse b" by simp
huffman@22987
   404
    also have "\<dots> = (inverse a * a) * (b * inverse b)"
huffman@22987
   405
      by (simp only: mult_assoc)
nipkow@29667
   406
    also have "\<dots> = 1" using a b by simp
nipkow@29667
   407
    finally show False by simp
huffman@22987
   408
  qed
huffman@22987
   409
qed
huffman@20496
   410
haftmann@26274
   411
lemma nonzero_imp_inverse_nonzero:
haftmann@26274
   412
  "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
haftmann@26274
   413
proof
haftmann@26274
   414
  assume ianz: "inverse a = 0"
haftmann@26274
   415
  assume "a \<noteq> 0"
haftmann@26274
   416
  hence "1 = a * inverse a" by simp
haftmann@26274
   417
  also have "... = 0" by (simp add: ianz)
haftmann@26274
   418
  finally have "1 = 0" .
haftmann@26274
   419
  thus False by (simp add: eq_commute)
haftmann@26274
   420
qed
haftmann@26274
   421
haftmann@26274
   422
lemma inverse_zero_imp_zero:
haftmann@26274
   423
  "inverse a = 0 \<Longrightarrow> a = 0"
haftmann@26274
   424
apply (rule classical)
haftmann@26274
   425
apply (drule nonzero_imp_inverse_nonzero)
haftmann@26274
   426
apply auto
haftmann@26274
   427
done
haftmann@26274
   428
haftmann@26274
   429
lemma inverse_unique: 
haftmann@26274
   430
  assumes ab: "a * b = 1"
haftmann@26274
   431
  shows "inverse a = b"
haftmann@26274
   432
proof -
haftmann@26274
   433
  have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
huffman@29406
   434
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
huffman@29406
   435
  ultimately show ?thesis by (simp add: mult_assoc [symmetric])
haftmann@26274
   436
qed
haftmann@26274
   437
huffman@29406
   438
lemma nonzero_inverse_minus_eq:
huffman@29406
   439
  "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
nipkow@29667
   440
by (rule inverse_unique) simp
huffman@29406
   441
huffman@29406
   442
lemma nonzero_inverse_inverse_eq:
huffman@29406
   443
  "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
nipkow@29667
   444
by (rule inverse_unique) simp
huffman@29406
   445
huffman@29406
   446
lemma nonzero_inverse_eq_imp_eq:
huffman@29406
   447
  assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
huffman@29406
   448
  shows "a = b"
huffman@29406
   449
proof -
huffman@29406
   450
  from `inverse a = inverse b`
nipkow@29667
   451
  have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
huffman@29406
   452
  with `a \<noteq> 0` and `b \<noteq> 0` show "a = b"
huffman@29406
   453
    by (simp add: nonzero_inverse_inverse_eq)
huffman@29406
   454
qed
huffman@29406
   455
huffman@29406
   456
lemma inverse_1 [simp]: "inverse 1 = 1"
nipkow@29667
   457
by (rule inverse_unique) simp
huffman@29406
   458
haftmann@26274
   459
lemma nonzero_inverse_mult_distrib: 
huffman@29406
   460
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@26274
   461
  shows "inverse (a * b) = inverse b * inverse a"
haftmann@26274
   462
proof -
nipkow@29667
   463
  have "a * (b * inverse b) * inverse a = 1" using assms by simp
nipkow@29667
   464
  hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc)
nipkow@29667
   465
  thus ?thesis by (rule inverse_unique)
haftmann@26274
   466
qed
haftmann@26274
   467
haftmann@26274
   468
lemma division_ring_inverse_add:
haftmann@26274
   469
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
nipkow@29667
   470
by (simp add: algebra_simps)
haftmann@26274
   471
haftmann@26274
   472
lemma division_ring_inverse_diff:
haftmann@26274
   473
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
nipkow@29667
   474
by (simp add: algebra_simps)
haftmann@26274
   475
haftmann@25186
   476
end
haftmann@25152
   477
huffman@22987
   478
class field = comm_ring_1 + inverse +
haftmann@25062
   479
  assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
haftmann@25062
   480
  assumes divide_inverse: "a / b = a * inverse b"
haftmann@25267
   481
begin
huffman@20496
   482
haftmann@25267
   483
subclass division_ring
haftmann@28823
   484
proof
huffman@22987
   485
  fix a :: 'a
huffman@22987
   486
  assume "a \<noteq> 0"
huffman@22987
   487
  thus "inverse a * a = 1" by (rule field_inverse)
huffman@22987
   488
  thus "a * inverse a = 1" by (simp only: mult_commute)
obua@14738
   489
qed
haftmann@25230
   490
huffman@27516
   491
subclass idom ..
haftmann@25230
   492
haftmann@25230
   493
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
haftmann@25230
   494
proof
haftmann@25230
   495
  assume neq: "b \<noteq> 0"
haftmann@25230
   496
  {
haftmann@25230
   497
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
haftmann@25230
   498
    also assume "a / b = 1"
haftmann@25230
   499
    finally show "a = b" by simp
haftmann@25230
   500
  next
haftmann@25230
   501
    assume "a = b"
haftmann@25230
   502
    with neq show "a / b = 1" by (simp add: divide_inverse)
haftmann@25230
   503
  }
haftmann@25230
   504
qed
haftmann@25230
   505
haftmann@25230
   506
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
nipkow@29667
   507
by (simp add: divide_inverse)
haftmann@25230
   508
haftmann@25230
   509
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
nipkow@29667
   510
by (simp add: divide_inverse)
haftmann@25230
   511
haftmann@25230
   512
lemma divide_zero_left [simp]: "0 / a = 0"
nipkow@29667
   513
by (simp add: divide_inverse)
haftmann@25230
   514
haftmann@25230
   515
lemma inverse_eq_divide: "inverse a = 1 / a"
nipkow@29667
   516
by (simp add: divide_inverse)
haftmann@25230
   517
haftmann@25230
   518
lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
nipkow@29667
   519
by (simp add: divide_inverse algebra_simps) 
haftmann@25230
   520
haftmann@25230
   521
end
haftmann@25230
   522
haftmann@22390
   523
class division_by_zero = zero + inverse +
haftmann@25062
   524
  assumes inverse_zero [simp]: "inverse 0 = 0"
paulson@14265
   525
haftmann@25230
   526
lemma divide_zero [simp]:
haftmann@25230
   527
  "a / 0 = (0::'a::{field,division_by_zero})"
nipkow@29667
   528
by (simp add: divide_inverse)
haftmann@25230
   529
haftmann@25230
   530
lemma divide_self_if [simp]:
haftmann@25230
   531
  "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
nipkow@29667
   532
by simp
haftmann@25230
   533
haftmann@22390
   534
class mult_mono = times + zero + ord +
haftmann@25062
   535
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25062
   536
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
paulson@14267
   537
haftmann@22390
   538
class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
haftmann@25230
   539
begin
haftmann@25230
   540
haftmann@25230
   541
lemma mult_mono:
haftmann@25230
   542
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c
haftmann@25230
   543
     \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   544
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   545
apply (erule mult_left_mono, assumption)
haftmann@25230
   546
done
haftmann@25230
   547
haftmann@25230
   548
lemma mult_mono':
haftmann@25230
   549
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c
haftmann@25230
   550
     \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   551
apply (rule mult_mono)
haftmann@25230
   552
apply (fast intro: order_trans)+
haftmann@25230
   553
done
haftmann@25230
   554
haftmann@25230
   555
end
krauss@21199
   556
haftmann@22390
   557
class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
huffman@29904
   558
  + semiring + cancel_comm_monoid_add
haftmann@25267
   559
begin
paulson@14268
   560
huffman@27516
   561
subclass semiring_0_cancel ..
huffman@27516
   562
subclass pordered_semiring ..
obua@23521
   563
haftmann@25230
   564
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
nipkow@29667
   565
by (drule mult_left_mono [of zero b], auto)
haftmann@25230
   566
haftmann@25230
   567
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
nipkow@29667
   568
by (drule mult_left_mono [of b zero], auto)
haftmann@25230
   569
haftmann@25230
   570
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
nipkow@29667
   571
by (drule mult_right_mono [of b zero], auto)
haftmann@25230
   572
haftmann@26234
   573
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
nipkow@29667
   574
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   575
haftmann@25230
   576
end
haftmann@25230
   577
haftmann@25230
   578
class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
haftmann@25267
   579
begin
haftmann@25230
   580
huffman@27516
   581
subclass pordered_cancel_semiring ..
haftmann@25512
   582
huffman@27516
   583
subclass pordered_comm_monoid_add ..
haftmann@25304
   584
haftmann@25230
   585
lemma mult_left_less_imp_less:
haftmann@25230
   586
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   587
by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   588
 
haftmann@25230
   589
lemma mult_right_less_imp_less:
haftmann@25230
   590
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   591
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   592
haftmann@25186
   593
end
haftmann@25152
   594
haftmann@22390
   595
class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
haftmann@25062
   596
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   597
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   598
begin
paulson@14341
   599
huffman@27516
   600
subclass semiring_0_cancel ..
obua@14940
   601
haftmann@25267
   602
subclass ordered_semiring
haftmann@28823
   603
proof
huffman@23550
   604
  fix a b c :: 'a
huffman@23550
   605
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   606
  from A show "c * a \<le> c * b"
haftmann@25186
   607
    unfolding le_less
haftmann@25186
   608
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   609
  from A show "a * c \<le> b * c"
haftmann@25152
   610
    unfolding le_less
haftmann@25186
   611
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   612
qed
haftmann@25152
   613
haftmann@25230
   614
lemma mult_left_le_imp_le:
haftmann@25230
   615
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   616
by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   617
 
haftmann@25230
   618
lemma mult_right_le_imp_le:
haftmann@25230
   619
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   620
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   621
haftmann@25230
   622
lemma mult_pos_pos:
haftmann@25230
   623
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
nipkow@29667
   624
by (drule mult_strict_left_mono [of zero b], auto)
haftmann@25230
   625
haftmann@25230
   626
lemma mult_pos_neg:
haftmann@25230
   627
  "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
nipkow@29667
   628
by (drule mult_strict_left_mono [of b zero], auto)
haftmann@25230
   629
haftmann@25230
   630
lemma mult_pos_neg2:
haftmann@25230
   631
  "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
nipkow@29667
   632
by (drule mult_strict_right_mono [of b zero], auto)
haftmann@25230
   633
haftmann@25230
   634
lemma zero_less_mult_pos:
haftmann@25230
   635
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
haftmann@25230
   636
apply (cases "b\<le>0") 
haftmann@25230
   637
 apply (auto simp add: le_less not_less)
haftmann@25230
   638
apply (drule_tac mult_pos_neg [of a b]) 
haftmann@25230
   639
 apply (auto dest: less_not_sym)
haftmann@25230
   640
done
haftmann@25230
   641
haftmann@25230
   642
lemma zero_less_mult_pos2:
haftmann@25230
   643
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
haftmann@25230
   644
apply (cases "b\<le>0") 
haftmann@25230
   645
 apply (auto simp add: le_less not_less)
haftmann@25230
   646
apply (drule_tac mult_pos_neg2 [of a b]) 
haftmann@25230
   647
 apply (auto dest: less_not_sym)
haftmann@25230
   648
done
haftmann@25230
   649
haftmann@26193
   650
text{*Strict monotonicity in both arguments*}
haftmann@26193
   651
lemma mult_strict_mono:
haftmann@26193
   652
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
   653
  shows "a * c < b * d"
haftmann@26193
   654
  using assms apply (cases "c=0")
haftmann@26193
   655
  apply (simp add: mult_pos_pos) 
haftmann@26193
   656
  apply (erule mult_strict_right_mono [THEN less_trans])
haftmann@26193
   657
  apply (force simp add: le_less) 
haftmann@26193
   658
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
   659
  done
haftmann@26193
   660
haftmann@26193
   661
text{*This weaker variant has more natural premises*}
haftmann@26193
   662
lemma mult_strict_mono':
haftmann@26193
   663
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
   664
  shows "a * c < b * d"
nipkow@29667
   665
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
   666
haftmann@26193
   667
lemma mult_less_le_imp_less:
haftmann@26193
   668
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
   669
  shows "a * c < b * d"
haftmann@26193
   670
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
   671
  apply (erule less_le_trans)
haftmann@26193
   672
  apply (erule mult_left_mono)
haftmann@26193
   673
  apply simp
haftmann@26193
   674
  apply (erule mult_strict_right_mono)
haftmann@26193
   675
  apply assumption
haftmann@26193
   676
  done
haftmann@26193
   677
haftmann@26193
   678
lemma mult_le_less_imp_less:
haftmann@26193
   679
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
   680
  shows "a * c < b * d"
haftmann@26193
   681
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
   682
  apply (erule le_less_trans)
haftmann@26193
   683
  apply (erule mult_strict_left_mono)
haftmann@26193
   684
  apply simp
haftmann@26193
   685
  apply (erule mult_right_mono)
haftmann@26193
   686
  apply simp
haftmann@26193
   687
  done
haftmann@26193
   688
haftmann@26193
   689
lemma mult_less_imp_less_left:
haftmann@26193
   690
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
haftmann@26193
   691
  shows "a < b"
haftmann@26193
   692
proof (rule ccontr)
haftmann@26193
   693
  assume "\<not>  a < b"
haftmann@26193
   694
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   695
  hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)
nipkow@29667
   696
  with this and less show False by (simp add: not_less [symmetric])
haftmann@26193
   697
qed
haftmann@26193
   698
haftmann@26193
   699
lemma mult_less_imp_less_right:
haftmann@26193
   700
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
haftmann@26193
   701
  shows "a < b"
haftmann@26193
   702
proof (rule ccontr)
haftmann@26193
   703
  assume "\<not> a < b"
haftmann@26193
   704
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   705
  hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)
nipkow@29667
   706
  with this and less show False by (simp add: not_less [symmetric])
haftmann@26193
   707
qed  
haftmann@26193
   708
haftmann@25230
   709
end
haftmann@25230
   710
haftmann@22390
   711
class mult_mono1 = times + zero + ord +
haftmann@25230
   712
  assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
paulson@14270
   713
haftmann@22390
   714
class pordered_comm_semiring = comm_semiring_0
haftmann@22390
   715
  + pordered_ab_semigroup_add + mult_mono1
haftmann@25186
   716
begin
haftmann@25152
   717
haftmann@25267
   718
subclass pordered_semiring
haftmann@28823
   719
proof
krauss@21199
   720
  fix a b c :: 'a
huffman@23550
   721
  assume "a \<le> b" "0 \<le> c"
haftmann@25230
   722
  thus "c * a \<le> c * b" by (rule mult_mono1)
huffman@23550
   723
  thus "a * c \<le> b * c" by (simp only: mult_commute)
krauss@21199
   724
qed
paulson@14265
   725
haftmann@25267
   726
end
haftmann@25267
   727
haftmann@25267
   728
class pordered_cancel_comm_semiring = comm_semiring_0_cancel
haftmann@25267
   729
  + pordered_ab_semigroup_add + mult_mono1
haftmann@25267
   730
begin
paulson@14265
   731
huffman@27516
   732
subclass pordered_comm_semiring ..
huffman@27516
   733
subclass pordered_cancel_semiring ..
haftmann@25267
   734
haftmann@25267
   735
end
haftmann@25267
   736
haftmann@25267
   737
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
haftmann@26193
   738
  assumes mult_strict_left_mono_comm: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
   739
begin
haftmann@25267
   740
haftmann@25267
   741
subclass ordered_semiring_strict
haftmann@28823
   742
proof
huffman@23550
   743
  fix a b c :: 'a
huffman@23550
   744
  assume "a < b" "0 < c"
haftmann@26193
   745
  thus "c * a < c * b" by (rule mult_strict_left_mono_comm)
huffman@23550
   746
  thus "a * c < b * c" by (simp only: mult_commute)
huffman@23550
   747
qed
paulson@14272
   748
haftmann@25267
   749
subclass pordered_cancel_comm_semiring
haftmann@28823
   750
proof
huffman@23550
   751
  fix a b c :: 'a
huffman@23550
   752
  assume "a \<le> b" "0 \<le> c"
huffman@23550
   753
  thus "c * a \<le> c * b"
haftmann@25186
   754
    unfolding le_less
haftmann@26193
   755
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   756
qed
paulson@14272
   757
haftmann@25267
   758
end
haftmann@25230
   759
haftmann@25267
   760
class pordered_ring = ring + pordered_cancel_semiring 
haftmann@25267
   761
begin
haftmann@25230
   762
huffman@27516
   763
subclass pordered_ab_group_add ..
paulson@14270
   764
nipkow@29667
   765
text{*Legacy - use @{text algebra_simps} *}
nipkow@29833
   766
lemmas ring_simps[noatp] = algebra_simps
haftmann@25230
   767
haftmann@25230
   768
lemma less_add_iff1:
haftmann@25230
   769
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
   770
by (simp add: algebra_simps)
haftmann@25230
   771
haftmann@25230
   772
lemma less_add_iff2:
haftmann@25230
   773
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
   774
by (simp add: algebra_simps)
haftmann@25230
   775
haftmann@25230
   776
lemma le_add_iff1:
haftmann@25230
   777
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
   778
by (simp add: algebra_simps)
haftmann@25230
   779
haftmann@25230
   780
lemma le_add_iff2:
haftmann@25230
   781
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
   782
by (simp add: algebra_simps)
haftmann@25230
   783
haftmann@25230
   784
lemma mult_left_mono_neg:
haftmann@25230
   785
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@25230
   786
  apply (drule mult_left_mono [of _ _ "uminus c"])
haftmann@25230
   787
  apply (simp_all add: minus_mult_left [symmetric]) 
haftmann@25230
   788
  done
haftmann@25230
   789
haftmann@25230
   790
lemma mult_right_mono_neg:
haftmann@25230
   791
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   792
  apply (drule mult_right_mono [of _ _ "uminus c"])
haftmann@25230
   793
  apply (simp_all add: minus_mult_right [symmetric]) 
haftmann@25230
   794
  done
haftmann@25230
   795
haftmann@25230
   796
lemma mult_nonpos_nonpos:
haftmann@25230
   797
  "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
nipkow@29667
   798
by (drule mult_right_mono_neg [of a zero b]) auto
haftmann@25230
   799
haftmann@25230
   800
lemma split_mult_pos_le:
haftmann@25230
   801
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@29667
   802
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
haftmann@25186
   803
haftmann@25186
   804
end
paulson@14270
   805
haftmann@25762
   806
class abs_if = minus + uminus + ord + zero + abs +
haftmann@25762
   807
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@25762
   808
haftmann@25762
   809
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@25186
   810
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
nipkow@24506
   811
nipkow@25564
   812
lemma (in sgn_if) sgn0[simp]: "sgn 0 = 0"
nipkow@25564
   813
by(simp add:sgn_if)
nipkow@25564
   814
haftmann@25230
   815
class ordered_ring = ring + ordered_semiring
haftmann@25304
   816
  + ordered_ab_group_add + abs_if
haftmann@25304
   817
begin
haftmann@25304
   818
huffman@27516
   819
subclass pordered_ring ..
haftmann@25304
   820
haftmann@25304
   821
subclass pordered_ab_group_add_abs
haftmann@28823
   822
proof
haftmann@25304
   823
  fix a b
haftmann@25304
   824
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
nipkow@29667
   825
by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
haftmann@25304
   826
   (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]
haftmann@25304
   827
     neg_less_eq_nonneg less_eq_neg_nonpos, auto intro: add_nonneg_nonneg,
haftmann@25304
   828
      auto intro!: less_imp_le add_neg_neg)
haftmann@25304
   829
qed (auto simp add: abs_if less_eq_neg_nonpos neg_equal_zero)
haftmann@25304
   830
haftmann@25304
   831
end
obua@23521
   832
haftmann@25230
   833
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
haftmann@25230
   834
   Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
haftmann@25230
   835
 *)
haftmann@25230
   836
class ordered_ring_strict = ring + ordered_semiring_strict
haftmann@25304
   837
  + ordered_ab_group_add + abs_if
haftmann@25230
   838
begin
paulson@14348
   839
huffman@27516
   840
subclass ordered_ring ..
haftmann@25304
   841
paulson@14265
   842
lemma mult_strict_left_mono_neg:
haftmann@25230
   843
  "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
haftmann@25230
   844
  apply (drule mult_strict_left_mono [of _ _ "uminus c"])
haftmann@25230
   845
  apply (simp_all add: minus_mult_left [symmetric]) 
haftmann@25230
   846
  done
obua@14738
   847
paulson@14265
   848
lemma mult_strict_right_mono_neg:
haftmann@25230
   849
  "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
haftmann@25230
   850
  apply (drule mult_strict_right_mono [of _ _ "uminus c"])
haftmann@25230
   851
  apply (simp_all add: minus_mult_right [symmetric]) 
haftmann@25230
   852
  done
obua@14738
   853
haftmann@25230
   854
lemma mult_neg_neg:
haftmann@25230
   855
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
nipkow@29667
   856
by (drule mult_strict_right_mono_neg, auto)
obua@14738
   857
haftmann@25917
   858
subclass ring_no_zero_divisors
haftmann@28823
   859
proof
haftmann@25917
   860
  fix a b
haftmann@25917
   861
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
   862
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
   863
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
   864
  proof (cases "a < 0")
haftmann@25917
   865
    case True note A' = this
haftmann@25917
   866
    show ?thesis proof (cases "b < 0")
haftmann@25917
   867
      case True with A'
haftmann@25917
   868
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
   869
    next
haftmann@25917
   870
      case False with B have "0 < b" by auto
haftmann@25917
   871
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
   872
    qed
haftmann@25917
   873
  next
haftmann@25917
   874
    case False with A have A': "0 < a" by auto
haftmann@25917
   875
    show ?thesis proof (cases "b < 0")
haftmann@25917
   876
      case True with A'
haftmann@25917
   877
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
   878
    next
haftmann@25917
   879
      case False with B have "0 < b" by auto
haftmann@25917
   880
      with A' show ?thesis by (auto dest: mult_pos_pos)
haftmann@25917
   881
    qed
haftmann@25917
   882
  qed
haftmann@25917
   883
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
   884
qed
haftmann@25304
   885
paulson@14265
   886
lemma zero_less_mult_iff:
haftmann@25917
   887
  "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
haftmann@25917
   888
  apply (auto simp add: mult_pos_pos mult_neg_neg)
haftmann@25917
   889
  apply (simp_all add: not_less le_less)
haftmann@25917
   890
  apply (erule disjE) apply assumption defer
haftmann@25917
   891
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
   892
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
   893
  apply (erule disjE) apply assumption apply (drule sym) apply simp
haftmann@25917
   894
  apply (drule sym) apply simp
haftmann@25917
   895
  apply (blast dest: zero_less_mult_pos)
haftmann@25230
   896
  apply (blast dest: zero_less_mult_pos2)
haftmann@25230
   897
  done
huffman@22990
   898
paulson@14265
   899
lemma zero_le_mult_iff:
haftmann@25917
   900
  "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
nipkow@29667
   901
by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
   902
paulson@14265
   903
lemma mult_less_0_iff:
haftmann@25917
   904
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
haftmann@25917
   905
  apply (insert zero_less_mult_iff [of "-a" b]) 
haftmann@25917
   906
  apply (force simp add: minus_mult_left[symmetric]) 
haftmann@25917
   907
  done
paulson@14265
   908
paulson@14265
   909
lemma mult_le_0_iff:
haftmann@25917
   910
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
   911
  apply (insert zero_le_mult_iff [of "-a" b]) 
haftmann@25917
   912
  apply (force simp add: minus_mult_left[symmetric]) 
haftmann@25917
   913
  done
haftmann@25917
   914
haftmann@25917
   915
lemma zero_le_square [simp]: "0 \<le> a * a"
nipkow@29667
   916
by (simp add: zero_le_mult_iff linear)
haftmann@25917
   917
haftmann@25917
   918
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
nipkow@29667
   919
by (simp add: not_less)
haftmann@25917
   920
haftmann@26193
   921
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
   922
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
   923
haftmann@26193
   924
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
   925
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
   926
haftmann@26193
   927
lemma mult_less_cancel_right_disj:
haftmann@26193
   928
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   929
  apply (cases "c = 0")
haftmann@26193
   930
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
   931
                      mult_strict_right_mono_neg)
haftmann@26193
   932
  apply (auto simp add: not_less 
haftmann@26193
   933
                      not_le [symmetric, of "a*c"]
haftmann@26193
   934
                      not_le [symmetric, of a])
haftmann@26193
   935
  apply (erule_tac [!] notE)
haftmann@26193
   936
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
   937
                      mult_right_mono_neg)
haftmann@26193
   938
  done
haftmann@26193
   939
haftmann@26193
   940
lemma mult_less_cancel_left_disj:
haftmann@26193
   941
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   942
  apply (cases "c = 0")
haftmann@26193
   943
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
   944
                      mult_strict_left_mono_neg)
haftmann@26193
   945
  apply (auto simp add: not_less 
haftmann@26193
   946
                      not_le [symmetric, of "c*a"]
haftmann@26193
   947
                      not_le [symmetric, of a])
haftmann@26193
   948
  apply (erule_tac [!] notE)
haftmann@26193
   949
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
   950
                      mult_left_mono_neg)
haftmann@26193
   951
  done
haftmann@26193
   952
haftmann@26193
   953
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
   954
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
   955
haftmann@26193
   956
lemma mult_less_cancel_right:
haftmann@26193
   957
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   958
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
   959
haftmann@26193
   960
lemma mult_less_cancel_left:
haftmann@26193
   961
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   962
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
   963
haftmann@26193
   964
lemma mult_le_cancel_right:
haftmann@26193
   965
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
   966
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
   967
haftmann@26193
   968
lemma mult_le_cancel_left:
haftmann@26193
   969
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
   970
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
   971
haftmann@25917
   972
end
paulson@14265
   973
nipkow@29667
   974
text{*Legacy - use @{text algebra_simps} *}
nipkow@29833
   975
lemmas ring_simps[noatp] = algebra_simps
haftmann@25230
   976
haftmann@25230
   977
haftmann@25230
   978
class pordered_comm_ring = comm_ring + pordered_comm_semiring
haftmann@25267
   979
begin
haftmann@25230
   980
huffman@27516
   981
subclass pordered_ring ..
huffman@27516
   982
subclass pordered_cancel_comm_semiring ..
haftmann@25230
   983
haftmann@25267
   984
end
haftmann@25230
   985
haftmann@25230
   986
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
haftmann@25230
   987
  (*previously ordered_semiring*)
haftmann@25230
   988
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
   989
begin
haftmann@25230
   990
haftmann@25230
   991
lemma pos_add_strict:
haftmann@25230
   992
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@25230
   993
  using add_strict_mono [of zero a b c] by simp
haftmann@25230
   994
haftmann@26193
   995
lemma zero_le_one [simp]: "0 \<le> 1"
nipkow@29667
   996
by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
   997
haftmann@26193
   998
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
nipkow@29667
   999
by (simp add: not_le) 
haftmann@26193
  1000
haftmann@26193
  1001
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
nipkow@29667
  1002
by (simp add: not_less) 
haftmann@26193
  1003
haftmann@26193
  1004
lemma less_1_mult:
haftmann@26193
  1005
  assumes "1 < m" and "1 < n"
haftmann@26193
  1006
  shows "1 < m * n"
haftmann@26193
  1007
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
  1008
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
  1009
haftmann@25230
  1010
end
haftmann@25230
  1011
haftmann@26193
  1012
class ordered_idom = comm_ring_1 +
haftmann@26193
  1013
  ordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1014
  abs_if + sgn_if
haftmann@25230
  1015
  (*previously ordered_ring*)
haftmann@25917
  1016
begin
haftmann@25917
  1017
huffman@27516
  1018
subclass ordered_ring_strict ..
huffman@27516
  1019
subclass pordered_comm_ring ..
huffman@27516
  1020
subclass idom ..
haftmann@25917
  1021
haftmann@25917
  1022
subclass ordered_semidom
haftmann@28823
  1023
proof
haftmann@26193
  1024
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1025
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
  1026
qed 
haftmann@25917
  1027
haftmann@26193
  1028
lemma linorder_neqE_ordered_idom:
haftmann@26193
  1029
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1030
  using assms by (rule neqE)
haftmann@26193
  1031
haftmann@26274
  1032
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1033
haftmann@26274
  1034
lemma mult_le_cancel_right1:
haftmann@26274
  1035
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1036
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1037
haftmann@26274
  1038
lemma mult_le_cancel_right2:
haftmann@26274
  1039
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1040
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1041
haftmann@26274
  1042
lemma mult_le_cancel_left1:
haftmann@26274
  1043
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1044
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1045
haftmann@26274
  1046
lemma mult_le_cancel_left2:
haftmann@26274
  1047
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1048
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1049
haftmann@26274
  1050
lemma mult_less_cancel_right1:
haftmann@26274
  1051
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1052
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1053
haftmann@26274
  1054
lemma mult_less_cancel_right2:
haftmann@26274
  1055
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1056
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1057
haftmann@26274
  1058
lemma mult_less_cancel_left1:
haftmann@26274
  1059
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1060
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1061
haftmann@26274
  1062
lemma mult_less_cancel_left2:
haftmann@26274
  1063
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1064
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1065
haftmann@27651
  1066
lemma sgn_sgn [simp]:
haftmann@27651
  1067
  "sgn (sgn a) = sgn a"
nipkow@29700
  1068
unfolding sgn_if by simp
haftmann@27651
  1069
haftmann@27651
  1070
lemma sgn_0_0:
haftmann@27651
  1071
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1072
unfolding sgn_if by simp
haftmann@27651
  1073
haftmann@27651
  1074
lemma sgn_1_pos:
haftmann@27651
  1075
  "sgn a = 1 \<longleftrightarrow> a > 0"
nipkow@29700
  1076
unfolding sgn_if by (simp add: neg_equal_zero)
haftmann@27651
  1077
haftmann@27651
  1078
lemma sgn_1_neg:
haftmann@27651
  1079
  "sgn a = - 1 \<longleftrightarrow> a < 0"
nipkow@29700
  1080
unfolding sgn_if by (auto simp add: equal_neg_zero)
haftmann@27651
  1081
haftmann@27651
  1082
lemma sgn_times:
haftmann@27651
  1083
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1084
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1085
haftmann@29653
  1086
lemma abs_sgn: "abs k = k * sgn k"
nipkow@29700
  1087
unfolding sgn_if abs_if by auto
nipkow@29700
  1088
nipkow@29700
  1089
(* The int instances are proved, these generic ones are tedious to prove here.
nipkow@29700
  1090
And not very useful, as int seems to be the only instance.
nipkow@29700
  1091
If needed, they should be proved later, when metis is available.
nipkow@29700
  1092
lemma dvd_abs[simp]: "(abs m) dvd k \<longleftrightarrow> m dvd k"
nipkow@29700
  1093
proof-
nipkow@29700
  1094
  have "\<forall>k.\<exists>ka. - (m * k) = m * ka"
nipkow@29700
  1095
    by(simp add: mult_minus_right[symmetric] del: mult_minus_right)
nipkow@29700
  1096
  moreover
nipkow@29700
  1097
  have "\<forall>k.\<exists>ka. m * k = - (m * ka)"
nipkow@29700
  1098
    by(auto intro!: minus_minus[symmetric]
nipkow@29700
  1099
         simp add: mult_minus_right[symmetric] simp del: mult_minus_right)
nipkow@29700
  1100
  ultimately show ?thesis by (auto simp: abs_if dvd_def)
nipkow@29700
  1101
qed
nipkow@29700
  1102
nipkow@29700
  1103
lemma dvd_abs2[simp]: "m dvd (abs k) \<longleftrightarrow> m dvd k"
nipkow@29700
  1104
proof-
nipkow@29700
  1105
  have "\<forall>k.\<exists>ka. - (m * k) = m * ka"
nipkow@29700
  1106
    by(simp add: mult_minus_right[symmetric] del: mult_minus_right)
nipkow@29700
  1107
  moreover
nipkow@29700
  1108
  have "\<forall>k.\<exists>ka. - (m * ka) = m * k"
nipkow@29700
  1109
    by(auto intro!: minus_minus
nipkow@29700
  1110
         simp add: mult_minus_right[symmetric] simp del: mult_minus_right)
nipkow@29700
  1111
  ultimately show ?thesis
nipkow@29700
  1112
    by (auto simp add:abs_if dvd_def minus_equation_iff[of k])
nipkow@29700
  1113
qed
nipkow@29700
  1114
*)
haftmann@29653
  1115
haftmann@25917
  1116
end
haftmann@25230
  1117
haftmann@25230
  1118
class ordered_field = field + ordered_idom
haftmann@25230
  1119
haftmann@26274
  1120
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1121
nipkow@29833
  1122
lemmas mult_compare_simps[noatp] =
paulson@15234
  1123
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1124
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1125
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1126
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1127
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1128
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1129
    mult_cancel_right mult_cancel_left
paulson@15234
  1130
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1131
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1132
haftmann@26274
  1133
-- {* FIXME continue localization here *}
paulson@14268
  1134
paulson@14268
  1135
lemma inverse_nonzero_iff_nonzero [simp]:
huffman@20496
  1136
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
haftmann@26274
  1137
by (force dest: inverse_zero_imp_zero) 
paulson@14268
  1138
paulson@14268
  1139
lemma inverse_minus_eq [simp]:
huffman@20496
  1140
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
paulson@14377
  1141
proof cases
paulson@14377
  1142
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
  1143
next
paulson@14377
  1144
  assume "a\<noteq>0" 
paulson@14377
  1145
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
  1146
qed
paulson@14268
  1147
paulson@14268
  1148
lemma inverse_eq_imp_eq:
huffman@20496
  1149
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
haftmann@21328
  1150
apply (cases "a=0 | b=0") 
paulson@14268
  1151
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
  1152
              simp add: eq_commute [of "0::'a"])
paulson@14268
  1153
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
  1154
done
paulson@14268
  1155
paulson@14268
  1156
lemma inverse_eq_iff_eq [simp]:
huffman@20496
  1157
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
huffman@20496
  1158
by (force dest!: inverse_eq_imp_eq)
paulson@14268
  1159
paulson@14270
  1160
lemma inverse_inverse_eq [simp]:
huffman@20496
  1161
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
paulson@14270
  1162
  proof cases
paulson@14270
  1163
    assume "a=0" thus ?thesis by simp
paulson@14270
  1164
  next
paulson@14270
  1165
    assume "a\<noteq>0" 
paulson@14270
  1166
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
  1167
  qed
paulson@14270
  1168
paulson@14270
  1169
text{*This version builds in division by zero while also re-orienting
paulson@14270
  1170
      the right-hand side.*}
paulson@14270
  1171
lemma inverse_mult_distrib [simp]:
paulson@14270
  1172
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
  1173
  proof cases
paulson@14270
  1174
    assume "a \<noteq> 0 & b \<noteq> 0" 
nipkow@29667
  1175
    thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
  1176
  next
paulson@14270
  1177
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
nipkow@29667
  1178
    thus ?thesis by force
paulson@14270
  1179
  qed
paulson@14270
  1180
paulson@14270
  1181
text{*There is no slick version using division by zero.*}
paulson@14270
  1182
lemma inverse_add:
nipkow@23477
  1183
  "[|a \<noteq> 0;  b \<noteq> 0|]
nipkow@23477
  1184
   ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
huffman@20496
  1185
by (simp add: division_ring_inverse_add mult_ac)
paulson@14270
  1186
paulson@14365
  1187
lemma inverse_divide [simp]:
nipkow@23477
  1188
  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
nipkow@23477
  1189
by (simp add: divide_inverse mult_commute)
paulson@14365
  1190
wenzelm@23389
  1191
avigad@16775
  1192
subsection {* Calculations with fractions *}
avigad@16775
  1193
nipkow@23413
  1194
text{* There is a whole bunch of simp-rules just for class @{text
nipkow@23413
  1195
field} but none for class @{text field} and @{text nonzero_divides}
nipkow@23413
  1196
because the latter are covered by a simproc. *}
nipkow@23413
  1197
paulson@24427
  1198
lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]:
nipkow@23477
  1199
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
  1200
proof -
paulson@14277
  1201
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
nipkow@23482
  1202
    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
paulson@14277
  1203
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
  1204
    by (simp only: mult_ac)
nipkow@29667
  1205
  also have "... =  a * inverse b" by simp
nipkow@29667
  1206
    finally show ?thesis by (simp add: divide_inverse)
paulson@14277
  1207
qed
paulson@14277
  1208
nipkow@23413
  1209
lemma mult_divide_mult_cancel_left:
nipkow@23477
  1210
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1211
apply (cases "b = 0")
nipkow@23413
  1212
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
paulson@14277
  1213
done
paulson@14277
  1214
paulson@24427
  1215
lemma nonzero_mult_divide_mult_cancel_right [noatp]:
nipkow@23477
  1216
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1217
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
paulson@14321
  1218
nipkow@23413
  1219
lemma mult_divide_mult_cancel_right:
nipkow@23477
  1220
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1221
apply (cases "b = 0")
nipkow@23413
  1222
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
paulson@14321
  1223
done
nipkow@23413
  1224
paulson@14284
  1225
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
nipkow@23477
  1226
by (simp add: divide_inverse)
paulson@14284
  1227
paulson@15234
  1228
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
  1229
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1230
paulson@14430
  1231
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
  1232
by (simp add: divide_inverse mult_ac)
paulson@14288
  1233
nipkow@29833
  1234
lemmas times_divide_eq[noatp] = times_divide_eq_right times_divide_eq_left
nipkow@23482
  1235
paulson@24286
  1236
lemma divide_divide_eq_right [simp,noatp]:
nipkow@23477
  1237
  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
  1238
by (simp add: divide_inverse mult_ac)
paulson@14288
  1239
paulson@24286
  1240
lemma divide_divide_eq_left [simp,noatp]:
nipkow@23477
  1241
  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
  1242
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1243
avigad@16775
  1244
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1245
    x / y + w / z = (x * z + w * y) / (y * z)"
nipkow@23477
  1246
apply (subgoal_tac "x / y = (x * z) / (y * z)")
nipkow@23477
  1247
apply (erule ssubst)
nipkow@23477
  1248
apply (subgoal_tac "w / z = (w * y) / (y * z)")
nipkow@23477
  1249
apply (erule ssubst)
nipkow@23477
  1250
apply (rule add_divide_distrib [THEN sym])
nipkow@23477
  1251
apply (subst mult_commute)
nipkow@23477
  1252
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
nipkow@23477
  1253
apply assumption
nipkow@23477
  1254
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
nipkow@23477
  1255
apply assumption
avigad@16775
  1256
done
paulson@14268
  1257
wenzelm@23389
  1258
paulson@15234
  1259
subsubsection{*Special Cancellation Simprules for Division*}
paulson@15234
  1260
paulson@24427
  1261
lemma mult_divide_mult_cancel_left_if[simp,noatp]:
nipkow@23477
  1262
fixes c :: "'a :: {field,division_by_zero}"
nipkow@23477
  1263
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
nipkow@23413
  1264
by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
  1265
paulson@24427
  1266
lemma nonzero_mult_divide_cancel_right[simp,noatp]:
nipkow@23413
  1267
  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
nipkow@23413
  1268
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
nipkow@23413
  1269
paulson@24427
  1270
lemma nonzero_mult_divide_cancel_left[simp,noatp]:
nipkow@23413
  1271
  "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
nipkow@23413
  1272
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
nipkow@23413
  1273
nipkow@23413
  1274
paulson@24427
  1275
lemma nonzero_divide_mult_cancel_right[simp,noatp]:
nipkow@23413
  1276
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
nipkow@23413
  1277
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
nipkow@23413
  1278
paulson@24427
  1279
lemma nonzero_divide_mult_cancel_left[simp,noatp]:
nipkow@23413
  1280
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
nipkow@23413
  1281
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
nipkow@23413
  1282
nipkow@23413
  1283
paulson@24427
  1284
lemma nonzero_mult_divide_mult_cancel_left2[simp,noatp]:
nipkow@23477
  1285
  "[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1286
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
nipkow@23413
  1287
paulson@24427
  1288
lemma nonzero_mult_divide_mult_cancel_right2[simp,noatp]:
nipkow@23477
  1289
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
nipkow@23413
  1290
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
nipkow@23413
  1291
paulson@15234
  1292
paulson@14293
  1293
subsection {* Division and Unary Minus *}
paulson@14293
  1294
paulson@14293
  1295
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
huffman@29407
  1296
by (simp add: divide_inverse)
paulson@14293
  1297
paulson@14293
  1298
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
huffman@29407
  1299
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
  1300
paulson@14293
  1301
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
  1302
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
  1303
paulson@14430
  1304
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
huffman@29407
  1305
by (simp add: divide_inverse)
paulson@14293
  1306
paulson@14293
  1307
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
huffman@29407
  1308
by (simp add: divide_inverse)
paulson@14430
  1309
paulson@14293
  1310
paulson@14293
  1311
text{*The effect is to extract signs from divisions*}
nipkow@29833
  1312
lemmas divide_minus_left[noatp] = minus_divide_left [symmetric]
nipkow@29833
  1313
lemmas divide_minus_right[noatp] = minus_divide_right [symmetric]
paulson@17085
  1314
declare divide_minus_left [simp]   divide_minus_right [simp]
paulson@14293
  1315
paulson@14293
  1316
lemma minus_divide_divide [simp]:
nipkow@23477
  1317
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1318
apply (cases "b=0", simp) 
paulson@14293
  1319
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
  1320
done
paulson@14293
  1321
paulson@14430
  1322
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
  1323
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
  1324
nipkow@23482
  1325
lemma add_divide_eq_iff:
nipkow@23482
  1326
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
nipkow@23482
  1327
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1328
nipkow@23482
  1329
lemma divide_add_eq_iff:
nipkow@23482
  1330
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
nipkow@23482
  1331
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1332
nipkow@23482
  1333
lemma diff_divide_eq_iff:
nipkow@23482
  1334
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
nipkow@23482
  1335
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1336
nipkow@23482
  1337
lemma divide_diff_eq_iff:
nipkow@23482
  1338
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
nipkow@23482
  1339
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
nipkow@23482
  1340
nipkow@23482
  1341
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
nipkow@23482
  1342
proof -
nipkow@23482
  1343
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1344
  have "(a = b/c) = (a*c = (b/c)*c)" by simp
nipkow@23496
  1345
  also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
nipkow@23482
  1346
  finally show ?thesis .
nipkow@23482
  1347
qed
nipkow@23482
  1348
nipkow@23482
  1349
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
nipkow@23482
  1350
proof -
nipkow@23482
  1351
  assume [simp]: "c\<noteq>0"
nipkow@23496
  1352
  have "(b/c = a) = ((b/c)*c = a*c)"  by simp
nipkow@23496
  1353
  also have "... = (b = a*c)"  by (simp add: divide_inverse mult_assoc) 
nipkow@23482
  1354
  finally show ?thesis .
nipkow@23482
  1355
qed
nipkow@23482
  1356
nipkow@23482
  1357
lemma eq_divide_eq:
nipkow@23482
  1358
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
nipkow@23482
  1359
by (simp add: nonzero_eq_divide_eq) 
nipkow@23482
  1360
nipkow@23482
  1361
lemma divide_eq_eq:
nipkow@23482
  1362
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
nipkow@23482
  1363
by (force simp add: nonzero_divide_eq_eq) 
nipkow@23482
  1364
nipkow@23482
  1365
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1366
    b = a * c ==> b / c = a"
nipkow@29667
  1367
by (subst divide_eq_eq, simp)
nipkow@23482
  1368
nipkow@23482
  1369
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
nipkow@23482
  1370
    a * c = b ==> a = b / c"
nipkow@29667
  1371
by (subst eq_divide_eq, simp)
nipkow@29667
  1372
nipkow@29667
  1373
nipkow@29833
  1374
lemmas field_eq_simps[noatp] = algebra_simps
nipkow@23482
  1375
  (* pull / out*)
nipkow@23482
  1376
  add_divide_eq_iff divide_add_eq_iff
nipkow@23482
  1377
  diff_divide_eq_iff divide_diff_eq_iff
nipkow@23482
  1378
  (* multiply eqn *)
nipkow@23482
  1379
  nonzero_eq_divide_eq nonzero_divide_eq_eq
nipkow@23482
  1380
(* is added later:
nipkow@23482
  1381
  times_divide_eq_left times_divide_eq_right
nipkow@23482
  1382
*)
nipkow@23482
  1383
nipkow@23482
  1384
text{*An example:*}
nipkow@23482
  1385
lemma fixes a b c d e f :: "'a::field"
nipkow@23482
  1386
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
  1387
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
nipkow@23482
  1388
 apply(simp add:field_eq_simps)
nipkow@23482
  1389
apply(simp)
nipkow@23482
  1390
done
nipkow@23482
  1391
nipkow@23482
  1392
avigad@16775
  1393
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1394
    x / y - w / z = (x * z - w * y) / (y * z)"
nipkow@23482
  1395
by (simp add:field_eq_simps times_divide_eq)
nipkow@23482
  1396
nipkow@23482
  1397
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
nipkow@23482
  1398
    (x / y = w / z) = (x * z = w * y)"
nipkow@23482
  1399
by (simp add:field_eq_simps times_divide_eq)
paulson@14293
  1400
wenzelm@23389
  1401
paulson@14268
  1402
subsection {* Ordered Fields *}
paulson@14268
  1403
paulson@14277
  1404
lemma positive_imp_inverse_positive: 
nipkow@23482
  1405
assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
nipkow@23482
  1406
proof -
paulson@14268
  1407
  have "0 < a * inverse a" 
paulson@14268
  1408
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
  1409
  thus "0 < inverse a" 
paulson@14268
  1410
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
nipkow@23482
  1411
qed
paulson@14268
  1412
paulson@14277
  1413
lemma negative_imp_inverse_negative:
nipkow@23482
  1414
  "a < 0 ==> inverse a < (0::'a::ordered_field)"
nipkow@23482
  1415
by (insert positive_imp_inverse_positive [of "-a"], 
nipkow@23482
  1416
    simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
paulson@14268
  1417
paulson@14268
  1418
lemma inverse_le_imp_le:
nipkow@23482
  1419
assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
nipkow@23482
  1420
shows "b \<le> (a::'a::ordered_field)"
nipkow@23482
  1421
proof (rule classical)
paulson@14268
  1422
  assume "~ b \<le> a"
nipkow@23482
  1423
  hence "a < b"  by (simp add: linorder_not_le)
nipkow@23482
  1424
  hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
paulson@14268
  1425
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
  1426
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
  1427
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
  1428
    by (simp add: bpos order_less_imp_le mult_right_mono)
nipkow@23482
  1429
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
nipkow@23482
  1430
qed
paulson@14268
  1431
paulson@14277
  1432
lemma inverse_positive_imp_positive:
nipkow@23482
  1433
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
nipkow@23482
  1434
shows "0 < (a::'a::ordered_field)"
wenzelm@23389
  1435
proof -
paulson@14277
  1436
  have "0 < inverse (inverse a)"
wenzelm@23389
  1437
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
  1438
  thus "0 < a"
wenzelm@23389
  1439
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1440
qed
paulson@14277
  1441
paulson@14277
  1442
lemma inverse_positive_iff_positive [simp]:
nipkow@23482
  1443
  "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1444
apply (cases "a = 0", simp)
paulson@14277
  1445
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
  1446
done
paulson@14277
  1447
paulson@14277
  1448
lemma inverse_negative_imp_negative:
nipkow@23482
  1449
assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
nipkow@23482
  1450
shows "a < (0::'a::ordered_field)"
wenzelm@23389
  1451
proof -
paulson@14277
  1452
  have "inverse (inverse a) < 0"
wenzelm@23389
  1453
    using inv_less_0 by (rule negative_imp_inverse_negative)
nipkow@23482
  1454
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1455
qed
paulson@14277
  1456
paulson@14277
  1457
lemma inverse_negative_iff_negative [simp]:
nipkow@23482
  1458
  "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1459
apply (cases "a = 0", simp)
paulson@14277
  1460
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
  1461
done
paulson@14277
  1462
paulson@14277
  1463
lemma inverse_nonnegative_iff_nonnegative [simp]:
nipkow@23482
  1464
  "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1465
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1466
paulson@14277
  1467
lemma inverse_nonpositive_iff_nonpositive [simp]:
nipkow@23482
  1468
  "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1469
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1470
chaieb@23406
  1471
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
chaieb@23406
  1472
proof
chaieb@23406
  1473
  fix x::'a
chaieb@23406
  1474
  have m1: "- (1::'a) < 0" by simp
chaieb@23406
  1475
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1476
  have "(- 1) + x < x" by simp
chaieb@23406
  1477
  thus "\<exists>y. y < x" by blast
chaieb@23406
  1478
qed
chaieb@23406
  1479
chaieb@23406
  1480
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
chaieb@23406
  1481
proof
chaieb@23406
  1482
  fix x::'a
chaieb@23406
  1483
  have m1: " (1::'a) > 0" by simp
chaieb@23406
  1484
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1485
  have "1 + x > x" by simp
chaieb@23406
  1486
  thus "\<exists>y. y > x" by blast
chaieb@23406
  1487
qed
paulson@14277
  1488
paulson@14277
  1489
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
  1490
paulson@14268
  1491
lemma less_imp_inverse_less:
nipkow@23482
  1492
assumes less: "a < b" and apos:  "0 < a"
nipkow@23482
  1493
shows "inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1494
proof (rule ccontr)
paulson@14268
  1495
  assume "~ inverse b < inverse a"
nipkow@29667
  1496
  hence "inverse a \<le> inverse b" by (simp add: linorder_not_less)
paulson@14268
  1497
  hence "~ (a < b)"
paulson@14268
  1498
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
nipkow@29667
  1499
  thus False by (rule notE [OF _ less])
nipkow@23482
  1500
qed
paulson@14268
  1501
paulson@14268
  1502
lemma inverse_less_imp_less:
nipkow@23482
  1503
  "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1504
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
  1505
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
  1506
done
paulson@14268
  1507
paulson@14268
  1508
text{*Both premises are essential. Consider -1 and 1.*}
paulson@24286
  1509
lemma inverse_less_iff_less [simp,noatp]:
nipkow@23482
  1510
  "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1511
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
  1512
paulson@14268
  1513
lemma le_imp_inverse_le:
nipkow@23482
  1514
  "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1515
by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
  1516
paulson@24286
  1517
lemma inverse_le_iff_le [simp,noatp]:
nipkow@23482
  1518
 "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1519
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
  1520
paulson@14268
  1521
paulson@14268
  1522
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
  1523
case is trivial, since inverse preserves signs.*}
paulson@14268
  1524
lemma inverse_le_imp_le_neg:
nipkow@23482
  1525
  "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
nipkow@23482
  1526
apply (rule classical) 
nipkow@23482
  1527
apply (subgoal_tac "a < 0") 
nipkow@23482
  1528
 prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
nipkow@23482
  1529
apply (insert inverse_le_imp_le [of "-b" "-a"])
nipkow@23482
  1530
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1531
done
paulson@14268
  1532
paulson@14268
  1533
lemma less_imp_inverse_less_neg:
paulson@14268
  1534
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1535
apply (subgoal_tac "a < 0") 
nipkow@23482
  1536
 prefer 2 apply (blast intro: order_less_trans) 
nipkow@23482
  1537
apply (insert less_imp_inverse_less [of "-b" "-a"])
nipkow@23482
  1538
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1539
done
paulson@14268
  1540
paulson@14268
  1541
lemma inverse_less_imp_less_neg:
paulson@14268
  1542
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
nipkow@23482
  1543
apply (rule classical) 
nipkow@23482
  1544
apply (subgoal_tac "a < 0") 
nipkow@23482
  1545
 prefer 2
nipkow@23482
  1546
 apply (force simp add: linorder_not_less intro: order_le_less_trans) 
nipkow@23482
  1547
apply (insert inverse_less_imp_less [of "-b" "-a"])
nipkow@23482
  1548
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1549
done
paulson@14268
  1550
paulson@24286
  1551
lemma inverse_less_iff_less_neg [simp,noatp]:
nipkow@23482
  1552
  "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
nipkow@23482
  1553
apply (insert inverse_less_iff_less [of "-b" "-a"])
nipkow@23482
  1554
apply (simp del: inverse_less_iff_less 
nipkow@23482
  1555
            add: order_less_imp_not_eq nonzero_inverse_minus_eq)
nipkow@23482
  1556
done
paulson@14268
  1557
paulson@14268
  1558
lemma le_imp_inverse_le_neg:
nipkow@23482
  1559
  "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1560
by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1561
paulson@24286
  1562
lemma inverse_le_iff_le_neg [simp,noatp]:
nipkow@23482
  1563
 "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1564
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1565
paulson@14277
  1566
paulson@14365
  1567
subsection{*Inverses and the Number One*}
paulson@14365
  1568
paulson@14365
  1569
lemma one_less_inverse_iff:
nipkow@23482
  1570
  "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
nipkow@23482
  1571
proof cases
paulson@14365
  1572
  assume "0 < x"
paulson@14365
  1573
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1574
    show ?thesis by simp
paulson@14365
  1575
next
paulson@14365
  1576
  assume notless: "~ (0 < x)"
paulson@14365
  1577
  have "~ (1 < inverse x)"
paulson@14365
  1578
  proof
paulson@14365
  1579
    assume "1 < inverse x"
paulson@14365
  1580
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1581
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1582
    finally show False by auto
paulson@14365
  1583
  qed
paulson@14365
  1584
  with notless show ?thesis by simp
paulson@14365
  1585
qed
paulson@14365
  1586
paulson@14365
  1587
lemma inverse_eq_1_iff [simp]:
nipkow@23482
  1588
  "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1589
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1590
paulson@14365
  1591
lemma one_le_inverse_iff:
nipkow@23482
  1592
  "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1593
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1594
                    eq_commute [of 1]) 
paulson@14365
  1595
paulson@14365
  1596
lemma inverse_less_1_iff:
nipkow@23482
  1597
  "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1598
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1599
paulson@14365
  1600
lemma inverse_le_1_iff:
nipkow@23482
  1601
  "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1602
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1603
wenzelm@23389
  1604
paulson@14288
  1605
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1606
paulson@14288
  1607
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1608
proof -
paulson@14288
  1609
  assume less: "0<c"
paulson@14288
  1610
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1611
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1612
  also have "... = (a*c \<le> b)"
paulson@14288
  1613
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1614
  finally show ?thesis .
paulson@14288
  1615
qed
paulson@14288
  1616
paulson@14288
  1617
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1618
proof -
paulson@14288
  1619
  assume less: "c<0"
paulson@14288
  1620
  hence "(a \<le> b/c) = ((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_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1624
  finally show ?thesis .
paulson@14288
  1625
qed
paulson@14288
  1626
paulson@14288
  1627
lemma le_divide_eq:
paulson@14288
  1628
  "(a \<le> b/c) = 
paulson@14288
  1629
   (if 0 < c then a*c \<le> b
paulson@14288
  1630
             else if c < 0 then b \<le> a*c
paulson@14288
  1631
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1632
apply (cases "c=0", simp) 
paulson@14288
  1633
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1634
done
paulson@14288
  1635
paulson@14288
  1636
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1637
proof -
paulson@14288
  1638
  assume less: "0<c"
paulson@14288
  1639
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1640
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1641
  also have "... = (b \<le> a*c)"
paulson@14288
  1642
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1643
  finally show ?thesis .
paulson@14288
  1644
qed
paulson@14288
  1645
paulson@14288
  1646
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1647
proof -
paulson@14288
  1648
  assume less: "c<0"
paulson@14288
  1649
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1650
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1651
  also have "... = (a*c \<le> b)"
paulson@14288
  1652
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1653
  finally show ?thesis .
paulson@14288
  1654
qed
paulson@14288
  1655
paulson@14288
  1656
lemma divide_le_eq:
paulson@14288
  1657
  "(b/c \<le> a) = 
paulson@14288
  1658
   (if 0 < c then b \<le> a*c
paulson@14288
  1659
             else if c < 0 then a*c \<le> b
paulson@14288
  1660
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1661
apply (cases "c=0", simp) 
paulson@14288
  1662
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1663
done
paulson@14288
  1664
paulson@14288
  1665
lemma pos_less_divide_eq:
paulson@14288
  1666
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1667
proof -
paulson@14288
  1668
  assume less: "0<c"
paulson@14288
  1669
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@15234
  1670
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1671
  also have "... = (a*c < b)"
paulson@14288
  1672
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1673
  finally show ?thesis .
paulson@14288
  1674
qed
paulson@14288
  1675
paulson@14288
  1676
lemma neg_less_divide_eq:
paulson@14288
  1677
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1678
proof -
paulson@14288
  1679
  assume less: "c<0"
paulson@14288
  1680
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@15234
  1681
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1682
  also have "... = (b < a*c)"
paulson@14288
  1683
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1684
  finally show ?thesis .
paulson@14288
  1685
qed
paulson@14288
  1686
paulson@14288
  1687
lemma less_divide_eq:
paulson@14288
  1688
  "(a < b/c) = 
paulson@14288
  1689
   (if 0 < c then a*c < b
paulson@14288
  1690
             else if c < 0 then b < a*c
paulson@14288
  1691
             else  a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1692
apply (cases "c=0", simp) 
paulson@14288
  1693
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1694
done
paulson@14288
  1695
paulson@14288
  1696
lemma pos_divide_less_eq:
paulson@14288
  1697
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1698
proof -
paulson@14288
  1699
  assume less: "0<c"
paulson@14288
  1700
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@15234
  1701
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1702
  also have "... = (b < a*c)"
paulson@14288
  1703
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1704
  finally show ?thesis .
paulson@14288
  1705
qed
paulson@14288
  1706
paulson@14288
  1707
lemma neg_divide_less_eq:
paulson@14288
  1708
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1709
proof -
paulson@14288
  1710
  assume less: "c<0"
paulson@14288
  1711
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@15234
  1712
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1713
  also have "... = (a*c < b)"
paulson@14288
  1714
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1715
  finally show ?thesis .
paulson@14288
  1716
qed
paulson@14288
  1717
paulson@14288
  1718
lemma divide_less_eq:
paulson@14288
  1719
  "(b/c < a) = 
paulson@14288
  1720
   (if 0 < c then b < a*c
paulson@14288
  1721
             else if c < 0 then a*c < b
paulson@14288
  1722
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1723
apply (cases "c=0", simp) 
paulson@14288
  1724
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1725
done
paulson@14288
  1726
nipkow@23482
  1727
nipkow@23482
  1728
subsection{*Field simplification*}
nipkow@23482
  1729
nipkow@29667
  1730
text{* Lemmas @{text field_simps} multiply with denominators in in(equations)
nipkow@29667
  1731
if they can be proved to be non-zero (for equations) or positive/negative
nipkow@29667
  1732
(for inequations). Can be too aggressive and is therefore separate from the
nipkow@29667
  1733
more benign @{text algebra_simps}. *}
paulson@14288
  1734
nipkow@29833
  1735
lemmas field_simps[noatp] = field_eq_simps
nipkow@23482
  1736
  (* multiply ineqn *)
nipkow@23482
  1737
  pos_divide_less_eq neg_divide_less_eq
nipkow@23482
  1738
  pos_less_divide_eq neg_less_divide_eq
nipkow@23482
  1739
  pos_divide_le_eq neg_divide_le_eq
nipkow@23482
  1740
  pos_le_divide_eq neg_le_divide_eq
paulson@14288
  1741
nipkow@23482
  1742
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
nipkow@23483
  1743
of positivity/negativity needed for @{text field_simps}. Have not added @{text
nipkow@23482
  1744
sign_simps} to @{text field_simps} because the former can lead to case
nipkow@23482
  1745
explosions. *}
paulson@14288
  1746
nipkow@29833
  1747
lemmas sign_simps[noatp] = group_simps
nipkow@23482
  1748
  zero_less_mult_iff  mult_less_0_iff
paulson@14288
  1749
nipkow@23482
  1750
(* Only works once linear arithmetic is installed:
nipkow@23482
  1751
text{*An example:*}
nipkow@23482
  1752
lemma fixes a b c d e f :: "'a::ordered_field"
nipkow@23482
  1753
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
nipkow@23482
  1754
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
nipkow@23482
  1755
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
nipkow@23482
  1756
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
nipkow@23482
  1757
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1758
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
nipkow@23482
  1759
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1760
apply(simp add:field_simps)
avigad@16775
  1761
done
nipkow@23482
  1762
*)
avigad@16775
  1763
wenzelm@23389
  1764
avigad@16775
  1765
subsection{*Division and Signs*}
avigad@16775
  1766
avigad@16775
  1767
lemma zero_less_divide_iff:
avigad@16775
  1768
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
  1769
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
  1770
avigad@16775
  1771
lemma divide_less_0_iff:
avigad@16775
  1772
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
avigad@16775
  1773
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
  1774
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
  1775
avigad@16775
  1776
lemma zero_le_divide_iff:
avigad@16775
  1777
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
avigad@16775
  1778
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
  1779
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
  1780
avigad@16775
  1781
lemma divide_le_0_iff:
avigad@16775
  1782
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
avigad@16775
  1783
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
  1784
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
  1785
paulson@24286
  1786
lemma divide_eq_0_iff [simp,noatp]:
avigad@16775
  1787
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
nipkow@23482
  1788
by (simp add: divide_inverse)
avigad@16775
  1789
nipkow@23482
  1790
lemma divide_pos_pos:
nipkow@23482
  1791
  "0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
nipkow@23482
  1792
by(simp add:field_simps)
nipkow@23482
  1793
avigad@16775
  1794
nipkow@23482
  1795
lemma divide_nonneg_pos:
nipkow@23482
  1796
  "0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
nipkow@23482
  1797
by(simp add:field_simps)
avigad@16775
  1798
nipkow@23482
  1799
lemma divide_neg_pos:
nipkow@23482
  1800
  "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
nipkow@23482
  1801
by(simp add:field_simps)
avigad@16775
  1802
nipkow@23482
  1803
lemma divide_nonpos_pos:
nipkow@23482
  1804
  "(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
nipkow@23482
  1805
by(simp add:field_simps)
avigad@16775
  1806
nipkow@23482
  1807
lemma divide_pos_neg:
nipkow@23482
  1808
  "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
nipkow@23482
  1809
by(simp add:field_simps)
avigad@16775
  1810
nipkow@23482
  1811
lemma divide_nonneg_neg:
nipkow@23482
  1812
  "0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" 
nipkow@23482
  1813
by(simp add:field_simps)
avigad@16775
  1814
nipkow@23482
  1815
lemma divide_neg_neg:
nipkow@23482
  1816
  "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
nipkow@23482
  1817
by(simp add:field_simps)
avigad@16775
  1818
nipkow@23482
  1819
lemma divide_nonpos_neg:
nipkow@23482
  1820
  "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
nipkow@23482
  1821
by(simp add:field_simps)
paulson@15234
  1822
wenzelm@23389
  1823
paulson@14288
  1824
subsection{*Cancellation Laws for Division*}
paulson@14288
  1825
paulson@24286
  1826
lemma divide_cancel_right [simp,noatp]:
paulson@14288
  1827
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1828
apply (cases "c=0", simp)
nipkow@23496
  1829
apply (simp add: divide_inverse)
paulson@14288
  1830
done
paulson@14288
  1831
paulson@24286
  1832
lemma divide_cancel_left [simp,noatp]:
paulson@14288
  1833
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
nipkow@23482
  1834
apply (cases "c=0", simp)
nipkow@23496
  1835
apply (simp add: divide_inverse)
paulson@14288
  1836
done
paulson@14288
  1837
wenzelm@23389
  1838
paulson@14353
  1839
subsection {* Division and the Number One *}
paulson@14353
  1840
paulson@14353
  1841
text{*Simplify expressions equated with 1*}
paulson@24286
  1842
lemma divide_eq_1_iff [simp,noatp]:
paulson@14353
  1843
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1844
apply (cases "b=0", simp)
nipkow@23482
  1845
apply (simp add: right_inverse_eq)
paulson@14353
  1846
done
paulson@14353
  1847
paulson@24286
  1848
lemma one_eq_divide_iff [simp,noatp]:
paulson@14353
  1849
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1850
by (simp add: eq_commute [of 1])
paulson@14353
  1851
paulson@24286
  1852
lemma zero_eq_1_divide_iff [simp,noatp]:
paulson@14353
  1853
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
nipkow@23482
  1854
apply (cases "a=0", simp)
nipkow@23482
  1855
apply (auto simp add: nonzero_eq_divide_eq)
paulson@14353
  1856
done
paulson@14353
  1857
paulson@24286
  1858
lemma one_divide_eq_0_iff [simp,noatp]:
paulson@14353
  1859
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
nipkow@23482
  1860
apply (cases "a=0", simp)
nipkow@23482
  1861
apply (insert zero_neq_one [THEN not_sym])
nipkow@23482
  1862
apply (auto simp add: nonzero_divide_eq_eq)
paulson@14353
  1863
done
paulson@14353
  1864
paulson@14353
  1865
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@18623
  1866
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
paulson@18623
  1867
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
paulson@18623
  1868
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
paulson@18623
  1869
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
paulson@17085
  1870
nipkow@29833
  1871
declare zero_less_divide_1_iff [simp,noatp]
paulson@24286
  1872
declare divide_less_0_1_iff [simp,noatp]
nipkow@29833
  1873
declare zero_le_divide_1_iff [simp,noatp]
paulson@24286
  1874
declare divide_le_0_1_iff [simp,noatp]
paulson@14353
  1875
wenzelm@23389
  1876
paulson@14293
  1877
subsection {* Ordering Rules for Division *}
paulson@14293
  1878
paulson@14293
  1879
lemma divide_strict_right_mono:
paulson@14293
  1880
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1881
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
nipkow@23482
  1882
              positive_imp_inverse_positive)
paulson@14293
  1883
paulson@14293
  1884
lemma divide_right_mono:
paulson@14293
  1885
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
nipkow@23482
  1886
by (force simp add: divide_strict_right_mono order_le_less)
paulson@14293
  1887
avigad@16775
  1888
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1889
    ==> c <= 0 ==> b / c <= a / c"
nipkow@23482
  1890
apply (drule divide_right_mono [of _ _ "- c"])
nipkow@23482
  1891
apply auto
avigad@16775
  1892
done
avigad@16775
  1893
avigad@16775
  1894
lemma divide_strict_right_mono_neg:
avigad@16775
  1895
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
nipkow@23482
  1896
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
nipkow@23482
  1897
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
avigad@16775
  1898
done
paulson@14293
  1899
paulson@14293
  1900
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1901
      have the same sign*}
paulson@14293
  1902
lemma divide_strict_left_mono:
nipkow@23482
  1903
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1904
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
paulson@14293
  1905
paulson@14293
  1906
lemma divide_left_mono:
nipkow@23482
  1907
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
nipkow@23482
  1908
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
paulson@14293
  1909
avigad@16775
  1910
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1911
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1912
  apply (drule divide_left_mono [of _ _ "- c"])
avigad@16775
  1913
  apply (auto simp add: mult_commute)
avigad@16775
  1914
done
avigad@16775
  1915
paulson@14293
  1916
lemma divide_strict_left_mono_neg:
nipkow@23482
  1917
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1918
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
nipkow@23482
  1919
paulson@14293
  1920
avigad@16775
  1921
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1922
paulson@24286
  1923
lemma le_divide_eq_1 [noatp]:
avigad@16775
  1924
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1925
  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1926
by (auto simp add: le_divide_eq)
avigad@16775
  1927
paulson@24286
  1928
lemma divide_le_eq_1 [noatp]:
avigad@16775
  1929
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1930
  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1931
by (auto simp add: divide_le_eq)
avigad@16775
  1932
paulson@24286
  1933
lemma less_divide_eq_1 [noatp]:
avigad@16775
  1934
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1935
  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1936
by (auto simp add: less_divide_eq)
avigad@16775
  1937
paulson@24286
  1938
lemma divide_less_eq_1 [noatp]:
avigad@16775
  1939
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1940
  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1941
by (auto simp add: divide_less_eq)
avigad@16775
  1942
wenzelm@23389
  1943
avigad@16775
  1944
subsection{*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1945
paulson@24286
  1946
lemma le_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 \<le> b/a) = (a \<le> b)"
avigad@16775
  1949
by (auto simp add: le_divide_eq)
avigad@16775
  1950
paulson@24286
  1951
lemma le_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 \<le> b/a) = (b \<le> a)"
avigad@16775
  1954
by (auto simp add: le_divide_eq)
avigad@16775
  1955
paulson@24286
  1956
lemma divide_le_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 \<le> 1) = (b \<le> a)"
avigad@16775
  1959
by (auto simp add: divide_le_eq)
avigad@16775
  1960
paulson@24286
  1961
lemma divide_le_eq_1_neg [simp,noatp]:
avigad@16775
  1962
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1963
  shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1964
by (auto simp add: divide_le_eq)
avigad@16775
  1965
paulson@24286
  1966
lemma less_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1967
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1968
  shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1969
by (auto simp add: less_divide_eq)
avigad@16775
  1970
paulson@24286
  1971
lemma less_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1972
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1973
  shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1974
by (auto simp add: less_divide_eq)
avigad@16775
  1975
paulson@24286
  1976
lemma divide_less_eq_1_pos [simp,noatp]:
avigad@16775
  1977
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1978
  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1979
by (auto simp add: divide_less_eq)
paulson@18649
  1980
paulson@24286
  1981
lemma divide_less_eq_1_neg [simp,noatp]:
paulson@18649
  1982
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1983
  shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1984
by (auto simp add: divide_less_eq)
avigad@16775
  1985
paulson@24286
  1986
lemma eq_divide_eq_1 [simp,noatp]:
avigad@16775
  1987
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1988
  shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1989
by (auto simp add: eq_divide_eq)
avigad@16775
  1990
paulson@24286
  1991
lemma divide_eq_eq_1 [simp,noatp]:
avigad@16775
  1992
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1993
  shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1994
by (auto simp add: divide_eq_eq)
avigad@16775
  1995
wenzelm@23389
  1996
avigad@16775
  1997
subsection {* Reasoning about inequalities with division *}
avigad@16775
  1998
avigad@16775
  1999
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  2000
    ==> x * y <= x"
nipkow@29667
  2001
by (auto simp add: mult_compare_simps);
avigad@16775
  2002
avigad@16775
  2003
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  2004
    ==> y * x <= x"
nipkow@29667
  2005
by (auto simp add: mult_compare_simps);
avigad@16775
  2006
avigad@16775
  2007
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
avigad@16775
  2008
    x / y <= z";
nipkow@29667
  2009
by (subst pos_divide_le_eq, assumption+);
avigad@16775
  2010
avigad@16775
  2011
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
nipkow@23482
  2012
    z <= x / y"
nipkow@23482
  2013
by(simp add:field_simps)
avigad@16775
  2014
avigad@16775
  2015
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
avigad@16775
  2016
    x / y < z"
nipkow@23482
  2017
by(simp add:field_simps)
avigad@16775
  2018
avigad@16775
  2019
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
avigad@16775
  2020
    z < x / y"
nipkow@23482
  2021
by(simp add:field_simps)
avigad@16775
  2022
avigad@16775
  2023
lemma frac_le: "(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_le)
haftmann@25230
  2026
  apply simp
haftmann@25230
  2027
  apply (subst times_divide_eq_left)
avigad@16775
  2028
  apply (rule mult_imp_le_div_pos, assumption)
avigad@16775
  2029
  apply (rule mult_mono)
avigad@16775
  2030
  apply simp_all
paulson@14293
  2031
done
paulson@14293
  2032
avigad@16775
  2033
lemma frac_less: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  2034
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
avigad@16775
  2035
  apply (rule mult_imp_div_pos_less)
avigad@16775
  2036
  apply simp;
avigad@16775
  2037
  apply (subst times_divide_eq_left);
avigad@16775
  2038
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  2039
  apply (erule mult_less_le_imp_less)
avigad@16775
  2040
  apply simp_all
avigad@16775
  2041
done
avigad@16775
  2042
avigad@16775
  2043
lemma frac_less2: "(0::'a::ordered_field) < x ==> 
avigad@16775
  2044
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
avigad@16775
  2045
  apply (rule mult_imp_div_pos_less)
avigad@16775
  2046
  apply simp_all
avigad@16775
  2047
  apply (subst times_divide_eq_left);
avigad@16775
  2048
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  2049
  apply (erule mult_le_less_imp_less)
avigad@16775
  2050
  apply simp_all
avigad@16775
  2051
done
avigad@16775
  2052
avigad@16775
  2053
text{*It's not obvious whether these should be simprules or not. 
avigad@16775
  2054
  Their effect is to gather terms into one big fraction, like
avigad@16775
  2055
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
avigad@16775
  2056
  seem to need them.*}
avigad@16775
  2057
avigad@16775
  2058
declare times_divide_eq [simp]
paulson@14293
  2059
wenzelm@23389
  2060
paulson@14293
  2061
subsection {* Ordered Fields are Dense *}
paulson@14293
  2062
haftmann@25193
  2063
context ordered_semidom
haftmann@25193
  2064
begin
haftmann@25193
  2065
haftmann@25193
  2066
lemma less_add_one: "a < a + 1"
paulson@14293
  2067
proof -
haftmann@25193
  2068
  have "a + 0 < a + 1"
nipkow@23482
  2069
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  2070
  thus ?thesis by simp
paulson@14293
  2071
qed
paulson@14293
  2072
haftmann@25193
  2073
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  2074
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  2075
haftmann@25193
  2076
end
paulson@14365
  2077
paulson@14293
  2078
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
nipkow@23482
  2079
by (simp add: field_simps zero_less_two)
paulson@14293
  2080
paulson@14293
  2081
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
nipkow@23482
  2082
by (simp add: field_simps zero_less_two)
paulson@14293
  2083
haftmann@24422
  2084
instance ordered_field < dense_linear_order
haftmann@24422
  2085
proof
haftmann@24422
  2086
  fix x y :: 'a
haftmann@24422
  2087
  have "x < x + 1" by simp
haftmann@24422
  2088
  then show "\<exists>y. x < y" .. 
haftmann@24422
  2089
  have "x - 1 < x" by simp
haftmann@24422
  2090
  then show "\<exists>y. y < x" ..
haftmann@24422
  2091
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
haftmann@24422
  2092
qed
paulson@14293
  2093
paulson@15234
  2094
paulson@14293
  2095
subsection {* Absolute Value *}
paulson@14293
  2096
haftmann@25304
  2097
context ordered_idom
haftmann@25304
  2098
begin
haftmann@25304
  2099
haftmann@25304
  2100
lemma mult_sgn_abs: "sgn x * abs x = x"
haftmann@25304
  2101
  unfolding abs_if sgn_if by auto
haftmann@25304
  2102
haftmann@25304
  2103
end
nipkow@24491
  2104
obua@14738
  2105
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
nipkow@29667
  2106
by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
haftmann@25304
  2107
haftmann@25304
  2108
class pordered_ring_abs = pordered_ring + pordered_ab_group_add_abs +
haftmann@25304
  2109
  assumes abs_eq_mult:
haftmann@25304
  2110
    "(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
  2111
haftmann@25304
  2112
haftmann@25304
  2113
class lordered_ring = pordered_ring + lordered_ab_group_add_abs
haftmann@25304
  2114
begin
haftmann@25304
  2115
huffman@27516
  2116
subclass lordered_ab_group_add_meet ..
huffman@27516
  2117
subclass lordered_ab_group_add_join ..
haftmann@25304
  2118
haftmann@25304
  2119
end
paulson@14294
  2120
obua@14738
  2121
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  2122
proof -
obua@14738
  2123
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  2124
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  2125
  have a: "(abs a) * (abs b) = ?x"
nipkow@29667
  2126
    by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
obua@14738
  2127
  {
obua@14738
  2128
    fix u v :: 'a
paulson@15481
  2129
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
paulson@15481
  2130
              u * v = pprt a * pprt b + pprt a * nprt b + 
paulson@15481
  2131
                      nprt a * pprt b + nprt a * nprt b"
obua@14738
  2132
      apply (subst prts[of u], subst prts[of v])
nipkow@29667
  2133
      apply (simp add: algebra_simps) 
obua@14738
  2134
      done
obua@14738
  2135
  }
obua@14738
  2136
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  2137
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  2138
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  2139
  have xy: "- ?x <= ?y"
obua@14754
  2140
    apply (simp)
obua@14754
  2141
    apply (rule_tac y="0::'a" in order_trans)
nipkow@16568
  2142
    apply (rule addm2)
avigad@16775
  2143
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
nipkow@16568
  2144
    apply (rule addm)
avigad@16775
  2145
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14754
  2146
    done
obua@14738
  2147
  have yx: "?y <= ?x"
nipkow@16568
  2148
    apply (simp add:diff_def)
obua@14754
  2149
    apply (rule_tac y=0 in order_trans)
avigad@16775
  2150
    apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
avigad@16775
  2151
    apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
obua@14738
  2152
    done
obua@14738
  2153
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  2154
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  2155
  show ?thesis
obua@14738
  2156
    apply (rule abs_leI)
obua@14738
  2157
    apply (simp add: i1)
obua@14738
  2158
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  2159
    done
obua@14738
  2160
qed
paulson@14294
  2161
haftmann@25304
  2162
instance lordered_ring \<subseteq> pordered_ring_abs
haftmann@25304
  2163
proof
haftmann@25304
  2164
  fix a b :: "'a\<Colon> lordered_ring"
haftmann@25304
  2165
  assume "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
haftmann@25304
  2166
  show "abs (a*b) = abs a * abs b"
obua@14738
  2167
proof -
obua@14738
  2168
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  2169
    apply (auto)    
obua@14738
  2170
    apply (rule_tac split_mult_pos_le)
obua@14738
  2171
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  2172
    apply (simp)
obua@14738
  2173
    apply (rule_tac split_mult_neg_le)
obua@14738
  2174
    apply (insert prems)
obua@14738
  2175
    apply (blast)
obua@14738
  2176
    done
obua@14738
  2177
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  2178
    by (simp add: prts[symmetric])
obua@14738
  2179
  show ?thesis
obua@14738
  2180
  proof cases
obua@14738
  2181
    assume "0 <= a * b"
obua@14738
  2182
    then show ?thesis
obua@14738
  2183
      apply (simp_all add: mulprts abs_prts)
obua@14738
  2184
      apply (insert prems)
obua@14754
  2185
      apply (auto simp add: 
nipkow@29667
  2186
	algebra_simps 
haftmann@25078
  2187
	iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
haftmann@25078
  2188
	iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
avigad@16775
  2189
	apply(drule (1) mult_nonneg_nonpos[of a b], simp)
avigad@16775
  2190
	apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
obua@14738
  2191
      done
obua@14738
  2192
  next
obua@14738
  2193
    assume "~(0 <= a*b)"
obua@14738
  2194
    with s have "a*b <= 0" by simp
obua@14738
  2195
    then show ?thesis
obua@14738
  2196
      apply (simp_all add: mulprts abs_prts)
obua@14738
  2197
      apply (insert prems)
nipkow@29667
  2198
      apply (auto simp add: algebra_simps)
avigad@16775
  2199
      apply(drule (1) mult_nonneg_nonneg[of a b],simp)
avigad@16775
  2200
      apply(drule (1) mult_nonpos_nonpos[of a b],simp)
obua@14738
  2201
      done
obua@14738
  2202
  qed
obua@14738
  2203
qed
haftmann@25304
  2204
qed
haftmann@25304
  2205
haftmann@25304
  2206
instance ordered_idom \<subseteq> pordered_ring_abs
haftmann@25304
  2207
by default (auto simp add: abs_if not_less
haftmann@25304
  2208
  equal_neg_zero neg_equal_zero mult_less_0_iff)
paulson@14294
  2209
obua@14738
  2210
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
nipkow@29667
  2211
by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  2212
obua@14738
  2213
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
nipkow@29667
  2214
by (simp add: abs_if) 
paulson@14294
  2215
paulson@14294
  2216
lemma nonzero_abs_inverse:
paulson@14294
  2217
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  2218
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  2219
                      negative_imp_inverse_negative)
paulson@14294
  2220
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  2221
done
paulson@14294
  2222
paulson@14294
  2223
lemma abs_inverse [simp]:
paulson@14294
  2224
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  2225
      inverse (abs a)"
haftmann@21328
  2226
apply (cases "a=0", simp) 
paulson@14294
  2227
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  2228
done
paulson@14294
  2229
paulson@14294
  2230
lemma nonzero_abs_divide:
paulson@14294
  2231
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  2232
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  2233
paulson@15234
  2234
lemma abs_divide [simp]:
paulson@14294
  2235
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
haftmann@21328
  2236
apply (cases "b=0", simp) 
paulson@14294
  2237
apply (simp add: nonzero_abs_divide) 
paulson@14294
  2238
done
paulson@14294
  2239
paulson@14294
  2240
lemma abs_mult_less:
obua@14738
  2241
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  2242
proof -
paulson@14294
  2243
  assume ac: "abs a < c"
paulson@14294
  2244
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  2245
  assume "abs b < d"
paulson@14294
  2246
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  2247
qed
paulson@14293
  2248
nipkow@29833
  2249
lemmas eq_minus_self_iff[noatp] = equal_neg_zero
obua@14738
  2250
obua@14738
  2251
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
haftmann@25304
  2252
  unfolding order_less_le less_eq_neg_nonpos equal_neg_zero ..
obua@14738
  2253
obua@14738
  2254
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  2255
apply (simp add: order_less_le abs_le_iff)  
haftmann@25304
  2256
apply (auto simp add: abs_if neg_less_eq_nonneg less_eq_neg_nonpos)
obua@14738
  2257
done
obua@14738
  2258
avigad@16775
  2259
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
haftmann@25304
  2260
    (abs y) * x = abs (y * x)"
haftmann@25304
  2261
  apply (subst abs_mult)
haftmann@25304
  2262
  apply simp
haftmann@25304
  2263
done
avigad@16775
  2264
avigad@16775
  2265
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
haftmann@25304
  2266
    abs x / y = abs (x / y)"
haftmann@25304
  2267
  apply (subst abs_divide)
haftmann@25304
  2268
  apply (simp add: order_less_imp_le)
haftmann@25304
  2269
done
avigad@16775
  2270
wenzelm@23389
  2271
obua@19404
  2272
subsection {* Bounds of products via negative and positive Part *}
obua@15178
  2273
obua@15580
  2274
lemma mult_le_prts:
obua@15580
  2275
  assumes
obua@15580
  2276
  "a1 <= (a::'a::lordered_ring)"
obua@15580
  2277
  "a <= a2"
obua@15580
  2278
  "b1 <= b"
obua@15580
  2279
  "b <= b2"
obua@15580
  2280
  shows
obua@15580
  2281
  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
obua@15580
  2282
proof - 
obua@15580
  2283
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
obua@15580
  2284
    apply (subst prts[symmetric])+
obua@15580
  2285
    apply simp
obua@15580
  2286
    done
obua@15580
  2287
  then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
nipkow@29667
  2288
    by (simp add: algebra_simps)
obua@15580
  2289
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
obua@15580
  2290
    by (simp_all add: prems mult_mono)
obua@15580
  2291
  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
obua@15580
  2292
  proof -
obua@15580
  2293
    have "pprt a * nprt b <= pprt a * nprt b2"
obua@15580
  2294
      by (simp add: mult_left_mono prems)
obua@15580
  2295
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
obua@15580
  2296
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2297
    ultimately show ?thesis
obua@15580
  2298
      by simp
obua@15580
  2299
  qed
obua@15580
  2300
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
obua@15580
  2301
  proof - 
obua@15580
  2302
    have "nprt a * pprt b <= nprt a2 * pprt b"
obua@15580
  2303
      by (simp add: mult_right_mono prems)
obua@15580
  2304
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
obua@15580
  2305
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2306
    ultimately show ?thesis
obua@15580
  2307
      by simp
obua@15580
  2308
  qed
obua@15580
  2309
  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
obua@15580
  2310
  proof -
obua@15580
  2311
    have "nprt a * nprt b <= nprt a * nprt b1"
obua@15580
  2312
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2313
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
obua@15580
  2314
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2315
    ultimately show ?thesis
obua@15580
  2316
      by simp
obua@15580
  2317
  qed
obua@15580
  2318
  ultimately show ?thesis
obua@15580
  2319
    by - (rule add_mono | simp)+
obua@15580
  2320
qed
obua@19404
  2321
obua@19404
  2322
lemma mult_ge_prts:
obua@15178
  2323
  assumes
obua@19404
  2324
  "a1 <= (a::'a::lordered_ring)"
obua@19404
  2325
  "a <= a2"
obua@19404
  2326
  "b1 <= b"
obua@19404
  2327
  "b <= b2"
obua@15178
  2328
  shows
obua@19404
  2329
  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
obua@19404
  2330
proof - 
obua@19404
  2331
  from prems have a1:"- a2 <= -a" by auto
obua@19404
  2332
  from prems have a2: "-a <= -a1" by auto
obua@19404
  2333
  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
  2334
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
obua@19404
  2335
  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
obua@19404
  2336
    by (simp only: minus_le_iff)
obua@19404
  2337
  then show ?thesis by simp
obua@15178
  2338
qed
obua@15178
  2339
paulson@14265
  2340
end