src/HOL/Ring_and_Field.thy
author wenzelm
Sat Oct 17 14:43:18 2009 +0200 (2009-10-17)
changeset 32960 69916a850301
parent 30961 541bfff659af
child 33319 74f0dcc0b5fb
permissions -rw-r--r--
eliminated hard tabulators, guessing at each author's individual tab-width;
tuned headers;
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(*  Title:      HOL/Ring_and_Field.thy
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    Author:     Gertrud Bauer
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    Author:     Steven Obua
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Markus Wenzel
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    Author:     Jeremy Avigad
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*)
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header {* (Ordered) Rings and Fields *}
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theory Ring_and_Field
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imports OrderedGroup
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begin
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text {*
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  The theory of partially ordered rings is taken from the books:
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  \begin{itemize}
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  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
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  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
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  \end{itemize}
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  Most of the used notions can also be looked up in 
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  \begin{itemize}
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  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
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  \item \emph{Algebra I} by van der Waerden, Springer.
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  \end{itemize}
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*}
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes left_distrib[algebra_simps]: "(a + b) * c = a * c + b * c"
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  assumes right_distrib[algebra_simps]: "a * (b + c) = a * b + a * c"
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begin
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor:
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  "a * e + (b * e + c) = (a + b) * e + c"
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by (simp add: left_distrib add_ac)
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end
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "0 * a = 0"
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  assumes mult_zero_right [simp]: "a * 0 = 0"
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0" by (simp add: left_distrib [symmetric])
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  thus "0 * a = 0" by (simp only: add_left_cancel)
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next
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  fix a :: 'a
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  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: right_distrib [symmetric])
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  thus "a * 0 = 0" by (simp only: add_left_cancel)
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qed
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end
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a + b) * c = a * c + b * c"
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begin
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subclass semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: mult_ac)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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end
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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begin
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subclass semiring_0 ..
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end
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class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0_cancel ..
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subclass comm_semiring_0 ..
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end
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "0 \<noteq> 1"
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begin
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lemma one_neq_zero [simp]: "1 \<noteq> 0"
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by (rule not_sym) (rule zero_neq_one)
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end
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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text {* Abstract divisibility *}
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class dvd = times
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begin
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definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where
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  [code del]: "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
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lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
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  unfolding dvd_def ..
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding dvd_def by blast 
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end
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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd
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  (*previously almost_semiring*)
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begin
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subclass semiring_1 ..
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lemma dvd_refl[simp]: "a dvd a"
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proof
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  show "a = a * 1" by simp
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qed
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lemma dvd_trans:
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  assumes "a dvd b" and "b dvd c"
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  shows "a dvd c"
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proof -
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  from assms obtain v where "b = a * v" by (auto elim!: dvdE)
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  moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
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  ultimately have "c = a * (v * w)" by (simp add: mult_assoc)
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  then show ?thesis ..
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qed
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lemma dvd_0_left_iff [noatp, simp]: "0 dvd a \<longleftrightarrow> a = 0"
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by (auto intro: dvd_refl elim!: dvdE)
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lemma dvd_0_right [iff]: "a dvd 0"
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proof
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  show "0 = a * 0" by simp
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qed
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lemma one_dvd [simp]: "1 dvd a"
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by (auto intro!: dvdI)
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lemma dvd_mult[simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
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by (auto intro!: mult_left_commute dvdI elim!: dvdE)
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lemma dvd_mult2[simp]: "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 "a dvd b"
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    and "c dvd d"
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  shows "a * c dvd b * d"
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proof -
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  from `a dvd b` obtain b' where "b = a * b'" ..
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  moreover from `c dvd d` 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[simp]:
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  assumes "a dvd b" and "a dvd c" shows "a dvd (b + c)"
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proof -
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  from `a dvd b` obtain b' where "b = a * b'" ..
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  moreover from `a dvd c` 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[simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
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by (simp add: diff_minus 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]:
haftmann@26274
   357
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   358
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   359
haftmann@26274
   360
lemma mult_cancel_right2 [simp]:
haftmann@26274
   361
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   362
by (insert mult_cancel_right [of a c 1], simp)
haftmann@26274
   363
 
haftmann@26274
   364
lemma mult_cancel_left1 [simp]:
haftmann@26274
   365
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   366
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   367
haftmann@26274
   368
lemma mult_cancel_left2 [simp]:
haftmann@26274
   369
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   370
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   371
haftmann@26274
   372
end
huffman@22990
   373
haftmann@22390
   374
class idom = comm_ring_1 + no_zero_divisors
haftmann@25186
   375
begin
paulson@14421
   376
huffman@27516
   377
subclass ring_1_no_zero_divisors ..
huffman@22990
   378
huffman@29915
   379
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)"
huffman@29915
   380
proof
huffman@29915
   381
  assume "a * a = b * b"
huffman@29915
   382
  then have "(a - b) * (a + b) = 0"
huffman@29915
   383
    by (simp add: algebra_simps)
huffman@29915
   384
  then show "a = b \<or> a = - b"
huffman@29915
   385
    by (simp add: right_minus_eq eq_neg_iff_add_eq_0)
huffman@29915
   386
next
huffman@29915
   387
  assume "a = b \<or> a = - b"
huffman@29915
   388
  then show "a * a = b * b" by auto
huffman@29915
   389
qed
huffman@29915
   390
huffman@29981
   391
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   392
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   393
proof -
huffman@29981
   394
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
huffman@29981
   395
    unfolding dvd_def by (simp add: mult_ac)
huffman@29981
   396
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   397
    unfolding dvd_def by simp
huffman@29981
   398
  finally show ?thesis .
huffman@29981
   399
qed
huffman@29981
   400
huffman@29981
   401
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   402
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   403
proof -
huffman@29981
   404
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
huffman@29981
   405
    unfolding dvd_def by (simp add: mult_ac)
huffman@29981
   406
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   407
    unfolding dvd_def by simp
huffman@29981
   408
  finally show ?thesis .
huffman@29981
   409
qed
huffman@29981
   410
haftmann@25186
   411
end
haftmann@25152
   412
haftmann@22390
   413
class division_ring = ring_1 + inverse +
haftmann@25062
   414
  assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
haftmann@25062
   415
  assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
haftmann@25186
   416
begin
huffman@20496
   417
haftmann@25186
   418
subclass ring_1_no_zero_divisors
haftmann@28823
   419
proof
huffman@22987
   420
  fix a b :: 'a
huffman@22987
   421
  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
huffman@22987
   422
  show "a * b \<noteq> 0"
huffman@22987
   423
  proof
huffman@22987
   424
    assume ab: "a * b = 0"
nipkow@29667
   425
    hence "0 = inverse a * (a * b) * inverse b" by simp
huffman@22987
   426
    also have "\<dots> = (inverse a * a) * (b * inverse b)"
huffman@22987
   427
      by (simp only: mult_assoc)
nipkow@29667
   428
    also have "\<dots> = 1" using a b by simp
nipkow@29667
   429
    finally show False by simp
huffman@22987
   430
  qed
huffman@22987
   431
qed
huffman@20496
   432
haftmann@26274
   433
lemma nonzero_imp_inverse_nonzero:
haftmann@26274
   434
  "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
haftmann@26274
   435
proof
haftmann@26274
   436
  assume ianz: "inverse a = 0"
haftmann@26274
   437
  assume "a \<noteq> 0"
haftmann@26274
   438
  hence "1 = a * inverse a" by simp
haftmann@26274
   439
  also have "... = 0" by (simp add: ianz)
haftmann@26274
   440
  finally have "1 = 0" .
haftmann@26274
   441
  thus False by (simp add: eq_commute)
haftmann@26274
   442
qed
haftmann@26274
   443
haftmann@26274
   444
lemma inverse_zero_imp_zero:
haftmann@26274
   445
  "inverse a = 0 \<Longrightarrow> a = 0"
haftmann@26274
   446
apply (rule classical)
haftmann@26274
   447
apply (drule nonzero_imp_inverse_nonzero)
haftmann@26274
   448
apply auto
haftmann@26274
   449
done
haftmann@26274
   450
haftmann@26274
   451
lemma inverse_unique: 
haftmann@26274
   452
  assumes ab: "a * b = 1"
haftmann@26274
   453
  shows "inverse a = b"
haftmann@26274
   454
proof -
haftmann@26274
   455
  have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
huffman@29406
   456
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
huffman@29406
   457
  ultimately show ?thesis by (simp add: mult_assoc [symmetric])
haftmann@26274
   458
qed
haftmann@26274
   459
huffman@29406
   460
lemma nonzero_inverse_minus_eq:
huffman@29406
   461
  "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
nipkow@29667
   462
by (rule inverse_unique) simp
huffman@29406
   463
huffman@29406
   464
lemma nonzero_inverse_inverse_eq:
huffman@29406
   465
  "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
nipkow@29667
   466
by (rule inverse_unique) simp
huffman@29406
   467
huffman@29406
   468
lemma nonzero_inverse_eq_imp_eq:
huffman@29406
   469
  assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
huffman@29406
   470
  shows "a = b"
huffman@29406
   471
proof -
huffman@29406
   472
  from `inverse a = inverse b`
nipkow@29667
   473
  have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
huffman@29406
   474
  with `a \<noteq> 0` and `b \<noteq> 0` show "a = b"
huffman@29406
   475
    by (simp add: nonzero_inverse_inverse_eq)
huffman@29406
   476
qed
huffman@29406
   477
huffman@29406
   478
lemma inverse_1 [simp]: "inverse 1 = 1"
nipkow@29667
   479
by (rule inverse_unique) simp
huffman@29406
   480
haftmann@26274
   481
lemma nonzero_inverse_mult_distrib: 
huffman@29406
   482
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@26274
   483
  shows "inverse (a * b) = inverse b * inverse a"
haftmann@26274
   484
proof -
nipkow@29667
   485
  have "a * (b * inverse b) * inverse a = 1" using assms by simp
nipkow@29667
   486
  hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc)
nipkow@29667
   487
  thus ?thesis by (rule inverse_unique)
haftmann@26274
   488
qed
haftmann@26274
   489
haftmann@26274
   490
lemma division_ring_inverse_add:
haftmann@26274
   491
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
nipkow@29667
   492
by (simp add: algebra_simps)
haftmann@26274
   493
haftmann@26274
   494
lemma division_ring_inverse_diff:
haftmann@26274
   495
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
nipkow@29667
   496
by (simp add: algebra_simps)
haftmann@26274
   497
haftmann@25186
   498
end
haftmann@25152
   499
huffman@22987
   500
class field = comm_ring_1 + inverse +
haftmann@25062
   501
  assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
haftmann@25062
   502
  assumes divide_inverse: "a / b = a * inverse b"
haftmann@25267
   503
begin
huffman@20496
   504
haftmann@25267
   505
subclass division_ring
haftmann@28823
   506
proof
huffman@22987
   507
  fix a :: 'a
huffman@22987
   508
  assume "a \<noteq> 0"
huffman@22987
   509
  thus "inverse a * a = 1" by (rule field_inverse)
huffman@22987
   510
  thus "a * inverse a = 1" by (simp only: mult_commute)
obua@14738
   511
qed
haftmann@25230
   512
huffman@27516
   513
subclass idom ..
haftmann@25230
   514
haftmann@25230
   515
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
haftmann@25230
   516
proof
haftmann@25230
   517
  assume neq: "b \<noteq> 0"
haftmann@25230
   518
  {
haftmann@25230
   519
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
haftmann@25230
   520
    also assume "a / b = 1"
haftmann@25230
   521
    finally show "a = b" by simp
haftmann@25230
   522
  next
haftmann@25230
   523
    assume "a = b"
haftmann@25230
   524
    with neq show "a / b = 1" by (simp add: divide_inverse)
haftmann@25230
   525
  }
haftmann@25230
   526
qed
haftmann@25230
   527
haftmann@25230
   528
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
nipkow@29667
   529
by (simp add: divide_inverse)
haftmann@25230
   530
haftmann@25230
   531
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
nipkow@29667
   532
by (simp add: divide_inverse)
haftmann@25230
   533
haftmann@25230
   534
lemma divide_zero_left [simp]: "0 / a = 0"
nipkow@29667
   535
by (simp add: divide_inverse)
haftmann@25230
   536
haftmann@25230
   537
lemma inverse_eq_divide: "inverse a = 1 / a"
nipkow@29667
   538
by (simp add: divide_inverse)
haftmann@25230
   539
haftmann@25230
   540
lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
huffman@30630
   541
by (simp add: divide_inverse algebra_simps)
huffman@30630
   542
huffman@30630
   543
text{*There is no slick version using division by zero.*}
huffman@30630
   544
lemma inverse_add:
huffman@30630
   545
  "[| a \<noteq> 0;  b \<noteq> 0 |]
huffman@30630
   546
   ==> inverse a + inverse b = (a + b) * inverse a * inverse b"
huffman@30630
   547
by (simp add: division_ring_inverse_add mult_ac)
huffman@30630
   548
huffman@30630
   549
lemma nonzero_mult_divide_mult_cancel_left [simp, noatp]:
huffman@30630
   550
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b"
huffman@30630
   551
proof -
huffman@30630
   552
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
huffman@30630
   553
    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
huffman@30630
   554
  also have "... =  a * inverse b * (inverse c * c)"
huffman@30630
   555
    by (simp only: mult_ac)
huffman@30630
   556
  also have "... =  a * inverse b" by simp
huffman@30630
   557
    finally show ?thesis by (simp add: divide_inverse)
huffman@30630
   558
qed
huffman@30630
   559
huffman@30630
   560
lemma nonzero_mult_divide_mult_cancel_right [simp, noatp]:
huffman@30630
   561
  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b"
huffman@30630
   562
by (simp add: mult_commute [of _ c])
huffman@30630
   563
huffman@30630
   564
lemma divide_1 [simp]: "a / 1 = a"
huffman@30630
   565
by (simp add: divide_inverse)
huffman@30630
   566
huffman@30630
   567
lemma times_divide_eq_right: "a * (b / c) = (a * b) / c"
huffman@30630
   568
by (simp add: divide_inverse mult_assoc)
huffman@30630
   569
huffman@30630
   570
lemma times_divide_eq_left: "(b / c) * a = (b * a) / c"
huffman@30630
   571
by (simp add: divide_inverse mult_ac)
huffman@30630
   572
huffman@30630
   573
text {* These are later declared as simp rules. *}
huffman@30630
   574
lemmas times_divide_eq [noatp] = times_divide_eq_right times_divide_eq_left
huffman@30630
   575
huffman@30630
   576
lemma add_frac_eq:
huffman@30630
   577
  assumes "y \<noteq> 0" and "z \<noteq> 0"
huffman@30630
   578
  shows "x / y + w / z = (x * z + w * y) / (y * z)"
huffman@30630
   579
proof -
huffman@30630
   580
  have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
huffman@30630
   581
    using assms by simp
huffman@30630
   582
  also have "\<dots> = (x * z + y * w) / (y * z)"
huffman@30630
   583
    by (simp only: add_divide_distrib)
huffman@30630
   584
  finally show ?thesis
huffman@30630
   585
    by (simp only: mult_commute)
huffman@30630
   586
qed
huffman@30630
   587
huffman@30630
   588
text{*Special Cancellation Simprules for Division*}
huffman@30630
   589
huffman@30630
   590
lemma nonzero_mult_divide_cancel_right [simp, noatp]:
huffman@30630
   591
  "b \<noteq> 0 \<Longrightarrow> a * b / b = a"
huffman@30630
   592
using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp
huffman@30630
   593
huffman@30630
   594
lemma nonzero_mult_divide_cancel_left [simp, noatp]:
huffman@30630
   595
  "a \<noteq> 0 \<Longrightarrow> a * b / a = b"
huffman@30630
   596
using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp
huffman@30630
   597
huffman@30630
   598
lemma nonzero_divide_mult_cancel_right [simp, noatp]:
huffman@30630
   599
  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a"
huffman@30630
   600
using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp
huffman@30630
   601
huffman@30630
   602
lemma nonzero_divide_mult_cancel_left [simp, noatp]:
huffman@30630
   603
  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b"
huffman@30630
   604
using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp
huffman@30630
   605
huffman@30630
   606
lemma nonzero_mult_divide_mult_cancel_left2 [simp, noatp]:
huffman@30630
   607
  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b"
huffman@30630
   608
using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac)
huffman@30630
   609
huffman@30630
   610
lemma nonzero_mult_divide_mult_cancel_right2 [simp, noatp]:
huffman@30630
   611
  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b"
huffman@30630
   612
using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac)
huffman@30630
   613
huffman@30630
   614
lemma minus_divide_left: "- (a / b) = (-a) / b"
huffman@30630
   615
by (simp add: divide_inverse)
huffman@30630
   616
huffman@30630
   617
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
huffman@30630
   618
by (simp add: divide_inverse nonzero_inverse_minus_eq)
huffman@30630
   619
huffman@30630
   620
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
huffman@30630
   621
by (simp add: divide_inverse nonzero_inverse_minus_eq)
huffman@30630
   622
huffman@30630
   623
lemma divide_minus_left [simp, noatp]: "(-a) / b = - (a / b)"
huffman@30630
   624
by (simp add: divide_inverse)
huffman@30630
   625
huffman@30630
   626
lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
huffman@30630
   627
by (simp add: diff_minus add_divide_distrib)
huffman@30630
   628
huffman@30630
   629
lemma add_divide_eq_iff:
huffman@30630
   630
  "z \<noteq> 0 \<Longrightarrow> x + y / z = (z * x + y) / z"
huffman@30630
   631
by (simp add: add_divide_distrib)
huffman@30630
   632
huffman@30630
   633
lemma divide_add_eq_iff:
huffman@30630
   634
  "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + z * y) / z"
huffman@30630
   635
by (simp add: add_divide_distrib)
huffman@30630
   636
huffman@30630
   637
lemma diff_divide_eq_iff:
huffman@30630
   638
  "z \<noteq> 0 \<Longrightarrow> x - y / z = (z * x - y) / z"
huffman@30630
   639
by (simp add: diff_divide_distrib)
huffman@30630
   640
huffman@30630
   641
lemma divide_diff_eq_iff:
huffman@30630
   642
  "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - z * y) / z"
huffman@30630
   643
by (simp add: diff_divide_distrib)
huffman@30630
   644
huffman@30630
   645
lemma nonzero_eq_divide_eq: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
huffman@30630
   646
proof -
huffman@30630
   647
  assume [simp]: "c \<noteq> 0"
huffman@30630
   648
  have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
huffman@30630
   649
  also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc)
huffman@30630
   650
  finally show ?thesis .
huffman@30630
   651
qed
huffman@30630
   652
huffman@30630
   653
lemma nonzero_divide_eq_eq: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
huffman@30630
   654
proof -
huffman@30630
   655
  assume [simp]: "c \<noteq> 0"
huffman@30630
   656
  have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
huffman@30630
   657
  also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc) 
huffman@30630
   658
  finally show ?thesis .
huffman@30630
   659
qed
huffman@30630
   660
huffman@30630
   661
lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
huffman@30630
   662
by simp
huffman@30630
   663
huffman@30630
   664
lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
huffman@30630
   665
by (erule subst, simp)
huffman@30630
   666
huffman@30630
   667
lemmas field_eq_simps[noatp] = algebra_simps
huffman@30630
   668
  (* pull / out*)
huffman@30630
   669
  add_divide_eq_iff divide_add_eq_iff
huffman@30630
   670
  diff_divide_eq_iff divide_diff_eq_iff
huffman@30630
   671
  (* multiply eqn *)
huffman@30630
   672
  nonzero_eq_divide_eq nonzero_divide_eq_eq
huffman@30630
   673
(* is added later:
huffman@30630
   674
  times_divide_eq_left times_divide_eq_right
huffman@30630
   675
*)
huffman@30630
   676
huffman@30630
   677
text{*An example:*}
huffman@30630
   678
lemma "\<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"
huffman@30630
   679
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
huffman@30630
   680
 apply(simp add:field_eq_simps)
huffman@30630
   681
apply(simp)
huffman@30630
   682
done
huffman@30630
   683
huffman@30630
   684
lemma diff_frac_eq:
huffman@30630
   685
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
huffman@30630
   686
by (simp add: field_eq_simps times_divide_eq)
huffman@30630
   687
huffman@30630
   688
lemma frac_eq_eq:
huffman@30630
   689
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
huffman@30630
   690
by (simp add: field_eq_simps times_divide_eq)
haftmann@25230
   691
haftmann@25230
   692
end
haftmann@25230
   693
haftmann@22390
   694
class division_by_zero = zero + inverse +
haftmann@25062
   695
  assumes inverse_zero [simp]: "inverse 0 = 0"
paulson@14265
   696
haftmann@25230
   697
lemma divide_zero [simp]:
haftmann@25230
   698
  "a / 0 = (0::'a::{field,division_by_zero})"
nipkow@29667
   699
by (simp add: divide_inverse)
haftmann@25230
   700
haftmann@25230
   701
lemma divide_self_if [simp]:
haftmann@25230
   702
  "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
nipkow@29667
   703
by simp
haftmann@25230
   704
haftmann@22390
   705
class mult_mono = times + zero + ord +
haftmann@25062
   706
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25062
   707
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
paulson@14267
   708
haftmann@22390
   709
class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
haftmann@25230
   710
begin
haftmann@25230
   711
haftmann@25230
   712
lemma mult_mono:
haftmann@25230
   713
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c
haftmann@25230
   714
     \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   715
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   716
apply (erule mult_left_mono, assumption)
haftmann@25230
   717
done
haftmann@25230
   718
haftmann@25230
   719
lemma mult_mono':
haftmann@25230
   720
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c
haftmann@25230
   721
     \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   722
apply (rule mult_mono)
haftmann@25230
   723
apply (fast intro: order_trans)+
haftmann@25230
   724
done
haftmann@25230
   725
haftmann@25230
   726
end
krauss@21199
   727
haftmann@22390
   728
class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
huffman@29904
   729
  + semiring + cancel_comm_monoid_add
haftmann@25267
   730
begin
paulson@14268
   731
huffman@27516
   732
subclass semiring_0_cancel ..
huffman@27516
   733
subclass pordered_semiring ..
obua@23521
   734
haftmann@25230
   735
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
huffman@30692
   736
using mult_left_mono [of zero b a] by simp
haftmann@25230
   737
haftmann@25230
   738
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
huffman@30692
   739
using mult_left_mono [of b zero a] by simp
huffman@30692
   740
huffman@30692
   741
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
huffman@30692
   742
using mult_right_mono [of a zero b] by simp
huffman@30692
   743
huffman@30692
   744
text {* Legacy - use @{text mult_nonpos_nonneg} *}
haftmann@25230
   745
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
nipkow@29667
   746
by (drule mult_right_mono [of b zero], auto)
haftmann@25230
   747
haftmann@26234
   748
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
nipkow@29667
   749
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   750
haftmann@25230
   751
end
haftmann@25230
   752
haftmann@25230
   753
class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
haftmann@25267
   754
begin
haftmann@25230
   755
huffman@27516
   756
subclass pordered_cancel_semiring ..
haftmann@25512
   757
huffman@27516
   758
subclass pordered_comm_monoid_add ..
haftmann@25304
   759
haftmann@25230
   760
lemma mult_left_less_imp_less:
haftmann@25230
   761
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   762
by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   763
 
haftmann@25230
   764
lemma mult_right_less_imp_less:
haftmann@25230
   765
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   766
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   767
haftmann@25186
   768
end
haftmann@25152
   769
haftmann@22390
   770
class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
haftmann@25062
   771
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   772
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   773
begin
paulson@14341
   774
huffman@27516
   775
subclass semiring_0_cancel ..
obua@14940
   776
haftmann@25267
   777
subclass ordered_semiring
haftmann@28823
   778
proof
huffman@23550
   779
  fix a b c :: 'a
huffman@23550
   780
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   781
  from A show "c * a \<le> c * b"
haftmann@25186
   782
    unfolding le_less
haftmann@25186
   783
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   784
  from A show "a * c \<le> b * c"
haftmann@25152
   785
    unfolding le_less
haftmann@25186
   786
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   787
qed
haftmann@25152
   788
haftmann@25230
   789
lemma mult_left_le_imp_le:
haftmann@25230
   790
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   791
by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   792
 
haftmann@25230
   793
lemma mult_right_le_imp_le:
haftmann@25230
   794
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   795
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   796
huffman@30692
   797
lemma mult_pos_pos: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
huffman@30692
   798
using mult_strict_left_mono [of zero b a] by simp
huffman@30692
   799
huffman@30692
   800
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
huffman@30692
   801
using mult_strict_left_mono [of b zero a] by simp
huffman@30692
   802
huffman@30692
   803
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
huffman@30692
   804
using mult_strict_right_mono [of a zero b] by simp
huffman@30692
   805
huffman@30692
   806
text {* Legacy - use @{text mult_neg_pos} *}
huffman@30692
   807
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
nipkow@29667
   808
by (drule mult_strict_right_mono [of b zero], auto)
haftmann@25230
   809
haftmann@25230
   810
lemma zero_less_mult_pos:
haftmann@25230
   811
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   812
apply (cases "b\<le>0")
haftmann@25230
   813
 apply (auto simp add: le_less not_less)
huffman@30692
   814
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
   815
 apply (auto dest: less_not_sym)
haftmann@25230
   816
done
haftmann@25230
   817
haftmann@25230
   818
lemma zero_less_mult_pos2:
haftmann@25230
   819
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   820
apply (cases "b\<le>0")
haftmann@25230
   821
 apply (auto simp add: le_less not_less)
huffman@30692
   822
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
   823
 apply (auto dest: less_not_sym)
haftmann@25230
   824
done
haftmann@25230
   825
haftmann@26193
   826
text{*Strict monotonicity in both arguments*}
haftmann@26193
   827
lemma mult_strict_mono:
haftmann@26193
   828
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
   829
  shows "a * c < b * d"
haftmann@26193
   830
  using assms apply (cases "c=0")
huffman@30692
   831
  apply (simp add: mult_pos_pos)
haftmann@26193
   832
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
   833
  apply (force simp add: le_less)
haftmann@26193
   834
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
   835
  done
haftmann@26193
   836
haftmann@26193
   837
text{*This weaker variant has more natural premises*}
haftmann@26193
   838
lemma mult_strict_mono':
haftmann@26193
   839
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
   840
  shows "a * c < b * d"
nipkow@29667
   841
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
   842
haftmann@26193
   843
lemma mult_less_le_imp_less:
haftmann@26193
   844
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
   845
  shows "a * c < b * d"
haftmann@26193
   846
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
   847
  apply (erule less_le_trans)
haftmann@26193
   848
  apply (erule mult_left_mono)
haftmann@26193
   849
  apply simp
haftmann@26193
   850
  apply (erule mult_strict_right_mono)
haftmann@26193
   851
  apply assumption
haftmann@26193
   852
  done
haftmann@26193
   853
haftmann@26193
   854
lemma mult_le_less_imp_less:
haftmann@26193
   855
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
   856
  shows "a * c < b * d"
haftmann@26193
   857
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
   858
  apply (erule le_less_trans)
haftmann@26193
   859
  apply (erule mult_strict_left_mono)
haftmann@26193
   860
  apply simp
haftmann@26193
   861
  apply (erule mult_right_mono)
haftmann@26193
   862
  apply simp
haftmann@26193
   863
  done
haftmann@26193
   864
haftmann@26193
   865
lemma mult_less_imp_less_left:
haftmann@26193
   866
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
haftmann@26193
   867
  shows "a < b"
haftmann@26193
   868
proof (rule ccontr)
haftmann@26193
   869
  assume "\<not>  a < b"
haftmann@26193
   870
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   871
  hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)
nipkow@29667
   872
  with this and less show False by (simp add: not_less [symmetric])
haftmann@26193
   873
qed
haftmann@26193
   874
haftmann@26193
   875
lemma mult_less_imp_less_right:
haftmann@26193
   876
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
haftmann@26193
   877
  shows "a < b"
haftmann@26193
   878
proof (rule ccontr)
haftmann@26193
   879
  assume "\<not> a < b"
haftmann@26193
   880
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   881
  hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)
nipkow@29667
   882
  with this and less show False by (simp add: not_less [symmetric])
haftmann@26193
   883
qed  
haftmann@26193
   884
haftmann@25230
   885
end
haftmann@25230
   886
haftmann@22390
   887
class mult_mono1 = times + zero + ord +
haftmann@25230
   888
  assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
paulson@14270
   889
haftmann@22390
   890
class pordered_comm_semiring = comm_semiring_0
haftmann@22390
   891
  + pordered_ab_semigroup_add + mult_mono1
haftmann@25186
   892
begin
haftmann@25152
   893
haftmann@25267
   894
subclass pordered_semiring
haftmann@28823
   895
proof
krauss@21199
   896
  fix a b c :: 'a
huffman@23550
   897
  assume "a \<le> b" "0 \<le> c"
haftmann@25230
   898
  thus "c * a \<le> c * b" by (rule mult_mono1)
huffman@23550
   899
  thus "a * c \<le> b * c" by (simp only: mult_commute)
krauss@21199
   900
qed
paulson@14265
   901
haftmann@25267
   902
end
haftmann@25267
   903
haftmann@25267
   904
class pordered_cancel_comm_semiring = comm_semiring_0_cancel
haftmann@25267
   905
  + pordered_ab_semigroup_add + mult_mono1
haftmann@25267
   906
begin
paulson@14265
   907
huffman@27516
   908
subclass pordered_comm_semiring ..
huffman@27516
   909
subclass pordered_cancel_semiring ..
haftmann@25267
   910
haftmann@25267
   911
end
haftmann@25267
   912
haftmann@25267
   913
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
haftmann@26193
   914
  assumes mult_strict_left_mono_comm: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
   915
begin
haftmann@25267
   916
haftmann@25267
   917
subclass ordered_semiring_strict
haftmann@28823
   918
proof
huffman@23550
   919
  fix a b c :: 'a
huffman@23550
   920
  assume "a < b" "0 < c"
haftmann@26193
   921
  thus "c * a < c * b" by (rule mult_strict_left_mono_comm)
huffman@23550
   922
  thus "a * c < b * c" by (simp only: mult_commute)
huffman@23550
   923
qed
paulson@14272
   924
haftmann@25267
   925
subclass pordered_cancel_comm_semiring
haftmann@28823
   926
proof
huffman@23550
   927
  fix a b c :: 'a
huffman@23550
   928
  assume "a \<le> b" "0 \<le> c"
huffman@23550
   929
  thus "c * a \<le> c * b"
haftmann@25186
   930
    unfolding le_less
haftmann@26193
   931
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   932
qed
paulson@14272
   933
haftmann@25267
   934
end
haftmann@25230
   935
haftmann@25267
   936
class pordered_ring = ring + pordered_cancel_semiring 
haftmann@25267
   937
begin
haftmann@25230
   938
huffman@27516
   939
subclass pordered_ab_group_add ..
paulson@14270
   940
nipkow@29667
   941
text{*Legacy - use @{text algebra_simps} *}
nipkow@29833
   942
lemmas ring_simps[noatp] = algebra_simps
haftmann@25230
   943
haftmann@25230
   944
lemma less_add_iff1:
haftmann@25230
   945
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
   946
by (simp add: algebra_simps)
haftmann@25230
   947
haftmann@25230
   948
lemma less_add_iff2:
haftmann@25230
   949
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
   950
by (simp add: algebra_simps)
haftmann@25230
   951
haftmann@25230
   952
lemma le_add_iff1:
haftmann@25230
   953
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
   954
by (simp add: algebra_simps)
haftmann@25230
   955
haftmann@25230
   956
lemma le_add_iff2:
haftmann@25230
   957
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
   958
by (simp add: algebra_simps)
haftmann@25230
   959
haftmann@25230
   960
lemma mult_left_mono_neg:
haftmann@25230
   961
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@25230
   962
  apply (drule mult_left_mono [of _ _ "uminus c"])
haftmann@25230
   963
  apply (simp_all add: minus_mult_left [symmetric]) 
haftmann@25230
   964
  done
haftmann@25230
   965
haftmann@25230
   966
lemma mult_right_mono_neg:
haftmann@25230
   967
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   968
  apply (drule mult_right_mono [of _ _ "uminus c"])
haftmann@25230
   969
  apply (simp_all add: minus_mult_right [symmetric]) 
haftmann@25230
   970
  done
haftmann@25230
   971
huffman@30692
   972
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
huffman@30692
   973
using mult_right_mono_neg [of a zero b] by simp
haftmann@25230
   974
haftmann@25230
   975
lemma split_mult_pos_le:
haftmann@25230
   976
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@29667
   977
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
haftmann@25186
   978
haftmann@25186
   979
end
paulson@14270
   980
haftmann@25762
   981
class abs_if = minus + uminus + ord + zero + abs +
haftmann@25762
   982
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@25762
   983
haftmann@25762
   984
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@25186
   985
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
nipkow@24506
   986
nipkow@25564
   987
lemma (in sgn_if) sgn0[simp]: "sgn 0 = 0"
nipkow@25564
   988
by(simp add:sgn_if)
nipkow@25564
   989
haftmann@25230
   990
class ordered_ring = ring + ordered_semiring
haftmann@25304
   991
  + ordered_ab_group_add + abs_if
haftmann@25304
   992
begin
haftmann@25304
   993
huffman@27516
   994
subclass pordered_ring ..
haftmann@25304
   995
haftmann@25304
   996
subclass pordered_ab_group_add_abs
haftmann@28823
   997
proof
haftmann@25304
   998
  fix a b
haftmann@25304
   999
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
nipkow@29667
  1000
by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
haftmann@25304
  1001
   (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]
haftmann@25304
  1002
     neg_less_eq_nonneg less_eq_neg_nonpos, auto intro: add_nonneg_nonneg,
haftmann@25304
  1003
      auto intro!: less_imp_le add_neg_neg)
haftmann@25304
  1004
qed (auto simp add: abs_if less_eq_neg_nonpos neg_equal_zero)
haftmann@25304
  1005
haftmann@25304
  1006
end
obua@23521
  1007
haftmann@25230
  1008
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
haftmann@25230
  1009
   Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
haftmann@25230
  1010
 *)
haftmann@25230
  1011
class ordered_ring_strict = ring + ordered_semiring_strict
haftmann@25304
  1012
  + ordered_ab_group_add + abs_if
haftmann@25230
  1013
begin
paulson@14348
  1014
huffman@27516
  1015
subclass ordered_ring ..
haftmann@25304
  1016
huffman@30692
  1017
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
  1018
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
  1019
huffman@30692
  1020
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
  1021
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
  1022
huffman@30692
  1023
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
huffman@30692
  1024
using mult_strict_right_mono_neg [of a zero b] by simp
obua@14738
  1025
haftmann@25917
  1026
subclass ring_no_zero_divisors
haftmann@28823
  1027
proof
haftmann@25917
  1028
  fix a b
haftmann@25917
  1029
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
  1030
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
  1031
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
  1032
  proof (cases "a < 0")
haftmann@25917
  1033
    case True note A' = this
haftmann@25917
  1034
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1035
      case True with A'
haftmann@25917
  1036
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
  1037
    next
haftmann@25917
  1038
      case False with B have "0 < b" by auto
haftmann@25917
  1039
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
  1040
    qed
haftmann@25917
  1041
  next
haftmann@25917
  1042
    case False with A have A': "0 < a" by auto
haftmann@25917
  1043
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1044
      case True with A'
haftmann@25917
  1045
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
  1046
    next
haftmann@25917
  1047
      case False with B have "0 < b" by auto
haftmann@25917
  1048
      with A' show ?thesis by (auto dest: mult_pos_pos)
haftmann@25917
  1049
    qed
haftmann@25917
  1050
  qed
haftmann@25917
  1051
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
  1052
qed
haftmann@25304
  1053
paulson@14265
  1054
lemma zero_less_mult_iff:
haftmann@25917
  1055
  "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
haftmann@25917
  1056
  apply (auto simp add: mult_pos_pos mult_neg_neg)
haftmann@25917
  1057
  apply (simp_all add: not_less le_less)
haftmann@25917
  1058
  apply (erule disjE) apply assumption defer
haftmann@25917
  1059
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
  1060
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
  1061
  apply (erule disjE) apply assumption apply (drule sym) apply simp
haftmann@25917
  1062
  apply (drule sym) apply simp
haftmann@25917
  1063
  apply (blast dest: zero_less_mult_pos)
haftmann@25230
  1064
  apply (blast dest: zero_less_mult_pos2)
haftmann@25230
  1065
  done
huffman@22990
  1066
paulson@14265
  1067
lemma zero_le_mult_iff:
haftmann@25917
  1068
  "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
nipkow@29667
  1069
by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
  1070
paulson@14265
  1071
lemma mult_less_0_iff:
haftmann@25917
  1072
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
haftmann@25917
  1073
  apply (insert zero_less_mult_iff [of "-a" b]) 
haftmann@25917
  1074
  apply (force simp add: minus_mult_left[symmetric]) 
haftmann@25917
  1075
  done
paulson@14265
  1076
paulson@14265
  1077
lemma mult_le_0_iff:
haftmann@25917
  1078
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
  1079
  apply (insert zero_le_mult_iff [of "-a" b]) 
haftmann@25917
  1080
  apply (force simp add: minus_mult_left[symmetric]) 
haftmann@25917
  1081
  done
haftmann@25917
  1082
haftmann@25917
  1083
lemma zero_le_square [simp]: "0 \<le> a * a"
nipkow@29667
  1084
by (simp add: zero_le_mult_iff linear)
haftmann@25917
  1085
haftmann@25917
  1086
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
nipkow@29667
  1087
by (simp add: not_less)
haftmann@25917
  1088
haftmann@26193
  1089
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
  1090
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
  1091
haftmann@26193
  1092
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
  1093
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
  1094
haftmann@26193
  1095
lemma mult_less_cancel_right_disj:
haftmann@26193
  1096
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1097
  apply (cases "c = 0")
haftmann@26193
  1098
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
  1099
                      mult_strict_right_mono_neg)
haftmann@26193
  1100
  apply (auto simp add: not_less 
haftmann@26193
  1101
                      not_le [symmetric, of "a*c"]
haftmann@26193
  1102
                      not_le [symmetric, of a])
haftmann@26193
  1103
  apply (erule_tac [!] notE)
haftmann@26193
  1104
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
  1105
                      mult_right_mono_neg)
haftmann@26193
  1106
  done
haftmann@26193
  1107
haftmann@26193
  1108
lemma mult_less_cancel_left_disj:
haftmann@26193
  1109
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1110
  apply (cases "c = 0")
haftmann@26193
  1111
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
  1112
                      mult_strict_left_mono_neg)
haftmann@26193
  1113
  apply (auto simp add: not_less 
haftmann@26193
  1114
                      not_le [symmetric, of "c*a"]
haftmann@26193
  1115
                      not_le [symmetric, of a])
haftmann@26193
  1116
  apply (erule_tac [!] notE)
haftmann@26193
  1117
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
  1118
                      mult_left_mono_neg)
haftmann@26193
  1119
  done
haftmann@26193
  1120
haftmann@26193
  1121
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
  1122
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
  1123
haftmann@26193
  1124
lemma mult_less_cancel_right:
haftmann@26193
  1125
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1126
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
  1127
haftmann@26193
  1128
lemma mult_less_cancel_left:
haftmann@26193
  1129
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1130
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
  1131
haftmann@26193
  1132
lemma mult_le_cancel_right:
haftmann@26193
  1133
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1134
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
  1135
haftmann@26193
  1136
lemma mult_le_cancel_left:
haftmann@26193
  1137
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1138
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
  1139
nipkow@30649
  1140
lemma mult_le_cancel_left_pos:
nipkow@30649
  1141
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
  1142
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1143
nipkow@30649
  1144
lemma mult_le_cancel_left_neg:
nipkow@30649
  1145
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
  1146
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1147
nipkow@30649
  1148
lemma mult_less_cancel_left_pos:
nipkow@30649
  1149
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
  1150
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1151
nipkow@30649
  1152
lemma mult_less_cancel_left_neg:
nipkow@30649
  1153
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
  1154
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1155
haftmann@25917
  1156
end
paulson@14265
  1157
nipkow@29667
  1158
text{*Legacy - use @{text algebra_simps} *}
nipkow@29833
  1159
lemmas ring_simps[noatp] = algebra_simps
haftmann@25230
  1160
huffman@30692
  1161
lemmas mult_sign_intros =
huffman@30692
  1162
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
  1163
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
  1164
  mult_pos_pos mult_pos_neg
huffman@30692
  1165
  mult_neg_pos mult_neg_neg
haftmann@25230
  1166
haftmann@25230
  1167
class pordered_comm_ring = comm_ring + pordered_comm_semiring
haftmann@25267
  1168
begin
haftmann@25230
  1169
huffman@27516
  1170
subclass pordered_ring ..
huffman@27516
  1171
subclass pordered_cancel_comm_semiring ..
haftmann@25230
  1172
haftmann@25267
  1173
end
haftmann@25230
  1174
haftmann@25230
  1175
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
haftmann@25230
  1176
  (*previously ordered_semiring*)
haftmann@25230
  1177
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
  1178
begin
haftmann@25230
  1179
haftmann@25230
  1180
lemma pos_add_strict:
haftmann@25230
  1181
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@25230
  1182
  using add_strict_mono [of zero a b c] by simp
haftmann@25230
  1183
haftmann@26193
  1184
lemma zero_le_one [simp]: "0 \<le> 1"
nipkow@29667
  1185
by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
  1186
haftmann@26193
  1187
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
nipkow@29667
  1188
by (simp add: not_le) 
haftmann@26193
  1189
haftmann@26193
  1190
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
nipkow@29667
  1191
by (simp add: not_less) 
haftmann@26193
  1192
haftmann@26193
  1193
lemma less_1_mult:
haftmann@26193
  1194
  assumes "1 < m" and "1 < n"
haftmann@26193
  1195
  shows "1 < m * n"
haftmann@26193
  1196
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
  1197
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
  1198
haftmann@25230
  1199
end
haftmann@25230
  1200
haftmann@26193
  1201
class ordered_idom = comm_ring_1 +
haftmann@26193
  1202
  ordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1203
  abs_if + sgn_if
haftmann@25230
  1204
  (*previously ordered_ring*)
haftmann@25917
  1205
begin
haftmann@25917
  1206
huffman@27516
  1207
subclass ordered_ring_strict ..
huffman@27516
  1208
subclass pordered_comm_ring ..
huffman@27516
  1209
subclass idom ..
haftmann@25917
  1210
haftmann@25917
  1211
subclass ordered_semidom
haftmann@28823
  1212
proof
haftmann@26193
  1213
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1214
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
  1215
qed 
haftmann@25917
  1216
haftmann@26193
  1217
lemma linorder_neqE_ordered_idom:
haftmann@26193
  1218
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1219
  using assms by (rule neqE)
haftmann@26193
  1220
haftmann@26274
  1221
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1222
haftmann@26274
  1223
lemma mult_le_cancel_right1:
haftmann@26274
  1224
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1225
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1226
haftmann@26274
  1227
lemma mult_le_cancel_right2:
haftmann@26274
  1228
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1229
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1230
haftmann@26274
  1231
lemma mult_le_cancel_left1:
haftmann@26274
  1232
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1233
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1234
haftmann@26274
  1235
lemma mult_le_cancel_left2:
haftmann@26274
  1236
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1237
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1238
haftmann@26274
  1239
lemma mult_less_cancel_right1:
haftmann@26274
  1240
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1241
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1242
haftmann@26274
  1243
lemma mult_less_cancel_right2:
haftmann@26274
  1244
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1245
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1246
haftmann@26274
  1247
lemma mult_less_cancel_left1:
haftmann@26274
  1248
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1249
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1250
haftmann@26274
  1251
lemma mult_less_cancel_left2:
haftmann@26274
  1252
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1253
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1254
haftmann@27651
  1255
lemma sgn_sgn [simp]:
haftmann@27651
  1256
  "sgn (sgn a) = sgn a"
nipkow@29700
  1257
unfolding sgn_if by simp
haftmann@27651
  1258
haftmann@27651
  1259
lemma sgn_0_0:
haftmann@27651
  1260
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1261
unfolding sgn_if by simp
haftmann@27651
  1262
haftmann@27651
  1263
lemma sgn_1_pos:
haftmann@27651
  1264
  "sgn a = 1 \<longleftrightarrow> a > 0"
nipkow@29700
  1265
unfolding sgn_if by (simp add: neg_equal_zero)
haftmann@27651
  1266
haftmann@27651
  1267
lemma sgn_1_neg:
haftmann@27651
  1268
  "sgn a = - 1 \<longleftrightarrow> a < 0"
nipkow@29700
  1269
unfolding sgn_if by (auto simp add: equal_neg_zero)
haftmann@27651
  1270
haftmann@29940
  1271
lemma sgn_pos [simp]:
haftmann@29940
  1272
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1273
unfolding sgn_1_pos .
haftmann@29940
  1274
haftmann@29940
  1275
lemma sgn_neg [simp]:
haftmann@29940
  1276
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1277
unfolding sgn_1_neg .
haftmann@29940
  1278
haftmann@27651
  1279
lemma sgn_times:
haftmann@27651
  1280
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1281
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1282
haftmann@29653
  1283
lemma abs_sgn: "abs k = k * sgn k"
nipkow@29700
  1284
unfolding sgn_if abs_if by auto
nipkow@29700
  1285
haftmann@29940
  1286
lemma sgn_greater [simp]:
haftmann@29940
  1287
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1288
  unfolding sgn_if by auto
haftmann@29940
  1289
haftmann@29940
  1290
lemma sgn_less [simp]:
haftmann@29940
  1291
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1292
  unfolding sgn_if by auto
haftmann@29940
  1293
huffman@29949
  1294
lemma abs_dvd_iff [simp]: "(abs m) dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1295
  by (simp add: abs_if)
huffman@29949
  1296
huffman@29949
  1297
lemma dvd_abs_iff [simp]: "m dvd (abs k) \<longleftrightarrow> m dvd k"
huffman@29949
  1298
  by (simp add: abs_if)
haftmann@29653
  1299
haftmann@25917
  1300
end
haftmann@25230
  1301
haftmann@25230
  1302
class ordered_field = field + ordered_idom
haftmann@25230
  1303
haftmann@26274
  1304
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1305
nipkow@29833
  1306
lemmas mult_compare_simps[noatp] =
paulson@15234
  1307
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1308
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1309
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1310
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1311
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1312
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1313
    mult_cancel_right mult_cancel_left
paulson@15234
  1314
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1315
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1316
haftmann@26274
  1317
-- {* FIXME continue localization here *}
paulson@14268
  1318
paulson@14268
  1319
lemma inverse_nonzero_iff_nonzero [simp]:
huffman@20496
  1320
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
haftmann@26274
  1321
by (force dest: inverse_zero_imp_zero) 
paulson@14268
  1322
paulson@14268
  1323
lemma inverse_minus_eq [simp]:
huffman@20496
  1324
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
paulson@14377
  1325
proof cases
paulson@14377
  1326
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
  1327
next
paulson@14377
  1328
  assume "a\<noteq>0" 
paulson@14377
  1329
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
  1330
qed
paulson@14268
  1331
paulson@14268
  1332
lemma inverse_eq_imp_eq:
huffman@20496
  1333
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
haftmann@21328
  1334
apply (cases "a=0 | b=0") 
paulson@14268
  1335
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
  1336
              simp add: eq_commute [of "0::'a"])
paulson@14268
  1337
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
  1338
done
paulson@14268
  1339
paulson@14268
  1340
lemma inverse_eq_iff_eq [simp]:
huffman@20496
  1341
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
huffman@20496
  1342
by (force dest!: inverse_eq_imp_eq)
paulson@14268
  1343
paulson@14270
  1344
lemma inverse_inverse_eq [simp]:
huffman@20496
  1345
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
paulson@14270
  1346
  proof cases
paulson@14270
  1347
    assume "a=0" thus ?thesis by simp
paulson@14270
  1348
  next
paulson@14270
  1349
    assume "a\<noteq>0" 
paulson@14270
  1350
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
  1351
  qed
paulson@14270
  1352
paulson@14270
  1353
text{*This version builds in division by zero while also re-orienting
paulson@14270
  1354
      the right-hand side.*}
paulson@14270
  1355
lemma inverse_mult_distrib [simp]:
paulson@14270
  1356
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
  1357
  proof cases
paulson@14270
  1358
    assume "a \<noteq> 0 & b \<noteq> 0" 
nipkow@29667
  1359
    thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
  1360
  next
paulson@14270
  1361
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
nipkow@29667
  1362
    thus ?thesis by force
paulson@14270
  1363
  qed
paulson@14270
  1364
paulson@14365
  1365
lemma inverse_divide [simp]:
nipkow@23477
  1366
  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
nipkow@23477
  1367
by (simp add: divide_inverse mult_commute)
paulson@14365
  1368
wenzelm@23389
  1369
avigad@16775
  1370
subsection {* Calculations with fractions *}
avigad@16775
  1371
nipkow@23413
  1372
text{* There is a whole bunch of simp-rules just for class @{text
nipkow@23413
  1373
field} but none for class @{text field} and @{text nonzero_divides}
nipkow@23413
  1374
because the latter are covered by a simproc. *}
nipkow@23413
  1375
nipkow@23413
  1376
lemma mult_divide_mult_cancel_left:
nipkow@23477
  1377
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1378
apply (cases "b = 0")
nipkow@23413
  1379
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
paulson@14277
  1380
done
paulson@14277
  1381
nipkow@23413
  1382
lemma mult_divide_mult_cancel_right:
nipkow@23477
  1383
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1384
apply (cases "b = 0")
nipkow@23413
  1385
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
paulson@14321
  1386
done
nipkow@23413
  1387
paulson@24286
  1388
lemma divide_divide_eq_right [simp,noatp]:
nipkow@23477
  1389
  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
  1390
by (simp add: divide_inverse mult_ac)
paulson@14288
  1391
paulson@24286
  1392
lemma divide_divide_eq_left [simp,noatp]:
nipkow@23477
  1393
  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
  1394
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1395
wenzelm@23389
  1396
paulson@15234
  1397
subsubsection{*Special Cancellation Simprules for Division*}
paulson@15234
  1398
paulson@24427
  1399
lemma mult_divide_mult_cancel_left_if[simp,noatp]:
nipkow@23477
  1400
fixes c :: "'a :: {field,division_by_zero}"
nipkow@23477
  1401
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
nipkow@23413
  1402
by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
  1403
paulson@15234
  1404
paulson@14293
  1405
subsection {* Division and Unary Minus *}
paulson@14293
  1406
paulson@14293
  1407
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
huffman@29407
  1408
by (simp add: divide_inverse)
paulson@14430
  1409
huffman@30630
  1410
lemma divide_minus_right [simp, noatp]:
huffman@30630
  1411
  "a / -(b::'a::{field,division_by_zero}) = -(a / b)"
huffman@30630
  1412
by (simp add: divide_inverse)
huffman@30630
  1413
huffman@30630
  1414
lemma minus_divide_divide:
nipkow@23477
  1415
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1416
apply (cases "b=0", simp) 
paulson@14293
  1417
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
  1418
done
paulson@14293
  1419
nipkow@23482
  1420
lemma eq_divide_eq:
nipkow@23482
  1421
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
huffman@30630
  1422
by (simp add: nonzero_eq_divide_eq)
nipkow@23482
  1423
nipkow@23482
  1424
lemma divide_eq_eq:
nipkow@23482
  1425
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
huffman@30630
  1426
by (force simp add: nonzero_divide_eq_eq)
paulson@14293
  1427
wenzelm@23389
  1428
paulson@14268
  1429
subsection {* Ordered Fields *}
paulson@14268
  1430
paulson@14277
  1431
lemma positive_imp_inverse_positive: 
nipkow@23482
  1432
assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
nipkow@23482
  1433
proof -
paulson@14268
  1434
  have "0 < a * inverse a" 
paulson@14268
  1435
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
  1436
  thus "0 < inverse a" 
paulson@14268
  1437
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
nipkow@23482
  1438
qed
paulson@14268
  1439
paulson@14277
  1440
lemma negative_imp_inverse_negative:
nipkow@23482
  1441
  "a < 0 ==> inverse a < (0::'a::ordered_field)"
nipkow@23482
  1442
by (insert positive_imp_inverse_positive [of "-a"], 
nipkow@23482
  1443
    simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
paulson@14268
  1444
paulson@14268
  1445
lemma inverse_le_imp_le:
nipkow@23482
  1446
assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
nipkow@23482
  1447
shows "b \<le> (a::'a::ordered_field)"
nipkow@23482
  1448
proof (rule classical)
paulson@14268
  1449
  assume "~ b \<le> a"
nipkow@23482
  1450
  hence "a < b"  by (simp add: linorder_not_le)
nipkow@23482
  1451
  hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
paulson@14268
  1452
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
  1453
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
  1454
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
  1455
    by (simp add: bpos order_less_imp_le mult_right_mono)
nipkow@23482
  1456
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
nipkow@23482
  1457
qed
paulson@14268
  1458
paulson@14277
  1459
lemma inverse_positive_imp_positive:
nipkow@23482
  1460
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
nipkow@23482
  1461
shows "0 < (a::'a::ordered_field)"
wenzelm@23389
  1462
proof -
paulson@14277
  1463
  have "0 < inverse (inverse a)"
wenzelm@23389
  1464
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
  1465
  thus "0 < a"
wenzelm@23389
  1466
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1467
qed
paulson@14277
  1468
paulson@14277
  1469
lemma inverse_positive_iff_positive [simp]:
nipkow@23482
  1470
  "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1471
apply (cases "a = 0", simp)
paulson@14277
  1472
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
  1473
done
paulson@14277
  1474
paulson@14277
  1475
lemma inverse_negative_imp_negative:
nipkow@23482
  1476
assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
nipkow@23482
  1477
shows "a < (0::'a::ordered_field)"
wenzelm@23389
  1478
proof -
paulson@14277
  1479
  have "inverse (inverse a) < 0"
wenzelm@23389
  1480
    using inv_less_0 by (rule negative_imp_inverse_negative)
nipkow@23482
  1481
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1482
qed
paulson@14277
  1483
paulson@14277
  1484
lemma inverse_negative_iff_negative [simp]:
nipkow@23482
  1485
  "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1486
apply (cases "a = 0", simp)
paulson@14277
  1487
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
  1488
done
paulson@14277
  1489
paulson@14277
  1490
lemma inverse_nonnegative_iff_nonnegative [simp]:
nipkow@23482
  1491
  "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1492
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1493
paulson@14277
  1494
lemma inverse_nonpositive_iff_nonpositive [simp]:
nipkow@23482
  1495
  "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1496
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1497
chaieb@23406
  1498
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
chaieb@23406
  1499
proof
chaieb@23406
  1500
  fix x::'a
chaieb@23406
  1501
  have m1: "- (1::'a) < 0" by simp
chaieb@23406
  1502
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1503
  have "(- 1) + x < x" by simp
chaieb@23406
  1504
  thus "\<exists>y. y < x" by blast
chaieb@23406
  1505
qed
chaieb@23406
  1506
chaieb@23406
  1507
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
chaieb@23406
  1508
proof
chaieb@23406
  1509
  fix x::'a
chaieb@23406
  1510
  have m1: " (1::'a) > 0" by simp
chaieb@23406
  1511
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1512
  have "1 + x > x" by simp
chaieb@23406
  1513
  thus "\<exists>y. y > x" by blast
chaieb@23406
  1514
qed
paulson@14277
  1515
paulson@14277
  1516
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
  1517
paulson@14268
  1518
lemma less_imp_inverse_less:
nipkow@23482
  1519
assumes less: "a < b" and apos:  "0 < a"
nipkow@23482
  1520
shows "inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1521
proof (rule ccontr)
paulson@14268
  1522
  assume "~ inverse b < inverse a"
nipkow@29667
  1523
  hence "inverse a \<le> inverse b" by (simp add: linorder_not_less)
paulson@14268
  1524
  hence "~ (a < b)"
paulson@14268
  1525
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
nipkow@29667
  1526
  thus False by (rule notE [OF _ less])
nipkow@23482
  1527
qed
paulson@14268
  1528
paulson@14268
  1529
lemma inverse_less_imp_less:
nipkow@23482
  1530
  "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1531
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
  1532
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
  1533
done
paulson@14268
  1534
paulson@14268
  1535
text{*Both premises are essential. Consider -1 and 1.*}
paulson@24286
  1536
lemma inverse_less_iff_less [simp,noatp]:
nipkow@23482
  1537
  "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1538
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
  1539
paulson@14268
  1540
lemma le_imp_inverse_le:
nipkow@23482
  1541
  "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1542
by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
  1543
paulson@24286
  1544
lemma inverse_le_iff_le [simp,noatp]:
nipkow@23482
  1545
 "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1546
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
  1547
paulson@14268
  1548
paulson@14268
  1549
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
  1550
case is trivial, since inverse preserves signs.*}
paulson@14268
  1551
lemma inverse_le_imp_le_neg:
nipkow@23482
  1552
  "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
nipkow@23482
  1553
apply (rule classical) 
nipkow@23482
  1554
apply (subgoal_tac "a < 0") 
nipkow@23482
  1555
 prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
nipkow@23482
  1556
apply (insert inverse_le_imp_le [of "-b" "-a"])
nipkow@23482
  1557
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1558
done
paulson@14268
  1559
paulson@14268
  1560
lemma less_imp_inverse_less_neg:
paulson@14268
  1561
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
nipkow@23482
  1562
apply (subgoal_tac "a < 0") 
nipkow@23482
  1563
 prefer 2 apply (blast intro: order_less_trans) 
nipkow@23482
  1564
apply (insert less_imp_inverse_less [of "-b" "-a"])
nipkow@23482
  1565
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1566
done
paulson@14268
  1567
paulson@14268
  1568
lemma inverse_less_imp_less_neg:
paulson@14268
  1569
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
nipkow@23482
  1570
apply (rule classical) 
nipkow@23482
  1571
apply (subgoal_tac "a < 0") 
nipkow@23482
  1572
 prefer 2
nipkow@23482
  1573
 apply (force simp add: linorder_not_less intro: order_le_less_trans) 
nipkow@23482
  1574
apply (insert inverse_less_imp_less [of "-b" "-a"])
nipkow@23482
  1575
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
nipkow@23482
  1576
done
paulson@14268
  1577
paulson@24286
  1578
lemma inverse_less_iff_less_neg [simp,noatp]:
nipkow@23482
  1579
  "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
nipkow@23482
  1580
apply (insert inverse_less_iff_less [of "-b" "-a"])
nipkow@23482
  1581
apply (simp del: inverse_less_iff_less 
nipkow@23482
  1582
            add: order_less_imp_not_eq nonzero_inverse_minus_eq)
nipkow@23482
  1583
done
paulson@14268
  1584
paulson@14268
  1585
lemma le_imp_inverse_le_neg:
nipkow@23482
  1586
  "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
nipkow@23482
  1587
by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1588
paulson@24286
  1589
lemma inverse_le_iff_le_neg [simp,noatp]:
nipkow@23482
  1590
 "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1591
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1592
paulson@14277
  1593
paulson@14365
  1594
subsection{*Inverses and the Number One*}
paulson@14365
  1595
paulson@14365
  1596
lemma one_less_inverse_iff:
nipkow@23482
  1597
  "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
nipkow@23482
  1598
proof cases
paulson@14365
  1599
  assume "0 < x"
paulson@14365
  1600
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1601
    show ?thesis by simp
paulson@14365
  1602
next
paulson@14365
  1603
  assume notless: "~ (0 < x)"
paulson@14365
  1604
  have "~ (1 < inverse x)"
paulson@14365
  1605
  proof
paulson@14365
  1606
    assume "1 < inverse x"
paulson@14365
  1607
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1608
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1609
    finally show False by auto
paulson@14365
  1610
  qed
paulson@14365
  1611
  with notless show ?thesis by simp
paulson@14365
  1612
qed
paulson@14365
  1613
paulson@14365
  1614
lemma inverse_eq_1_iff [simp]:
nipkow@23482
  1615
  "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1616
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1617
paulson@14365
  1618
lemma one_le_inverse_iff:
nipkow@23482
  1619
  "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1620
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1621
                    eq_commute [of 1]) 
paulson@14365
  1622
paulson@14365
  1623
lemma inverse_less_1_iff:
nipkow@23482
  1624
  "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1625
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1626
paulson@14365
  1627
lemma inverse_le_1_iff:
nipkow@23482
  1628
  "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1629
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1630
wenzelm@23389
  1631
paulson@14288
  1632
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1633
paulson@14288
  1634
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1635
proof -
paulson@14288
  1636
  assume less: "0<c"
paulson@14288
  1637
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1638
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1639
  also have "... = (a*c \<le> b)"
paulson@14288
  1640
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1641
  finally show ?thesis .
paulson@14288
  1642
qed
paulson@14288
  1643
paulson@14288
  1644
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1645
proof -
paulson@14288
  1646
  assume less: "c<0"
paulson@14288
  1647
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1648
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1649
  also have "... = (b \<le> a*c)"
paulson@14288
  1650
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1651
  finally show ?thesis .
paulson@14288
  1652
qed
paulson@14288
  1653
paulson@14288
  1654
lemma le_divide_eq:
paulson@14288
  1655
  "(a \<le> b/c) = 
paulson@14288
  1656
   (if 0 < c then a*c \<le> b
paulson@14288
  1657
             else if c < 0 then b \<le> a*c
paulson@14288
  1658
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1659
apply (cases "c=0", simp) 
paulson@14288
  1660
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1661
done
paulson@14288
  1662
paulson@14288
  1663
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1664
proof -
paulson@14288
  1665
  assume less: "0<c"
paulson@14288
  1666
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1667
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1668
  also have "... = (b \<le> a*c)"
paulson@14288
  1669
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1670
  finally show ?thesis .
paulson@14288
  1671
qed
paulson@14288
  1672
paulson@14288
  1673
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1674
proof -
paulson@14288
  1675
  assume less: "c<0"
paulson@14288
  1676
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1677
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1678
  also have "... = (a*c \<le> b)"
paulson@14288
  1679
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1680
  finally show ?thesis .
paulson@14288
  1681
qed
paulson@14288
  1682
paulson@14288
  1683
lemma divide_le_eq:
paulson@14288
  1684
  "(b/c \<le> a) = 
paulson@14288
  1685
   (if 0 < c then b \<le> a*c
paulson@14288
  1686
             else if c < 0 then a*c \<le> b
paulson@14288
  1687
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1688
apply (cases "c=0", simp) 
paulson@14288
  1689
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1690
done
paulson@14288
  1691
paulson@14288
  1692
lemma pos_less_divide_eq:
paulson@14288
  1693
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1694
proof -
paulson@14288
  1695
  assume less: "0<c"
paulson@14288
  1696
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@15234
  1697
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1698
  also have "... = (a*c < b)"
paulson@14288
  1699
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1700
  finally show ?thesis .
paulson@14288
  1701
qed
paulson@14288
  1702
paulson@14288
  1703
lemma neg_less_divide_eq:
paulson@14288
  1704
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1705
proof -
paulson@14288
  1706
  assume less: "c<0"
paulson@14288
  1707
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@15234
  1708
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1709
  also have "... = (b < a*c)"
paulson@14288
  1710
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1711
  finally show ?thesis .
paulson@14288
  1712
qed
paulson@14288
  1713
paulson@14288
  1714
lemma less_divide_eq:
paulson@14288
  1715
  "(a < b/c) = 
paulson@14288
  1716
   (if 0 < c then a*c < b
paulson@14288
  1717
             else if c < 0 then b < a*c
paulson@14288
  1718
             else  a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1719
apply (cases "c=0", simp) 
paulson@14288
  1720
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1721
done
paulson@14288
  1722
paulson@14288
  1723
lemma pos_divide_less_eq:
paulson@14288
  1724
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1725
proof -
paulson@14288
  1726
  assume less: "0<c"
paulson@14288
  1727
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@15234
  1728
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1729
  also have "... = (b < a*c)"
paulson@14288
  1730
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1731
  finally show ?thesis .
paulson@14288
  1732
qed
paulson@14288
  1733
paulson@14288
  1734
lemma neg_divide_less_eq:
paulson@14288
  1735
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1736
proof -
paulson@14288
  1737
  assume less: "c<0"
paulson@14288
  1738
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@15234
  1739
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1740
  also have "... = (a*c < b)"
paulson@14288
  1741
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1742
  finally show ?thesis .
paulson@14288
  1743
qed
paulson@14288
  1744
paulson@14288
  1745
lemma divide_less_eq:
paulson@14288
  1746
  "(b/c < a) = 
paulson@14288
  1747
   (if 0 < c then b < a*c
paulson@14288
  1748
             else if c < 0 then a*c < b
paulson@14288
  1749
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1750
apply (cases "c=0", simp) 
paulson@14288
  1751
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1752
done
paulson@14288
  1753
nipkow@23482
  1754
nipkow@23482
  1755
subsection{*Field simplification*}
nipkow@23482
  1756
nipkow@29667
  1757
text{* Lemmas @{text field_simps} multiply with denominators in in(equations)
nipkow@29667
  1758
if they can be proved to be non-zero (for equations) or positive/negative
nipkow@29667
  1759
(for inequations). Can be too aggressive and is therefore separate from the
nipkow@29667
  1760
more benign @{text algebra_simps}. *}
paulson@14288
  1761
nipkow@29833
  1762
lemmas field_simps[noatp] = field_eq_simps
nipkow@23482
  1763
  (* multiply ineqn *)
nipkow@23482
  1764
  pos_divide_less_eq neg_divide_less_eq
nipkow@23482
  1765
  pos_less_divide_eq neg_less_divide_eq
nipkow@23482
  1766
  pos_divide_le_eq neg_divide_le_eq
nipkow@23482
  1767
  pos_le_divide_eq neg_le_divide_eq
paulson@14288
  1768
nipkow@23482
  1769
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
nipkow@23483
  1770
of positivity/negativity needed for @{text field_simps}. Have not added @{text
nipkow@23482
  1771
sign_simps} to @{text field_simps} because the former can lead to case
nipkow@23482
  1772
explosions. *}
paulson@14288
  1773
nipkow@29833
  1774
lemmas sign_simps[noatp] = group_simps
nipkow@23482
  1775
  zero_less_mult_iff  mult_less_0_iff
paulson@14288
  1776
nipkow@23482
  1777
(* Only works once linear arithmetic is installed:
nipkow@23482
  1778
text{*An example:*}
nipkow@23482
  1779
lemma fixes a b c d e f :: "'a::ordered_field"
nipkow@23482
  1780
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
nipkow@23482
  1781
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
nipkow@23482
  1782
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
nipkow@23482
  1783
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
nipkow@23482
  1784
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1785
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
nipkow@23482
  1786
 prefer 2 apply(simp add:sign_simps)
nipkow@23482
  1787
apply(simp add:field_simps)
avigad@16775
  1788
done
nipkow@23482
  1789
*)
avigad@16775
  1790
wenzelm@23389
  1791
avigad@16775
  1792
subsection{*Division and Signs*}
avigad@16775
  1793
avigad@16775
  1794
lemma zero_less_divide_iff:
avigad@16775
  1795
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
  1796
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
  1797
avigad@16775
  1798
lemma divide_less_0_iff:
avigad@16775
  1799
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
avigad@16775
  1800
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
  1801
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
  1802
avigad@16775
  1803
lemma zero_le_divide_iff:
avigad@16775
  1804
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
avigad@16775
  1805
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
  1806
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
  1807
avigad@16775
  1808
lemma divide_le_0_iff:
avigad@16775
  1809
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
avigad@16775
  1810
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
  1811
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
  1812
paulson@24286
  1813
lemma divide_eq_0_iff [simp,noatp]:
avigad@16775
  1814
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
nipkow@23482
  1815
by (simp add: divide_inverse)
avigad@16775
  1816
nipkow@23482
  1817
lemma divide_pos_pos:
nipkow@23482
  1818
  "0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
nipkow@23482
  1819
by(simp add:field_simps)
nipkow@23482
  1820
avigad@16775
  1821
nipkow@23482
  1822
lemma divide_nonneg_pos:
nipkow@23482
  1823
  "0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
nipkow@23482
  1824
by(simp add:field_simps)
avigad@16775
  1825
nipkow@23482
  1826
lemma divide_neg_pos:
nipkow@23482
  1827
  "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
nipkow@23482
  1828
by(simp add:field_simps)
avigad@16775
  1829
nipkow@23482
  1830
lemma divide_nonpos_pos:
nipkow@23482
  1831
  "(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
nipkow@23482
  1832
by(simp add:field_simps)
avigad@16775
  1833
nipkow@23482
  1834
lemma divide_pos_neg:
nipkow@23482
  1835
  "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
nipkow@23482
  1836
by(simp add:field_simps)
avigad@16775
  1837
nipkow@23482
  1838
lemma divide_nonneg_neg:
nipkow@23482
  1839
  "0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0" 
nipkow@23482
  1840
by(simp add:field_simps)
avigad@16775
  1841
nipkow@23482
  1842
lemma divide_neg_neg:
nipkow@23482
  1843
  "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
nipkow@23482
  1844
by(simp add:field_simps)
avigad@16775
  1845
nipkow@23482
  1846
lemma divide_nonpos_neg:
nipkow@23482
  1847
  "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
nipkow@23482
  1848
by(simp add:field_simps)
paulson@15234
  1849
wenzelm@23389
  1850
paulson@14288
  1851
subsection{*Cancellation Laws for Division*}
paulson@14288
  1852
paulson@24286
  1853
lemma divide_cancel_right [simp,noatp]:
paulson@14288
  1854
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1855
apply (cases "c=0", simp)
nipkow@23496
  1856
apply (simp add: divide_inverse)
paulson@14288
  1857
done
paulson@14288
  1858
paulson@24286
  1859
lemma divide_cancel_left [simp,noatp]:
paulson@14288
  1860
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
nipkow@23482
  1861
apply (cases "c=0", simp)
nipkow@23496
  1862
apply (simp add: divide_inverse)
paulson@14288
  1863
done
paulson@14288
  1864
wenzelm@23389
  1865
paulson@14353
  1866
subsection {* Division and the Number One *}
paulson@14353
  1867
paulson@14353
  1868
text{*Simplify expressions equated with 1*}
paulson@24286
  1869
lemma divide_eq_1_iff [simp,noatp]:
paulson@14353
  1870
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1871
apply (cases "b=0", simp)
nipkow@23482
  1872
apply (simp add: right_inverse_eq)
paulson@14353
  1873
done
paulson@14353
  1874
paulson@24286
  1875
lemma one_eq_divide_iff [simp,noatp]:
paulson@14353
  1876
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
nipkow@23482
  1877
by (simp add: eq_commute [of 1])
paulson@14353
  1878
paulson@24286
  1879
lemma zero_eq_1_divide_iff [simp,noatp]:
paulson@14353
  1880
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
nipkow@23482
  1881
apply (cases "a=0", simp)
nipkow@23482
  1882
apply (auto simp add: nonzero_eq_divide_eq)
paulson@14353
  1883
done
paulson@14353
  1884
paulson@24286
  1885
lemma one_divide_eq_0_iff [simp,noatp]:
paulson@14353
  1886
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
nipkow@23482
  1887
apply (cases "a=0", simp)
nipkow@23482
  1888
apply (insert zero_neq_one [THEN not_sym])
nipkow@23482
  1889
apply (auto simp add: nonzero_divide_eq_eq)
paulson@14353
  1890
done
paulson@14353
  1891
paulson@14353
  1892
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@18623
  1893
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
paulson@18623
  1894
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
paulson@18623
  1895
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
paulson@18623
  1896
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
paulson@17085
  1897
nipkow@29833
  1898
declare zero_less_divide_1_iff [simp,noatp]
paulson@24286
  1899
declare divide_less_0_1_iff [simp,noatp]
nipkow@29833
  1900
declare zero_le_divide_1_iff [simp,noatp]
paulson@24286
  1901
declare divide_le_0_1_iff [simp,noatp]
paulson@14353
  1902
wenzelm@23389
  1903
paulson@14293
  1904
subsection {* Ordering Rules for Division *}
paulson@14293
  1905
paulson@14293
  1906
lemma divide_strict_right_mono:
paulson@14293
  1907
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1908
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
nipkow@23482
  1909
              positive_imp_inverse_positive)
paulson@14293
  1910
paulson@14293
  1911
lemma divide_right_mono:
paulson@14293
  1912
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
nipkow@23482
  1913
by (force simp add: divide_strict_right_mono order_le_less)
paulson@14293
  1914
avigad@16775
  1915
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1916
    ==> c <= 0 ==> b / c <= a / c"
nipkow@23482
  1917
apply (drule divide_right_mono [of _ _ "- c"])
nipkow@23482
  1918
apply auto
avigad@16775
  1919
done
avigad@16775
  1920
avigad@16775
  1921
lemma divide_strict_right_mono_neg:
avigad@16775
  1922
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
nipkow@23482
  1923
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
nipkow@23482
  1924
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
avigad@16775
  1925
done
paulson@14293
  1926
paulson@14293
  1927
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1928
      have the same sign*}
paulson@14293
  1929
lemma divide_strict_left_mono:
nipkow@23482
  1930
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1931
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
paulson@14293
  1932
paulson@14293
  1933
lemma divide_left_mono:
nipkow@23482
  1934
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
nipkow@23482
  1935
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
paulson@14293
  1936
avigad@16775
  1937
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1938
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1939
  apply (drule divide_left_mono [of _ _ "- c"])
avigad@16775
  1940
  apply (auto simp add: mult_commute)
avigad@16775
  1941
done
avigad@16775
  1942
paulson@14293
  1943
lemma divide_strict_left_mono_neg:
nipkow@23482
  1944
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
nipkow@23482
  1945
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
nipkow@23482
  1946
paulson@14293
  1947
avigad@16775
  1948
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1949
paulson@24286
  1950
lemma le_divide_eq_1 [noatp]:
avigad@16775
  1951
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1952
  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1953
by (auto simp add: le_divide_eq)
avigad@16775
  1954
paulson@24286
  1955
lemma divide_le_eq_1 [noatp]:
avigad@16775
  1956
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1957
  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1958
by (auto simp add: divide_le_eq)
avigad@16775
  1959
paulson@24286
  1960
lemma less_divide_eq_1 [noatp]:
avigad@16775
  1961
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1962
  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1963
by (auto simp add: less_divide_eq)
avigad@16775
  1964
paulson@24286
  1965
lemma divide_less_eq_1 [noatp]:
avigad@16775
  1966
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1967
  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1968
by (auto simp add: divide_less_eq)
avigad@16775
  1969
wenzelm@23389
  1970
avigad@16775
  1971
subsection{*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1972
paulson@24286
  1973
lemma le_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1974
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1975
  shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1976
by (auto simp add: le_divide_eq)
avigad@16775
  1977
paulson@24286
  1978
lemma le_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1979
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1980
  shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1981
by (auto simp add: le_divide_eq)
avigad@16775
  1982
paulson@24286
  1983
lemma divide_le_eq_1_pos [simp,noatp]:
avigad@16775
  1984
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1985
  shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1986
by (auto simp add: divide_le_eq)
avigad@16775
  1987
paulson@24286
  1988
lemma divide_le_eq_1_neg [simp,noatp]:
avigad@16775
  1989
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1990
  shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1991
by (auto simp add: divide_le_eq)
avigad@16775
  1992
paulson@24286
  1993
lemma less_divide_eq_1_pos [simp,noatp]:
avigad@16775
  1994
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1995
  shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1996
by (auto simp add: less_divide_eq)
avigad@16775
  1997
paulson@24286
  1998
lemma less_divide_eq_1_neg [simp,noatp]:
avigad@16775
  1999
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  2000
  shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  2001
by (auto simp add: less_divide_eq)
avigad@16775
  2002
paulson@24286
  2003
lemma divide_less_eq_1_pos [simp,noatp]:
avigad@16775
  2004
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  2005
  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  2006
by (auto simp add: divide_less_eq)
paulson@18649
  2007
paulson@24286
  2008
lemma divide_less_eq_1_neg [simp,noatp]:
paulson@18649
  2009
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  2010
  shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  2011
by (auto simp add: divide_less_eq)
avigad@16775
  2012
paulson@24286
  2013
lemma eq_divide_eq_1 [simp,noatp]:
avigad@16775
  2014
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  2015
  shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  2016
by (auto simp add: eq_divide_eq)
avigad@16775
  2017
paulson@24286
  2018
lemma divide_eq_eq_1 [simp,noatp]:
avigad@16775
  2019
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  2020
  shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  2021
by (auto simp add: divide_eq_eq)
avigad@16775
  2022
wenzelm@23389
  2023
avigad@16775
  2024
subsection {* Reasoning about inequalities with division *}
avigad@16775
  2025
avigad@16775
  2026
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  2027
    ==> x * y <= x"
nipkow@29667
  2028
by (auto simp add: mult_compare_simps);
avigad@16775
  2029
avigad@16775
  2030
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  2031
    ==> y * x <= x"
nipkow@29667
  2032
by (auto simp add: mult_compare_simps);
avigad@16775
  2033
avigad@16775
  2034
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
avigad@16775
  2035
    x / y <= z";
nipkow@29667
  2036
by (subst pos_divide_le_eq, assumption+);
avigad@16775
  2037
avigad@16775
  2038
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
nipkow@23482
  2039
    z <= x / y"
nipkow@23482
  2040
by(simp add:field_simps)
avigad@16775
  2041
avigad@16775
  2042
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
avigad@16775
  2043
    x / y < z"
nipkow@23482
  2044
by(simp add:field_simps)
avigad@16775
  2045
avigad@16775
  2046
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
avigad@16775
  2047
    z < x / y"
nipkow@23482
  2048
by(simp add:field_simps)
avigad@16775
  2049
avigad@16775
  2050
lemma frac_le: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  2051
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
avigad@16775
  2052
  apply (rule mult_imp_div_pos_le)
haftmann@25230
  2053
  apply simp
haftmann@25230
  2054
  apply (subst times_divide_eq_left)
avigad@16775
  2055
  apply (rule mult_imp_le_div_pos, assumption)
avigad@16775
  2056
  apply (rule mult_mono)
avigad@16775
  2057
  apply simp_all
paulson@14293
  2058
done
paulson@14293
  2059
avigad@16775
  2060
lemma frac_less: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  2061
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
avigad@16775
  2062
  apply (rule mult_imp_div_pos_less)
avigad@16775
  2063
  apply simp;
avigad@16775
  2064
  apply (subst times_divide_eq_left);
avigad@16775
  2065
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  2066
  apply (erule mult_less_le_imp_less)
avigad@16775
  2067
  apply simp_all
avigad@16775
  2068
done
avigad@16775
  2069
avigad@16775
  2070
lemma frac_less2: "(0::'a::ordered_field) < x ==> 
avigad@16775
  2071
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
avigad@16775
  2072
  apply (rule mult_imp_div_pos_less)
avigad@16775
  2073
  apply simp_all
avigad@16775
  2074
  apply (subst times_divide_eq_left);
avigad@16775
  2075
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  2076
  apply (erule mult_le_less_imp_less)
avigad@16775
  2077
  apply simp_all
avigad@16775
  2078
done
avigad@16775
  2079
avigad@16775
  2080
text{*It's not obvious whether these should be simprules or not. 
avigad@16775
  2081
  Their effect is to gather terms into one big fraction, like
avigad@16775
  2082
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
avigad@16775
  2083
  seem to need them.*}
avigad@16775
  2084
avigad@16775
  2085
declare times_divide_eq [simp]
paulson@14293
  2086
wenzelm@23389
  2087
paulson@14293
  2088
subsection {* Ordered Fields are Dense *}
paulson@14293
  2089
haftmann@25193
  2090
context ordered_semidom
haftmann@25193
  2091
begin
haftmann@25193
  2092
haftmann@25193
  2093
lemma less_add_one: "a < a + 1"
paulson@14293
  2094
proof -
haftmann@25193
  2095
  have "a + 0 < a + 1"
nipkow@23482
  2096
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  2097
  thus ?thesis by simp
paulson@14293
  2098
qed
paulson@14293
  2099
haftmann@25193
  2100
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  2101
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  2102
haftmann@25193
  2103
end
paulson@14365
  2104
paulson@14293
  2105
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
nipkow@23482
  2106
by (simp add: field_simps zero_less_two)
paulson@14293
  2107
paulson@14293
  2108
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
nipkow@23482
  2109
by (simp add: field_simps zero_less_two)
paulson@14293
  2110
haftmann@24422
  2111
instance ordered_field < dense_linear_order
haftmann@24422
  2112
proof
haftmann@24422
  2113
  fix x y :: 'a
haftmann@24422
  2114
  have "x < x + 1" by simp
haftmann@24422
  2115
  then show "\<exists>y. x < y" .. 
haftmann@24422
  2116
  have "x - 1 < x" by simp
haftmann@24422
  2117
  then show "\<exists>y. y < x" ..
haftmann@24422
  2118
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
haftmann@24422
  2119
qed
paulson@14293
  2120
paulson@15234
  2121
paulson@14293
  2122
subsection {* Absolute Value *}
paulson@14293
  2123
haftmann@25304
  2124
context ordered_idom
haftmann@25304
  2125
begin
haftmann@25304
  2126
haftmann@25304
  2127
lemma mult_sgn_abs: "sgn x * abs x = x"
haftmann@25304
  2128
  unfolding abs_if sgn_if by auto
haftmann@25304
  2129
haftmann@25304
  2130
end
nipkow@24491
  2131
obua@14738
  2132
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
nipkow@29667
  2133
by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
haftmann@25304
  2134
haftmann@25304
  2135
class pordered_ring_abs = pordered_ring + pordered_ab_group_add_abs +
haftmann@25304
  2136
  assumes abs_eq_mult:
haftmann@25304
  2137
    "(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
  2138
haftmann@25304
  2139
haftmann@25304
  2140
class lordered_ring = pordered_ring + lordered_ab_group_add_abs
haftmann@25304
  2141
begin
haftmann@25304
  2142
huffman@27516
  2143
subclass lordered_ab_group_add_meet ..
huffman@27516
  2144
subclass lordered_ab_group_add_join ..
haftmann@25304
  2145
haftmann@25304
  2146
end
paulson@14294
  2147
obua@14738
  2148
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  2149
proof -
obua@14738
  2150
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  2151
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  2152
  have a: "(abs a) * (abs b) = ?x"
nipkow@29667
  2153
    by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
obua@14738
  2154
  {
obua@14738
  2155
    fix u v :: 'a
paulson@15481
  2156
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
paulson@15481
  2157
              u * v = pprt a * pprt b + pprt a * nprt b + 
paulson@15481
  2158
                      nprt a * pprt b + nprt a * nprt b"
obua@14738
  2159
      apply (subst prts[of u], subst prts[of v])
nipkow@29667
  2160
      apply (simp add: algebra_simps) 
obua@14738
  2161
      done
obua@14738
  2162
  }
obua@14738
  2163
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  2164
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  2165
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  2166
  have xy: "- ?x <= ?y"
obua@14754
  2167
    apply (simp)
obua@14754
  2168
    apply (rule_tac y="0::'a" in order_trans)
nipkow@16568
  2169
    apply (rule addm2)
avigad@16775
  2170
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
nipkow@16568
  2171
    apply (rule addm)
avigad@16775
  2172
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14754
  2173
    done
obua@14738
  2174
  have yx: "?y <= ?x"
nipkow@16568
  2175
    apply (simp add:diff_def)
obua@14754
  2176
    apply (rule_tac y=0 in order_trans)
avigad@16775
  2177
    apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
avigad@16775
  2178
    apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
obua@14738
  2179
    done
obua@14738
  2180
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  2181
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  2182
  show ?thesis
obua@14738
  2183
    apply (rule abs_leI)
obua@14738
  2184
    apply (simp add: i1)
obua@14738
  2185
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  2186
    done
obua@14738
  2187
qed
paulson@14294
  2188
haftmann@25304
  2189
instance lordered_ring \<subseteq> pordered_ring_abs
haftmann@25304
  2190
proof
haftmann@25304
  2191
  fix a b :: "'a\<Colon> lordered_ring"
haftmann@25304
  2192
  assume "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
haftmann@25304
  2193
  show "abs (a*b) = abs a * abs b"
obua@14738
  2194
proof -
obua@14738
  2195
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  2196
    apply (auto)    
obua@14738
  2197
    apply (rule_tac split_mult_pos_le)
obua@14738
  2198
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  2199
    apply (simp)
obua@14738
  2200
    apply (rule_tac split_mult_neg_le)
obua@14738
  2201
    apply (insert prems)
obua@14738
  2202
    apply (blast)
obua@14738
  2203
    done
obua@14738
  2204
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  2205
    by (simp add: prts[symmetric])
obua@14738
  2206
  show ?thesis
obua@14738
  2207
  proof cases
obua@14738
  2208
    assume "0 <= a * b"
obua@14738
  2209
    then show ?thesis
obua@14738
  2210
      apply (simp_all add: mulprts abs_prts)
obua@14738
  2211
      apply (insert prems)
obua@14754
  2212
      apply (auto simp add: 
wenzelm@32960
  2213
        algebra_simps 
wenzelm@32960
  2214
        iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
wenzelm@32960
  2215
        iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
wenzelm@32960
  2216
        apply(drule (1) mult_nonneg_nonpos[of a b], simp)
wenzelm@32960
  2217
        apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
obua@14738
  2218
      done
obua@14738
  2219
  next
obua@14738
  2220
    assume "~(0 <= a*b)"
obua@14738
  2221
    with s have "a*b <= 0" by simp
obua@14738
  2222
    then show ?thesis
obua@14738
  2223
      apply (simp_all add: mulprts abs_prts)
obua@14738
  2224
      apply (insert prems)
nipkow@29667
  2225
      apply (auto simp add: algebra_simps)
avigad@16775
  2226
      apply(drule (1) mult_nonneg_nonneg[of a b],simp)
avigad@16775
  2227
      apply(drule (1) mult_nonpos_nonpos[of a b],simp)
obua@14738
  2228
      done
obua@14738
  2229
  qed
obua@14738
  2230
qed
haftmann@25304
  2231
qed
haftmann@25304
  2232
haftmann@30961
  2233
context ordered_idom
haftmann@30961
  2234
begin
haftmann@30961
  2235
haftmann@30961
  2236
subclass pordered_ring_abs proof
haftmann@30961
  2237
qed (auto simp add: abs_if not_less equal_neg_zero neg_equal_zero mult_less_0_iff)
haftmann@30961
  2238
haftmann@30961
  2239
lemma abs_mult:
haftmann@30961
  2240
  "abs (a * b) = abs a * abs b" 
haftmann@30961
  2241
  by (rule abs_eq_mult) auto
haftmann@30961
  2242
haftmann@30961
  2243
lemma abs_mult_self:
haftmann@30961
  2244
  "abs a * abs a = a * a"
haftmann@30961
  2245
  by (simp add: abs_if) 
haftmann@30961
  2246
haftmann@30961
  2247
end
paulson@14294
  2248
paulson@14294
  2249
lemma nonzero_abs_inverse:
paulson@14294
  2250
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  2251
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  2252
                      negative_imp_inverse_negative)
paulson@14294
  2253
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  2254
done
paulson@14294
  2255
paulson@14294
  2256
lemma abs_inverse [simp]:
paulson@14294
  2257
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  2258
      inverse (abs a)"
haftmann@21328
  2259
apply (cases "a=0", simp) 
paulson@14294
  2260
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  2261
done
paulson@14294
  2262
paulson@14294
  2263
lemma nonzero_abs_divide:
paulson@14294
  2264
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  2265
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  2266
paulson@15234
  2267
lemma abs_divide [simp]:
paulson@14294
  2268
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
haftmann@21328
  2269
apply (cases "b=0", simp) 
paulson@14294
  2270
apply (simp add: nonzero_abs_divide) 
paulson@14294
  2271
done
paulson@14294
  2272
paulson@14294
  2273
lemma abs_mult_less:
obua@14738
  2274
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  2275
proof -
paulson@14294
  2276
  assume ac: "abs a < c"
paulson@14294
  2277
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  2278
  assume "abs b < d"
paulson@14294
  2279
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  2280
qed
paulson@14293
  2281
nipkow@29833
  2282
lemmas eq_minus_self_iff[noatp] = equal_neg_zero
obua@14738
  2283
obua@14738
  2284
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
haftmann@25304
  2285
  unfolding order_less_le less_eq_neg_nonpos equal_neg_zero ..
obua@14738
  2286
obua@14738
  2287
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  2288
apply (simp add: order_less_le abs_le_iff)  
haftmann@25304
  2289
apply (auto simp add: abs_if neg_less_eq_nonneg less_eq_neg_nonpos)
obua@14738
  2290
done
obua@14738
  2291
avigad@16775
  2292
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
haftmann@25304
  2293
    (abs y) * x = abs (y * x)"
haftmann@25304
  2294
  apply (subst abs_mult)
haftmann@25304
  2295
  apply simp
haftmann@25304
  2296
done
avigad@16775
  2297
avigad@16775
  2298
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
haftmann@25304
  2299
    abs x / y = abs (x / y)"
haftmann@25304
  2300
  apply (subst abs_divide)
haftmann@25304
  2301
  apply (simp add: order_less_imp_le)
haftmann@25304
  2302
done
avigad@16775
  2303
wenzelm@23389
  2304
obua@19404
  2305
subsection {* Bounds of products via negative and positive Part *}
obua@15178
  2306
obua@15580
  2307
lemma mult_le_prts:
obua@15580
  2308
  assumes
obua@15580
  2309
  "a1 <= (a::'a::lordered_ring)"
obua@15580
  2310
  "a <= a2"
obua@15580
  2311
  "b1 <= b"
obua@15580
  2312
  "b <= b2"
obua@15580
  2313
  shows
obua@15580
  2314
  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
obua@15580
  2315
proof - 
obua@15580
  2316
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
obua@15580
  2317
    apply (subst prts[symmetric])+
obua@15580
  2318
    apply simp
obua@15580
  2319
    done
obua@15580
  2320
  then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
nipkow@29667
  2321
    by (simp add: algebra_simps)
obua@15580
  2322
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
obua@15580
  2323
    by (simp_all add: prems mult_mono)
obua@15580
  2324
  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
obua@15580
  2325
  proof -
obua@15580
  2326
    have "pprt a * nprt b <= pprt a * nprt b2"
obua@15580
  2327
      by (simp add: mult_left_mono prems)
obua@15580
  2328
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
obua@15580
  2329
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2330
    ultimately show ?thesis
obua@15580
  2331
      by simp
obua@15580
  2332
  qed
obua@15580
  2333
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
obua@15580
  2334
  proof - 
obua@15580
  2335
    have "nprt a * pprt b <= nprt a2 * pprt b"
obua@15580
  2336
      by (simp add: mult_right_mono prems)
obua@15580
  2337
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
obua@15580
  2338
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2339
    ultimately show ?thesis
obua@15580
  2340
      by simp
obua@15580
  2341
  qed
obua@15580
  2342
  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
obua@15580
  2343
  proof -
obua@15580
  2344
    have "nprt a * nprt b <= nprt a * nprt b1"
obua@15580
  2345
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2346
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
obua@15580
  2347
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2348
    ultimately show ?thesis
obua@15580
  2349
      by simp
obua@15580
  2350
  qed
obua@15580
  2351
  ultimately show ?thesis
obua@15580
  2352
    by - (rule add_mono | simp)+
obua@15580
  2353
qed
obua@19404
  2354
obua@19404
  2355
lemma mult_ge_prts:
obua@15178
  2356
  assumes
obua@19404
  2357
  "a1 <= (a::'a::lordered_ring)"
obua@19404
  2358
  "a <= a2"
obua@19404
  2359
  "b1 <= b"
obua@19404
  2360
  "b <= b2"
obua@15178
  2361
  shows
obua@19404
  2362
  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
obua@19404
  2363
proof - 
obua@19404
  2364
  from prems have a1:"- a2 <= -a" by auto
obua@19404
  2365
  from prems have a2: "-a <= -a1" by auto
obua@19404
  2366
  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
  2367
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
obua@19404
  2368
  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
obua@19404
  2369
    by (simp only: minus_le_iff)
obua@19404
  2370
  then show ?thesis by simp
obua@15178
  2371
qed
obua@15178
  2372
paulson@14265
  2373
end