src/HOL/Rings.thy
author wenzelm
Fri Dec 17 17:43:54 2010 +0100 (2010-12-17)
changeset 41229 d797baa3d57c
parent 38642 8fa437809c67
child 44064 5bce8ff0d9ae
permissions -rw-r--r--
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
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(*  Title:      HOL/Rings.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 {* Rings *}
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theory Rings
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imports Groups
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begin
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes left_distrib[algebra_simps, field_simps]: "(a + b) * c = a * c + b * c"
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  assumes right_distrib[algebra_simps, field_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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  "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 [no_atp, 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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begin
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lemma divisors_zero:
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  assumes "a * b = 0"
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  shows "a = 0 \<or> b = 0"
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proof (rule classical)
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  assume "\<not> (a = 0 \<or> b = 0)"
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  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
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  with no_zero_divisors have "a * b \<noteq> 0" by blast
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  with assms show ?thesis by simp
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qed
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end
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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 minus_unique) (simp add: left_distrib [symmetric]) 
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lemma minus_mult_right: "- (a * b) = a * - b"
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by (rule minus_unique) (simp add: right_distrib [symmetric]) 
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text{*Extract signs from products*}
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lemmas mult_minus_left [simp, no_atp] = minus_mult_left [symmetric]
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lemmas mult_minus_right [simp,no_atp] = 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, field_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, field_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[no_atp] =
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  right_distrib left_distrib left_diff_distrib right_diff_distrib
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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[no_atp] =
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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 only: diff_minus dvd_add dvd_minus_iff)
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end
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class ring_no_zero_divisors = ring + no_zero_divisors
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begin
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lemma mult_eq_0_iff [simp]:
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  shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)"
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proof (cases "a = 0 \<or> b = 0")
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  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
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    then show ?thesis using no_zero_divisors by simp
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next
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  case True then show ?thesis by auto
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qed
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text{*Cancellation of equalities with a common factor*}
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lemma mult_cancel_right [simp, no_atp]:
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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)
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  thus ?thesis by (simp add: disj_commute)
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qed
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lemma mult_cancel_left [simp, no_atp]:
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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)
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  thus ?thesis by simp
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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 square_eq_1_iff:
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  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
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proof -
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  have "(x - 1) * (x + 1) = x * x - 1"
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    by (simp add: algebra_simps)
huffman@36821
   357
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   358
    by simp
huffman@36821
   359
  thus ?thesis
huffman@36821
   360
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   361
qed
huffman@36821
   362
haftmann@26274
   363
lemma mult_cancel_right1 [simp]:
haftmann@26274
   364
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   365
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   366
haftmann@26274
   367
lemma mult_cancel_right2 [simp]:
haftmann@26274
   368
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   369
by (insert mult_cancel_right [of a c 1], simp)
haftmann@26274
   370
 
haftmann@26274
   371
lemma mult_cancel_left1 [simp]:
haftmann@26274
   372
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   373
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   374
haftmann@26274
   375
lemma mult_cancel_left2 [simp]:
haftmann@26274
   376
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   377
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   378
haftmann@26274
   379
end
huffman@22990
   380
haftmann@22390
   381
class idom = comm_ring_1 + no_zero_divisors
haftmann@25186
   382
begin
paulson@14421
   383
huffman@27516
   384
subclass ring_1_no_zero_divisors ..
huffman@22990
   385
huffman@29915
   386
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)"
huffman@29915
   387
proof
huffman@29915
   388
  assume "a * a = b * b"
huffman@29915
   389
  then have "(a - b) * (a + b) = 0"
huffman@29915
   390
    by (simp add: algebra_simps)
huffman@29915
   391
  then show "a = b \<or> a = - b"
huffman@35216
   392
    by (simp add: eq_neg_iff_add_eq_0)
huffman@29915
   393
next
huffman@29915
   394
  assume "a = b \<or> a = - b"
huffman@29915
   395
  then show "a * a = b * b" by auto
huffman@29915
   396
qed
huffman@29915
   397
huffman@29981
   398
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   399
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   400
proof -
huffman@29981
   401
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
huffman@29981
   402
    unfolding dvd_def by (simp add: mult_ac)
huffman@29981
   403
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   404
    unfolding dvd_def by simp
huffman@29981
   405
  finally show ?thesis .
huffman@29981
   406
qed
huffman@29981
   407
huffman@29981
   408
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   409
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   410
proof -
huffman@29981
   411
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
huffman@29981
   412
    unfolding dvd_def by (simp add: mult_ac)
huffman@29981
   413
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   414
    unfolding dvd_def by simp
huffman@29981
   415
  finally show ?thesis .
huffman@29981
   416
qed
huffman@29981
   417
haftmann@25186
   418
end
haftmann@25152
   419
haftmann@35083
   420
class inverse =
haftmann@35083
   421
  fixes inverse :: "'a \<Rightarrow> 'a"
haftmann@35083
   422
    and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)
haftmann@35083
   423
haftmann@22390
   424
class division_ring = ring_1 + inverse +
haftmann@25062
   425
  assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
haftmann@25062
   426
  assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
haftmann@35083
   427
  assumes divide_inverse: "a / b = a * inverse b"
haftmann@25186
   428
begin
huffman@20496
   429
haftmann@25186
   430
subclass ring_1_no_zero_divisors
haftmann@28823
   431
proof
huffman@22987
   432
  fix a b :: 'a
huffman@22987
   433
  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
huffman@22987
   434
  show "a * b \<noteq> 0"
huffman@22987
   435
  proof
huffman@22987
   436
    assume ab: "a * b = 0"
nipkow@29667
   437
    hence "0 = inverse a * (a * b) * inverse b" by simp
huffman@22987
   438
    also have "\<dots> = (inverse a * a) * (b * inverse b)"
huffman@22987
   439
      by (simp only: mult_assoc)
nipkow@29667
   440
    also have "\<dots> = 1" using a b by simp
nipkow@29667
   441
    finally show False by simp
huffman@22987
   442
  qed
huffman@22987
   443
qed
huffman@20496
   444
haftmann@26274
   445
lemma nonzero_imp_inverse_nonzero:
haftmann@26274
   446
  "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
haftmann@26274
   447
proof
haftmann@26274
   448
  assume ianz: "inverse a = 0"
haftmann@26274
   449
  assume "a \<noteq> 0"
haftmann@26274
   450
  hence "1 = a * inverse a" by simp
haftmann@26274
   451
  also have "... = 0" by (simp add: ianz)
haftmann@26274
   452
  finally have "1 = 0" .
haftmann@26274
   453
  thus False by (simp add: eq_commute)
haftmann@26274
   454
qed
haftmann@26274
   455
haftmann@26274
   456
lemma inverse_zero_imp_zero:
haftmann@26274
   457
  "inverse a = 0 \<Longrightarrow> a = 0"
haftmann@26274
   458
apply (rule classical)
haftmann@26274
   459
apply (drule nonzero_imp_inverse_nonzero)
haftmann@26274
   460
apply auto
haftmann@26274
   461
done
haftmann@26274
   462
haftmann@26274
   463
lemma inverse_unique: 
haftmann@26274
   464
  assumes ab: "a * b = 1"
haftmann@26274
   465
  shows "inverse a = b"
haftmann@26274
   466
proof -
haftmann@26274
   467
  have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
huffman@29406
   468
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
huffman@29406
   469
  ultimately show ?thesis by (simp add: mult_assoc [symmetric])
haftmann@26274
   470
qed
haftmann@26274
   471
huffman@29406
   472
lemma nonzero_inverse_minus_eq:
huffman@29406
   473
  "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
nipkow@29667
   474
by (rule inverse_unique) simp
huffman@29406
   475
huffman@29406
   476
lemma nonzero_inverse_inverse_eq:
huffman@29406
   477
  "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
nipkow@29667
   478
by (rule inverse_unique) simp
huffman@29406
   479
huffman@29406
   480
lemma nonzero_inverse_eq_imp_eq:
huffman@29406
   481
  assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
huffman@29406
   482
  shows "a = b"
huffman@29406
   483
proof -
huffman@29406
   484
  from `inverse a = inverse b`
nipkow@29667
   485
  have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
huffman@29406
   486
  with `a \<noteq> 0` and `b \<noteq> 0` show "a = b"
huffman@29406
   487
    by (simp add: nonzero_inverse_inverse_eq)
huffman@29406
   488
qed
huffman@29406
   489
huffman@29406
   490
lemma inverse_1 [simp]: "inverse 1 = 1"
nipkow@29667
   491
by (rule inverse_unique) simp
huffman@29406
   492
haftmann@26274
   493
lemma nonzero_inverse_mult_distrib: 
huffman@29406
   494
  assumes "a \<noteq> 0" and "b \<noteq> 0"
haftmann@26274
   495
  shows "inverse (a * b) = inverse b * inverse a"
haftmann@26274
   496
proof -
nipkow@29667
   497
  have "a * (b * inverse b) * inverse a = 1" using assms by simp
nipkow@29667
   498
  hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc)
nipkow@29667
   499
  thus ?thesis by (rule inverse_unique)
haftmann@26274
   500
qed
haftmann@26274
   501
haftmann@26274
   502
lemma division_ring_inverse_add:
haftmann@26274
   503
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
nipkow@29667
   504
by (simp add: algebra_simps)
haftmann@26274
   505
haftmann@26274
   506
lemma division_ring_inverse_diff:
haftmann@26274
   507
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
nipkow@29667
   508
by (simp add: algebra_simps)
haftmann@26274
   509
haftmann@36301
   510
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
haftmann@36301
   511
proof
haftmann@36301
   512
  assume neq: "b \<noteq> 0"
haftmann@36301
   513
  {
haftmann@36301
   514
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_assoc)
haftmann@36301
   515
    also assume "a / b = 1"
haftmann@36301
   516
    finally show "a = b" by simp
haftmann@36301
   517
  next
haftmann@36301
   518
    assume "a = b"
haftmann@36301
   519
    with neq show "a / b = 1" by (simp add: divide_inverse)
haftmann@36301
   520
  }
haftmann@36301
   521
qed
haftmann@36301
   522
haftmann@36301
   523
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
haftmann@36301
   524
by (simp add: divide_inverse)
haftmann@36301
   525
haftmann@36301
   526
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
haftmann@36301
   527
by (simp add: divide_inverse)
haftmann@36301
   528
haftmann@36301
   529
lemma divide_zero_left [simp]: "0 / a = 0"
haftmann@36301
   530
by (simp add: divide_inverse)
haftmann@36301
   531
haftmann@36301
   532
lemma inverse_eq_divide: "inverse a = 1 / a"
haftmann@36301
   533
by (simp add: divide_inverse)
haftmann@36301
   534
haftmann@36301
   535
lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
haftmann@36301
   536
by (simp add: divide_inverse algebra_simps)
haftmann@36301
   537
haftmann@36301
   538
lemma divide_1 [simp]: "a / 1 = a"
haftmann@36301
   539
  by (simp add: divide_inverse)
haftmann@36301
   540
haftmann@36304
   541
lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c"
haftmann@36301
   542
  by (simp add: divide_inverse mult_assoc)
haftmann@36301
   543
haftmann@36301
   544
lemma minus_divide_left: "- (a / b) = (-a) / b"
haftmann@36301
   545
  by (simp add: divide_inverse)
haftmann@36301
   546
haftmann@36301
   547
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
haftmann@36301
   548
  by (simp add: divide_inverse nonzero_inverse_minus_eq)
haftmann@36301
   549
haftmann@36301
   550
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
haftmann@36301
   551
  by (simp add: divide_inverse nonzero_inverse_minus_eq)
haftmann@36301
   552
haftmann@36301
   553
lemma divide_minus_left [simp, no_atp]: "(-a) / b = - (a / b)"
haftmann@36301
   554
  by (simp add: divide_inverse)
haftmann@36301
   555
haftmann@36301
   556
lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
haftmann@36301
   557
  by (simp add: diff_minus add_divide_distrib)
haftmann@36301
   558
haftmann@36348
   559
lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
haftmann@36301
   560
proof -
haftmann@36301
   561
  assume [simp]: "c \<noteq> 0"
haftmann@36301
   562
  have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
haftmann@36301
   563
  also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc)
haftmann@36301
   564
  finally show ?thesis .
haftmann@36301
   565
qed
haftmann@36301
   566
haftmann@36348
   567
lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
haftmann@36301
   568
proof -
haftmann@36301
   569
  assume [simp]: "c \<noteq> 0"
haftmann@36301
   570
  have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
haftmann@36301
   571
  also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc) 
haftmann@36301
   572
  finally show ?thesis .
haftmann@36301
   573
qed
haftmann@36301
   574
haftmann@36301
   575
lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
haftmann@36301
   576
  by (simp add: divide_inverse mult_assoc)
haftmann@36301
   577
haftmann@36301
   578
lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
haftmann@36301
   579
  by (drule sym) (simp add: divide_inverse mult_assoc)
haftmann@36301
   580
haftmann@36301
   581
end
haftmann@36301
   582
haftmann@36348
   583
class division_ring_inverse_zero = division_ring +
haftmann@36301
   584
  assumes inverse_zero [simp]: "inverse 0 = 0"
haftmann@36301
   585
begin
haftmann@36301
   586
haftmann@36301
   587
lemma divide_zero [simp]:
haftmann@36301
   588
  "a / 0 = 0"
haftmann@36301
   589
  by (simp add: divide_inverse)
haftmann@36301
   590
haftmann@36301
   591
lemma divide_self_if [simp]:
haftmann@36301
   592
  "a / a = (if a = 0 then 0 else 1)"
haftmann@36301
   593
  by simp
haftmann@36301
   594
haftmann@36301
   595
lemma inverse_nonzero_iff_nonzero [simp]:
haftmann@36301
   596
  "inverse a = 0 \<longleftrightarrow> a = 0"
haftmann@36301
   597
  by rule (fact inverse_zero_imp_zero, simp)
haftmann@36301
   598
haftmann@36301
   599
lemma inverse_minus_eq [simp]:
haftmann@36301
   600
  "inverse (- a) = - inverse a"
haftmann@36301
   601
proof cases
haftmann@36301
   602
  assume "a=0" thus ?thesis by simp
haftmann@36301
   603
next
haftmann@36301
   604
  assume "a\<noteq>0" 
haftmann@36301
   605
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
haftmann@36301
   606
qed
haftmann@36301
   607
haftmann@36301
   608
lemma inverse_eq_imp_eq:
haftmann@36301
   609
  "inverse a = inverse b \<Longrightarrow> a = b"
haftmann@36301
   610
apply (cases "a=0 | b=0") 
haftmann@36301
   611
 apply (force dest!: inverse_zero_imp_zero
haftmann@36301
   612
              simp add: eq_commute [of "0::'a"])
haftmann@36301
   613
apply (force dest!: nonzero_inverse_eq_imp_eq) 
haftmann@36301
   614
done
haftmann@36301
   615
haftmann@36301
   616
lemma inverse_eq_iff_eq [simp]:
haftmann@36301
   617
  "inverse a = inverse b \<longleftrightarrow> a = b"
haftmann@36301
   618
  by (force dest!: inverse_eq_imp_eq)
haftmann@36301
   619
haftmann@36301
   620
lemma inverse_inverse_eq [simp]:
haftmann@36301
   621
  "inverse (inverse a) = a"
haftmann@36301
   622
proof cases
haftmann@36301
   623
  assume "a=0" thus ?thesis by simp
haftmann@36301
   624
next
haftmann@36301
   625
  assume "a\<noteq>0" 
haftmann@36301
   626
  thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
haftmann@36301
   627
qed
haftmann@36301
   628
haftmann@25186
   629
end
haftmann@25152
   630
haftmann@35302
   631
text {*
haftmann@35302
   632
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   633
  \begin{itemize}
haftmann@35302
   634
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35302
   635
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   636
  \end{itemize}
haftmann@35302
   637
  Most of the used notions can also be looked up in 
haftmann@35302
   638
  \begin{itemize}
haftmann@35302
   639
  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
haftmann@35302
   640
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   641
  \end{itemize}
haftmann@35302
   642
*}
haftmann@35302
   643
haftmann@38642
   644
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
   645
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
   646
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   647
begin
haftmann@25230
   648
haftmann@25230
   649
lemma mult_mono:
haftmann@38642
   650
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   651
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   652
apply (erule mult_left_mono, assumption)
haftmann@25230
   653
done
haftmann@25230
   654
haftmann@25230
   655
lemma mult_mono':
haftmann@38642
   656
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   657
apply (rule mult_mono)
haftmann@25230
   658
apply (fast intro: order_trans)+
haftmann@25230
   659
done
haftmann@25230
   660
haftmann@25230
   661
end
krauss@21199
   662
haftmann@38642
   663
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
   664
begin
paulson@14268
   665
huffman@27516
   666
subclass semiring_0_cancel ..
obua@23521
   667
haftmann@25230
   668
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   669
using mult_left_mono [of 0 b a] by simp
haftmann@25230
   670
haftmann@25230
   671
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   672
using mult_left_mono [of b 0 a] by simp
huffman@30692
   673
huffman@30692
   674
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   675
using mult_right_mono [of a 0 b] by simp
huffman@30692
   676
huffman@30692
   677
text {* Legacy - use @{text mult_nonpos_nonneg} *}
haftmann@25230
   678
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
haftmann@36301
   679
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
   680
haftmann@26234
   681
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
nipkow@29667
   682
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   683
haftmann@25230
   684
end
haftmann@25230
   685
haftmann@38642
   686
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
   687
begin
haftmann@25230
   688
haftmann@35028
   689
subclass ordered_cancel_semiring ..
haftmann@35028
   690
haftmann@35028
   691
subclass ordered_comm_monoid_add ..
haftmann@25304
   692
haftmann@25230
   693
lemma mult_left_less_imp_less:
haftmann@25230
   694
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   695
by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   696
 
haftmann@25230
   697
lemma mult_right_less_imp_less:
haftmann@25230
   698
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   699
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   700
haftmann@25186
   701
end
haftmann@25152
   702
haftmann@35043
   703
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
   704
begin
hoelzl@36622
   705
hoelzl@36622
   706
lemma convex_bound_le:
hoelzl@36622
   707
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   708
  shows "u * x + v * y \<le> a"
hoelzl@36622
   709
proof-
hoelzl@36622
   710
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
   711
    by (simp add: add_mono mult_left_mono)
hoelzl@36622
   712
  thus ?thesis using assms unfolding left_distrib[symmetric] by simp
hoelzl@36622
   713
qed
hoelzl@36622
   714
hoelzl@36622
   715
end
haftmann@35043
   716
haftmann@35043
   717
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
   718
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   719
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   720
begin
paulson@14341
   721
huffman@27516
   722
subclass semiring_0_cancel ..
obua@14940
   723
haftmann@35028
   724
subclass linordered_semiring
haftmann@28823
   725
proof
huffman@23550
   726
  fix a b c :: 'a
huffman@23550
   727
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   728
  from A show "c * a \<le> c * b"
haftmann@25186
   729
    unfolding le_less
haftmann@25186
   730
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   731
  from A show "a * c \<le> b * c"
haftmann@25152
   732
    unfolding le_less
haftmann@25186
   733
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   734
qed
haftmann@25152
   735
haftmann@25230
   736
lemma mult_left_le_imp_le:
haftmann@25230
   737
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   738
by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   739
 
haftmann@25230
   740
lemma mult_right_le_imp_le:
haftmann@25230
   741
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   742
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   743
huffman@30692
   744
lemma mult_pos_pos: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
   745
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
   746
huffman@30692
   747
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
   748
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
   749
huffman@30692
   750
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
   751
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
   752
huffman@30692
   753
text {* Legacy - use @{text mult_neg_pos} *}
huffman@30692
   754
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
haftmann@36301
   755
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
   756
haftmann@25230
   757
lemma zero_less_mult_pos:
haftmann@25230
   758
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   759
apply (cases "b\<le>0")
haftmann@25230
   760
 apply (auto simp add: le_less not_less)
huffman@30692
   761
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
   762
 apply (auto dest: less_not_sym)
haftmann@25230
   763
done
haftmann@25230
   764
haftmann@25230
   765
lemma zero_less_mult_pos2:
haftmann@25230
   766
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   767
apply (cases "b\<le>0")
haftmann@25230
   768
 apply (auto simp add: le_less not_less)
huffman@30692
   769
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
   770
 apply (auto dest: less_not_sym)
haftmann@25230
   771
done
haftmann@25230
   772
haftmann@26193
   773
text{*Strict monotonicity in both arguments*}
haftmann@26193
   774
lemma mult_strict_mono:
haftmann@26193
   775
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
   776
  shows "a * c < b * d"
haftmann@26193
   777
  using assms apply (cases "c=0")
huffman@30692
   778
  apply (simp add: mult_pos_pos)
haftmann@26193
   779
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
   780
  apply (force simp add: le_less)
haftmann@26193
   781
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
   782
  done
haftmann@26193
   783
haftmann@26193
   784
text{*This weaker variant has more natural premises*}
haftmann@26193
   785
lemma mult_strict_mono':
haftmann@26193
   786
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
   787
  shows "a * c < b * d"
nipkow@29667
   788
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
   789
haftmann@26193
   790
lemma mult_less_le_imp_less:
haftmann@26193
   791
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
   792
  shows "a * c < b * d"
haftmann@26193
   793
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
   794
  apply (erule less_le_trans)
haftmann@26193
   795
  apply (erule mult_left_mono)
haftmann@26193
   796
  apply simp
haftmann@26193
   797
  apply (erule mult_strict_right_mono)
haftmann@26193
   798
  apply assumption
haftmann@26193
   799
  done
haftmann@26193
   800
haftmann@26193
   801
lemma mult_le_less_imp_less:
haftmann@26193
   802
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
   803
  shows "a * c < b * d"
haftmann@26193
   804
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
   805
  apply (erule le_less_trans)
haftmann@26193
   806
  apply (erule mult_strict_left_mono)
haftmann@26193
   807
  apply simp
haftmann@26193
   808
  apply (erule mult_right_mono)
haftmann@26193
   809
  apply simp
haftmann@26193
   810
  done
haftmann@26193
   811
haftmann@26193
   812
lemma mult_less_imp_less_left:
haftmann@26193
   813
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
haftmann@26193
   814
  shows "a < b"
haftmann@26193
   815
proof (rule ccontr)
haftmann@26193
   816
  assume "\<not>  a < b"
haftmann@26193
   817
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   818
  hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)
nipkow@29667
   819
  with this and less show False by (simp add: not_less [symmetric])
haftmann@26193
   820
qed
haftmann@26193
   821
haftmann@26193
   822
lemma mult_less_imp_less_right:
haftmann@26193
   823
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
haftmann@26193
   824
  shows "a < b"
haftmann@26193
   825
proof (rule ccontr)
haftmann@26193
   826
  assume "\<not> a < b"
haftmann@26193
   827
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   828
  hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)
nipkow@29667
   829
  with this and less show False by (simp add: not_less [symmetric])
haftmann@26193
   830
qed  
haftmann@26193
   831
haftmann@25230
   832
end
haftmann@25230
   833
haftmann@35097
   834
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
   835
begin
hoelzl@36622
   836
hoelzl@36622
   837
subclass linordered_semiring_1 ..
hoelzl@36622
   838
hoelzl@36622
   839
lemma convex_bound_lt:
hoelzl@36622
   840
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   841
  shows "u * x + v * y < a"
hoelzl@36622
   842
proof -
hoelzl@36622
   843
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
   844
    by (cases "u = 0")
hoelzl@36622
   845
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
hoelzl@36622
   846
  thus ?thesis using assms unfolding left_distrib[symmetric] by simp
hoelzl@36622
   847
qed
hoelzl@36622
   848
hoelzl@36622
   849
end
haftmann@33319
   850
haftmann@38642
   851
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add + 
haftmann@38642
   852
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
   853
begin
haftmann@25152
   854
haftmann@35028
   855
subclass ordered_semiring
haftmann@28823
   856
proof
krauss@21199
   857
  fix a b c :: 'a
huffman@23550
   858
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
   859
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
huffman@23550
   860
  thus "a * c \<le> b * c" by (simp only: mult_commute)
krauss@21199
   861
qed
paulson@14265
   862
haftmann@25267
   863
end
haftmann@25267
   864
haftmann@38642
   865
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
   866
begin
paulson@14265
   867
haftmann@38642
   868
subclass comm_semiring_0_cancel ..
haftmann@35028
   869
subclass ordered_comm_semiring ..
haftmann@35028
   870
subclass ordered_cancel_semiring ..
haftmann@25267
   871
haftmann@25267
   872
end
haftmann@25267
   873
haftmann@35028
   874
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
   875
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
   876
begin
haftmann@25267
   877
haftmann@35043
   878
subclass linordered_semiring_strict
haftmann@28823
   879
proof
huffman@23550
   880
  fix a b c :: 'a
huffman@23550
   881
  assume "a < b" "0 < c"
haftmann@38642
   882
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
huffman@23550
   883
  thus "a * c < b * c" by (simp only: mult_commute)
huffman@23550
   884
qed
paulson@14272
   885
haftmann@35028
   886
subclass ordered_cancel_comm_semiring
haftmann@28823
   887
proof
huffman@23550
   888
  fix a b c :: 'a
huffman@23550
   889
  assume "a \<le> b" "0 \<le> c"
huffman@23550
   890
  thus "c * a \<le> c * b"
haftmann@25186
   891
    unfolding le_less
haftmann@26193
   892
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   893
qed
paulson@14272
   894
haftmann@25267
   895
end
haftmann@25230
   896
haftmann@35028
   897
class ordered_ring = ring + ordered_cancel_semiring 
haftmann@25267
   898
begin
haftmann@25230
   899
haftmann@35028
   900
subclass ordered_ab_group_add ..
paulson@14270
   901
haftmann@25230
   902
lemma less_add_iff1:
haftmann@25230
   903
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
   904
by (simp add: algebra_simps)
haftmann@25230
   905
haftmann@25230
   906
lemma less_add_iff2:
haftmann@25230
   907
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
   908
by (simp add: algebra_simps)
haftmann@25230
   909
haftmann@25230
   910
lemma le_add_iff1:
haftmann@25230
   911
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
   912
by (simp add: algebra_simps)
haftmann@25230
   913
haftmann@25230
   914
lemma le_add_iff2:
haftmann@25230
   915
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
   916
by (simp add: algebra_simps)
haftmann@25230
   917
haftmann@25230
   918
lemma mult_left_mono_neg:
haftmann@25230
   919
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
   920
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
   921
  apply simp_all
haftmann@25230
   922
  done
haftmann@25230
   923
haftmann@25230
   924
lemma mult_right_mono_neg:
haftmann@25230
   925
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
   926
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
   927
  apply simp_all
haftmann@25230
   928
  done
haftmann@25230
   929
huffman@30692
   930
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   931
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
   932
haftmann@25230
   933
lemma split_mult_pos_le:
haftmann@25230
   934
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@29667
   935
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
haftmann@25186
   936
haftmann@25186
   937
end
paulson@14270
   938
haftmann@35028
   939
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
   940
begin
haftmann@25304
   941
haftmann@35028
   942
subclass ordered_ring ..
haftmann@35028
   943
haftmann@35028
   944
subclass ordered_ab_group_add_abs
haftmann@28823
   945
proof
haftmann@25304
   946
  fix a b
haftmann@25304
   947
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
huffman@35216
   948
    by (auto simp add: abs_if not_less)
huffman@35216
   949
    (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric],
huffman@36977
   950
     auto intro!: less_imp_le add_neg_neg)
huffman@35216
   951
qed (auto simp add: abs_if)
haftmann@25304
   952
huffman@35631
   953
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
   954
  using linear [of 0 a]
huffman@35631
   955
  by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
huffman@35631
   956
huffman@35631
   957
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
   958
  by (simp add: not_less)
huffman@35631
   959
haftmann@25304
   960
end
obua@23521
   961
haftmann@35028
   962
(* The "strict" suffix can be seen as describing the combination of linordered_ring and no_zero_divisors.
haftmann@35043
   963
   Basically, linordered_ring + no_zero_divisors = linordered_ring_strict.
haftmann@25230
   964
 *)
haftmann@35043
   965
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
   966
  + ordered_ab_group_add + abs_if
haftmann@25230
   967
begin
paulson@14348
   968
haftmann@35028
   969
subclass linordered_ring ..
haftmann@25304
   970
huffman@30692
   971
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
   972
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
   973
huffman@30692
   974
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
   975
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
   976
huffman@30692
   977
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
   978
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
   979
haftmann@25917
   980
subclass ring_no_zero_divisors
haftmann@28823
   981
proof
haftmann@25917
   982
  fix a b
haftmann@25917
   983
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
   984
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
   985
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
   986
  proof (cases "a < 0")
haftmann@25917
   987
    case True note A' = this
haftmann@25917
   988
    show ?thesis proof (cases "b < 0")
haftmann@25917
   989
      case True with A'
haftmann@25917
   990
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
   991
    next
haftmann@25917
   992
      case False with B have "0 < b" by auto
haftmann@25917
   993
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
   994
    qed
haftmann@25917
   995
  next
haftmann@25917
   996
    case False with A have A': "0 < a" by auto
haftmann@25917
   997
    show ?thesis proof (cases "b < 0")
haftmann@25917
   998
      case True with A'
haftmann@25917
   999
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
  1000
    next
haftmann@25917
  1001
      case False with B have "0 < b" by auto
haftmann@25917
  1002
      with A' show ?thesis by (auto dest: mult_pos_pos)
haftmann@25917
  1003
    qed
haftmann@25917
  1004
  qed
haftmann@25917
  1005
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
  1006
qed
haftmann@25304
  1007
paulson@14265
  1008
lemma zero_less_mult_iff:
haftmann@25917
  1009
  "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
haftmann@25917
  1010
  apply (auto simp add: mult_pos_pos mult_neg_neg)
haftmann@25917
  1011
  apply (simp_all add: not_less le_less)
haftmann@25917
  1012
  apply (erule disjE) apply assumption defer
haftmann@25917
  1013
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
  1014
  apply (erule disjE) defer apply (drule sym) apply simp
haftmann@25917
  1015
  apply (erule disjE) apply assumption apply (drule sym) apply simp
haftmann@25917
  1016
  apply (drule sym) apply simp
haftmann@25917
  1017
  apply (blast dest: zero_less_mult_pos)
haftmann@25230
  1018
  apply (blast dest: zero_less_mult_pos2)
haftmann@25230
  1019
  done
huffman@22990
  1020
paulson@14265
  1021
lemma zero_le_mult_iff:
haftmann@25917
  1022
  "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
nipkow@29667
  1023
by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
  1024
paulson@14265
  1025
lemma mult_less_0_iff:
haftmann@25917
  1026
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
  1027
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
  1028
  apply force
haftmann@25917
  1029
  done
paulson@14265
  1030
paulson@14265
  1031
lemma mult_le_0_iff:
haftmann@25917
  1032
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
  1033
  apply (insert zero_le_mult_iff [of "-a" b]) 
huffman@35216
  1034
  apply force
haftmann@25917
  1035
  done
haftmann@25917
  1036
haftmann@26193
  1037
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
  1038
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
  1039
haftmann@26193
  1040
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
  1041
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
  1042
haftmann@26193
  1043
lemma mult_less_cancel_right_disj:
haftmann@26193
  1044
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1045
  apply (cases "c = 0")
haftmann@26193
  1046
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
  1047
                      mult_strict_right_mono_neg)
haftmann@26193
  1048
  apply (auto simp add: not_less 
haftmann@26193
  1049
                      not_le [symmetric, of "a*c"]
haftmann@26193
  1050
                      not_le [symmetric, of a])
haftmann@26193
  1051
  apply (erule_tac [!] notE)
haftmann@26193
  1052
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
  1053
                      mult_right_mono_neg)
haftmann@26193
  1054
  done
haftmann@26193
  1055
haftmann@26193
  1056
lemma mult_less_cancel_left_disj:
haftmann@26193
  1057
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1058
  apply (cases "c = 0")
haftmann@26193
  1059
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
  1060
                      mult_strict_left_mono_neg)
haftmann@26193
  1061
  apply (auto simp add: not_less 
haftmann@26193
  1062
                      not_le [symmetric, of "c*a"]
haftmann@26193
  1063
                      not_le [symmetric, of a])
haftmann@26193
  1064
  apply (erule_tac [!] notE)
haftmann@26193
  1065
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
  1066
                      mult_left_mono_neg)
haftmann@26193
  1067
  done
haftmann@26193
  1068
haftmann@26193
  1069
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
  1070
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
  1071
haftmann@26193
  1072
lemma mult_less_cancel_right:
haftmann@26193
  1073
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1074
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
  1075
haftmann@26193
  1076
lemma mult_less_cancel_left:
haftmann@26193
  1077
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1078
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
  1079
haftmann@26193
  1080
lemma mult_le_cancel_right:
haftmann@26193
  1081
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1082
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
  1083
haftmann@26193
  1084
lemma mult_le_cancel_left:
haftmann@26193
  1085
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1086
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
  1087
nipkow@30649
  1088
lemma mult_le_cancel_left_pos:
nipkow@30649
  1089
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
  1090
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1091
nipkow@30649
  1092
lemma mult_le_cancel_left_neg:
nipkow@30649
  1093
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
  1094
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1095
nipkow@30649
  1096
lemma mult_less_cancel_left_pos:
nipkow@30649
  1097
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
  1098
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1099
nipkow@30649
  1100
lemma mult_less_cancel_left_neg:
nipkow@30649
  1101
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
  1102
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1103
haftmann@25917
  1104
end
paulson@14265
  1105
huffman@30692
  1106
lemmas mult_sign_intros =
huffman@30692
  1107
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
  1108
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
  1109
  mult_pos_pos mult_pos_neg
huffman@30692
  1110
  mult_neg_pos mult_neg_neg
haftmann@25230
  1111
haftmann@35028
  1112
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
  1113
begin
haftmann@25230
  1114
haftmann@35028
  1115
subclass ordered_ring ..
haftmann@35028
  1116
subclass ordered_cancel_comm_semiring ..
haftmann@25230
  1117
haftmann@25267
  1118
end
haftmann@25230
  1119
haftmann@35028
  1120
class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +
haftmann@35028
  1121
  (*previously linordered_semiring*)
haftmann@25230
  1122
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
  1123
begin
haftmann@25230
  1124
haftmann@25230
  1125
lemma pos_add_strict:
haftmann@25230
  1126
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
  1127
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
  1128
haftmann@26193
  1129
lemma zero_le_one [simp]: "0 \<le> 1"
nipkow@29667
  1130
by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
  1131
haftmann@26193
  1132
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
nipkow@29667
  1133
by (simp add: not_le) 
haftmann@26193
  1134
haftmann@26193
  1135
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
nipkow@29667
  1136
by (simp add: not_less) 
haftmann@26193
  1137
haftmann@26193
  1138
lemma less_1_mult:
haftmann@26193
  1139
  assumes "1 < m" and "1 < n"
haftmann@26193
  1140
  shows "1 < m * n"
haftmann@26193
  1141
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
  1142
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
  1143
haftmann@25230
  1144
end
haftmann@25230
  1145
haftmann@35028
  1146
class linordered_idom = comm_ring_1 +
haftmann@35028
  1147
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1148
  abs_if + sgn_if
haftmann@35028
  1149
  (*previously linordered_ring*)
haftmann@25917
  1150
begin
haftmann@25917
  1151
hoelzl@36622
  1152
subclass linordered_semiring_1_strict ..
haftmann@35043
  1153
subclass linordered_ring_strict ..
haftmann@35028
  1154
subclass ordered_comm_ring ..
huffman@27516
  1155
subclass idom ..
haftmann@25917
  1156
haftmann@35028
  1157
subclass linordered_semidom
haftmann@28823
  1158
proof
haftmann@26193
  1159
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1160
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
  1161
qed 
haftmann@25917
  1162
haftmann@35028
  1163
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1164
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1165
  using assms by (rule neqE)
haftmann@26193
  1166
haftmann@26274
  1167
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1168
haftmann@26274
  1169
lemma mult_le_cancel_right1:
haftmann@26274
  1170
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1171
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1172
haftmann@26274
  1173
lemma mult_le_cancel_right2:
haftmann@26274
  1174
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1175
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1176
haftmann@26274
  1177
lemma mult_le_cancel_left1:
haftmann@26274
  1178
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1179
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1180
haftmann@26274
  1181
lemma mult_le_cancel_left2:
haftmann@26274
  1182
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1183
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1184
haftmann@26274
  1185
lemma mult_less_cancel_right1:
haftmann@26274
  1186
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1187
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1188
haftmann@26274
  1189
lemma mult_less_cancel_right2:
haftmann@26274
  1190
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1191
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1192
haftmann@26274
  1193
lemma mult_less_cancel_left1:
haftmann@26274
  1194
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1195
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1196
haftmann@26274
  1197
lemma mult_less_cancel_left2:
haftmann@26274
  1198
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1199
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1200
haftmann@27651
  1201
lemma sgn_sgn [simp]:
haftmann@27651
  1202
  "sgn (sgn a) = sgn a"
nipkow@29700
  1203
unfolding sgn_if by simp
haftmann@27651
  1204
haftmann@27651
  1205
lemma sgn_0_0:
haftmann@27651
  1206
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1207
unfolding sgn_if by simp
haftmann@27651
  1208
haftmann@27651
  1209
lemma sgn_1_pos:
haftmann@27651
  1210
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1211
unfolding sgn_if by simp
haftmann@27651
  1212
haftmann@27651
  1213
lemma sgn_1_neg:
haftmann@27651
  1214
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1215
unfolding sgn_if by auto
haftmann@27651
  1216
haftmann@29940
  1217
lemma sgn_pos [simp]:
haftmann@29940
  1218
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1219
unfolding sgn_1_pos .
haftmann@29940
  1220
haftmann@29940
  1221
lemma sgn_neg [simp]:
haftmann@29940
  1222
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1223
unfolding sgn_1_neg .
haftmann@29940
  1224
haftmann@27651
  1225
lemma sgn_times:
haftmann@27651
  1226
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1227
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1228
haftmann@36301
  1229
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1230
unfolding sgn_if abs_if by auto
nipkow@29700
  1231
haftmann@29940
  1232
lemma sgn_greater [simp]:
haftmann@29940
  1233
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1234
  unfolding sgn_if by auto
haftmann@29940
  1235
haftmann@29940
  1236
lemma sgn_less [simp]:
haftmann@29940
  1237
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1238
  unfolding sgn_if by auto
haftmann@29940
  1239
haftmann@36301
  1240
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1241
  by (simp add: abs_if)
huffman@29949
  1242
haftmann@36301
  1243
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1244
  by (simp add: abs_if)
haftmann@29653
  1245
nipkow@33676
  1246
lemma dvd_if_abs_eq:
haftmann@36301
  1247
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1248
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1249
haftmann@25917
  1250
end
haftmann@25230
  1251
haftmann@26274
  1252
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1253
blanchet@35828
  1254
lemmas mult_compare_simps[no_atp] =
paulson@15234
  1255
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1256
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1257
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1258
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1259
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1260
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1261
    mult_cancel_right mult_cancel_left
paulson@15234
  1262
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1263
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1264
haftmann@36301
  1265
text {* Reasoning about inequalities with division *}
avigad@16775
  1266
haftmann@35028
  1267
context linordered_semidom
haftmann@25193
  1268
begin
haftmann@25193
  1269
haftmann@25193
  1270
lemma less_add_one: "a < a + 1"
paulson@14293
  1271
proof -
haftmann@25193
  1272
  have "a + 0 < a + 1"
nipkow@23482
  1273
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1274
  thus ?thesis by simp
paulson@14293
  1275
qed
paulson@14293
  1276
haftmann@25193
  1277
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1278
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1279
haftmann@25193
  1280
end
paulson@14365
  1281
haftmann@36301
  1282
context linordered_idom
haftmann@36301
  1283
begin
paulson@15234
  1284
haftmann@36301
  1285
lemma mult_right_le_one_le:
haftmann@36301
  1286
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@36301
  1287
  by (auto simp add: mult_le_cancel_left2)
haftmann@36301
  1288
haftmann@36301
  1289
lemma mult_left_le_one_le:
haftmann@36301
  1290
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1291
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1292
haftmann@36301
  1293
end
haftmann@36301
  1294
haftmann@36301
  1295
text {* Absolute Value *}
paulson@14293
  1296
haftmann@35028
  1297
context linordered_idom
haftmann@25304
  1298
begin
haftmann@25304
  1299
haftmann@36301
  1300
lemma mult_sgn_abs:
haftmann@36301
  1301
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1302
  unfolding abs_if sgn_if by auto
haftmann@25304
  1303
haftmann@36301
  1304
lemma abs_one [simp]:
haftmann@36301
  1305
  "\<bar>1\<bar> = 1"
haftmann@36301
  1306
  by (simp add: abs_if zero_less_one [THEN less_not_sym])
haftmann@36301
  1307
haftmann@25304
  1308
end
nipkow@24491
  1309
haftmann@35028
  1310
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1311
  assumes abs_eq_mult:
haftmann@25304
  1312
    "(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
  1313
haftmann@35028
  1314
context linordered_idom
haftmann@30961
  1315
begin
haftmann@30961
  1316
haftmann@35028
  1317
subclass ordered_ring_abs proof
huffman@35216
  1318
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1319
haftmann@30961
  1320
lemma abs_mult:
haftmann@36301
  1321
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" 
haftmann@30961
  1322
  by (rule abs_eq_mult) auto
haftmann@30961
  1323
haftmann@30961
  1324
lemma abs_mult_self:
haftmann@36301
  1325
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
haftmann@30961
  1326
  by (simp add: abs_if) 
haftmann@30961
  1327
paulson@14294
  1328
lemma abs_mult_less:
haftmann@36301
  1329
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1330
proof -
haftmann@36301
  1331
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1332
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1333
  assume "\<bar>b\<bar> < d"
paulson@14294
  1334
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1335
qed
paulson@14293
  1336
haftmann@36301
  1337
lemma less_minus_self_iff:
haftmann@36301
  1338
  "a < - a \<longleftrightarrow> a < 0"
haftmann@36301
  1339
  by (simp only: less_le less_eq_neg_nonpos equal_neg_zero)
obua@14738
  1340
haftmann@36301
  1341
lemma abs_less_iff:
haftmann@36301
  1342
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" 
haftmann@36301
  1343
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1344
haftmann@36301
  1345
lemma abs_mult_pos:
haftmann@36301
  1346
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1347
  by (simp add: abs_mult)
haftmann@36301
  1348
haftmann@36301
  1349
end
avigad@16775
  1350
haftmann@33364
  1351
code_modulename SML
haftmann@35050
  1352
  Rings Arith
haftmann@33364
  1353
haftmann@33364
  1354
code_modulename OCaml
haftmann@35050
  1355
  Rings Arith
haftmann@33364
  1356
haftmann@33364
  1357
code_modulename Haskell
haftmann@35050
  1358
  Rings Arith
haftmann@33364
  1359
paulson@14265
  1360
end