src/HOL/Rings.thy
author haftmann
Sat Mar 28 20:43:19 2015 +0100 (2015-03-28)
changeset 59832 d5ccdca16cca
parent 59816 034b13f4efae
child 59833 ab828c2c5d67
permissions -rw-r--r--
dropped long-outdated comments
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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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section {* 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 distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c"
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  assumes distrib_left[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: distrib_right ac_simps)
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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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begin
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lemma mult_not_zero:
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  "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
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  by auto
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end
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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: distrib_right [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: distrib_left [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: ac_simps)
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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: ac_simps)
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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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definition of_bool :: "bool \<Rightarrow> 'a"
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where
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  "of_bool p = (if p then 1 else 0)" 
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lemma of_bool_eq [simp, code]:
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  "of_bool False = 0"
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  "of_bool True = 1"
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  by (simp_all add: of_bool_def)
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lemma of_bool_eq_iff:
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  "of_bool p = of_bool q \<longleftrightarrow> p = q"
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  by (simp add: of_bool_def)
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lemma split_of_bool [split]:
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  "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
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  by (cases p) simp_all
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lemma split_of_bool_asm:
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  "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
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  by (cases p) simp_all
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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" (infix "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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context comm_monoid_mult
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begin
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subclass dvd .
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lemma dvd_refl [simp]:
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  "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 one_dvd [simp]:
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  "1 dvd a"
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  by (auto intro!: dvdI)
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lemma dvd_mult [simp]:
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  "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]:
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  "a dvd b \<Longrightarrow> a dvd (b * c)"
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  using dvd_mult [of a b c] by (simp add: ac_simps) 
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lemma dvd_triv_right [simp]:
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  "a dvd b * a"
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  by (rule dvd_mult) (rule dvd_refl)
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lemma dvd_triv_left [simp]:
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  "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: ac_simps)
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  then show ?thesis ..
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qed
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lemma dvd_mult_left:
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  "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:
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  "a * b dvd c \<Longrightarrow> b dvd c"
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  using dvd_mult_left [of b a c] by (simp add: ac_simps)
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end
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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
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begin
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subclass semiring_1 ..
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lemma dvd_0_left_iff [simp]:
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  "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]:
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  "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 dvd_0_left:
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  "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"
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  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: distrib_left)
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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 comm_semiring_1_diff_distrib = comm_semiring_1_cancel +
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  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
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begin
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lemma left_diff_distrib' [algebra_simps]:
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  "(b - c) * a = b * a - c * a"
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  by (simp add: algebra_simps)
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lemma dvd_add_times_triv_left_iff [simp]:
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  "a dvd c * a + b \<longleftrightarrow> a dvd b"
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proof -
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  have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
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  proof
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    assume ?Q then show ?P by simp
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  next
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    assume ?P
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    then obtain d where "a * c + b = a * d" ..
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    then have "a * c + b - a * c = a * d - a * c" by simp
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    then have "b = a * d - a * c" by simp
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    then have "b = a * (d - c)" by (simp add: algebra_simps) 
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    then show ?Q ..
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  qed
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  then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
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qed
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lemma dvd_add_times_triv_right_iff [simp]:
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  "a dvd b + c * a \<longleftrightarrow> a dvd b"
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  using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)
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lemma dvd_add_triv_left_iff [simp]:
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  "a dvd a + b \<longleftrightarrow> a dvd b"
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  using dvd_add_times_triv_left_iff [of a 1 b] by simp
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lemma dvd_add_triv_right_iff [simp]:
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  "a dvd b + a \<longleftrightarrow> a dvd b"
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  using dvd_add_times_triv_right_iff [of a b 1] by simp
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lemma dvd_add_right_iff:
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  assumes "a dvd b"
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  shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then obtain d where "b + c = a * d" ..
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  moreover from `a dvd b` obtain e where "b = a * e" ..
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  ultimately have "a * e + c = a * d" by simp
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  then have "a * e + c - a * e = a * d - a * e" by simp
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  then have "c = a * d - a * e" by simp
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  then have "c = a * (d - e)" by (simp add: algebra_simps)
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  then show ?Q ..
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next
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  assume ?Q with assms show ?P by simp
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qed
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lemma dvd_add_left_iff:
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  assumes "a dvd c"
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  shows "a dvd b + c \<longleftrightarrow> a dvd b"
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  using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps)
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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: distrib_right [symmetric]) 
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lemma minus_mult_right: "- (a * b) = a * - b"
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by (rule minus_unique) (simp add: distrib_left [symmetric]) 
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text{*Extract signs from products*}
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lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
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lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
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lemma minus_mult_minus [simp]: "- a * - b = a * b"
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by simp
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lemma minus_mult_commute: "- a * b = a * - b"
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by simp
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lemma right_diff_distrib [algebra_simps]:
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  "a * (b - c) = a * b - a * c"
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  using distrib_left [of a b "-c "] by simp
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lemma left_diff_distrib [algebra_simps]:
haftmann@54230
   353
  "(a - b) * c = a * c - b * c"
haftmann@54230
   354
  using distrib_right [of a "- b" c] by simp
haftmann@25152
   355
blanchet@54147
   356
lemmas ring_distribs =
webertj@49962
   357
  distrib_left distrib_right left_diff_distrib right_diff_distrib
haftmann@25152
   358
haftmann@25230
   359
lemma eq_add_iff1:
haftmann@25230
   360
  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
nipkow@29667
   361
by (simp add: algebra_simps)
haftmann@25230
   362
haftmann@25230
   363
lemma eq_add_iff2:
haftmann@25230
   364
  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
nipkow@29667
   365
by (simp add: algebra_simps)
haftmann@25230
   366
haftmann@25152
   367
end
haftmann@25152
   368
blanchet@54147
   369
lemmas ring_distribs =
webertj@49962
   370
  distrib_left distrib_right left_diff_distrib right_diff_distrib
haftmann@25152
   371
haftmann@22390
   372
class comm_ring = comm_semiring + ab_group_add
haftmann@25267
   373
begin
obua@14738
   374
huffman@27516
   375
subclass ring ..
huffman@28141
   376
subclass comm_semiring_0_cancel ..
haftmann@25267
   377
huffman@44350
   378
lemma square_diff_square_factored:
huffman@44350
   379
  "x * x - y * y = (x + y) * (x - y)"
huffman@44350
   380
  by (simp add: algebra_simps)
huffman@44350
   381
haftmann@25267
   382
end
obua@14738
   383
haftmann@22390
   384
class ring_1 = ring + zero_neq_one + monoid_mult
haftmann@25267
   385
begin
paulson@14265
   386
huffman@27516
   387
subclass semiring_1_cancel ..
haftmann@25267
   388
huffman@44346
   389
lemma square_diff_one_factored:
huffman@44346
   390
  "x * x - 1 = (x + 1) * (x - 1)"
huffman@44346
   391
  by (simp add: algebra_simps)
huffman@44346
   392
haftmann@25267
   393
end
haftmann@25152
   394
haftmann@22390
   395
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
haftmann@25267
   396
begin
obua@14738
   397
huffman@27516
   398
subclass ring_1 ..
huffman@27516
   399
subclass comm_semiring_1_cancel ..
haftmann@25267
   400
haftmann@59816
   401
subclass comm_semiring_1_diff_distrib
haftmann@59816
   402
  by unfold_locales (simp add: algebra_simps)
haftmann@58647
   403
huffman@29465
   404
lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
huffman@29408
   405
proof
huffman@29408
   406
  assume "x dvd - y"
huffman@29408
   407
  then have "x dvd - 1 * - y" by (rule dvd_mult)
huffman@29408
   408
  then show "x dvd y" by simp
huffman@29408
   409
next
huffman@29408
   410
  assume "x dvd y"
huffman@29408
   411
  then have "x dvd - 1 * y" by (rule dvd_mult)
huffman@29408
   412
  then show "x dvd - y" by simp
huffman@29408
   413
qed
huffman@29408
   414
huffman@29465
   415
lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
huffman@29408
   416
proof
huffman@29408
   417
  assume "- x dvd y"
huffman@29408
   418
  then obtain k where "y = - x * k" ..
huffman@29408
   419
  then have "y = x * - k" by simp
huffman@29408
   420
  then show "x dvd y" ..
huffman@29408
   421
next
huffman@29408
   422
  assume "x dvd y"
huffman@29408
   423
  then obtain k where "y = x * k" ..
huffman@29408
   424
  then have "y = - x * - k" by simp
huffman@29408
   425
  then show "- x dvd y" ..
huffman@29408
   426
qed
huffman@29408
   427
haftmann@54230
   428
lemma dvd_diff [simp]:
haftmann@54230
   429
  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
haftmann@54230
   430
  using dvd_add [of x y "- z"] by simp
huffman@29409
   431
haftmann@25267
   432
end
haftmann@25152
   433
haftmann@58952
   434
class semiring_no_zero_divisors = semiring_0 + no_zero_divisors
haftmann@25230
   435
begin
haftmann@25230
   436
haftmann@25230
   437
lemma mult_eq_0_iff [simp]:
haftmann@58952
   438
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@25230
   439
proof (cases "a = 0 \<or> b = 0")
haftmann@25230
   440
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@25230
   441
    then show ?thesis using no_zero_divisors by simp
haftmann@25230
   442
next
haftmann@25230
   443
  case True then show ?thesis by auto
haftmann@25230
   444
qed
haftmann@25230
   445
haftmann@58952
   446
end
haftmann@58952
   447
haftmann@58952
   448
class ring_no_zero_divisors = ring + semiring_no_zero_divisors
haftmann@58952
   449
begin
haftmann@58952
   450
haftmann@26193
   451
text{*Cancellation of equalities with a common factor*}
blanchet@54147
   452
lemma mult_cancel_right [simp]:
haftmann@26193
   453
  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@26193
   454
proof -
haftmann@26193
   455
  have "(a * c = b * c) = ((a - b) * c = 0)"
huffman@35216
   456
    by (simp add: algebra_simps)
huffman@35216
   457
  thus ?thesis by (simp add: disj_commute)
haftmann@26193
   458
qed
haftmann@26193
   459
blanchet@54147
   460
lemma mult_cancel_left [simp]:
haftmann@26193
   461
  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@26193
   462
proof -
haftmann@26193
   463
  have "(c * a = c * b) = (c * (a - b) = 0)"
huffman@35216
   464
    by (simp add: algebra_simps)
huffman@35216
   465
  thus ?thesis by simp
haftmann@26193
   466
qed
haftmann@26193
   467
haftmann@58952
   468
lemma mult_left_cancel:
haftmann@58952
   469
  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
haftmann@58952
   470
  by simp 
lp15@56217
   471
haftmann@58952
   472
lemma mult_right_cancel:
haftmann@58952
   473
  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
haftmann@58952
   474
  by simp 
lp15@56217
   475
haftmann@25230
   476
end
huffman@22990
   477
huffman@23544
   478
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   479
begin
haftmann@26274
   480
huffman@36970
   481
lemma square_eq_1_iff:
huffman@36821
   482
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
huffman@36821
   483
proof -
huffman@36821
   484
  have "(x - 1) * (x + 1) = x * x - 1"
huffman@36821
   485
    by (simp add: algebra_simps)
huffman@36821
   486
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   487
    by simp
huffman@36821
   488
  thus ?thesis
huffman@36821
   489
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   490
qed
huffman@36821
   491
haftmann@26274
   492
lemma mult_cancel_right1 [simp]:
haftmann@26274
   493
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   494
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   495
haftmann@26274
   496
lemma mult_cancel_right2 [simp]:
haftmann@26274
   497
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   498
by (insert mult_cancel_right [of a c 1], simp)
haftmann@26274
   499
 
haftmann@26274
   500
lemma mult_cancel_left1 [simp]:
haftmann@26274
   501
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   502
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   503
haftmann@26274
   504
lemma mult_cancel_left2 [simp]:
haftmann@26274
   505
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   506
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   507
haftmann@26274
   508
end
huffman@22990
   509
haftmann@22390
   510
class idom = comm_ring_1 + no_zero_divisors
haftmann@25186
   511
begin
paulson@14421
   512
huffman@27516
   513
subclass ring_1_no_zero_divisors ..
huffman@22990
   514
huffman@29915
   515
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)"
huffman@29915
   516
proof
huffman@29915
   517
  assume "a * a = b * b"
huffman@29915
   518
  then have "(a - b) * (a + b) = 0"
huffman@29915
   519
    by (simp add: algebra_simps)
huffman@29915
   520
  then show "a = b \<or> a = - b"
huffman@35216
   521
    by (simp add: eq_neg_iff_add_eq_0)
huffman@29915
   522
next
huffman@29915
   523
  assume "a = b \<or> a = - b"
huffman@29915
   524
  then show "a * a = b * b" by auto
huffman@29915
   525
qed
huffman@29915
   526
huffman@29981
   527
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   528
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   529
proof -
huffman@29981
   530
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   531
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   532
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   533
    unfolding dvd_def by simp
huffman@29981
   534
  finally show ?thesis .
huffman@29981
   535
qed
huffman@29981
   536
huffman@29981
   537
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   538
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   539
proof -
huffman@29981
   540
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   541
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   542
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   543
    unfolding dvd_def by simp
huffman@29981
   544
  finally show ?thesis .
huffman@29981
   545
qed
huffman@29981
   546
haftmann@25186
   547
end
haftmann@25152
   548
haftmann@35302
   549
text {*
haftmann@35302
   550
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   551
  \begin{itemize}
haftmann@35302
   552
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35302
   553
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   554
  \end{itemize}
haftmann@35302
   555
  Most of the used notions can also be looked up in 
haftmann@35302
   556
  \begin{itemize}
wenzelm@54703
   557
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35302
   558
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   559
  \end{itemize}
haftmann@35302
   560
*}
haftmann@35302
   561
haftmann@38642
   562
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
   563
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
   564
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   565
begin
haftmann@25230
   566
haftmann@25230
   567
lemma mult_mono:
haftmann@38642
   568
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   569
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   570
apply (erule mult_left_mono, assumption)
haftmann@25230
   571
done
haftmann@25230
   572
haftmann@25230
   573
lemma mult_mono':
haftmann@38642
   574
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   575
apply (rule mult_mono)
haftmann@25230
   576
apply (fast intro: order_trans)+
haftmann@25230
   577
done
haftmann@25230
   578
haftmann@25230
   579
end
krauss@21199
   580
haftmann@38642
   581
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
   582
begin
paulson@14268
   583
huffman@27516
   584
subclass semiring_0_cancel ..
obua@23521
   585
nipkow@56536
   586
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   587
using mult_left_mono [of 0 b a] by simp
haftmann@25230
   588
haftmann@25230
   589
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   590
using mult_left_mono [of b 0 a] by simp
huffman@30692
   591
huffman@30692
   592
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   593
using mult_right_mono [of a 0 b] by simp
huffman@30692
   594
huffman@30692
   595
text {* Legacy - use @{text mult_nonpos_nonneg} *}
haftmann@25230
   596
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
haftmann@36301
   597
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
   598
haftmann@26234
   599
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
nipkow@29667
   600
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   601
haftmann@25230
   602
end
haftmann@25230
   603
haftmann@38642
   604
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
   605
begin
haftmann@25230
   606
haftmann@35028
   607
subclass ordered_cancel_semiring ..
haftmann@35028
   608
haftmann@35028
   609
subclass ordered_comm_monoid_add ..
haftmann@25304
   610
haftmann@25230
   611
lemma mult_left_less_imp_less:
haftmann@25230
   612
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   613
by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   614
 
haftmann@25230
   615
lemma mult_right_less_imp_less:
haftmann@25230
   616
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   617
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   618
haftmann@25186
   619
end
haftmann@25152
   620
haftmann@35043
   621
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
   622
begin
hoelzl@36622
   623
hoelzl@36622
   624
lemma convex_bound_le:
hoelzl@36622
   625
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   626
  shows "u * x + v * y \<le> a"
hoelzl@36622
   627
proof-
hoelzl@36622
   628
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
   629
    by (simp add: add_mono mult_left_mono)
webertj@49962
   630
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   631
qed
hoelzl@36622
   632
hoelzl@36622
   633
end
haftmann@35043
   634
haftmann@35043
   635
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
   636
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   637
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   638
begin
paulson@14341
   639
huffman@27516
   640
subclass semiring_0_cancel ..
obua@14940
   641
haftmann@35028
   642
subclass linordered_semiring
haftmann@28823
   643
proof
huffman@23550
   644
  fix a b c :: 'a
huffman@23550
   645
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   646
  from A show "c * a \<le> c * b"
haftmann@25186
   647
    unfolding le_less
haftmann@25186
   648
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   649
  from A show "a * c \<le> b * c"
haftmann@25152
   650
    unfolding le_less
haftmann@25186
   651
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   652
qed
haftmann@25152
   653
haftmann@25230
   654
lemma mult_left_le_imp_le:
haftmann@25230
   655
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   656
by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   657
 
haftmann@25230
   658
lemma mult_right_le_imp_le:
haftmann@25230
   659
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   660
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   661
nipkow@56544
   662
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
   663
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
   664
huffman@30692
   665
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
   666
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
   667
huffman@30692
   668
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
   669
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
   670
huffman@30692
   671
text {* Legacy - use @{text mult_neg_pos} *}
huffman@30692
   672
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
haftmann@36301
   673
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
   674
haftmann@25230
   675
lemma zero_less_mult_pos:
haftmann@25230
   676
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   677
apply (cases "b\<le>0")
haftmann@25230
   678
 apply (auto simp add: le_less not_less)
huffman@30692
   679
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
   680
 apply (auto dest: less_not_sym)
haftmann@25230
   681
done
haftmann@25230
   682
haftmann@25230
   683
lemma zero_less_mult_pos2:
haftmann@25230
   684
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   685
apply (cases "b\<le>0")
haftmann@25230
   686
 apply (auto simp add: le_less not_less)
huffman@30692
   687
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
   688
 apply (auto dest: less_not_sym)
haftmann@25230
   689
done
haftmann@25230
   690
haftmann@26193
   691
text{*Strict monotonicity in both arguments*}
haftmann@26193
   692
lemma mult_strict_mono:
haftmann@26193
   693
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
   694
  shows "a * c < b * d"
haftmann@26193
   695
  using assms apply (cases "c=0")
nipkow@56544
   696
  apply (simp)
haftmann@26193
   697
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
   698
  apply (force simp add: le_less)
haftmann@26193
   699
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
   700
  done
haftmann@26193
   701
haftmann@26193
   702
text{*This weaker variant has more natural premises*}
haftmann@26193
   703
lemma mult_strict_mono':
haftmann@26193
   704
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
   705
  shows "a * c < b * d"
nipkow@29667
   706
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
   707
haftmann@26193
   708
lemma mult_less_le_imp_less:
haftmann@26193
   709
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
   710
  shows "a * c < b * d"
haftmann@26193
   711
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
   712
  apply (erule less_le_trans)
haftmann@26193
   713
  apply (erule mult_left_mono)
haftmann@26193
   714
  apply simp
haftmann@26193
   715
  apply (erule mult_strict_right_mono)
haftmann@26193
   716
  apply assumption
haftmann@26193
   717
  done
haftmann@26193
   718
haftmann@26193
   719
lemma mult_le_less_imp_less:
haftmann@26193
   720
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
   721
  shows "a * c < b * d"
haftmann@26193
   722
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
   723
  apply (erule le_less_trans)
haftmann@26193
   724
  apply (erule mult_strict_left_mono)
haftmann@26193
   725
  apply simp
haftmann@26193
   726
  apply (erule mult_right_mono)
haftmann@26193
   727
  apply simp
haftmann@26193
   728
  done
haftmann@26193
   729
haftmann@25230
   730
end
haftmann@25230
   731
haftmann@35097
   732
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
   733
begin
hoelzl@36622
   734
hoelzl@36622
   735
subclass linordered_semiring_1 ..
hoelzl@36622
   736
hoelzl@36622
   737
lemma convex_bound_lt:
hoelzl@36622
   738
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   739
  shows "u * x + v * y < a"
hoelzl@36622
   740
proof -
hoelzl@36622
   741
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
   742
    by (cases "u = 0")
hoelzl@36622
   743
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
webertj@49962
   744
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   745
qed
hoelzl@36622
   746
hoelzl@36622
   747
end
haftmann@33319
   748
haftmann@38642
   749
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add + 
haftmann@38642
   750
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
   751
begin
haftmann@25152
   752
haftmann@35028
   753
subclass ordered_semiring
haftmann@28823
   754
proof
krauss@21199
   755
  fix a b c :: 'a
huffman@23550
   756
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
   757
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
haftmann@57512
   758
  thus "a * c \<le> b * c" by (simp only: mult.commute)
krauss@21199
   759
qed
paulson@14265
   760
haftmann@25267
   761
end
haftmann@25267
   762
haftmann@38642
   763
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
   764
begin
paulson@14265
   765
haftmann@38642
   766
subclass comm_semiring_0_cancel ..
haftmann@35028
   767
subclass ordered_comm_semiring ..
haftmann@35028
   768
subclass ordered_cancel_semiring ..
haftmann@25267
   769
haftmann@25267
   770
end
haftmann@25267
   771
haftmann@35028
   772
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
   773
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
   774
begin
haftmann@25267
   775
haftmann@35043
   776
subclass linordered_semiring_strict
haftmann@28823
   777
proof
huffman@23550
   778
  fix a b c :: 'a
huffman@23550
   779
  assume "a < b" "0 < c"
haftmann@38642
   780
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
haftmann@57512
   781
  thus "a * c < b * c" by (simp only: mult.commute)
huffman@23550
   782
qed
paulson@14272
   783
haftmann@35028
   784
subclass ordered_cancel_comm_semiring
haftmann@28823
   785
proof
huffman@23550
   786
  fix a b c :: 'a
huffman@23550
   787
  assume "a \<le> b" "0 \<le> c"
huffman@23550
   788
  thus "c * a \<le> c * b"
haftmann@25186
   789
    unfolding le_less
haftmann@26193
   790
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   791
qed
paulson@14272
   792
haftmann@25267
   793
end
haftmann@25230
   794
haftmann@35028
   795
class ordered_ring = ring + ordered_cancel_semiring 
haftmann@25267
   796
begin
haftmann@25230
   797
haftmann@35028
   798
subclass ordered_ab_group_add ..
paulson@14270
   799
haftmann@25230
   800
lemma less_add_iff1:
haftmann@25230
   801
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
   802
by (simp add: algebra_simps)
haftmann@25230
   803
haftmann@25230
   804
lemma less_add_iff2:
haftmann@25230
   805
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
   806
by (simp add: algebra_simps)
haftmann@25230
   807
haftmann@25230
   808
lemma le_add_iff1:
haftmann@25230
   809
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
   810
by (simp add: algebra_simps)
haftmann@25230
   811
haftmann@25230
   812
lemma le_add_iff2:
haftmann@25230
   813
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
   814
by (simp add: algebra_simps)
haftmann@25230
   815
haftmann@25230
   816
lemma mult_left_mono_neg:
haftmann@25230
   817
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
   818
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
   819
  apply simp_all
haftmann@25230
   820
  done
haftmann@25230
   821
haftmann@25230
   822
lemma mult_right_mono_neg:
haftmann@25230
   823
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
   824
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
   825
  apply simp_all
haftmann@25230
   826
  done
haftmann@25230
   827
huffman@30692
   828
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   829
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
   830
haftmann@25230
   831
lemma split_mult_pos_le:
haftmann@25230
   832
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@56536
   833
by (auto simp add: mult_nonpos_nonpos)
haftmann@25186
   834
haftmann@25186
   835
end
paulson@14270
   836
haftmann@35028
   837
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
   838
begin
haftmann@25304
   839
haftmann@35028
   840
subclass ordered_ring ..
haftmann@35028
   841
haftmann@35028
   842
subclass ordered_ab_group_add_abs
haftmann@28823
   843
proof
haftmann@25304
   844
  fix a b
haftmann@25304
   845
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@54230
   846
    by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
huffman@35216
   847
qed (auto simp add: abs_if)
haftmann@25304
   848
huffman@35631
   849
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
   850
  using linear [of 0 a]
nipkow@56536
   851
  by (auto simp add: mult_nonpos_nonpos)
huffman@35631
   852
huffman@35631
   853
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
   854
  by (simp add: not_less)
huffman@35631
   855
haftmann@25304
   856
end
obua@23521
   857
haftmann@35043
   858
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
   859
  + ordered_ab_group_add + abs_if
haftmann@25230
   860
begin
paulson@14348
   861
haftmann@35028
   862
subclass linordered_ring ..
haftmann@25304
   863
huffman@30692
   864
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
   865
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
   866
huffman@30692
   867
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
   868
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
   869
huffman@30692
   870
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
   871
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
   872
haftmann@25917
   873
subclass ring_no_zero_divisors
haftmann@28823
   874
proof
haftmann@25917
   875
  fix a b
haftmann@25917
   876
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
   877
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
   878
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
   879
  proof (cases "a < 0")
haftmann@25917
   880
    case True note A' = this
haftmann@25917
   881
    show ?thesis proof (cases "b < 0")
haftmann@25917
   882
      case True with A'
haftmann@25917
   883
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
   884
    next
haftmann@25917
   885
      case False with B have "0 < b" by auto
haftmann@25917
   886
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
   887
    qed
haftmann@25917
   888
  next
haftmann@25917
   889
    case False with A have A': "0 < a" by auto
haftmann@25917
   890
    show ?thesis proof (cases "b < 0")
haftmann@25917
   891
      case True with A'
haftmann@25917
   892
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
   893
    next
haftmann@25917
   894
      case False with B have "0 < b" by auto
nipkow@56544
   895
      with A' show ?thesis by auto
haftmann@25917
   896
    qed
haftmann@25917
   897
  qed
haftmann@25917
   898
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
   899
qed
haftmann@25304
   900
hoelzl@56480
   901
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
hoelzl@56480
   902
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
nipkow@56544
   903
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
huffman@22990
   904
hoelzl@56480
   905
lemma zero_le_mult_iff: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
hoelzl@56480
   906
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
   907
paulson@14265
   908
lemma mult_less_0_iff:
haftmann@25917
   909
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
   910
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
   911
  apply force
haftmann@25917
   912
  done
paulson@14265
   913
paulson@14265
   914
lemma mult_le_0_iff:
haftmann@25917
   915
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
   916
  apply (insert zero_le_mult_iff [of "-a" b]) 
huffman@35216
   917
  apply force
haftmann@25917
   918
  done
haftmann@25917
   919
haftmann@26193
   920
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
   921
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
   922
haftmann@26193
   923
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
   924
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
   925
haftmann@26193
   926
lemma mult_less_cancel_right_disj:
haftmann@26193
   927
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   928
  apply (cases "c = 0")
haftmann@26193
   929
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
   930
                      mult_strict_right_mono_neg)
haftmann@26193
   931
  apply (auto simp add: not_less 
haftmann@26193
   932
                      not_le [symmetric, of "a*c"]
haftmann@26193
   933
                      not_le [symmetric, of a])
haftmann@26193
   934
  apply (erule_tac [!] notE)
haftmann@26193
   935
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
   936
                      mult_right_mono_neg)
haftmann@26193
   937
  done
haftmann@26193
   938
haftmann@26193
   939
lemma mult_less_cancel_left_disj:
haftmann@26193
   940
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   941
  apply (cases "c = 0")
haftmann@26193
   942
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
   943
                      mult_strict_left_mono_neg)
haftmann@26193
   944
  apply (auto simp add: not_less 
haftmann@26193
   945
                      not_le [symmetric, of "c*a"]
haftmann@26193
   946
                      not_le [symmetric, of a])
haftmann@26193
   947
  apply (erule_tac [!] notE)
haftmann@26193
   948
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
   949
                      mult_left_mono_neg)
haftmann@26193
   950
  done
haftmann@26193
   951
haftmann@26193
   952
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
   953
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
   954
haftmann@26193
   955
lemma mult_less_cancel_right:
haftmann@26193
   956
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   957
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
   958
haftmann@26193
   959
lemma mult_less_cancel_left:
haftmann@26193
   960
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   961
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
   962
haftmann@26193
   963
lemma mult_le_cancel_right:
haftmann@26193
   964
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
   965
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
   966
haftmann@26193
   967
lemma mult_le_cancel_left:
haftmann@26193
   968
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
   969
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
   970
nipkow@30649
   971
lemma mult_le_cancel_left_pos:
nipkow@30649
   972
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
   973
by (auto simp: mult_le_cancel_left)
nipkow@30649
   974
nipkow@30649
   975
lemma mult_le_cancel_left_neg:
nipkow@30649
   976
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
   977
by (auto simp: mult_le_cancel_left)
nipkow@30649
   978
nipkow@30649
   979
lemma mult_less_cancel_left_pos:
nipkow@30649
   980
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
   981
by (auto simp: mult_less_cancel_left)
nipkow@30649
   982
nipkow@30649
   983
lemma mult_less_cancel_left_neg:
nipkow@30649
   984
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
   985
by (auto simp: mult_less_cancel_left)
nipkow@30649
   986
haftmann@25917
   987
end
paulson@14265
   988
huffman@30692
   989
lemmas mult_sign_intros =
huffman@30692
   990
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
   991
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
   992
  mult_pos_pos mult_pos_neg
huffman@30692
   993
  mult_neg_pos mult_neg_neg
haftmann@25230
   994
haftmann@35028
   995
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
   996
begin
haftmann@25230
   997
haftmann@35028
   998
subclass ordered_ring ..
haftmann@35028
   999
subclass ordered_cancel_comm_semiring ..
haftmann@25230
  1000
haftmann@25267
  1001
end
haftmann@25230
  1002
haftmann@35028
  1003
class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +
haftmann@25230
  1004
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
  1005
begin
haftmann@25230
  1006
haftmann@25230
  1007
lemma pos_add_strict:
haftmann@25230
  1008
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
  1009
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
  1010
haftmann@26193
  1011
lemma zero_le_one [simp]: "0 \<le> 1"
nipkow@29667
  1012
by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
  1013
haftmann@26193
  1014
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
nipkow@29667
  1015
by (simp add: not_le) 
haftmann@26193
  1016
haftmann@26193
  1017
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
nipkow@29667
  1018
by (simp add: not_less) 
haftmann@26193
  1019
haftmann@26193
  1020
lemma less_1_mult:
haftmann@26193
  1021
  assumes "1 < m" and "1 < n"
haftmann@26193
  1022
  shows "1 < m * n"
haftmann@26193
  1023
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
  1024
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
  1025
hoelzl@59000
  1026
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
hoelzl@59000
  1027
  using mult_left_mono[of c 1 a] by simp
hoelzl@59000
  1028
hoelzl@59000
  1029
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
hoelzl@59000
  1030
  using mult_mono[of a 1 b 1] by simp
hoelzl@59000
  1031
haftmann@25230
  1032
end
haftmann@25230
  1033
haftmann@35028
  1034
class linordered_idom = comm_ring_1 +
haftmann@35028
  1035
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1036
  abs_if + sgn_if
haftmann@25917
  1037
begin
haftmann@25917
  1038
hoelzl@36622
  1039
subclass linordered_semiring_1_strict ..
haftmann@35043
  1040
subclass linordered_ring_strict ..
haftmann@35028
  1041
subclass ordered_comm_ring ..
huffman@27516
  1042
subclass idom ..
haftmann@25917
  1043
haftmann@35028
  1044
subclass linordered_semidom
haftmann@28823
  1045
proof
haftmann@26193
  1046
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1047
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
  1048
qed 
haftmann@25917
  1049
haftmann@35028
  1050
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1051
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1052
  using assms by (rule neqE)
haftmann@26193
  1053
haftmann@26274
  1054
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1055
haftmann@26274
  1056
lemma mult_le_cancel_right1:
haftmann@26274
  1057
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1058
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1059
haftmann@26274
  1060
lemma mult_le_cancel_right2:
haftmann@26274
  1061
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1062
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1063
haftmann@26274
  1064
lemma mult_le_cancel_left1:
haftmann@26274
  1065
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1066
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1067
haftmann@26274
  1068
lemma mult_le_cancel_left2:
haftmann@26274
  1069
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1070
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1071
haftmann@26274
  1072
lemma mult_less_cancel_right1:
haftmann@26274
  1073
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1074
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1075
haftmann@26274
  1076
lemma mult_less_cancel_right2:
haftmann@26274
  1077
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1078
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1079
haftmann@26274
  1080
lemma mult_less_cancel_left1:
haftmann@26274
  1081
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1082
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1083
haftmann@26274
  1084
lemma mult_less_cancel_left2:
haftmann@26274
  1085
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1086
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1087
haftmann@27651
  1088
lemma sgn_sgn [simp]:
haftmann@27651
  1089
  "sgn (sgn a) = sgn a"
nipkow@29700
  1090
unfolding sgn_if by simp
haftmann@27651
  1091
haftmann@27651
  1092
lemma sgn_0_0:
haftmann@27651
  1093
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1094
unfolding sgn_if by simp
haftmann@27651
  1095
haftmann@27651
  1096
lemma sgn_1_pos:
haftmann@27651
  1097
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1098
unfolding sgn_if by simp
haftmann@27651
  1099
haftmann@27651
  1100
lemma sgn_1_neg:
haftmann@27651
  1101
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1102
unfolding sgn_if by auto
haftmann@27651
  1103
haftmann@29940
  1104
lemma sgn_pos [simp]:
haftmann@29940
  1105
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1106
unfolding sgn_1_pos .
haftmann@29940
  1107
haftmann@29940
  1108
lemma sgn_neg [simp]:
haftmann@29940
  1109
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1110
unfolding sgn_1_neg .
haftmann@29940
  1111
haftmann@27651
  1112
lemma sgn_times:
haftmann@27651
  1113
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1114
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1115
haftmann@36301
  1116
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1117
unfolding sgn_if abs_if by auto
nipkow@29700
  1118
haftmann@29940
  1119
lemma sgn_greater [simp]:
haftmann@29940
  1120
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1121
  unfolding sgn_if by auto
haftmann@29940
  1122
haftmann@29940
  1123
lemma sgn_less [simp]:
haftmann@29940
  1124
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1125
  unfolding sgn_if by auto
haftmann@29940
  1126
haftmann@36301
  1127
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1128
  by (simp add: abs_if)
huffman@29949
  1129
haftmann@36301
  1130
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1131
  by (simp add: abs_if)
haftmann@29653
  1132
nipkow@33676
  1133
lemma dvd_if_abs_eq:
haftmann@36301
  1134
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1135
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1136
huffman@55912
  1137
text {* The following lemmas can be proven in more general structures, but
haftmann@54489
  1138
are dangerous as simp rules in absence of @{thm neg_equal_zero}, 
haftmann@54489
  1139
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
haftmann@54489
  1140
haftmann@54489
  1141
lemma equation_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1142
  "1 = - a \<longleftrightarrow> a = - 1"
haftmann@54489
  1143
  by (fact equation_minus_iff)
haftmann@54489
  1144
haftmann@54489
  1145
lemma minus_equation_iff_1 [simp, no_atp]:
haftmann@54489
  1146
  "- a = 1 \<longleftrightarrow> a = - 1"
haftmann@54489
  1147
  by (subst minus_equation_iff, auto)
haftmann@54489
  1148
haftmann@54489
  1149
lemma le_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1150
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
haftmann@54489
  1151
  by (fact le_minus_iff)
haftmann@54489
  1152
haftmann@54489
  1153
lemma minus_le_iff_1 [simp, no_atp]:
haftmann@54489
  1154
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
haftmann@54489
  1155
  by (fact minus_le_iff)
haftmann@54489
  1156
haftmann@54489
  1157
lemma less_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1158
  "1 < - b \<longleftrightarrow> b < - 1"
haftmann@54489
  1159
  by (fact less_minus_iff)
haftmann@54489
  1160
haftmann@54489
  1161
lemma minus_less_iff_1 [simp, no_atp]:
haftmann@54489
  1162
  "- a < 1 \<longleftrightarrow> - 1 < a"
haftmann@54489
  1163
  by (fact minus_less_iff)
haftmann@54489
  1164
haftmann@25917
  1165
end
haftmann@25230
  1166
haftmann@26274
  1167
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1168
blanchet@54147
  1169
lemmas mult_compare_simps =
paulson@15234
  1170
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1171
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1172
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1173
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1174
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1175
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1176
    mult_cancel_right mult_cancel_left
paulson@15234
  1177
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1178
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1179
haftmann@36301
  1180
text {* Reasoning about inequalities with division *}
avigad@16775
  1181
haftmann@35028
  1182
context linordered_semidom
haftmann@25193
  1183
begin
haftmann@25193
  1184
haftmann@25193
  1185
lemma less_add_one: "a < a + 1"
paulson@14293
  1186
proof -
haftmann@25193
  1187
  have "a + 0 < a + 1"
nipkow@23482
  1188
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1189
  thus ?thesis by simp
paulson@14293
  1190
qed
paulson@14293
  1191
haftmann@25193
  1192
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1193
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1194
haftmann@25193
  1195
end
paulson@14365
  1196
haftmann@36301
  1197
context linordered_idom
haftmann@36301
  1198
begin
paulson@15234
  1199
haftmann@36301
  1200
lemma mult_right_le_one_le:
haftmann@36301
  1201
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@36301
  1202
  by (auto simp add: mult_le_cancel_left2)
haftmann@36301
  1203
haftmann@36301
  1204
lemma mult_left_le_one_le:
haftmann@36301
  1205
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1206
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1207
haftmann@36301
  1208
end
haftmann@36301
  1209
haftmann@36301
  1210
text {* Absolute Value *}
paulson@14293
  1211
haftmann@35028
  1212
context linordered_idom
haftmann@25304
  1213
begin
haftmann@25304
  1214
haftmann@36301
  1215
lemma mult_sgn_abs:
haftmann@36301
  1216
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1217
  unfolding abs_if sgn_if by auto
haftmann@25304
  1218
haftmann@36301
  1219
lemma abs_one [simp]:
haftmann@36301
  1220
  "\<bar>1\<bar> = 1"
huffman@44921
  1221
  by (simp add: abs_if)
haftmann@36301
  1222
haftmann@25304
  1223
end
nipkow@24491
  1224
haftmann@35028
  1225
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1226
  assumes abs_eq_mult:
haftmann@25304
  1227
    "(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
  1228
haftmann@35028
  1229
context linordered_idom
haftmann@30961
  1230
begin
haftmann@30961
  1231
haftmann@35028
  1232
subclass ordered_ring_abs proof
huffman@35216
  1233
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1234
haftmann@30961
  1235
lemma abs_mult:
haftmann@36301
  1236
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" 
haftmann@30961
  1237
  by (rule abs_eq_mult) auto
haftmann@30961
  1238
haftmann@30961
  1239
lemma abs_mult_self:
haftmann@36301
  1240
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
haftmann@30961
  1241
  by (simp add: abs_if) 
haftmann@30961
  1242
paulson@14294
  1243
lemma abs_mult_less:
haftmann@36301
  1244
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1245
proof -
haftmann@36301
  1246
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1247
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1248
  assume "\<bar>b\<bar> < d"
paulson@14294
  1249
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1250
qed
paulson@14293
  1251
haftmann@36301
  1252
lemma abs_less_iff:
haftmann@36301
  1253
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" 
haftmann@36301
  1254
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1255
haftmann@36301
  1256
lemma abs_mult_pos:
haftmann@36301
  1257
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1258
  by (simp add: abs_mult)
haftmann@36301
  1259
hoelzl@51520
  1260
lemma abs_diff_less_iff:
hoelzl@51520
  1261
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
hoelzl@51520
  1262
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
hoelzl@51520
  1263
haftmann@36301
  1264
end
avigad@16775
  1265
haftmann@59557
  1266
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
haftmann@59557
  1267
haftmann@52435
  1268
code_identifier
haftmann@52435
  1269
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1270
paulson@14265
  1271
end