src/HOL/Rings.thy
author haftmann
Mon Jun 01 18:59:21 2015 +0200 (2015-06-01)
changeset 60352 d46de31a50c4
parent 59910 815de5506080
child 60353 838025c6e278
permissions -rw-r--r--
separate class for division operator, with particular syntax added in more specific classes
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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 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]:
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  "(a - b) * c = a * c - b * c"
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  using distrib_right [of a "- b" c] by simp
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lemmas ring_distribs =
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  distrib_left distrib_right 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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blanchet@54147
   353
lemmas ring_distribs =
webertj@49962
   354
  distrib_left distrib_right left_diff_distrib right_diff_distrib
haftmann@25152
   355
haftmann@22390
   356
class comm_ring = comm_semiring + ab_group_add
haftmann@25267
   357
begin
obua@14738
   358
huffman@27516
   359
subclass ring ..
huffman@28141
   360
subclass comm_semiring_0_cancel ..
haftmann@25267
   361
huffman@44350
   362
lemma square_diff_square_factored:
huffman@44350
   363
  "x * x - y * y = (x + y) * (x - y)"
huffman@44350
   364
  by (simp add: algebra_simps)
huffman@44350
   365
haftmann@25267
   366
end
obua@14738
   367
haftmann@22390
   368
class ring_1 = ring + zero_neq_one + monoid_mult
haftmann@25267
   369
begin
paulson@14265
   370
huffman@27516
   371
subclass semiring_1_cancel ..
haftmann@25267
   372
huffman@44346
   373
lemma square_diff_one_factored:
huffman@44346
   374
  "x * x - 1 = (x + 1) * (x - 1)"
huffman@44346
   375
  by (simp add: algebra_simps)
huffman@44346
   376
haftmann@25267
   377
end
haftmann@25152
   378
haftmann@22390
   379
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
haftmann@25267
   380
begin
obua@14738
   381
huffman@27516
   382
subclass ring_1 ..
huffman@27516
   383
subclass comm_semiring_1_cancel ..
haftmann@25267
   384
haftmann@59816
   385
subclass comm_semiring_1_diff_distrib
haftmann@59816
   386
  by unfold_locales (simp add: algebra_simps)
haftmann@58647
   387
huffman@29465
   388
lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
huffman@29408
   389
proof
huffman@29408
   390
  assume "x dvd - y"
huffman@29408
   391
  then have "x dvd - 1 * - y" by (rule dvd_mult)
huffman@29408
   392
  then show "x dvd y" by simp
huffman@29408
   393
next
huffman@29408
   394
  assume "x dvd y"
huffman@29408
   395
  then have "x dvd - 1 * y" by (rule dvd_mult)
huffman@29408
   396
  then show "x dvd - y" by simp
huffman@29408
   397
qed
huffman@29408
   398
huffman@29465
   399
lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
huffman@29408
   400
proof
huffman@29408
   401
  assume "- x dvd y"
huffman@29408
   402
  then obtain k where "y = - x * k" ..
huffman@29408
   403
  then have "y = x * - k" by simp
huffman@29408
   404
  then show "x dvd y" ..
huffman@29408
   405
next
huffman@29408
   406
  assume "x dvd y"
huffman@29408
   407
  then obtain k where "y = x * k" ..
huffman@29408
   408
  then have "y = - x * - k" by simp
huffman@29408
   409
  then show "- x dvd y" ..
huffman@29408
   410
qed
huffman@29408
   411
haftmann@54230
   412
lemma dvd_diff [simp]:
haftmann@54230
   413
  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
haftmann@54230
   414
  using dvd_add [of x y "- z"] by simp
huffman@29409
   415
haftmann@25267
   416
end
haftmann@25152
   417
haftmann@60352
   418
class divide =
haftmann@60352
   419
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@60352
   420
haftmann@60352
   421
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60352
   422
haftmann@60352
   423
context semiring
haftmann@60352
   424
begin
haftmann@60352
   425
haftmann@60352
   426
lemma [field_simps]:
haftmann@60352
   427
  shows distrib_left_NO_MATCH: "NO_MATCH (divide x y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
haftmann@60352
   428
    and distrib_right_NO_MATCH: "NO_MATCH (divide x y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
haftmann@60352
   429
  by (rule distrib_left distrib_right)+
haftmann@60352
   430
haftmann@60352
   431
end
haftmann@60352
   432
haftmann@60352
   433
context ring
haftmann@60352
   434
begin
haftmann@60352
   435
haftmann@60352
   436
lemma [field_simps]:
haftmann@60352
   437
  shows left_diff_distrib_NO_MATCH: "NO_MATCH (divide x y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
haftmann@60352
   438
    and right_diff_distrib_NO_MATCH: "NO_MATCH (divide x y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
haftmann@60352
   439
  by (rule left_diff_distrib right_diff_distrib)+
haftmann@60352
   440
haftmann@60352
   441
end
haftmann@60352
   442
haftmann@60352
   443
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60352
   444
  
haftmann@59833
   445
class semiring_no_zero_divisors = semiring_0 +
haftmann@59833
   446
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
haftmann@25230
   447
begin
haftmann@25230
   448
haftmann@59833
   449
lemma divisors_zero:
haftmann@59833
   450
  assumes "a * b = 0"
haftmann@59833
   451
  shows "a = 0 \<or> b = 0"
haftmann@59833
   452
proof (rule classical)
haftmann@59833
   453
  assume "\<not> (a = 0 \<or> b = 0)"
haftmann@59833
   454
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@59833
   455
  with no_zero_divisors have "a * b \<noteq> 0" by blast
haftmann@59833
   456
  with assms show ?thesis by simp
haftmann@59833
   457
qed
haftmann@59833
   458
haftmann@25230
   459
lemma mult_eq_0_iff [simp]:
haftmann@58952
   460
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@25230
   461
proof (cases "a = 0 \<or> b = 0")
haftmann@25230
   462
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@25230
   463
    then show ?thesis using no_zero_divisors by simp
haftmann@25230
   464
next
haftmann@25230
   465
  case True then show ?thesis by auto
haftmann@25230
   466
qed
haftmann@25230
   467
haftmann@58952
   468
end
haftmann@58952
   469
haftmann@58952
   470
class ring_no_zero_divisors = ring + semiring_no_zero_divisors
haftmann@58952
   471
begin
haftmann@58952
   472
haftmann@26193
   473
text{*Cancellation of equalities with a common factor*}
blanchet@54147
   474
lemma mult_cancel_right [simp]:
haftmann@26193
   475
  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@26193
   476
proof -
haftmann@26193
   477
  have "(a * c = b * c) = ((a - b) * c = 0)"
huffman@35216
   478
    by (simp add: algebra_simps)
huffman@35216
   479
  thus ?thesis by (simp add: disj_commute)
haftmann@26193
   480
qed
haftmann@26193
   481
blanchet@54147
   482
lemma mult_cancel_left [simp]:
haftmann@26193
   483
  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@26193
   484
proof -
haftmann@26193
   485
  have "(c * a = c * b) = (c * (a - b) = 0)"
huffman@35216
   486
    by (simp add: algebra_simps)
huffman@35216
   487
  thus ?thesis by simp
haftmann@26193
   488
qed
haftmann@26193
   489
haftmann@58952
   490
lemma mult_left_cancel:
haftmann@58952
   491
  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
haftmann@58952
   492
  by simp 
lp15@56217
   493
haftmann@58952
   494
lemma mult_right_cancel:
haftmann@58952
   495
  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
haftmann@58952
   496
  by simp 
lp15@56217
   497
haftmann@25230
   498
end
huffman@22990
   499
huffman@23544
   500
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   501
begin
haftmann@26274
   502
huffman@36970
   503
lemma square_eq_1_iff:
huffman@36821
   504
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
huffman@36821
   505
proof -
huffman@36821
   506
  have "(x - 1) * (x + 1) = x * x - 1"
huffman@36821
   507
    by (simp add: algebra_simps)
huffman@36821
   508
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   509
    by simp
huffman@36821
   510
  thus ?thesis
huffman@36821
   511
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   512
qed
huffman@36821
   513
haftmann@26274
   514
lemma mult_cancel_right1 [simp]:
haftmann@26274
   515
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   516
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   517
haftmann@26274
   518
lemma mult_cancel_right2 [simp]:
haftmann@26274
   519
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   520
by (insert mult_cancel_right [of a c 1], simp)
haftmann@26274
   521
 
haftmann@26274
   522
lemma mult_cancel_left1 [simp]:
haftmann@26274
   523
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   524
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   525
haftmann@26274
   526
lemma mult_cancel_left2 [simp]:
haftmann@26274
   527
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   528
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   529
haftmann@26274
   530
end
huffman@22990
   531
haftmann@59910
   532
class semidom = comm_semiring_1_diff_distrib + semiring_no_zero_divisors
haftmann@59833
   533
haftmann@59833
   534
class idom = comm_ring_1 + semiring_no_zero_divisors
haftmann@25186
   535
begin
paulson@14421
   536
haftmann@59833
   537
subclass semidom ..
haftmann@59833
   538
huffman@27516
   539
subclass ring_1_no_zero_divisors ..
huffman@22990
   540
huffman@29981
   541
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   542
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   543
proof -
huffman@29981
   544
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   545
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   546
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   547
    unfolding dvd_def by simp
huffman@29981
   548
  finally show ?thesis .
huffman@29981
   549
qed
huffman@29981
   550
huffman@29981
   551
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   552
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   553
proof -
huffman@29981
   554
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   555
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   556
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   557
    unfolding dvd_def by simp
huffman@29981
   558
  finally show ?thesis .
huffman@29981
   559
qed
huffman@29981
   560
haftmann@59833
   561
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)"
haftmann@59833
   562
proof
haftmann@59833
   563
  assume "a * a = b * b"
haftmann@59833
   564
  then have "(a - b) * (a + b) = 0"
haftmann@59833
   565
    by (simp add: algebra_simps)
haftmann@59833
   566
  then show "a = b \<or> a = - b"
haftmann@59833
   567
    by (simp add: eq_neg_iff_add_eq_0)
haftmann@59833
   568
next
haftmann@59833
   569
  assume "a = b \<or> a = - b"
haftmann@59833
   570
  then show "a * a = b * b" by auto
haftmann@59833
   571
qed
haftmann@59833
   572
haftmann@25186
   573
end
haftmann@25152
   574
haftmann@35302
   575
text {*
haftmann@35302
   576
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   577
  \begin{itemize}
haftmann@35302
   578
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35302
   579
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   580
  \end{itemize}
haftmann@35302
   581
  Most of the used notions can also be looked up in 
haftmann@35302
   582
  \begin{itemize}
wenzelm@54703
   583
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35302
   584
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   585
  \end{itemize}
haftmann@35302
   586
*}
haftmann@35302
   587
haftmann@38642
   588
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
   589
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
   590
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   591
begin
haftmann@25230
   592
haftmann@25230
   593
lemma mult_mono:
haftmann@38642
   594
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   595
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   596
apply (erule mult_left_mono, assumption)
haftmann@25230
   597
done
haftmann@25230
   598
haftmann@25230
   599
lemma mult_mono':
haftmann@38642
   600
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   601
apply (rule mult_mono)
haftmann@25230
   602
apply (fast intro: order_trans)+
haftmann@25230
   603
done
haftmann@25230
   604
haftmann@25230
   605
end
krauss@21199
   606
haftmann@38642
   607
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
   608
begin
paulson@14268
   609
huffman@27516
   610
subclass semiring_0_cancel ..
obua@23521
   611
nipkow@56536
   612
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   613
using mult_left_mono [of 0 b a] by simp
haftmann@25230
   614
haftmann@25230
   615
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   616
using mult_left_mono [of b 0 a] by simp
huffman@30692
   617
huffman@30692
   618
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   619
using mult_right_mono [of a 0 b] by simp
huffman@30692
   620
huffman@30692
   621
text {* Legacy - use @{text mult_nonpos_nonneg} *}
haftmann@25230
   622
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
haftmann@36301
   623
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
   624
haftmann@26234
   625
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
nipkow@29667
   626
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   627
haftmann@25230
   628
end
haftmann@25230
   629
haftmann@38642
   630
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
   631
begin
haftmann@25230
   632
haftmann@35028
   633
subclass ordered_cancel_semiring ..
haftmann@35028
   634
haftmann@35028
   635
subclass ordered_comm_monoid_add ..
haftmann@25304
   636
haftmann@25230
   637
lemma mult_left_less_imp_less:
haftmann@25230
   638
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   639
by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   640
 
haftmann@25230
   641
lemma mult_right_less_imp_less:
haftmann@25230
   642
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   643
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   644
haftmann@25186
   645
end
haftmann@25152
   646
haftmann@35043
   647
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
   648
begin
hoelzl@36622
   649
hoelzl@36622
   650
lemma convex_bound_le:
hoelzl@36622
   651
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   652
  shows "u * x + v * y \<le> a"
hoelzl@36622
   653
proof-
hoelzl@36622
   654
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
   655
    by (simp add: add_mono mult_left_mono)
webertj@49962
   656
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   657
qed
hoelzl@36622
   658
hoelzl@36622
   659
end
haftmann@35043
   660
haftmann@35043
   661
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
   662
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   663
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   664
begin
paulson@14341
   665
huffman@27516
   666
subclass semiring_0_cancel ..
obua@14940
   667
haftmann@35028
   668
subclass linordered_semiring
haftmann@28823
   669
proof
huffman@23550
   670
  fix a b c :: 'a
huffman@23550
   671
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   672
  from A show "c * a \<le> c * b"
haftmann@25186
   673
    unfolding le_less
haftmann@25186
   674
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   675
  from A show "a * c \<le> b * c"
haftmann@25152
   676
    unfolding le_less
haftmann@25186
   677
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   678
qed
haftmann@25152
   679
haftmann@25230
   680
lemma mult_left_le_imp_le:
haftmann@25230
   681
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   682
by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   683
 
haftmann@25230
   684
lemma mult_right_le_imp_le:
haftmann@25230
   685
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   686
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   687
nipkow@56544
   688
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
   689
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
   690
huffman@30692
   691
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
   692
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
   693
huffman@30692
   694
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
   695
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
   696
huffman@30692
   697
text {* Legacy - use @{text mult_neg_pos} *}
huffman@30692
   698
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
haftmann@36301
   699
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
   700
haftmann@25230
   701
lemma zero_less_mult_pos:
haftmann@25230
   702
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   703
apply (cases "b\<le>0")
haftmann@25230
   704
 apply (auto simp add: le_less not_less)
huffman@30692
   705
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
   706
 apply (auto dest: less_not_sym)
haftmann@25230
   707
done
haftmann@25230
   708
haftmann@25230
   709
lemma zero_less_mult_pos2:
haftmann@25230
   710
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   711
apply (cases "b\<le>0")
haftmann@25230
   712
 apply (auto simp add: le_less not_less)
huffman@30692
   713
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
   714
 apply (auto dest: less_not_sym)
haftmann@25230
   715
done
haftmann@25230
   716
haftmann@26193
   717
text{*Strict monotonicity in both arguments*}
haftmann@26193
   718
lemma mult_strict_mono:
haftmann@26193
   719
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
   720
  shows "a * c < b * d"
haftmann@26193
   721
  using assms apply (cases "c=0")
nipkow@56544
   722
  apply (simp)
haftmann@26193
   723
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
   724
  apply (force simp add: le_less)
haftmann@26193
   725
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
   726
  done
haftmann@26193
   727
haftmann@26193
   728
text{*This weaker variant has more natural premises*}
haftmann@26193
   729
lemma mult_strict_mono':
haftmann@26193
   730
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
   731
  shows "a * c < b * d"
nipkow@29667
   732
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
   733
haftmann@26193
   734
lemma mult_less_le_imp_less:
haftmann@26193
   735
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
   736
  shows "a * c < b * d"
haftmann@26193
   737
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
   738
  apply (erule less_le_trans)
haftmann@26193
   739
  apply (erule mult_left_mono)
haftmann@26193
   740
  apply simp
haftmann@26193
   741
  apply (erule mult_strict_right_mono)
haftmann@26193
   742
  apply assumption
haftmann@26193
   743
  done
haftmann@26193
   744
haftmann@26193
   745
lemma mult_le_less_imp_less:
haftmann@26193
   746
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
   747
  shows "a * c < b * d"
haftmann@26193
   748
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
   749
  apply (erule le_less_trans)
haftmann@26193
   750
  apply (erule mult_strict_left_mono)
haftmann@26193
   751
  apply simp
haftmann@26193
   752
  apply (erule mult_right_mono)
haftmann@26193
   753
  apply simp
haftmann@26193
   754
  done
haftmann@26193
   755
haftmann@25230
   756
end
haftmann@25230
   757
haftmann@35097
   758
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
   759
begin
hoelzl@36622
   760
hoelzl@36622
   761
subclass linordered_semiring_1 ..
hoelzl@36622
   762
hoelzl@36622
   763
lemma convex_bound_lt:
hoelzl@36622
   764
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   765
  shows "u * x + v * y < a"
hoelzl@36622
   766
proof -
hoelzl@36622
   767
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
   768
    by (cases "u = 0")
hoelzl@36622
   769
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
webertj@49962
   770
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   771
qed
hoelzl@36622
   772
hoelzl@36622
   773
end
haftmann@33319
   774
haftmann@38642
   775
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add + 
haftmann@38642
   776
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
   777
begin
haftmann@25152
   778
haftmann@35028
   779
subclass ordered_semiring
haftmann@28823
   780
proof
krauss@21199
   781
  fix a b c :: 'a
huffman@23550
   782
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
   783
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
haftmann@57512
   784
  thus "a * c \<le> b * c" by (simp only: mult.commute)
krauss@21199
   785
qed
paulson@14265
   786
haftmann@25267
   787
end
haftmann@25267
   788
haftmann@38642
   789
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
   790
begin
paulson@14265
   791
haftmann@38642
   792
subclass comm_semiring_0_cancel ..
haftmann@35028
   793
subclass ordered_comm_semiring ..
haftmann@35028
   794
subclass ordered_cancel_semiring ..
haftmann@25267
   795
haftmann@25267
   796
end
haftmann@25267
   797
haftmann@35028
   798
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
   799
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
   800
begin
haftmann@25267
   801
haftmann@35043
   802
subclass linordered_semiring_strict
haftmann@28823
   803
proof
huffman@23550
   804
  fix a b c :: 'a
huffman@23550
   805
  assume "a < b" "0 < c"
haftmann@38642
   806
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
haftmann@57512
   807
  thus "a * c < b * c" by (simp only: mult.commute)
huffman@23550
   808
qed
paulson@14272
   809
haftmann@35028
   810
subclass ordered_cancel_comm_semiring
haftmann@28823
   811
proof
huffman@23550
   812
  fix a b c :: 'a
huffman@23550
   813
  assume "a \<le> b" "0 \<le> c"
huffman@23550
   814
  thus "c * a \<le> c * b"
haftmann@25186
   815
    unfolding le_less
haftmann@26193
   816
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   817
qed
paulson@14272
   818
haftmann@25267
   819
end
haftmann@25230
   820
haftmann@35028
   821
class ordered_ring = ring + ordered_cancel_semiring 
haftmann@25267
   822
begin
haftmann@25230
   823
haftmann@35028
   824
subclass ordered_ab_group_add ..
paulson@14270
   825
haftmann@25230
   826
lemma less_add_iff1:
haftmann@25230
   827
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
   828
by (simp add: algebra_simps)
haftmann@25230
   829
haftmann@25230
   830
lemma less_add_iff2:
haftmann@25230
   831
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
   832
by (simp add: algebra_simps)
haftmann@25230
   833
haftmann@25230
   834
lemma le_add_iff1:
haftmann@25230
   835
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
   836
by (simp add: algebra_simps)
haftmann@25230
   837
haftmann@25230
   838
lemma le_add_iff2:
haftmann@25230
   839
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
   840
by (simp add: algebra_simps)
haftmann@25230
   841
haftmann@25230
   842
lemma mult_left_mono_neg:
haftmann@25230
   843
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
   844
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
   845
  apply simp_all
haftmann@25230
   846
  done
haftmann@25230
   847
haftmann@25230
   848
lemma mult_right_mono_neg:
haftmann@25230
   849
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
   850
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
   851
  apply simp_all
haftmann@25230
   852
  done
haftmann@25230
   853
huffman@30692
   854
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   855
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
   856
haftmann@25230
   857
lemma split_mult_pos_le:
haftmann@25230
   858
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@56536
   859
by (auto simp add: mult_nonpos_nonpos)
haftmann@25186
   860
haftmann@25186
   861
end
paulson@14270
   862
haftmann@35028
   863
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
   864
begin
haftmann@25304
   865
haftmann@35028
   866
subclass ordered_ring ..
haftmann@35028
   867
haftmann@35028
   868
subclass ordered_ab_group_add_abs
haftmann@28823
   869
proof
haftmann@25304
   870
  fix a b
haftmann@25304
   871
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@54230
   872
    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
   873
qed (auto simp add: abs_if)
haftmann@25304
   874
huffman@35631
   875
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
   876
  using linear [of 0 a]
nipkow@56536
   877
  by (auto simp add: mult_nonpos_nonpos)
huffman@35631
   878
huffman@35631
   879
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
   880
  by (simp add: not_less)
huffman@35631
   881
haftmann@25304
   882
end
obua@23521
   883
haftmann@35043
   884
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
   885
  + ordered_ab_group_add + abs_if
haftmann@25230
   886
begin
paulson@14348
   887
haftmann@35028
   888
subclass linordered_ring ..
haftmann@25304
   889
huffman@30692
   890
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
   891
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
   892
huffman@30692
   893
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
   894
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
   895
huffman@30692
   896
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
   897
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
   898
haftmann@25917
   899
subclass ring_no_zero_divisors
haftmann@28823
   900
proof
haftmann@25917
   901
  fix a b
haftmann@25917
   902
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
   903
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
   904
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
   905
  proof (cases "a < 0")
haftmann@25917
   906
    case True note A' = this
haftmann@25917
   907
    show ?thesis proof (cases "b < 0")
haftmann@25917
   908
      case True with A'
haftmann@25917
   909
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
   910
    next
haftmann@25917
   911
      case False with B have "0 < b" by auto
haftmann@25917
   912
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
   913
    qed
haftmann@25917
   914
  next
haftmann@25917
   915
    case False with A have A': "0 < a" by auto
haftmann@25917
   916
    show ?thesis proof (cases "b < 0")
haftmann@25917
   917
      case True with A'
haftmann@25917
   918
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
   919
    next
haftmann@25917
   920
      case False with B have "0 < b" by auto
nipkow@56544
   921
      with A' show ?thesis by auto
haftmann@25917
   922
    qed
haftmann@25917
   923
  qed
haftmann@25917
   924
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
   925
qed
haftmann@25304
   926
hoelzl@56480
   927
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
hoelzl@56480
   928
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
nipkow@56544
   929
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
huffman@22990
   930
hoelzl@56480
   931
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
   932
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
   933
paulson@14265
   934
lemma mult_less_0_iff:
haftmann@25917
   935
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
   936
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
   937
  apply force
haftmann@25917
   938
  done
paulson@14265
   939
paulson@14265
   940
lemma mult_le_0_iff:
haftmann@25917
   941
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
   942
  apply (insert zero_le_mult_iff [of "-a" b]) 
huffman@35216
   943
  apply force
haftmann@25917
   944
  done
haftmann@25917
   945
haftmann@26193
   946
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
   947
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
   948
haftmann@26193
   949
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
   950
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
   951
haftmann@26193
   952
lemma mult_less_cancel_right_disj:
haftmann@26193
   953
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   954
  apply (cases "c = 0")
haftmann@26193
   955
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
   956
                      mult_strict_right_mono_neg)
haftmann@26193
   957
  apply (auto simp add: not_less 
haftmann@26193
   958
                      not_le [symmetric, of "a*c"]
haftmann@26193
   959
                      not_le [symmetric, of a])
haftmann@26193
   960
  apply (erule_tac [!] notE)
haftmann@26193
   961
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
   962
                      mult_right_mono_neg)
haftmann@26193
   963
  done
haftmann@26193
   964
haftmann@26193
   965
lemma mult_less_cancel_left_disj:
haftmann@26193
   966
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   967
  apply (cases "c = 0")
haftmann@26193
   968
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
   969
                      mult_strict_left_mono_neg)
haftmann@26193
   970
  apply (auto simp add: not_less 
haftmann@26193
   971
                      not_le [symmetric, of "c*a"]
haftmann@26193
   972
                      not_le [symmetric, of a])
haftmann@26193
   973
  apply (erule_tac [!] notE)
haftmann@26193
   974
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
   975
                      mult_left_mono_neg)
haftmann@26193
   976
  done
haftmann@26193
   977
haftmann@26193
   978
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
   979
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
   980
haftmann@26193
   981
lemma mult_less_cancel_right:
haftmann@26193
   982
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   983
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
   984
haftmann@26193
   985
lemma mult_less_cancel_left:
haftmann@26193
   986
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   987
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
   988
haftmann@26193
   989
lemma mult_le_cancel_right:
haftmann@26193
   990
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
   991
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
   992
haftmann@26193
   993
lemma mult_le_cancel_left:
haftmann@26193
   994
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
   995
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
   996
nipkow@30649
   997
lemma mult_le_cancel_left_pos:
nipkow@30649
   998
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
   999
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1000
nipkow@30649
  1001
lemma mult_le_cancel_left_neg:
nipkow@30649
  1002
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
  1003
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1004
nipkow@30649
  1005
lemma mult_less_cancel_left_pos:
nipkow@30649
  1006
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
  1007
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1008
nipkow@30649
  1009
lemma mult_less_cancel_left_neg:
nipkow@30649
  1010
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
  1011
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1012
haftmann@25917
  1013
end
paulson@14265
  1014
huffman@30692
  1015
lemmas mult_sign_intros =
huffman@30692
  1016
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
  1017
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
  1018
  mult_pos_pos mult_pos_neg
huffman@30692
  1019
  mult_neg_pos mult_neg_neg
haftmann@25230
  1020
haftmann@35028
  1021
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
  1022
begin
haftmann@25230
  1023
haftmann@35028
  1024
subclass ordered_ring ..
haftmann@35028
  1025
subclass ordered_cancel_comm_semiring ..
haftmann@25230
  1026
haftmann@25267
  1027
end
haftmann@25230
  1028
haftmann@59833
  1029
class linordered_semidom = semidom + linordered_comm_semiring_strict +
haftmann@25230
  1030
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
  1031
begin
haftmann@25230
  1032
haftmann@25230
  1033
lemma pos_add_strict:
haftmann@25230
  1034
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
  1035
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
  1036
haftmann@26193
  1037
lemma zero_le_one [simp]: "0 \<le> 1"
nipkow@29667
  1038
by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
  1039
haftmann@26193
  1040
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
nipkow@29667
  1041
by (simp add: not_le) 
haftmann@26193
  1042
haftmann@26193
  1043
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
nipkow@29667
  1044
by (simp add: not_less) 
haftmann@26193
  1045
haftmann@26193
  1046
lemma less_1_mult:
haftmann@26193
  1047
  assumes "1 < m" and "1 < n"
haftmann@26193
  1048
  shows "1 < m * n"
haftmann@26193
  1049
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
  1050
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
  1051
hoelzl@59000
  1052
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
hoelzl@59000
  1053
  using mult_left_mono[of c 1 a] by simp
hoelzl@59000
  1054
hoelzl@59000
  1055
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
hoelzl@59000
  1056
  using mult_mono[of a 1 b 1] by simp
hoelzl@59000
  1057
haftmann@25230
  1058
end
haftmann@25230
  1059
haftmann@35028
  1060
class linordered_idom = comm_ring_1 +
haftmann@35028
  1061
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1062
  abs_if + sgn_if
haftmann@25917
  1063
begin
haftmann@25917
  1064
hoelzl@36622
  1065
subclass linordered_semiring_1_strict ..
haftmann@35043
  1066
subclass linordered_ring_strict ..
haftmann@35028
  1067
subclass ordered_comm_ring ..
huffman@27516
  1068
subclass idom ..
haftmann@25917
  1069
haftmann@35028
  1070
subclass linordered_semidom
haftmann@28823
  1071
proof
haftmann@26193
  1072
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1073
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
  1074
qed 
haftmann@25917
  1075
haftmann@35028
  1076
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1077
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1078
  using assms by (rule neqE)
haftmann@26193
  1079
haftmann@26274
  1080
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1081
haftmann@26274
  1082
lemma mult_le_cancel_right1:
haftmann@26274
  1083
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1084
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1085
haftmann@26274
  1086
lemma mult_le_cancel_right2:
haftmann@26274
  1087
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1088
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1089
haftmann@26274
  1090
lemma mult_le_cancel_left1:
haftmann@26274
  1091
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1092
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1093
haftmann@26274
  1094
lemma mult_le_cancel_left2:
haftmann@26274
  1095
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1096
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1097
haftmann@26274
  1098
lemma mult_less_cancel_right1:
haftmann@26274
  1099
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1100
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1101
haftmann@26274
  1102
lemma mult_less_cancel_right2:
haftmann@26274
  1103
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1104
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1105
haftmann@26274
  1106
lemma mult_less_cancel_left1:
haftmann@26274
  1107
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1108
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1109
haftmann@26274
  1110
lemma mult_less_cancel_left2:
haftmann@26274
  1111
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1112
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1113
haftmann@27651
  1114
lemma sgn_sgn [simp]:
haftmann@27651
  1115
  "sgn (sgn a) = sgn a"
nipkow@29700
  1116
unfolding sgn_if by simp
haftmann@27651
  1117
haftmann@27651
  1118
lemma sgn_0_0:
haftmann@27651
  1119
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1120
unfolding sgn_if by simp
haftmann@27651
  1121
haftmann@27651
  1122
lemma sgn_1_pos:
haftmann@27651
  1123
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1124
unfolding sgn_if by simp
haftmann@27651
  1125
haftmann@27651
  1126
lemma sgn_1_neg:
haftmann@27651
  1127
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1128
unfolding sgn_if by auto
haftmann@27651
  1129
haftmann@29940
  1130
lemma sgn_pos [simp]:
haftmann@29940
  1131
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1132
unfolding sgn_1_pos .
haftmann@29940
  1133
haftmann@29940
  1134
lemma sgn_neg [simp]:
haftmann@29940
  1135
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1136
unfolding sgn_1_neg .
haftmann@29940
  1137
haftmann@27651
  1138
lemma sgn_times:
haftmann@27651
  1139
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1140
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1141
haftmann@36301
  1142
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1143
unfolding sgn_if abs_if by auto
nipkow@29700
  1144
haftmann@29940
  1145
lemma sgn_greater [simp]:
haftmann@29940
  1146
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1147
  unfolding sgn_if by auto
haftmann@29940
  1148
haftmann@29940
  1149
lemma sgn_less [simp]:
haftmann@29940
  1150
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1151
  unfolding sgn_if by auto
haftmann@29940
  1152
haftmann@36301
  1153
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1154
  by (simp add: abs_if)
huffman@29949
  1155
haftmann@36301
  1156
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1157
  by (simp add: abs_if)
haftmann@29653
  1158
nipkow@33676
  1159
lemma dvd_if_abs_eq:
haftmann@36301
  1160
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1161
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1162
huffman@55912
  1163
text {* The following lemmas can be proven in more general structures, but
haftmann@54489
  1164
are dangerous as simp rules in absence of @{thm neg_equal_zero}, 
haftmann@54489
  1165
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
haftmann@54489
  1166
haftmann@54489
  1167
lemma equation_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1168
  "1 = - a \<longleftrightarrow> a = - 1"
haftmann@54489
  1169
  by (fact equation_minus_iff)
haftmann@54489
  1170
haftmann@54489
  1171
lemma minus_equation_iff_1 [simp, no_atp]:
haftmann@54489
  1172
  "- a = 1 \<longleftrightarrow> a = - 1"
haftmann@54489
  1173
  by (subst minus_equation_iff, auto)
haftmann@54489
  1174
haftmann@54489
  1175
lemma le_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1176
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
haftmann@54489
  1177
  by (fact le_minus_iff)
haftmann@54489
  1178
haftmann@54489
  1179
lemma minus_le_iff_1 [simp, no_atp]:
haftmann@54489
  1180
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
haftmann@54489
  1181
  by (fact minus_le_iff)
haftmann@54489
  1182
haftmann@54489
  1183
lemma less_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1184
  "1 < - b \<longleftrightarrow> b < - 1"
haftmann@54489
  1185
  by (fact less_minus_iff)
haftmann@54489
  1186
haftmann@54489
  1187
lemma minus_less_iff_1 [simp, no_atp]:
haftmann@54489
  1188
  "- a < 1 \<longleftrightarrow> - 1 < a"
haftmann@54489
  1189
  by (fact minus_less_iff)
haftmann@54489
  1190
haftmann@25917
  1191
end
haftmann@25230
  1192
haftmann@26274
  1193
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1194
blanchet@54147
  1195
lemmas mult_compare_simps =
paulson@15234
  1196
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1197
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1198
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1199
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1200
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1201
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1202
    mult_cancel_right mult_cancel_left
paulson@15234
  1203
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1204
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1205
haftmann@36301
  1206
text {* Reasoning about inequalities with division *}
avigad@16775
  1207
haftmann@35028
  1208
context linordered_semidom
haftmann@25193
  1209
begin
haftmann@25193
  1210
haftmann@25193
  1211
lemma less_add_one: "a < a + 1"
paulson@14293
  1212
proof -
haftmann@25193
  1213
  have "a + 0 < a + 1"
nipkow@23482
  1214
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1215
  thus ?thesis by simp
paulson@14293
  1216
qed
paulson@14293
  1217
haftmann@25193
  1218
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1219
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1220
haftmann@25193
  1221
end
paulson@14365
  1222
haftmann@36301
  1223
context linordered_idom
haftmann@36301
  1224
begin
paulson@15234
  1225
haftmann@36301
  1226
lemma mult_right_le_one_le:
haftmann@36301
  1227
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@59833
  1228
  by (rule mult_left_le)
haftmann@36301
  1229
haftmann@36301
  1230
lemma mult_left_le_one_le:
haftmann@36301
  1231
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1232
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1233
haftmann@36301
  1234
end
haftmann@36301
  1235
haftmann@36301
  1236
text {* Absolute Value *}
paulson@14293
  1237
haftmann@35028
  1238
context linordered_idom
haftmann@25304
  1239
begin
haftmann@25304
  1240
haftmann@36301
  1241
lemma mult_sgn_abs:
haftmann@36301
  1242
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1243
  unfolding abs_if sgn_if by auto
haftmann@25304
  1244
haftmann@36301
  1245
lemma abs_one [simp]:
haftmann@36301
  1246
  "\<bar>1\<bar> = 1"
huffman@44921
  1247
  by (simp add: abs_if)
haftmann@36301
  1248
haftmann@25304
  1249
end
nipkow@24491
  1250
haftmann@35028
  1251
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1252
  assumes abs_eq_mult:
haftmann@25304
  1253
    "(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
  1254
haftmann@35028
  1255
context linordered_idom
haftmann@30961
  1256
begin
haftmann@30961
  1257
haftmann@35028
  1258
subclass ordered_ring_abs proof
huffman@35216
  1259
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1260
haftmann@30961
  1261
lemma abs_mult:
haftmann@36301
  1262
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" 
haftmann@30961
  1263
  by (rule abs_eq_mult) auto
haftmann@30961
  1264
haftmann@30961
  1265
lemma abs_mult_self:
haftmann@36301
  1266
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
haftmann@30961
  1267
  by (simp add: abs_if) 
haftmann@30961
  1268
paulson@14294
  1269
lemma abs_mult_less:
haftmann@36301
  1270
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1271
proof -
haftmann@36301
  1272
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1273
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1274
  assume "\<bar>b\<bar> < d"
paulson@14294
  1275
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1276
qed
paulson@14293
  1277
haftmann@36301
  1278
lemma abs_less_iff:
haftmann@36301
  1279
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" 
haftmann@36301
  1280
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1281
haftmann@36301
  1282
lemma abs_mult_pos:
haftmann@36301
  1283
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1284
  by (simp add: abs_mult)
haftmann@36301
  1285
hoelzl@51520
  1286
lemma abs_diff_less_iff:
hoelzl@51520
  1287
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
hoelzl@51520
  1288
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
hoelzl@51520
  1289
lp15@59865
  1290
lemma abs_diff_le_iff:
lp15@59865
  1291
   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
lp15@59865
  1292
  by (auto simp add: diff_le_eq ac_simps abs_le_iff)
lp15@59865
  1293
haftmann@36301
  1294
end
avigad@16775
  1295
haftmann@59557
  1296
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
haftmann@59557
  1297
haftmann@52435
  1298
code_identifier
haftmann@52435
  1299
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1300
paulson@14265
  1301
end