src/HOL/Rings.thy
author paulson <lp15@cam.ac.uk>
Tue Jun 23 16:55:28 2015 +0100 (2015-06-23)
changeset 60562 24af00b010cf
parent 60529 24c2ef12318b
child 60570 7ed2cde6806d
permissions -rw-r--r--
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
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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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  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
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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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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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lemmas ring_distribs =
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  distrib_left distrib_right left_diff_distrib right_diff_distrib
haftmann@25152
   350
haftmann@22390
   351
class comm_ring = comm_semiring + ab_group_add
haftmann@25267
   352
begin
obua@14738
   353
huffman@27516
   354
subclass ring ..
huffman@28141
   355
subclass comm_semiring_0_cancel ..
haftmann@25267
   356
huffman@44350
   357
lemma square_diff_square_factored:
huffman@44350
   358
  "x * x - y * y = (x + y) * (x - y)"
huffman@44350
   359
  by (simp add: algebra_simps)
huffman@44350
   360
haftmann@25267
   361
end
obua@14738
   362
haftmann@22390
   363
class ring_1 = ring + zero_neq_one + monoid_mult
haftmann@25267
   364
begin
paulson@14265
   365
huffman@27516
   366
subclass semiring_1_cancel ..
haftmann@25267
   367
huffman@44346
   368
lemma square_diff_one_factored:
huffman@44346
   369
  "x * x - 1 = (x + 1) * (x - 1)"
huffman@44346
   370
  by (simp add: algebra_simps)
huffman@44346
   371
haftmann@25267
   372
end
haftmann@25152
   373
haftmann@22390
   374
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
haftmann@25267
   375
begin
obua@14738
   376
huffman@27516
   377
subclass ring_1 ..
lp15@60562
   378
subclass comm_semiring_1_cancel
haftmann@59816
   379
  by unfold_locales (simp add: algebra_simps)
haftmann@58647
   380
huffman@29465
   381
lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
huffman@29408
   382
proof
huffman@29408
   383
  assume "x dvd - y"
huffman@29408
   384
  then have "x dvd - 1 * - y" by (rule dvd_mult)
huffman@29408
   385
  then show "x dvd y" by simp
huffman@29408
   386
next
huffman@29408
   387
  assume "x dvd y"
huffman@29408
   388
  then have "x dvd - 1 * y" by (rule dvd_mult)
huffman@29408
   389
  then show "x dvd - y" by simp
huffman@29408
   390
qed
huffman@29408
   391
huffman@29465
   392
lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
huffman@29408
   393
proof
huffman@29408
   394
  assume "- x dvd y"
huffman@29408
   395
  then obtain k where "y = - x * k" ..
huffman@29408
   396
  then have "y = x * - k" by simp
huffman@29408
   397
  then show "x dvd y" ..
huffman@29408
   398
next
huffman@29408
   399
  assume "x dvd y"
huffman@29408
   400
  then obtain k where "y = x * k" ..
huffman@29408
   401
  then have "y = - x * - k" by simp
huffman@29408
   402
  then show "- x dvd y" ..
huffman@29408
   403
qed
huffman@29408
   404
haftmann@54230
   405
lemma dvd_diff [simp]:
haftmann@54230
   406
  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
haftmann@54230
   407
  using dvd_add [of x y "- z"] by simp
huffman@29409
   408
haftmann@25267
   409
end
haftmann@25152
   410
haftmann@59833
   411
class semiring_no_zero_divisors = semiring_0 +
haftmann@59833
   412
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
haftmann@25230
   413
begin
haftmann@25230
   414
haftmann@59833
   415
lemma divisors_zero:
haftmann@59833
   416
  assumes "a * b = 0"
haftmann@59833
   417
  shows "a = 0 \<or> b = 0"
haftmann@59833
   418
proof (rule classical)
haftmann@59833
   419
  assume "\<not> (a = 0 \<or> b = 0)"
haftmann@59833
   420
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@59833
   421
  with no_zero_divisors have "a * b \<noteq> 0" by blast
haftmann@59833
   422
  with assms show ?thesis by simp
haftmann@59833
   423
qed
haftmann@59833
   424
haftmann@25230
   425
lemma mult_eq_0_iff [simp]:
haftmann@58952
   426
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@25230
   427
proof (cases "a = 0 \<or> b = 0")
haftmann@25230
   428
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@25230
   429
    then show ?thesis using no_zero_divisors by simp
haftmann@25230
   430
next
haftmann@25230
   431
  case True then show ?thesis by auto
haftmann@25230
   432
qed
haftmann@25230
   433
haftmann@58952
   434
end
haftmann@58952
   435
haftmann@60516
   436
class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +
haftmann@60516
   437
  assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   438
    and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@58952
   439
begin
haftmann@58952
   440
haftmann@58952
   441
lemma mult_left_cancel:
haftmann@58952
   442
  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
lp15@60562
   443
  by simp
lp15@56217
   444
haftmann@58952
   445
lemma mult_right_cancel:
haftmann@58952
   446
  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
lp15@60562
   447
  by simp
lp15@56217
   448
haftmann@25230
   449
end
huffman@22990
   450
haftmann@60516
   451
class ring_no_zero_divisors = ring + semiring_no_zero_divisors
haftmann@60516
   452
begin
haftmann@60516
   453
haftmann@60516
   454
subclass semiring_no_zero_divisors_cancel
haftmann@60516
   455
proof
haftmann@60516
   456
  fix a b c
haftmann@60516
   457
  have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"
haftmann@60516
   458
    by (simp add: algebra_simps)
haftmann@60516
   459
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   460
    by auto
haftmann@60516
   461
  finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .
haftmann@60516
   462
  have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"
haftmann@60516
   463
    by (simp add: algebra_simps)
haftmann@60516
   464
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   465
    by auto
haftmann@60516
   466
  finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .
haftmann@60516
   467
qed
haftmann@60516
   468
haftmann@60516
   469
end
haftmann@60516
   470
huffman@23544
   471
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   472
begin
haftmann@26274
   473
huffman@36970
   474
lemma square_eq_1_iff:
huffman@36821
   475
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
huffman@36821
   476
proof -
huffman@36821
   477
  have "(x - 1) * (x + 1) = x * x - 1"
huffman@36821
   478
    by (simp add: algebra_simps)
huffman@36821
   479
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   480
    by simp
huffman@36821
   481
  thus ?thesis
huffman@36821
   482
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   483
qed
huffman@36821
   484
haftmann@26274
   485
lemma mult_cancel_right1 [simp]:
haftmann@26274
   486
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   487
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   488
haftmann@26274
   489
lemma mult_cancel_right2 [simp]:
haftmann@26274
   490
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   491
by (insert mult_cancel_right [of a c 1], simp)
lp15@60562
   492
haftmann@26274
   493
lemma mult_cancel_left1 [simp]:
haftmann@26274
   494
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   495
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   496
haftmann@26274
   497
lemma mult_cancel_left2 [simp]:
haftmann@26274
   498
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   499
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   500
haftmann@26274
   501
end
huffman@22990
   502
lp15@60562
   503
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors
haftmann@59833
   504
haftmann@59833
   505
class idom = comm_ring_1 + semiring_no_zero_divisors
haftmann@25186
   506
begin
paulson@14421
   507
haftmann@59833
   508
subclass semidom ..
haftmann@59833
   509
huffman@27516
   510
subclass ring_1_no_zero_divisors ..
huffman@22990
   511
huffman@29981
   512
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   513
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   514
proof -
huffman@29981
   515
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   516
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   517
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   518
    unfolding dvd_def by simp
huffman@29981
   519
  finally show ?thesis .
huffman@29981
   520
qed
huffman@29981
   521
huffman@29981
   522
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   523
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   524
proof -
huffman@29981
   525
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   526
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   527
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   528
    unfolding dvd_def by simp
huffman@29981
   529
  finally show ?thesis .
huffman@29981
   530
qed
huffman@29981
   531
haftmann@60516
   532
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"
haftmann@59833
   533
proof
haftmann@59833
   534
  assume "a * a = b * b"
haftmann@59833
   535
  then have "(a - b) * (a + b) = 0"
haftmann@59833
   536
    by (simp add: algebra_simps)
haftmann@59833
   537
  then show "a = b \<or> a = - b"
haftmann@59833
   538
    by (simp add: eq_neg_iff_add_eq_0)
haftmann@59833
   539
next
haftmann@59833
   540
  assume "a = b \<or> a = - b"
haftmann@59833
   541
  then show "a * a = b * b" by auto
haftmann@59833
   542
qed
haftmann@59833
   543
haftmann@25186
   544
end
haftmann@25152
   545
haftmann@35302
   546
text {*
haftmann@35302
   547
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   548
  \begin{itemize}
lp15@60562
   549
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
haftmann@35302
   550
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   551
  \end{itemize}
lp15@60562
   552
  Most of the used notions can also be looked up in
haftmann@35302
   553
  \begin{itemize}
wenzelm@54703
   554
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35302
   555
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   556
  \end{itemize}
haftmann@35302
   557
*}
haftmann@35302
   558
haftmann@60353
   559
class divide =
haftmann@60429
   560
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
haftmann@60353
   561
haftmann@60353
   562
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60353
   563
haftmann@60353
   564
context semiring
haftmann@60353
   565
begin
haftmann@60353
   566
haftmann@60353
   567
lemma [field_simps]:
haftmann@60429
   568
  shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
haftmann@60429
   569
    and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
haftmann@60353
   570
  by (rule distrib_left distrib_right)+
haftmann@60353
   571
haftmann@60353
   572
end
haftmann@60353
   573
haftmann@60353
   574
context ring
haftmann@60353
   575
begin
haftmann@60353
   576
haftmann@60353
   577
lemma [field_simps]:
haftmann@60429
   578
  shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
haftmann@60429
   579
    and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
haftmann@60353
   580
  by (rule left_diff_distrib right_diff_distrib)+
haftmann@60353
   581
haftmann@60353
   582
end
haftmann@60353
   583
haftmann@60353
   584
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60353
   585
haftmann@60353
   586
class semidom_divide = semidom + divide +
haftmann@60429
   587
  assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
haftmann@60429
   588
  assumes divide_zero [simp]: "a div 0 = 0"
haftmann@60353
   589
begin
haftmann@60353
   590
haftmann@60353
   591
lemma nonzero_mult_divide_cancel_left [simp]:
haftmann@60429
   592
  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
haftmann@60353
   593
  using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
haftmann@60353
   594
haftmann@60516
   595
subclass semiring_no_zero_divisors_cancel
haftmann@60516
   596
proof
haftmann@60516
   597
  fix a b c
haftmann@60516
   598
  { fix a b c
haftmann@60516
   599
    show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   600
    proof (cases "c = 0")
haftmann@60516
   601
      case True then show ?thesis by simp
haftmann@60516
   602
    next
haftmann@60516
   603
      case False
haftmann@60516
   604
      { assume "a * c = b * c"
haftmann@60516
   605
        then have "a * c div c = b * c div c"
haftmann@60516
   606
          by simp
haftmann@60516
   607
        with False have "a = b"
haftmann@60516
   608
          by simp
haftmann@60516
   609
      } then show ?thesis by auto
haftmann@60516
   610
    qed
haftmann@60516
   611
  }
haftmann@60516
   612
  from this [of a c b]
haftmann@60516
   613
  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   614
    by (simp add: ac_simps)
haftmann@60516
   615
qed
haftmann@60516
   616
haftmann@60516
   617
lemma div_self [simp]:
haftmann@60516
   618
  assumes "a \<noteq> 0"
haftmann@60516
   619
  shows "a div a = 1"
haftmann@60516
   620
  using assms nonzero_mult_divide_cancel_left [of a 1] by simp
haftmann@60516
   621
haftmann@60353
   622
end
haftmann@60353
   623
haftmann@60353
   624
class idom_divide = idom + semidom_divide
haftmann@60353
   625
haftmann@60517
   626
class algebraic_semidom = semidom_divide
haftmann@60517
   627
begin
haftmann@60517
   628
haftmann@60517
   629
lemma dvd_div_mult_self [simp]:
haftmann@60517
   630
  "a dvd b \<Longrightarrow> b div a * a = b"
haftmann@60517
   631
  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
haftmann@60517
   632
haftmann@60517
   633
lemma dvd_mult_div_cancel [simp]:
haftmann@60517
   634
  "a dvd b \<Longrightarrow> a * (b div a) = b"
haftmann@60517
   635
  using dvd_div_mult_self [of a b] by (simp add: ac_simps)
lp15@60562
   636
haftmann@60517
   637
lemma div_mult_swap:
haftmann@60517
   638
  assumes "c dvd b"
haftmann@60517
   639
  shows "a * (b div c) = (a * b) div c"
haftmann@60517
   640
proof (cases "c = 0")
haftmann@60517
   641
  case True then show ?thesis by simp
haftmann@60517
   642
next
haftmann@60517
   643
  case False from assms obtain d where "b = c * d" ..
haftmann@60517
   644
  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
haftmann@60517
   645
    by simp
haftmann@60517
   646
  ultimately show ?thesis by (simp add: ac_simps)
haftmann@60517
   647
qed
haftmann@60517
   648
haftmann@60517
   649
lemma dvd_div_mult:
haftmann@60517
   650
  assumes "c dvd b"
haftmann@60517
   651
  shows "b div c * a = (b * a) div c"
haftmann@60517
   652
  using assms div_mult_swap [of c b a] by (simp add: ac_simps)
haftmann@60517
   653
lp15@60562
   654
haftmann@60517
   655
text \<open>Units: invertible elements in a ring\<close>
haftmann@60517
   656
haftmann@60517
   657
abbreviation is_unit :: "'a \<Rightarrow> bool"
haftmann@60517
   658
where
haftmann@60517
   659
  "is_unit a \<equiv> a dvd 1"
haftmann@60517
   660
haftmann@60517
   661
lemma not_is_unit_0 [simp]:
haftmann@60517
   662
  "\<not> is_unit 0"
haftmann@60517
   663
  by simp
haftmann@60517
   664
lp15@60562
   665
lemma unit_imp_dvd [dest]:
haftmann@60517
   666
  "is_unit b \<Longrightarrow> b dvd a"
haftmann@60517
   667
  by (rule dvd_trans [of _ 1]) simp_all
haftmann@60517
   668
haftmann@60517
   669
lemma unit_dvdE:
haftmann@60517
   670
  assumes "is_unit a"
haftmann@60517
   671
  obtains c where "a \<noteq> 0" and "b = a * c"
haftmann@60517
   672
proof -
haftmann@60517
   673
  from assms have "a dvd b" by auto
haftmann@60517
   674
  then obtain c where "b = a * c" ..
haftmann@60517
   675
  moreover from assms have "a \<noteq> 0" by auto
haftmann@60517
   676
  ultimately show thesis using that by blast
haftmann@60517
   677
qed
haftmann@60517
   678
haftmann@60517
   679
lemma dvd_unit_imp_unit:
haftmann@60517
   680
  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
haftmann@60517
   681
  by (rule dvd_trans)
haftmann@60517
   682
haftmann@60517
   683
lemma unit_div_1_unit [simp, intro]:
haftmann@60517
   684
  assumes "is_unit a"
haftmann@60517
   685
  shows "is_unit (1 div a)"
haftmann@60517
   686
proof -
haftmann@60517
   687
  from assms have "1 = 1 div a * a" by simp
haftmann@60517
   688
  then show "is_unit (1 div a)" by (rule dvdI)
haftmann@60517
   689
qed
haftmann@60517
   690
haftmann@60517
   691
lemma is_unitE [elim?]:
haftmann@60517
   692
  assumes "is_unit a"
haftmann@60517
   693
  obtains b where "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60517
   694
    and "is_unit b" and "1 div a = b" and "1 div b = a"
haftmann@60517
   695
    and "a * b = 1" and "c div a = c * b"
haftmann@60517
   696
proof (rule that)
haftmann@60517
   697
  def b \<equiv> "1 div a"
haftmann@60517
   698
  then show "1 div a = b" by simp
haftmann@60517
   699
  from b_def `is_unit a` show "is_unit b" by simp
haftmann@60517
   700
  from `is_unit a` and `is_unit b` show "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@60517
   701
  from b_def `is_unit a` show "a * b = 1" by simp
haftmann@60517
   702
  then have "1 = a * b" ..
haftmann@60517
   703
  with b_def `b \<noteq> 0` show "1 div b = a" by simp
haftmann@60517
   704
  from `is_unit a` have "a dvd c" ..
haftmann@60517
   705
  then obtain d where "c = a * d" ..
haftmann@60517
   706
  with `a \<noteq> 0` `a * b = 1` show "c div a = c * b"
haftmann@60517
   707
    by (simp add: mult.assoc mult.left_commute [of a])
haftmann@60517
   708
qed
haftmann@60517
   709
haftmann@60517
   710
lemma unit_prod [intro]:
haftmann@60517
   711
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
lp15@60562
   712
  by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
lp15@60562
   713
haftmann@60517
   714
lemma unit_div [intro]:
haftmann@60517
   715
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
haftmann@60517
   716
  by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
haftmann@60517
   717
haftmann@60517
   718
lemma mult_unit_dvd_iff:
haftmann@60517
   719
  assumes "is_unit b"
haftmann@60517
   720
  shows "a * b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   721
proof
haftmann@60517
   722
  assume "a * b dvd c"
haftmann@60517
   723
  with assms show "a dvd c"
haftmann@60517
   724
    by (simp add: dvd_mult_left)
haftmann@60517
   725
next
haftmann@60517
   726
  assume "a dvd c"
haftmann@60517
   727
  then obtain k where "c = a * k" ..
haftmann@60517
   728
  with assms have "c = (a * b) * (1 div b * k)"
haftmann@60517
   729
    by (simp add: mult_ac)
haftmann@60517
   730
  then show "a * b dvd c" by (rule dvdI)
haftmann@60517
   731
qed
haftmann@60517
   732
haftmann@60517
   733
lemma dvd_mult_unit_iff:
haftmann@60517
   734
  assumes "is_unit b"
haftmann@60517
   735
  shows "a dvd c * b \<longleftrightarrow> a dvd c"
haftmann@60517
   736
proof
haftmann@60517
   737
  assume "a dvd c * b"
haftmann@60517
   738
  with assms have "c * b dvd c * (b * (1 div b))"
haftmann@60517
   739
    by (subst mult_assoc [symmetric]) simp
haftmann@60517
   740
  also from `is_unit b` have "b * (1 div b) = 1" by (rule is_unitE) simp
haftmann@60517
   741
  finally have "c * b dvd c" by simp
haftmann@60517
   742
  with `a dvd c * b` show "a dvd c" by (rule dvd_trans)
haftmann@60517
   743
next
haftmann@60517
   744
  assume "a dvd c"
haftmann@60517
   745
  then show "a dvd c * b" by simp
haftmann@60517
   746
qed
haftmann@60517
   747
haftmann@60517
   748
lemma div_unit_dvd_iff:
haftmann@60517
   749
  "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   750
  by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
haftmann@60517
   751
haftmann@60517
   752
lemma dvd_div_unit_iff:
haftmann@60517
   753
  "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
haftmann@60517
   754
  by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
haftmann@60517
   755
haftmann@60517
   756
lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
haftmann@60517
   757
  dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
haftmann@60517
   758
haftmann@60517
   759
lemma unit_mult_div_div [simp]:
haftmann@60517
   760
  "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
haftmann@60517
   761
  by (erule is_unitE [of _ b]) simp
haftmann@60517
   762
haftmann@60517
   763
lemma unit_div_mult_self [simp]:
haftmann@60517
   764
  "is_unit a \<Longrightarrow> b div a * a = b"
haftmann@60517
   765
  by (rule dvd_div_mult_self) auto
haftmann@60517
   766
haftmann@60517
   767
lemma unit_div_1_div_1 [simp]:
haftmann@60517
   768
  "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
haftmann@60517
   769
  by (erule is_unitE) simp
haftmann@60517
   770
haftmann@60517
   771
lemma unit_div_mult_swap:
haftmann@60517
   772
  "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
haftmann@60517
   773
  by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
haftmann@60517
   774
haftmann@60517
   775
lemma unit_div_commute:
haftmann@60517
   776
  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
haftmann@60517
   777
  using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
haftmann@60517
   778
haftmann@60517
   779
lemma unit_eq_div1:
haftmann@60517
   780
  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
haftmann@60517
   781
  by (auto elim: is_unitE)
haftmann@60517
   782
haftmann@60517
   783
lemma unit_eq_div2:
haftmann@60517
   784
  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
haftmann@60517
   785
  using unit_eq_div1 [of b c a] by auto
haftmann@60517
   786
haftmann@60517
   787
lemma unit_mult_left_cancel:
haftmann@60517
   788
  assumes "is_unit a"
haftmann@60517
   789
  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
lp15@60562
   790
  using assms mult_cancel_left [of a b c] by auto
haftmann@60517
   791
haftmann@60517
   792
lemma unit_mult_right_cancel:
haftmann@60517
   793
  "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
haftmann@60517
   794
  using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
haftmann@60517
   795
haftmann@60517
   796
lemma unit_div_cancel:
haftmann@60517
   797
  assumes "is_unit a"
haftmann@60517
   798
  shows "b div a = c div a \<longleftrightarrow> b = c"
haftmann@60517
   799
proof -
haftmann@60517
   800
  from assms have "is_unit (1 div a)" by simp
haftmann@60517
   801
  then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
haftmann@60517
   802
    by (rule unit_mult_right_cancel)
haftmann@60517
   803
  with assms show ?thesis by simp
haftmann@60517
   804
qed
lp15@60562
   805
haftmann@60517
   806
wenzelm@60529
   807
text \<open>Associated elements in a ring --- an equivalence relation induced
wenzelm@60529
   808
  by the quasi-order divisibility.\<close>
haftmann@60517
   809
lp15@60562
   810
definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
haftmann@60517
   811
where
haftmann@60517
   812
  "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"
haftmann@60517
   813
haftmann@60517
   814
lemma associatedI:
haftmann@60517
   815
  "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated a b"
haftmann@60517
   816
  by (simp add: associated_def)
haftmann@60517
   817
haftmann@60517
   818
lemma associatedD1:
haftmann@60517
   819
  "associated a b \<Longrightarrow> a dvd b"
haftmann@60517
   820
  by (simp add: associated_def)
haftmann@60517
   821
haftmann@60517
   822
lemma associatedD2:
haftmann@60517
   823
  "associated a b \<Longrightarrow> b dvd a"
haftmann@60517
   824
  by (simp add: associated_def)
haftmann@60517
   825
haftmann@60517
   826
lemma associated_refl [simp]:
haftmann@60517
   827
  "associated a a"
haftmann@60517
   828
  by (auto intro: associatedI)
haftmann@60517
   829
haftmann@60517
   830
lemma associated_sym:
haftmann@60517
   831
  "associated b a \<longleftrightarrow> associated a b"
haftmann@60517
   832
  by (auto intro: associatedI dest: associatedD1 associatedD2)
haftmann@60517
   833
haftmann@60517
   834
lemma associated_trans:
haftmann@60517
   835
  "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated a c"
haftmann@60517
   836
  by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)
haftmann@60517
   837
haftmann@60517
   838
lemma associated_0 [simp]:
haftmann@60517
   839
  "associated 0 b \<longleftrightarrow> b = 0"
haftmann@60517
   840
  "associated a 0 \<longleftrightarrow> a = 0"
haftmann@60517
   841
  by (auto dest: associatedD1 associatedD2)
haftmann@60517
   842
haftmann@60517
   843
lemma associated_unit:
haftmann@60517
   844
  "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
haftmann@60517
   845
  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
haftmann@60517
   846
haftmann@60517
   847
lemma is_unit_associatedI:
haftmann@60517
   848
  assumes "is_unit c" and "a = c * b"
haftmann@60517
   849
  shows "associated a b"
haftmann@60517
   850
proof (rule associatedI)
haftmann@60517
   851
  from `a = c * b` show "b dvd a" by auto
haftmann@60517
   852
  from `is_unit c` obtain d where "c * d = 1" by (rule is_unitE)
haftmann@60517
   853
  moreover from `a = c * b` have "d * a = d * (c * b)" by simp
haftmann@60517
   854
  ultimately have "b = a * d" by (simp add: ac_simps)
haftmann@60517
   855
  then show "a dvd b" ..
haftmann@60517
   856
qed
haftmann@60517
   857
haftmann@60517
   858
lemma associated_is_unitE:
haftmann@60517
   859
  assumes "associated a b"
haftmann@60517
   860
  obtains c where "is_unit c" and "a = c * b"
haftmann@60517
   861
proof (cases "b = 0")
haftmann@60517
   862
  case True with assms have "is_unit 1" and "a = 1 * b" by simp_all
haftmann@60517
   863
  with that show thesis .
haftmann@60517
   864
next
haftmann@60517
   865
  case False
haftmann@60517
   866
  from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
haftmann@60517
   867
  then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
haftmann@60517
   868
  then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)
haftmann@60517
   869
  with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp
haftmann@60517
   870
  then have "is_unit c" by auto
haftmann@60517
   871
  with `a = c * b` that show thesis by blast
haftmann@60517
   872
qed
lp15@60562
   873
lp15@60562
   874
lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
haftmann@60517
   875
  dvd_div_unit_iff unit_div_mult_swap unit_div_commute
lp15@60562
   876
  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
haftmann@60517
   877
  unit_eq_div1 unit_eq_div2
haftmann@60517
   878
haftmann@60517
   879
end
haftmann@60517
   880
haftmann@38642
   881
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
   882
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
   883
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   884
begin
haftmann@25230
   885
haftmann@25230
   886
lemma mult_mono:
haftmann@38642
   887
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   888
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   889
apply (erule mult_left_mono, assumption)
haftmann@25230
   890
done
haftmann@25230
   891
haftmann@25230
   892
lemma mult_mono':
haftmann@38642
   893
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   894
apply (rule mult_mono)
haftmann@25230
   895
apply (fast intro: order_trans)+
haftmann@25230
   896
done
haftmann@25230
   897
haftmann@25230
   898
end
krauss@21199
   899
haftmann@38642
   900
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
   901
begin
paulson@14268
   902
huffman@27516
   903
subclass semiring_0_cancel ..
obua@23521
   904
nipkow@56536
   905
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   906
using mult_left_mono [of 0 b a] by simp
haftmann@25230
   907
haftmann@25230
   908
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   909
using mult_left_mono [of b 0 a] by simp
huffman@30692
   910
huffman@30692
   911
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   912
using mult_right_mono [of a 0 b] by simp
huffman@30692
   913
huffman@30692
   914
text {* Legacy - use @{text mult_nonpos_nonneg} *}
lp15@60562
   915
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
haftmann@36301
   916
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
   917
lp15@60562
   918
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
nipkow@29667
   919
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   920
haftmann@25230
   921
end
haftmann@25230
   922
haftmann@38642
   923
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
   924
begin
haftmann@25230
   925
haftmann@35028
   926
subclass ordered_cancel_semiring ..
haftmann@35028
   927
haftmann@35028
   928
subclass ordered_comm_monoid_add ..
haftmann@25304
   929
haftmann@25230
   930
lemma mult_left_less_imp_less:
haftmann@25230
   931
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   932
by (force simp add: mult_left_mono not_le [symmetric])
lp15@60562
   933
haftmann@25230
   934
lemma mult_right_less_imp_less:
haftmann@25230
   935
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   936
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   937
haftmann@25186
   938
end
haftmann@25152
   939
haftmann@35043
   940
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
   941
begin
hoelzl@36622
   942
hoelzl@36622
   943
lemma convex_bound_le:
hoelzl@36622
   944
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   945
  shows "u * x + v * y \<le> a"
hoelzl@36622
   946
proof-
hoelzl@36622
   947
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
   948
    by (simp add: add_mono mult_left_mono)
webertj@49962
   949
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   950
qed
hoelzl@36622
   951
hoelzl@36622
   952
end
haftmann@35043
   953
haftmann@35043
   954
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
   955
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   956
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   957
begin
paulson@14341
   958
huffman@27516
   959
subclass semiring_0_cancel ..
obua@14940
   960
haftmann@35028
   961
subclass linordered_semiring
haftmann@28823
   962
proof
huffman@23550
   963
  fix a b c :: 'a
huffman@23550
   964
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   965
  from A show "c * a \<le> c * b"
haftmann@25186
   966
    unfolding le_less
haftmann@25186
   967
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   968
  from A show "a * c \<le> b * c"
haftmann@25152
   969
    unfolding le_less
haftmann@25186
   970
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   971
qed
haftmann@25152
   972
haftmann@25230
   973
lemma mult_left_le_imp_le:
haftmann@25230
   974
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   975
by (force simp add: mult_strict_left_mono _not_less [symmetric])
lp15@60562
   976
haftmann@25230
   977
lemma mult_right_le_imp_le:
haftmann@25230
   978
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   979
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   980
nipkow@56544
   981
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
   982
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
   983
huffman@30692
   984
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
   985
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
   986
huffman@30692
   987
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
   988
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
   989
huffman@30692
   990
text {* Legacy - use @{text mult_neg_pos} *}
lp15@60562
   991
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
haftmann@36301
   992
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
   993
haftmann@25230
   994
lemma zero_less_mult_pos:
haftmann@25230
   995
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   996
apply (cases "b\<le>0")
haftmann@25230
   997
 apply (auto simp add: le_less not_less)
huffman@30692
   998
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
   999
 apply (auto dest: less_not_sym)
haftmann@25230
  1000
done
haftmann@25230
  1001
haftmann@25230
  1002
lemma zero_less_mult_pos2:
haftmann@25230
  1003
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1004
apply (cases "b\<le>0")
haftmann@25230
  1005
 apply (auto simp add: le_less not_less)
huffman@30692
  1006
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
  1007
 apply (auto dest: less_not_sym)
haftmann@25230
  1008
done
haftmann@25230
  1009
haftmann@26193
  1010
text{*Strict monotonicity in both arguments*}
haftmann@26193
  1011
lemma mult_strict_mono:
haftmann@26193
  1012
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
  1013
  shows "a * c < b * d"
haftmann@26193
  1014
  using assms apply (cases "c=0")
nipkow@56544
  1015
  apply (simp)
haftmann@26193
  1016
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
  1017
  apply (force simp add: le_less)
haftmann@26193
  1018
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
  1019
  done
haftmann@26193
  1020
haftmann@26193
  1021
text{*This weaker variant has more natural premises*}
haftmann@26193
  1022
lemma mult_strict_mono':
haftmann@26193
  1023
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
  1024
  shows "a * c < b * d"
nipkow@29667
  1025
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
  1026
haftmann@26193
  1027
lemma mult_less_le_imp_less:
haftmann@26193
  1028
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
  1029
  shows "a * c < b * d"
haftmann@26193
  1030
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
  1031
  apply (erule less_le_trans)
haftmann@26193
  1032
  apply (erule mult_left_mono)
haftmann@26193
  1033
  apply simp
haftmann@26193
  1034
  apply (erule mult_strict_right_mono)
haftmann@26193
  1035
  apply assumption
haftmann@26193
  1036
  done
haftmann@26193
  1037
haftmann@26193
  1038
lemma mult_le_less_imp_less:
haftmann@26193
  1039
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
  1040
  shows "a * c < b * d"
haftmann@26193
  1041
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
  1042
  apply (erule le_less_trans)
haftmann@26193
  1043
  apply (erule mult_strict_left_mono)
haftmann@26193
  1044
  apply simp
haftmann@26193
  1045
  apply (erule mult_right_mono)
haftmann@26193
  1046
  apply simp
haftmann@26193
  1047
  done
haftmann@26193
  1048
haftmann@25230
  1049
end
haftmann@25230
  1050
haftmann@35097
  1051
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
  1052
begin
hoelzl@36622
  1053
hoelzl@36622
  1054
subclass linordered_semiring_1 ..
hoelzl@36622
  1055
hoelzl@36622
  1056
lemma convex_bound_lt:
hoelzl@36622
  1057
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
  1058
  shows "u * x + v * y < a"
hoelzl@36622
  1059
proof -
hoelzl@36622
  1060
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
  1061
    by (cases "u = 0")
hoelzl@36622
  1062
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
webertj@49962
  1063
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
  1064
qed
hoelzl@36622
  1065
hoelzl@36622
  1066
end
haftmann@33319
  1067
lp15@60562
  1068
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +
haftmann@38642
  1069
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
  1070
begin
haftmann@25152
  1071
haftmann@35028
  1072
subclass ordered_semiring
haftmann@28823
  1073
proof
krauss@21199
  1074
  fix a b c :: 'a
huffman@23550
  1075
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
  1076
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
haftmann@57512
  1077
  thus "a * c \<le> b * c" by (simp only: mult.commute)
krauss@21199
  1078
qed
paulson@14265
  1079
haftmann@25267
  1080
end
haftmann@25267
  1081
haftmann@38642
  1082
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
  1083
begin
paulson@14265
  1084
haftmann@38642
  1085
subclass comm_semiring_0_cancel ..
haftmann@35028
  1086
subclass ordered_comm_semiring ..
haftmann@35028
  1087
subclass ordered_cancel_semiring ..
haftmann@25267
  1088
haftmann@25267
  1089
end
haftmann@25267
  1090
haftmann@35028
  1091
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
  1092
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
  1093
begin
haftmann@25267
  1094
haftmann@35043
  1095
subclass linordered_semiring_strict
haftmann@28823
  1096
proof
huffman@23550
  1097
  fix a b c :: 'a
huffman@23550
  1098
  assume "a < b" "0 < c"
haftmann@38642
  1099
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
haftmann@57512
  1100
  thus "a * c < b * c" by (simp only: mult.commute)
huffman@23550
  1101
qed
paulson@14272
  1102
haftmann@35028
  1103
subclass ordered_cancel_comm_semiring
haftmann@28823
  1104
proof
huffman@23550
  1105
  fix a b c :: 'a
huffman@23550
  1106
  assume "a \<le> b" "0 \<le> c"
huffman@23550
  1107
  thus "c * a \<le> c * b"
haftmann@25186
  1108
    unfolding le_less
haftmann@26193
  1109
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
  1110
qed
paulson@14272
  1111
haftmann@25267
  1112
end
haftmann@25230
  1113
lp15@60562
  1114
class ordered_ring = ring + ordered_cancel_semiring
haftmann@25267
  1115
begin
haftmann@25230
  1116
haftmann@35028
  1117
subclass ordered_ab_group_add ..
paulson@14270
  1118
haftmann@25230
  1119
lemma less_add_iff1:
haftmann@25230
  1120
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
  1121
by (simp add: algebra_simps)
haftmann@25230
  1122
haftmann@25230
  1123
lemma less_add_iff2:
haftmann@25230
  1124
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
  1125
by (simp add: algebra_simps)
haftmann@25230
  1126
haftmann@25230
  1127
lemma le_add_iff1:
haftmann@25230
  1128
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
  1129
by (simp add: algebra_simps)
haftmann@25230
  1130
haftmann@25230
  1131
lemma le_add_iff2:
haftmann@25230
  1132
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
  1133
by (simp add: algebra_simps)
haftmann@25230
  1134
haftmann@25230
  1135
lemma mult_left_mono_neg:
haftmann@25230
  1136
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
  1137
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
  1138
  apply simp_all
haftmann@25230
  1139
  done
haftmann@25230
  1140
haftmann@25230
  1141
lemma mult_right_mono_neg:
haftmann@25230
  1142
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
  1143
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
  1144
  apply simp_all
haftmann@25230
  1145
  done
haftmann@25230
  1146
huffman@30692
  1147
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
  1148
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
  1149
haftmann@25230
  1150
lemma split_mult_pos_le:
haftmann@25230
  1151
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@56536
  1152
by (auto simp add: mult_nonpos_nonpos)
haftmann@25186
  1153
haftmann@25186
  1154
end
paulson@14270
  1155
haftmann@35028
  1156
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
  1157
begin
haftmann@25304
  1158
haftmann@35028
  1159
subclass ordered_ring ..
haftmann@35028
  1160
haftmann@35028
  1161
subclass ordered_ab_group_add_abs
haftmann@28823
  1162
proof
haftmann@25304
  1163
  fix a b
haftmann@25304
  1164
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@54230
  1165
    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
  1166
qed (auto simp add: abs_if)
haftmann@25304
  1167
huffman@35631
  1168
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
  1169
  using linear [of 0 a]
nipkow@56536
  1170
  by (auto simp add: mult_nonpos_nonpos)
huffman@35631
  1171
huffman@35631
  1172
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
  1173
  by (simp add: not_less)
huffman@35631
  1174
haftmann@25304
  1175
end
obua@23521
  1176
haftmann@35043
  1177
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
  1178
  + ordered_ab_group_add + abs_if
haftmann@25230
  1179
begin
paulson@14348
  1180
haftmann@35028
  1181
subclass linordered_ring ..
haftmann@25304
  1182
huffman@30692
  1183
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
  1184
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
  1185
huffman@30692
  1186
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
  1187
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
  1188
huffman@30692
  1189
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
  1190
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
  1191
haftmann@25917
  1192
subclass ring_no_zero_divisors
haftmann@28823
  1193
proof
haftmann@25917
  1194
  fix a b
haftmann@25917
  1195
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
  1196
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
  1197
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
  1198
  proof (cases "a < 0")
haftmann@25917
  1199
    case True note A' = this
haftmann@25917
  1200
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1201
      case True with A'
haftmann@25917
  1202
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
  1203
    next
haftmann@25917
  1204
      case False with B have "0 < b" by auto
haftmann@25917
  1205
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
  1206
    qed
haftmann@25917
  1207
  next
haftmann@25917
  1208
    case False with A have A': "0 < a" by auto
haftmann@25917
  1209
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1210
      case True with A'
haftmann@25917
  1211
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
  1212
    next
haftmann@25917
  1213
      case False with B have "0 < b" by auto
nipkow@56544
  1214
      with A' show ?thesis by auto
haftmann@25917
  1215
    qed
haftmann@25917
  1216
  qed
haftmann@25917
  1217
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
  1218
qed
haftmann@25304
  1219
hoelzl@56480
  1220
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
hoelzl@56480
  1221
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
nipkow@56544
  1222
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
huffman@22990
  1223
hoelzl@56480
  1224
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
  1225
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
  1226
paulson@14265
  1227
lemma mult_less_0_iff:
haftmann@25917
  1228
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
  1229
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
  1230
  apply force
haftmann@25917
  1231
  done
paulson@14265
  1232
paulson@14265
  1233
lemma mult_le_0_iff:
haftmann@25917
  1234
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
lp15@60562
  1235
  apply (insert zero_le_mult_iff [of "-a" b])
huffman@35216
  1236
  apply force
haftmann@25917
  1237
  done
haftmann@25917
  1238
haftmann@26193
  1239
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
  1240
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
  1241
haftmann@26193
  1242
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
  1243
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
  1244
haftmann@26193
  1245
lemma mult_less_cancel_right_disj:
haftmann@26193
  1246
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1247
  apply (cases "c = 0")
lp15@60562
  1248
  apply (auto simp add: neq_iff mult_strict_right_mono
haftmann@26193
  1249
                      mult_strict_right_mono_neg)
lp15@60562
  1250
  apply (auto simp add: not_less
haftmann@26193
  1251
                      not_le [symmetric, of "a*c"]
haftmann@26193
  1252
                      not_le [symmetric, of a])
haftmann@26193
  1253
  apply (erule_tac [!] notE)
lp15@60562
  1254
  apply (auto simp add: less_imp_le mult_right_mono
haftmann@26193
  1255
                      mult_right_mono_neg)
haftmann@26193
  1256
  done
haftmann@26193
  1257
haftmann@26193
  1258
lemma mult_less_cancel_left_disj:
haftmann@26193
  1259
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1260
  apply (cases "c = 0")
lp15@60562
  1261
  apply (auto simp add: neq_iff mult_strict_left_mono
haftmann@26193
  1262
                      mult_strict_left_mono_neg)
lp15@60562
  1263
  apply (auto simp add: not_less
haftmann@26193
  1264
                      not_le [symmetric, of "c*a"]
haftmann@26193
  1265
                      not_le [symmetric, of a])
haftmann@26193
  1266
  apply (erule_tac [!] notE)
lp15@60562
  1267
  apply (auto simp add: less_imp_le mult_left_mono
haftmann@26193
  1268
                      mult_left_mono_neg)
haftmann@26193
  1269
  done
haftmann@26193
  1270
haftmann@26193
  1271
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
  1272
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
  1273
haftmann@26193
  1274
lemma mult_less_cancel_right:
haftmann@26193
  1275
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1276
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
  1277
haftmann@26193
  1278
lemma mult_less_cancel_left:
haftmann@26193
  1279
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1280
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
  1281
haftmann@26193
  1282
lemma mult_le_cancel_right:
haftmann@26193
  1283
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1284
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
  1285
haftmann@26193
  1286
lemma mult_le_cancel_left:
haftmann@26193
  1287
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1288
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
  1289
nipkow@30649
  1290
lemma mult_le_cancel_left_pos:
nipkow@30649
  1291
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
  1292
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1293
nipkow@30649
  1294
lemma mult_le_cancel_left_neg:
nipkow@30649
  1295
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
  1296
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1297
nipkow@30649
  1298
lemma mult_less_cancel_left_pos:
nipkow@30649
  1299
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
  1300
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1301
nipkow@30649
  1302
lemma mult_less_cancel_left_neg:
nipkow@30649
  1303
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
  1304
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1305
haftmann@25917
  1306
end
paulson@14265
  1307
huffman@30692
  1308
lemmas mult_sign_intros =
huffman@30692
  1309
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
  1310
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
  1311
  mult_pos_pos mult_pos_neg
huffman@30692
  1312
  mult_neg_pos mult_neg_neg
haftmann@25230
  1313
haftmann@35028
  1314
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
  1315
begin
haftmann@25230
  1316
haftmann@35028
  1317
subclass ordered_ring ..
haftmann@35028
  1318
subclass ordered_cancel_comm_semiring ..
haftmann@25230
  1319
haftmann@25267
  1320
end
haftmann@25230
  1321
haftmann@59833
  1322
class linordered_semidom = semidom + linordered_comm_semiring_strict +
haftmann@25230
  1323
  assumes zero_less_one [simp]: "0 < 1"
lp15@60562
  1324
  assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
haftmann@25230
  1325
begin
haftmann@25230
  1326
lp15@60562
  1327
text {* Addition is the inverse of subtraction. *}
lp15@60562
  1328
lp15@60562
  1329
lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"
lp15@60562
  1330
  by (frule le_add_diff_inverse2) (simp add: add.commute)
lp15@60562
  1331
lp15@60562
  1332
lemma add_diff_inverse: "~ a<b \<Longrightarrow> b + (a - b) = a"
lp15@60562
  1333
  by simp
lp15@60562
  1334
  
haftmann@25230
  1335
lemma pos_add_strict:
haftmann@25230
  1336
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
  1337
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
  1338
haftmann@26193
  1339
lemma zero_le_one [simp]: "0 \<le> 1"
lp15@60562
  1340
by (rule zero_less_one [THEN less_imp_le])
haftmann@26193
  1341
haftmann@26193
  1342
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
lp15@60562
  1343
by (simp add: not_le)
haftmann@26193
  1344
haftmann@26193
  1345
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
lp15@60562
  1346
by (simp add: not_less)
haftmann@26193
  1347
haftmann@26193
  1348
lemma less_1_mult:
haftmann@26193
  1349
  assumes "1 < m" and "1 < n"
haftmann@26193
  1350
  shows "1 < m * n"
haftmann@26193
  1351
  using assms mult_strict_mono [of 1 m 1 n]
lp15@60562
  1352
    by (simp add:  less_trans [OF zero_less_one])
haftmann@26193
  1353
hoelzl@59000
  1354
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
hoelzl@59000
  1355
  using mult_left_mono[of c 1 a] by simp
hoelzl@59000
  1356
hoelzl@59000
  1357
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
hoelzl@59000
  1358
  using mult_mono[of a 1 b 1] by simp
hoelzl@59000
  1359
haftmann@25230
  1360
end
haftmann@25230
  1361
haftmann@35028
  1362
class linordered_idom = comm_ring_1 +
haftmann@35028
  1363
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1364
  abs_if + sgn_if
haftmann@25917
  1365
begin
haftmann@25917
  1366
hoelzl@36622
  1367
subclass linordered_semiring_1_strict ..
haftmann@35043
  1368
subclass linordered_ring_strict ..
haftmann@35028
  1369
subclass ordered_comm_ring ..
huffman@27516
  1370
subclass idom ..
haftmann@25917
  1371
haftmann@35028
  1372
subclass linordered_semidom
haftmann@28823
  1373
proof
haftmann@26193
  1374
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1375
  thus "0 < 1" by (simp add: le_less)
lp15@60562
  1376
  show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"
lp15@60562
  1377
    by simp
lp15@60562
  1378
qed
haftmann@25917
  1379
haftmann@35028
  1380
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1381
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1382
  using assms by (rule neqE)
haftmann@26193
  1383
haftmann@26274
  1384
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1385
haftmann@26274
  1386
lemma mult_le_cancel_right1:
haftmann@26274
  1387
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1388
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1389
haftmann@26274
  1390
lemma mult_le_cancel_right2:
haftmann@26274
  1391
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1392
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1393
haftmann@26274
  1394
lemma mult_le_cancel_left1:
haftmann@26274
  1395
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1396
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1397
haftmann@26274
  1398
lemma mult_le_cancel_left2:
haftmann@26274
  1399
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1400
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1401
haftmann@26274
  1402
lemma mult_less_cancel_right1:
haftmann@26274
  1403
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1404
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1405
haftmann@26274
  1406
lemma mult_less_cancel_right2:
haftmann@26274
  1407
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1408
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1409
haftmann@26274
  1410
lemma mult_less_cancel_left1:
haftmann@26274
  1411
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1412
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1413
haftmann@26274
  1414
lemma mult_less_cancel_left2:
haftmann@26274
  1415
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1416
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1417
haftmann@27651
  1418
lemma sgn_sgn [simp]:
haftmann@27651
  1419
  "sgn (sgn a) = sgn a"
nipkow@29700
  1420
unfolding sgn_if by simp
haftmann@27651
  1421
haftmann@27651
  1422
lemma sgn_0_0:
haftmann@27651
  1423
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1424
unfolding sgn_if by simp
haftmann@27651
  1425
haftmann@27651
  1426
lemma sgn_1_pos:
haftmann@27651
  1427
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1428
unfolding sgn_if by simp
haftmann@27651
  1429
haftmann@27651
  1430
lemma sgn_1_neg:
haftmann@27651
  1431
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1432
unfolding sgn_if by auto
haftmann@27651
  1433
haftmann@29940
  1434
lemma sgn_pos [simp]:
haftmann@29940
  1435
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1436
unfolding sgn_1_pos .
haftmann@29940
  1437
haftmann@29940
  1438
lemma sgn_neg [simp]:
haftmann@29940
  1439
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1440
unfolding sgn_1_neg .
haftmann@29940
  1441
haftmann@27651
  1442
lemma sgn_times:
haftmann@27651
  1443
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1444
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1445
haftmann@36301
  1446
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1447
unfolding sgn_if abs_if by auto
nipkow@29700
  1448
haftmann@29940
  1449
lemma sgn_greater [simp]:
haftmann@29940
  1450
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1451
  unfolding sgn_if by auto
haftmann@29940
  1452
haftmann@29940
  1453
lemma sgn_less [simp]:
haftmann@29940
  1454
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1455
  unfolding sgn_if by auto
haftmann@29940
  1456
haftmann@36301
  1457
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1458
  by (simp add: abs_if)
huffman@29949
  1459
haftmann@36301
  1460
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1461
  by (simp add: abs_if)
haftmann@29653
  1462
nipkow@33676
  1463
lemma dvd_if_abs_eq:
haftmann@36301
  1464
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1465
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1466
huffman@55912
  1467
text {* The following lemmas can be proven in more general structures, but
lp15@60562
  1468
are dangerous as simp rules in absence of @{thm neg_equal_zero},
haftmann@54489
  1469
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
haftmann@54489
  1470
haftmann@54489
  1471
lemma equation_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1472
  "1 = - a \<longleftrightarrow> a = - 1"
haftmann@54489
  1473
  by (fact equation_minus_iff)
haftmann@54489
  1474
haftmann@54489
  1475
lemma minus_equation_iff_1 [simp, no_atp]:
haftmann@54489
  1476
  "- a = 1 \<longleftrightarrow> a = - 1"
haftmann@54489
  1477
  by (subst minus_equation_iff, auto)
haftmann@54489
  1478
haftmann@54489
  1479
lemma le_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1480
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
haftmann@54489
  1481
  by (fact le_minus_iff)
haftmann@54489
  1482
haftmann@54489
  1483
lemma minus_le_iff_1 [simp, no_atp]:
haftmann@54489
  1484
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
haftmann@54489
  1485
  by (fact minus_le_iff)
haftmann@54489
  1486
haftmann@54489
  1487
lemma less_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1488
  "1 < - b \<longleftrightarrow> b < - 1"
haftmann@54489
  1489
  by (fact less_minus_iff)
haftmann@54489
  1490
haftmann@54489
  1491
lemma minus_less_iff_1 [simp, no_atp]:
haftmann@54489
  1492
  "- a < 1 \<longleftrightarrow> - 1 < a"
haftmann@54489
  1493
  by (fact minus_less_iff)
haftmann@54489
  1494
haftmann@25917
  1495
end
haftmann@25230
  1496
haftmann@26274
  1497
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1498
blanchet@54147
  1499
lemmas mult_compare_simps =
paulson@15234
  1500
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1501
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1502
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1503
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1504
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1505
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1506
    mult_cancel_right mult_cancel_left
paulson@15234
  1507
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1508
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1509
haftmann@36301
  1510
text {* Reasoning about inequalities with division *}
avigad@16775
  1511
haftmann@35028
  1512
context linordered_semidom
haftmann@25193
  1513
begin
haftmann@25193
  1514
haftmann@25193
  1515
lemma less_add_one: "a < a + 1"
paulson@14293
  1516
proof -
haftmann@25193
  1517
  have "a + 0 < a + 1"
nipkow@23482
  1518
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1519
  thus ?thesis by simp
paulson@14293
  1520
qed
paulson@14293
  1521
haftmann@25193
  1522
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1523
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1524
haftmann@25193
  1525
end
paulson@14365
  1526
haftmann@36301
  1527
context linordered_idom
haftmann@36301
  1528
begin
paulson@15234
  1529
haftmann@36301
  1530
lemma mult_right_le_one_le:
haftmann@36301
  1531
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@59833
  1532
  by (rule mult_left_le)
haftmann@36301
  1533
haftmann@36301
  1534
lemma mult_left_le_one_le:
haftmann@36301
  1535
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1536
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1537
haftmann@36301
  1538
end
haftmann@36301
  1539
haftmann@36301
  1540
text {* Absolute Value *}
paulson@14293
  1541
haftmann@35028
  1542
context linordered_idom
haftmann@25304
  1543
begin
haftmann@25304
  1544
haftmann@36301
  1545
lemma mult_sgn_abs:
haftmann@36301
  1546
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1547
  unfolding abs_if sgn_if by auto
haftmann@25304
  1548
haftmann@36301
  1549
lemma abs_one [simp]:
haftmann@36301
  1550
  "\<bar>1\<bar> = 1"
huffman@44921
  1551
  by (simp add: abs_if)
haftmann@36301
  1552
haftmann@25304
  1553
end
nipkow@24491
  1554
haftmann@35028
  1555
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1556
  assumes abs_eq_mult:
haftmann@25304
  1557
    "(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
  1558
haftmann@35028
  1559
context linordered_idom
haftmann@30961
  1560
begin
haftmann@30961
  1561
haftmann@35028
  1562
subclass ordered_ring_abs proof
huffman@35216
  1563
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1564
haftmann@30961
  1565
lemma abs_mult:
lp15@60562
  1566
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
haftmann@30961
  1567
  by (rule abs_eq_mult) auto
haftmann@30961
  1568
haftmann@30961
  1569
lemma abs_mult_self:
haftmann@36301
  1570
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
lp15@60562
  1571
  by (simp add: abs_if)
haftmann@30961
  1572
paulson@14294
  1573
lemma abs_mult_less:
haftmann@36301
  1574
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1575
proof -
haftmann@36301
  1576
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1577
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1578
  assume "\<bar>b\<bar> < d"
lp15@60562
  1579
  thus ?thesis by (simp add: ac cpos mult_strict_mono)
paulson@14294
  1580
qed
paulson@14293
  1581
haftmann@36301
  1582
lemma abs_less_iff:
lp15@60562
  1583
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
haftmann@36301
  1584
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1585
haftmann@36301
  1586
lemma abs_mult_pos:
haftmann@36301
  1587
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1588
  by (simp add: abs_mult)
haftmann@36301
  1589
hoelzl@51520
  1590
lemma abs_diff_less_iff:
hoelzl@51520
  1591
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
hoelzl@51520
  1592
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
hoelzl@51520
  1593
lp15@59865
  1594
lemma abs_diff_le_iff:
lp15@59865
  1595
   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
lp15@59865
  1596
  by (auto simp add: diff_le_eq ac_simps abs_le_iff)
lp15@59865
  1597
haftmann@36301
  1598
end
avigad@16775
  1599
haftmann@59557
  1600
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
haftmann@59557
  1601
haftmann@52435
  1602
code_identifier
haftmann@52435
  1603
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1604
paulson@14265
  1605
end