src/HOL/Rings.thy
author haftmann
Fri Jun 19 07:53:35 2015 +0200 (2015-06-19)
changeset 60517 f16e4fb20652
parent 60516 0826b7025d07
child 60529 24c2ef12318b
permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
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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@59833
   418
class semiring_no_zero_divisors = semiring_0 +
haftmann@59833
   419
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
haftmann@25230
   420
begin
haftmann@25230
   421
haftmann@59833
   422
lemma divisors_zero:
haftmann@59833
   423
  assumes "a * b = 0"
haftmann@59833
   424
  shows "a = 0 \<or> b = 0"
haftmann@59833
   425
proof (rule classical)
haftmann@59833
   426
  assume "\<not> (a = 0 \<or> b = 0)"
haftmann@59833
   427
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@59833
   428
  with no_zero_divisors have "a * b \<noteq> 0" by blast
haftmann@59833
   429
  with assms show ?thesis by simp
haftmann@59833
   430
qed
haftmann@59833
   431
haftmann@25230
   432
lemma mult_eq_0_iff [simp]:
haftmann@58952
   433
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@25230
   434
proof (cases "a = 0 \<or> b = 0")
haftmann@25230
   435
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@25230
   436
    then show ?thesis using no_zero_divisors by simp
haftmann@25230
   437
next
haftmann@25230
   438
  case True then show ?thesis by auto
haftmann@25230
   439
qed
haftmann@25230
   440
haftmann@58952
   441
end
haftmann@58952
   442
haftmann@60516
   443
class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +
haftmann@60516
   444
  assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   445
    and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@58952
   446
begin
haftmann@58952
   447
haftmann@58952
   448
lemma mult_left_cancel:
haftmann@58952
   449
  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
haftmann@58952
   450
  by simp 
lp15@56217
   451
haftmann@58952
   452
lemma mult_right_cancel:
haftmann@58952
   453
  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
haftmann@58952
   454
  by simp 
lp15@56217
   455
haftmann@25230
   456
end
huffman@22990
   457
haftmann@60516
   458
class ring_no_zero_divisors = ring + semiring_no_zero_divisors
haftmann@60516
   459
begin
haftmann@60516
   460
haftmann@60516
   461
subclass semiring_no_zero_divisors_cancel
haftmann@60516
   462
proof
haftmann@60516
   463
  fix a b c
haftmann@60516
   464
  have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"
haftmann@60516
   465
    by (simp add: algebra_simps)
haftmann@60516
   466
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   467
    by auto
haftmann@60516
   468
  finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .
haftmann@60516
   469
  have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"
haftmann@60516
   470
    by (simp add: algebra_simps)
haftmann@60516
   471
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   472
    by auto
haftmann@60516
   473
  finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .
haftmann@60516
   474
qed
haftmann@60516
   475
haftmann@60516
   476
end
haftmann@60516
   477
huffman@23544
   478
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   479
begin
haftmann@26274
   480
huffman@36970
   481
lemma square_eq_1_iff:
huffman@36821
   482
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
huffman@36821
   483
proof -
huffman@36821
   484
  have "(x - 1) * (x + 1) = x * x - 1"
huffman@36821
   485
    by (simp add: algebra_simps)
huffman@36821
   486
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   487
    by simp
huffman@36821
   488
  thus ?thesis
huffman@36821
   489
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   490
qed
huffman@36821
   491
haftmann@26274
   492
lemma mult_cancel_right1 [simp]:
haftmann@26274
   493
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   494
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   495
haftmann@26274
   496
lemma mult_cancel_right2 [simp]:
haftmann@26274
   497
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   498
by (insert mult_cancel_right [of a c 1], simp)
haftmann@26274
   499
 
haftmann@26274
   500
lemma mult_cancel_left1 [simp]:
haftmann@26274
   501
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   502
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   503
haftmann@26274
   504
lemma mult_cancel_left2 [simp]:
haftmann@26274
   505
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   506
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   507
haftmann@26274
   508
end
huffman@22990
   509
haftmann@59910
   510
class semidom = comm_semiring_1_diff_distrib + semiring_no_zero_divisors
haftmann@59833
   511
haftmann@59833
   512
class idom = comm_ring_1 + semiring_no_zero_divisors
haftmann@25186
   513
begin
paulson@14421
   514
haftmann@59833
   515
subclass semidom ..
haftmann@59833
   516
huffman@27516
   517
subclass ring_1_no_zero_divisors ..
huffman@22990
   518
huffman@29981
   519
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   520
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   521
proof -
huffman@29981
   522
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   523
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   524
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   525
    unfolding dvd_def by simp
huffman@29981
   526
  finally show ?thesis .
huffman@29981
   527
qed
huffman@29981
   528
huffman@29981
   529
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   530
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   531
proof -
huffman@29981
   532
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   533
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   534
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   535
    unfolding dvd_def by simp
huffman@29981
   536
  finally show ?thesis .
huffman@29981
   537
qed
huffman@29981
   538
haftmann@60516
   539
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"
haftmann@59833
   540
proof
haftmann@59833
   541
  assume "a * a = b * b"
haftmann@59833
   542
  then have "(a - b) * (a + b) = 0"
haftmann@59833
   543
    by (simp add: algebra_simps)
haftmann@59833
   544
  then show "a = b \<or> a = - b"
haftmann@59833
   545
    by (simp add: eq_neg_iff_add_eq_0)
haftmann@59833
   546
next
haftmann@59833
   547
  assume "a = b \<or> a = - b"
haftmann@59833
   548
  then show "a * a = b * b" by auto
haftmann@59833
   549
qed
haftmann@59833
   550
haftmann@25186
   551
end
haftmann@25152
   552
haftmann@35302
   553
text {*
haftmann@35302
   554
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   555
  \begin{itemize}
haftmann@35302
   556
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35302
   557
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   558
  \end{itemize}
haftmann@35302
   559
  Most of the used notions can also be looked up in 
haftmann@35302
   560
  \begin{itemize}
wenzelm@54703
   561
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35302
   562
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   563
  \end{itemize}
haftmann@35302
   564
*}
haftmann@35302
   565
haftmann@60353
   566
class divide =
haftmann@60429
   567
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
haftmann@60353
   568
haftmann@60353
   569
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60353
   570
haftmann@60353
   571
context semiring
haftmann@60353
   572
begin
haftmann@60353
   573
haftmann@60353
   574
lemma [field_simps]:
haftmann@60429
   575
  shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
haftmann@60429
   576
    and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
haftmann@60353
   577
  by (rule distrib_left distrib_right)+
haftmann@60353
   578
haftmann@60353
   579
end
haftmann@60353
   580
haftmann@60353
   581
context ring
haftmann@60353
   582
begin
haftmann@60353
   583
haftmann@60353
   584
lemma [field_simps]:
haftmann@60429
   585
  shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
haftmann@60429
   586
    and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
haftmann@60353
   587
  by (rule left_diff_distrib right_diff_distrib)+
haftmann@60353
   588
haftmann@60353
   589
end
haftmann@60353
   590
haftmann@60353
   591
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60353
   592
haftmann@60353
   593
class semidom_divide = semidom + divide +
haftmann@60429
   594
  assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
haftmann@60429
   595
  assumes divide_zero [simp]: "a div 0 = 0"
haftmann@60353
   596
begin
haftmann@60353
   597
haftmann@60353
   598
lemma nonzero_mult_divide_cancel_left [simp]:
haftmann@60429
   599
  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
haftmann@60353
   600
  using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
haftmann@60353
   601
haftmann@60516
   602
subclass semiring_no_zero_divisors_cancel
haftmann@60516
   603
proof
haftmann@60516
   604
  fix a b c
haftmann@60516
   605
  { fix a b c
haftmann@60516
   606
    show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   607
    proof (cases "c = 0")
haftmann@60516
   608
      case True then show ?thesis by simp
haftmann@60516
   609
    next
haftmann@60516
   610
      case False
haftmann@60516
   611
      { assume "a * c = b * c"
haftmann@60516
   612
        then have "a * c div c = b * c div c"
haftmann@60516
   613
          by simp
haftmann@60516
   614
        with False have "a = b"
haftmann@60516
   615
          by simp
haftmann@60516
   616
      } then show ?thesis by auto
haftmann@60516
   617
    qed
haftmann@60516
   618
  }
haftmann@60516
   619
  from this [of a c b]
haftmann@60516
   620
  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   621
    by (simp add: ac_simps)
haftmann@60516
   622
qed
haftmann@60516
   623
haftmann@60516
   624
lemma div_self [simp]:
haftmann@60516
   625
  assumes "a \<noteq> 0"
haftmann@60516
   626
  shows "a div a = 1"
haftmann@60516
   627
  using assms nonzero_mult_divide_cancel_left [of a 1] by simp
haftmann@60516
   628
haftmann@60353
   629
end
haftmann@60353
   630
haftmann@60353
   631
class idom_divide = idom + semidom_divide
haftmann@60353
   632
haftmann@60517
   633
class algebraic_semidom = semidom_divide
haftmann@60517
   634
begin
haftmann@60517
   635
haftmann@60517
   636
lemma dvd_div_mult_self [simp]:
haftmann@60517
   637
  "a dvd b \<Longrightarrow> b div a * a = b"
haftmann@60517
   638
  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
haftmann@60517
   639
haftmann@60517
   640
lemma dvd_mult_div_cancel [simp]:
haftmann@60517
   641
  "a dvd b \<Longrightarrow> a * (b div a) = b"
haftmann@60517
   642
  using dvd_div_mult_self [of a b] by (simp add: ac_simps)
haftmann@60517
   643
  
haftmann@60517
   644
lemma div_mult_swap:
haftmann@60517
   645
  assumes "c dvd b"
haftmann@60517
   646
  shows "a * (b div c) = (a * b) div c"
haftmann@60517
   647
proof (cases "c = 0")
haftmann@60517
   648
  case True then show ?thesis by simp
haftmann@60517
   649
next
haftmann@60517
   650
  case False from assms obtain d where "b = c * d" ..
haftmann@60517
   651
  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
haftmann@60517
   652
    by simp
haftmann@60517
   653
  ultimately show ?thesis by (simp add: ac_simps)
haftmann@60517
   654
qed
haftmann@60517
   655
haftmann@60517
   656
lemma dvd_div_mult:
haftmann@60517
   657
  assumes "c dvd b"
haftmann@60517
   658
  shows "b div c * a = (b * a) div c"
haftmann@60517
   659
  using assms div_mult_swap [of c b a] by (simp add: ac_simps)
haftmann@60517
   660
haftmann@60517
   661
  
haftmann@60517
   662
text \<open>Units: invertible elements in a ring\<close>
haftmann@60517
   663
haftmann@60517
   664
abbreviation is_unit :: "'a \<Rightarrow> bool"
haftmann@60517
   665
where
haftmann@60517
   666
  "is_unit a \<equiv> a dvd 1"
haftmann@60517
   667
haftmann@60517
   668
lemma not_is_unit_0 [simp]:
haftmann@60517
   669
  "\<not> is_unit 0"
haftmann@60517
   670
  by simp
haftmann@60517
   671
haftmann@60517
   672
lemma unit_imp_dvd [dest]: 
haftmann@60517
   673
  "is_unit b \<Longrightarrow> b dvd a"
haftmann@60517
   674
  by (rule dvd_trans [of _ 1]) simp_all
haftmann@60517
   675
haftmann@60517
   676
lemma unit_dvdE:
haftmann@60517
   677
  assumes "is_unit a"
haftmann@60517
   678
  obtains c where "a \<noteq> 0" and "b = a * c"
haftmann@60517
   679
proof -
haftmann@60517
   680
  from assms have "a dvd b" by auto
haftmann@60517
   681
  then obtain c where "b = a * c" ..
haftmann@60517
   682
  moreover from assms have "a \<noteq> 0" by auto
haftmann@60517
   683
  ultimately show thesis using that by blast
haftmann@60517
   684
qed
haftmann@60517
   685
haftmann@60517
   686
lemma dvd_unit_imp_unit:
haftmann@60517
   687
  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
haftmann@60517
   688
  by (rule dvd_trans)
haftmann@60517
   689
haftmann@60517
   690
lemma unit_div_1_unit [simp, intro]:
haftmann@60517
   691
  assumes "is_unit a"
haftmann@60517
   692
  shows "is_unit (1 div a)"
haftmann@60517
   693
proof -
haftmann@60517
   694
  from assms have "1 = 1 div a * a" by simp
haftmann@60517
   695
  then show "is_unit (1 div a)" by (rule dvdI)
haftmann@60517
   696
qed
haftmann@60517
   697
haftmann@60517
   698
lemma is_unitE [elim?]:
haftmann@60517
   699
  assumes "is_unit a"
haftmann@60517
   700
  obtains b where "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60517
   701
    and "is_unit b" and "1 div a = b" and "1 div b = a"
haftmann@60517
   702
    and "a * b = 1" and "c div a = c * b"
haftmann@60517
   703
proof (rule that)
haftmann@60517
   704
  def b \<equiv> "1 div a"
haftmann@60517
   705
  then show "1 div a = b" by simp
haftmann@60517
   706
  from b_def `is_unit a` show "is_unit b" by simp
haftmann@60517
   707
  from `is_unit a` and `is_unit b` show "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@60517
   708
  from b_def `is_unit a` show "a * b = 1" by simp
haftmann@60517
   709
  then have "1 = a * b" ..
haftmann@60517
   710
  with b_def `b \<noteq> 0` show "1 div b = a" by simp
haftmann@60517
   711
  from `is_unit a` have "a dvd c" ..
haftmann@60517
   712
  then obtain d where "c = a * d" ..
haftmann@60517
   713
  with `a \<noteq> 0` `a * b = 1` show "c div a = c * b"
haftmann@60517
   714
    by (simp add: mult.assoc mult.left_commute [of a])
haftmann@60517
   715
qed
haftmann@60517
   716
haftmann@60517
   717
lemma unit_prod [intro]:
haftmann@60517
   718
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
haftmann@60517
   719
  by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono) 
haftmann@60517
   720
  
haftmann@60517
   721
lemma unit_div [intro]:
haftmann@60517
   722
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
haftmann@60517
   723
  by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
haftmann@60517
   724
haftmann@60517
   725
lemma mult_unit_dvd_iff:
haftmann@60517
   726
  assumes "is_unit b"
haftmann@60517
   727
  shows "a * b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   728
proof
haftmann@60517
   729
  assume "a * b dvd c"
haftmann@60517
   730
  with assms show "a dvd c"
haftmann@60517
   731
    by (simp add: dvd_mult_left)
haftmann@60517
   732
next
haftmann@60517
   733
  assume "a dvd c"
haftmann@60517
   734
  then obtain k where "c = a * k" ..
haftmann@60517
   735
  with assms have "c = (a * b) * (1 div b * k)"
haftmann@60517
   736
    by (simp add: mult_ac)
haftmann@60517
   737
  then show "a * b dvd c" by (rule dvdI)
haftmann@60517
   738
qed
haftmann@60517
   739
haftmann@60517
   740
lemma dvd_mult_unit_iff:
haftmann@60517
   741
  assumes "is_unit b"
haftmann@60517
   742
  shows "a dvd c * b \<longleftrightarrow> a dvd c"
haftmann@60517
   743
proof
haftmann@60517
   744
  assume "a dvd c * b"
haftmann@60517
   745
  with assms have "c * b dvd c * (b * (1 div b))"
haftmann@60517
   746
    by (subst mult_assoc [symmetric]) simp
haftmann@60517
   747
  also from `is_unit b` have "b * (1 div b) = 1" by (rule is_unitE) simp
haftmann@60517
   748
  finally have "c * b dvd c" by simp
haftmann@60517
   749
  with `a dvd c * b` show "a dvd c" by (rule dvd_trans)
haftmann@60517
   750
next
haftmann@60517
   751
  assume "a dvd c"
haftmann@60517
   752
  then show "a dvd c * b" by simp
haftmann@60517
   753
qed
haftmann@60517
   754
haftmann@60517
   755
lemma div_unit_dvd_iff:
haftmann@60517
   756
  "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   757
  by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
haftmann@60517
   758
haftmann@60517
   759
lemma dvd_div_unit_iff:
haftmann@60517
   760
  "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
haftmann@60517
   761
  by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
haftmann@60517
   762
haftmann@60517
   763
lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
haftmann@60517
   764
  dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
haftmann@60517
   765
haftmann@60517
   766
lemma unit_mult_div_div [simp]:
haftmann@60517
   767
  "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
haftmann@60517
   768
  by (erule is_unitE [of _ b]) simp
haftmann@60517
   769
haftmann@60517
   770
lemma unit_div_mult_self [simp]:
haftmann@60517
   771
  "is_unit a \<Longrightarrow> b div a * a = b"
haftmann@60517
   772
  by (rule dvd_div_mult_self) auto
haftmann@60517
   773
haftmann@60517
   774
lemma unit_div_1_div_1 [simp]:
haftmann@60517
   775
  "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
haftmann@60517
   776
  by (erule is_unitE) simp
haftmann@60517
   777
haftmann@60517
   778
lemma unit_div_mult_swap:
haftmann@60517
   779
  "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
haftmann@60517
   780
  by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
haftmann@60517
   781
haftmann@60517
   782
lemma unit_div_commute:
haftmann@60517
   783
  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
haftmann@60517
   784
  using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
haftmann@60517
   785
haftmann@60517
   786
lemma unit_eq_div1:
haftmann@60517
   787
  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
haftmann@60517
   788
  by (auto elim: is_unitE)
haftmann@60517
   789
haftmann@60517
   790
lemma unit_eq_div2:
haftmann@60517
   791
  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
haftmann@60517
   792
  using unit_eq_div1 [of b c a] by auto
haftmann@60517
   793
haftmann@60517
   794
lemma unit_mult_left_cancel:
haftmann@60517
   795
  assumes "is_unit a"
haftmann@60517
   796
  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
haftmann@60517
   797
  using assms mult_cancel_left [of a b c] by auto 
haftmann@60517
   798
haftmann@60517
   799
lemma unit_mult_right_cancel:
haftmann@60517
   800
  "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
haftmann@60517
   801
  using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
haftmann@60517
   802
haftmann@60517
   803
lemma unit_div_cancel:
haftmann@60517
   804
  assumes "is_unit a"
haftmann@60517
   805
  shows "b div a = c div a \<longleftrightarrow> b = c"
haftmann@60517
   806
proof -
haftmann@60517
   807
  from assms have "is_unit (1 div a)" by simp
haftmann@60517
   808
  then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
haftmann@60517
   809
    by (rule unit_mult_right_cancel)
haftmann@60517
   810
  with assms show ?thesis by simp
haftmann@60517
   811
qed
haftmann@60517
   812
  
haftmann@60517
   813
haftmann@60517
   814
text \<open>Associated elements in a ring – an equivalence relation induced by the quasi-order divisibility \<close>
haftmann@60517
   815
haftmann@60517
   816
definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool" 
haftmann@60517
   817
where
haftmann@60517
   818
  "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"
haftmann@60517
   819
haftmann@60517
   820
lemma associatedI:
haftmann@60517
   821
  "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated a b"
haftmann@60517
   822
  by (simp add: associated_def)
haftmann@60517
   823
haftmann@60517
   824
lemma associatedD1:
haftmann@60517
   825
  "associated a b \<Longrightarrow> a dvd b"
haftmann@60517
   826
  by (simp add: associated_def)
haftmann@60517
   827
haftmann@60517
   828
lemma associatedD2:
haftmann@60517
   829
  "associated a b \<Longrightarrow> b dvd a"
haftmann@60517
   830
  by (simp add: associated_def)
haftmann@60517
   831
haftmann@60517
   832
lemma associated_refl [simp]:
haftmann@60517
   833
  "associated a a"
haftmann@60517
   834
  by (auto intro: associatedI)
haftmann@60517
   835
haftmann@60517
   836
lemma associated_sym:
haftmann@60517
   837
  "associated b a \<longleftrightarrow> associated a b"
haftmann@60517
   838
  by (auto intro: associatedI dest: associatedD1 associatedD2)
haftmann@60517
   839
haftmann@60517
   840
lemma associated_trans:
haftmann@60517
   841
  "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated a c"
haftmann@60517
   842
  by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)
haftmann@60517
   843
haftmann@60517
   844
lemma associated_0 [simp]:
haftmann@60517
   845
  "associated 0 b \<longleftrightarrow> b = 0"
haftmann@60517
   846
  "associated a 0 \<longleftrightarrow> a = 0"
haftmann@60517
   847
  by (auto dest: associatedD1 associatedD2)
haftmann@60517
   848
haftmann@60517
   849
lemma associated_unit:
haftmann@60517
   850
  "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
haftmann@60517
   851
  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
haftmann@60517
   852
haftmann@60517
   853
lemma is_unit_associatedI:
haftmann@60517
   854
  assumes "is_unit c" and "a = c * b"
haftmann@60517
   855
  shows "associated a b"
haftmann@60517
   856
proof (rule associatedI)
haftmann@60517
   857
  from `a = c * b` show "b dvd a" by auto
haftmann@60517
   858
  from `is_unit c` obtain d where "c * d = 1" by (rule is_unitE)
haftmann@60517
   859
  moreover from `a = c * b` have "d * a = d * (c * b)" by simp
haftmann@60517
   860
  ultimately have "b = a * d" by (simp add: ac_simps)
haftmann@60517
   861
  then show "a dvd b" ..
haftmann@60517
   862
qed
haftmann@60517
   863
haftmann@60517
   864
lemma associated_is_unitE:
haftmann@60517
   865
  assumes "associated a b"
haftmann@60517
   866
  obtains c where "is_unit c" and "a = c * b"
haftmann@60517
   867
proof (cases "b = 0")
haftmann@60517
   868
  case True with assms have "is_unit 1" and "a = 1 * b" by simp_all
haftmann@60517
   869
  with that show thesis .
haftmann@60517
   870
next
haftmann@60517
   871
  case False
haftmann@60517
   872
  from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
haftmann@60517
   873
  then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
haftmann@60517
   874
  then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)
haftmann@60517
   875
  with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp
haftmann@60517
   876
  then have "is_unit c" by auto
haftmann@60517
   877
  with `a = c * b` that show thesis by blast
haftmann@60517
   878
qed
haftmann@60517
   879
  
haftmann@60517
   880
lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff 
haftmann@60517
   881
  dvd_div_unit_iff unit_div_mult_swap unit_div_commute
haftmann@60517
   882
  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel 
haftmann@60517
   883
  unit_eq_div1 unit_eq_div2
haftmann@60517
   884
haftmann@60517
   885
end
haftmann@60517
   886
haftmann@38642
   887
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
   888
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
   889
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   890
begin
haftmann@25230
   891
haftmann@25230
   892
lemma mult_mono:
haftmann@38642
   893
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   894
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   895
apply (erule mult_left_mono, assumption)
haftmann@25230
   896
done
haftmann@25230
   897
haftmann@25230
   898
lemma mult_mono':
haftmann@38642
   899
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   900
apply (rule mult_mono)
haftmann@25230
   901
apply (fast intro: order_trans)+
haftmann@25230
   902
done
haftmann@25230
   903
haftmann@25230
   904
end
krauss@21199
   905
haftmann@38642
   906
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
   907
begin
paulson@14268
   908
huffman@27516
   909
subclass semiring_0_cancel ..
obua@23521
   910
nipkow@56536
   911
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   912
using mult_left_mono [of 0 b a] by simp
haftmann@25230
   913
haftmann@25230
   914
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   915
using mult_left_mono [of b 0 a] by simp
huffman@30692
   916
huffman@30692
   917
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   918
using mult_right_mono [of a 0 b] by simp
huffman@30692
   919
huffman@30692
   920
text {* Legacy - use @{text mult_nonpos_nonneg} *}
haftmann@25230
   921
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
haftmann@36301
   922
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
   923
haftmann@26234
   924
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
nipkow@29667
   925
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   926
haftmann@25230
   927
end
haftmann@25230
   928
haftmann@38642
   929
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
   930
begin
haftmann@25230
   931
haftmann@35028
   932
subclass ordered_cancel_semiring ..
haftmann@35028
   933
haftmann@35028
   934
subclass ordered_comm_monoid_add ..
haftmann@25304
   935
haftmann@25230
   936
lemma mult_left_less_imp_less:
haftmann@25230
   937
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   938
by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   939
 
haftmann@25230
   940
lemma mult_right_less_imp_less:
haftmann@25230
   941
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   942
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   943
haftmann@25186
   944
end
haftmann@25152
   945
haftmann@35043
   946
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
   947
begin
hoelzl@36622
   948
hoelzl@36622
   949
lemma convex_bound_le:
hoelzl@36622
   950
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   951
  shows "u * x + v * y \<le> a"
hoelzl@36622
   952
proof-
hoelzl@36622
   953
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
   954
    by (simp add: add_mono mult_left_mono)
webertj@49962
   955
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   956
qed
hoelzl@36622
   957
hoelzl@36622
   958
end
haftmann@35043
   959
haftmann@35043
   960
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
   961
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   962
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   963
begin
paulson@14341
   964
huffman@27516
   965
subclass semiring_0_cancel ..
obua@14940
   966
haftmann@35028
   967
subclass linordered_semiring
haftmann@28823
   968
proof
huffman@23550
   969
  fix a b c :: 'a
huffman@23550
   970
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   971
  from A show "c * a \<le> c * b"
haftmann@25186
   972
    unfolding le_less
haftmann@25186
   973
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   974
  from A show "a * c \<le> b * c"
haftmann@25152
   975
    unfolding le_less
haftmann@25186
   976
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   977
qed
haftmann@25152
   978
haftmann@25230
   979
lemma mult_left_le_imp_le:
haftmann@25230
   980
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   981
by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   982
 
haftmann@25230
   983
lemma mult_right_le_imp_le:
haftmann@25230
   984
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   985
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   986
nipkow@56544
   987
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
   988
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
   989
huffman@30692
   990
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
   991
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
   992
huffman@30692
   993
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
   994
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
   995
huffman@30692
   996
text {* Legacy - use @{text mult_neg_pos} *}
huffman@30692
   997
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
haftmann@36301
   998
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
   999
haftmann@25230
  1000
lemma zero_less_mult_pos:
haftmann@25230
  1001
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1002
apply (cases "b\<le>0")
haftmann@25230
  1003
 apply (auto simp add: le_less not_less)
huffman@30692
  1004
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
  1005
 apply (auto dest: less_not_sym)
haftmann@25230
  1006
done
haftmann@25230
  1007
haftmann@25230
  1008
lemma zero_less_mult_pos2:
haftmann@25230
  1009
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1010
apply (cases "b\<le>0")
haftmann@25230
  1011
 apply (auto simp add: le_less not_less)
huffman@30692
  1012
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
  1013
 apply (auto dest: less_not_sym)
haftmann@25230
  1014
done
haftmann@25230
  1015
haftmann@26193
  1016
text{*Strict monotonicity in both arguments*}
haftmann@26193
  1017
lemma mult_strict_mono:
haftmann@26193
  1018
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
  1019
  shows "a * c < b * d"
haftmann@26193
  1020
  using assms apply (cases "c=0")
nipkow@56544
  1021
  apply (simp)
haftmann@26193
  1022
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
  1023
  apply (force simp add: le_less)
haftmann@26193
  1024
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
  1025
  done
haftmann@26193
  1026
haftmann@26193
  1027
text{*This weaker variant has more natural premises*}
haftmann@26193
  1028
lemma mult_strict_mono':
haftmann@26193
  1029
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
  1030
  shows "a * c < b * d"
nipkow@29667
  1031
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
  1032
haftmann@26193
  1033
lemma mult_less_le_imp_less:
haftmann@26193
  1034
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
  1035
  shows "a * c < b * d"
haftmann@26193
  1036
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
  1037
  apply (erule less_le_trans)
haftmann@26193
  1038
  apply (erule mult_left_mono)
haftmann@26193
  1039
  apply simp
haftmann@26193
  1040
  apply (erule mult_strict_right_mono)
haftmann@26193
  1041
  apply assumption
haftmann@26193
  1042
  done
haftmann@26193
  1043
haftmann@26193
  1044
lemma mult_le_less_imp_less:
haftmann@26193
  1045
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
  1046
  shows "a * c < b * d"
haftmann@26193
  1047
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
  1048
  apply (erule le_less_trans)
haftmann@26193
  1049
  apply (erule mult_strict_left_mono)
haftmann@26193
  1050
  apply simp
haftmann@26193
  1051
  apply (erule mult_right_mono)
haftmann@26193
  1052
  apply simp
haftmann@26193
  1053
  done
haftmann@26193
  1054
haftmann@25230
  1055
end
haftmann@25230
  1056
haftmann@35097
  1057
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
  1058
begin
hoelzl@36622
  1059
hoelzl@36622
  1060
subclass linordered_semiring_1 ..
hoelzl@36622
  1061
hoelzl@36622
  1062
lemma convex_bound_lt:
hoelzl@36622
  1063
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
  1064
  shows "u * x + v * y < a"
hoelzl@36622
  1065
proof -
hoelzl@36622
  1066
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
  1067
    by (cases "u = 0")
hoelzl@36622
  1068
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
webertj@49962
  1069
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
  1070
qed
hoelzl@36622
  1071
hoelzl@36622
  1072
end
haftmann@33319
  1073
haftmann@38642
  1074
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add + 
haftmann@38642
  1075
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
  1076
begin
haftmann@25152
  1077
haftmann@35028
  1078
subclass ordered_semiring
haftmann@28823
  1079
proof
krauss@21199
  1080
  fix a b c :: 'a
huffman@23550
  1081
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
  1082
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
haftmann@57512
  1083
  thus "a * c \<le> b * c" by (simp only: mult.commute)
krauss@21199
  1084
qed
paulson@14265
  1085
haftmann@25267
  1086
end
haftmann@25267
  1087
haftmann@38642
  1088
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
  1089
begin
paulson@14265
  1090
haftmann@38642
  1091
subclass comm_semiring_0_cancel ..
haftmann@35028
  1092
subclass ordered_comm_semiring ..
haftmann@35028
  1093
subclass ordered_cancel_semiring ..
haftmann@25267
  1094
haftmann@25267
  1095
end
haftmann@25267
  1096
haftmann@35028
  1097
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
  1098
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
  1099
begin
haftmann@25267
  1100
haftmann@35043
  1101
subclass linordered_semiring_strict
haftmann@28823
  1102
proof
huffman@23550
  1103
  fix a b c :: 'a
huffman@23550
  1104
  assume "a < b" "0 < c"
haftmann@38642
  1105
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
haftmann@57512
  1106
  thus "a * c < b * c" by (simp only: mult.commute)
huffman@23550
  1107
qed
paulson@14272
  1108
haftmann@35028
  1109
subclass ordered_cancel_comm_semiring
haftmann@28823
  1110
proof
huffman@23550
  1111
  fix a b c :: 'a
huffman@23550
  1112
  assume "a \<le> b" "0 \<le> c"
huffman@23550
  1113
  thus "c * a \<le> c * b"
haftmann@25186
  1114
    unfolding le_less
haftmann@26193
  1115
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
  1116
qed
paulson@14272
  1117
haftmann@25267
  1118
end
haftmann@25230
  1119
haftmann@35028
  1120
class ordered_ring = ring + ordered_cancel_semiring 
haftmann@25267
  1121
begin
haftmann@25230
  1122
haftmann@35028
  1123
subclass ordered_ab_group_add ..
paulson@14270
  1124
haftmann@25230
  1125
lemma less_add_iff1:
haftmann@25230
  1126
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
  1127
by (simp add: algebra_simps)
haftmann@25230
  1128
haftmann@25230
  1129
lemma less_add_iff2:
haftmann@25230
  1130
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
  1131
by (simp add: algebra_simps)
haftmann@25230
  1132
haftmann@25230
  1133
lemma le_add_iff1:
haftmann@25230
  1134
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
  1135
by (simp add: algebra_simps)
haftmann@25230
  1136
haftmann@25230
  1137
lemma le_add_iff2:
haftmann@25230
  1138
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
  1139
by (simp add: algebra_simps)
haftmann@25230
  1140
haftmann@25230
  1141
lemma mult_left_mono_neg:
haftmann@25230
  1142
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
  1143
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
  1144
  apply simp_all
haftmann@25230
  1145
  done
haftmann@25230
  1146
haftmann@25230
  1147
lemma mult_right_mono_neg:
haftmann@25230
  1148
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
  1149
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
  1150
  apply simp_all
haftmann@25230
  1151
  done
haftmann@25230
  1152
huffman@30692
  1153
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
  1154
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
  1155
haftmann@25230
  1156
lemma split_mult_pos_le:
haftmann@25230
  1157
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@56536
  1158
by (auto simp add: mult_nonpos_nonpos)
haftmann@25186
  1159
haftmann@25186
  1160
end
paulson@14270
  1161
haftmann@35028
  1162
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
  1163
begin
haftmann@25304
  1164
haftmann@35028
  1165
subclass ordered_ring ..
haftmann@35028
  1166
haftmann@35028
  1167
subclass ordered_ab_group_add_abs
haftmann@28823
  1168
proof
haftmann@25304
  1169
  fix a b
haftmann@25304
  1170
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@54230
  1171
    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
  1172
qed (auto simp add: abs_if)
haftmann@25304
  1173
huffman@35631
  1174
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
  1175
  using linear [of 0 a]
nipkow@56536
  1176
  by (auto simp add: mult_nonpos_nonpos)
huffman@35631
  1177
huffman@35631
  1178
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
  1179
  by (simp add: not_less)
huffman@35631
  1180
haftmann@25304
  1181
end
obua@23521
  1182
haftmann@35043
  1183
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
  1184
  + ordered_ab_group_add + abs_if
haftmann@25230
  1185
begin
paulson@14348
  1186
haftmann@35028
  1187
subclass linordered_ring ..
haftmann@25304
  1188
huffman@30692
  1189
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
  1190
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
  1191
huffman@30692
  1192
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
  1193
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
  1194
huffman@30692
  1195
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
  1196
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
  1197
haftmann@25917
  1198
subclass ring_no_zero_divisors
haftmann@28823
  1199
proof
haftmann@25917
  1200
  fix a b
haftmann@25917
  1201
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
  1202
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
  1203
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
  1204
  proof (cases "a < 0")
haftmann@25917
  1205
    case True note A' = this
haftmann@25917
  1206
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1207
      case True with A'
haftmann@25917
  1208
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
  1209
    next
haftmann@25917
  1210
      case False with B have "0 < b" by auto
haftmann@25917
  1211
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
  1212
    qed
haftmann@25917
  1213
  next
haftmann@25917
  1214
    case False with A have A': "0 < a" by auto
haftmann@25917
  1215
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1216
      case True with A'
haftmann@25917
  1217
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
  1218
    next
haftmann@25917
  1219
      case False with B have "0 < b" by auto
nipkow@56544
  1220
      with A' show ?thesis by auto
haftmann@25917
  1221
    qed
haftmann@25917
  1222
  qed
haftmann@25917
  1223
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
  1224
qed
haftmann@25304
  1225
hoelzl@56480
  1226
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
hoelzl@56480
  1227
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
nipkow@56544
  1228
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
huffman@22990
  1229
hoelzl@56480
  1230
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
  1231
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
  1232
paulson@14265
  1233
lemma mult_less_0_iff:
haftmann@25917
  1234
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
  1235
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
  1236
  apply force
haftmann@25917
  1237
  done
paulson@14265
  1238
paulson@14265
  1239
lemma mult_le_0_iff:
haftmann@25917
  1240
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
  1241
  apply (insert zero_le_mult_iff [of "-a" b]) 
huffman@35216
  1242
  apply force
haftmann@25917
  1243
  done
haftmann@25917
  1244
haftmann@26193
  1245
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
  1246
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
  1247
haftmann@26193
  1248
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
  1249
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
  1250
haftmann@26193
  1251
lemma mult_less_cancel_right_disj:
haftmann@26193
  1252
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1253
  apply (cases "c = 0")
haftmann@26193
  1254
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
  1255
                      mult_strict_right_mono_neg)
haftmann@26193
  1256
  apply (auto simp add: not_less 
haftmann@26193
  1257
                      not_le [symmetric, of "a*c"]
haftmann@26193
  1258
                      not_le [symmetric, of a])
haftmann@26193
  1259
  apply (erule_tac [!] notE)
haftmann@26193
  1260
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
  1261
                      mult_right_mono_neg)
haftmann@26193
  1262
  done
haftmann@26193
  1263
haftmann@26193
  1264
lemma mult_less_cancel_left_disj:
haftmann@26193
  1265
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1266
  apply (cases "c = 0")
haftmann@26193
  1267
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
  1268
                      mult_strict_left_mono_neg)
haftmann@26193
  1269
  apply (auto simp add: not_less 
haftmann@26193
  1270
                      not_le [symmetric, of "c*a"]
haftmann@26193
  1271
                      not_le [symmetric, of a])
haftmann@26193
  1272
  apply (erule_tac [!] notE)
haftmann@26193
  1273
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
  1274
                      mult_left_mono_neg)
haftmann@26193
  1275
  done
haftmann@26193
  1276
haftmann@26193
  1277
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
  1278
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
  1279
haftmann@26193
  1280
lemma mult_less_cancel_right:
haftmann@26193
  1281
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1282
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
  1283
haftmann@26193
  1284
lemma mult_less_cancel_left:
haftmann@26193
  1285
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1286
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
  1287
haftmann@26193
  1288
lemma mult_le_cancel_right:
haftmann@26193
  1289
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1290
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
  1291
haftmann@26193
  1292
lemma mult_le_cancel_left:
haftmann@26193
  1293
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1294
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
  1295
nipkow@30649
  1296
lemma mult_le_cancel_left_pos:
nipkow@30649
  1297
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
  1298
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1299
nipkow@30649
  1300
lemma mult_le_cancel_left_neg:
nipkow@30649
  1301
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
  1302
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1303
nipkow@30649
  1304
lemma mult_less_cancel_left_pos:
nipkow@30649
  1305
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
  1306
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1307
nipkow@30649
  1308
lemma mult_less_cancel_left_neg:
nipkow@30649
  1309
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
  1310
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1311
haftmann@25917
  1312
end
paulson@14265
  1313
huffman@30692
  1314
lemmas mult_sign_intros =
huffman@30692
  1315
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
  1316
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
  1317
  mult_pos_pos mult_pos_neg
huffman@30692
  1318
  mult_neg_pos mult_neg_neg
haftmann@25230
  1319
haftmann@35028
  1320
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
  1321
begin
haftmann@25230
  1322
haftmann@35028
  1323
subclass ordered_ring ..
haftmann@35028
  1324
subclass ordered_cancel_comm_semiring ..
haftmann@25230
  1325
haftmann@25267
  1326
end
haftmann@25230
  1327
haftmann@59833
  1328
class linordered_semidom = semidom + linordered_comm_semiring_strict +
haftmann@25230
  1329
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
  1330
begin
haftmann@25230
  1331
haftmann@25230
  1332
lemma pos_add_strict:
haftmann@25230
  1333
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
  1334
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
  1335
haftmann@26193
  1336
lemma zero_le_one [simp]: "0 \<le> 1"
nipkow@29667
  1337
by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
  1338
haftmann@26193
  1339
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
nipkow@29667
  1340
by (simp add: not_le) 
haftmann@26193
  1341
haftmann@26193
  1342
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
nipkow@29667
  1343
by (simp add: not_less) 
haftmann@26193
  1344
haftmann@26193
  1345
lemma less_1_mult:
haftmann@26193
  1346
  assumes "1 < m" and "1 < n"
haftmann@26193
  1347
  shows "1 < m * n"
haftmann@26193
  1348
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
  1349
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
  1350
hoelzl@59000
  1351
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
hoelzl@59000
  1352
  using mult_left_mono[of c 1 a] by simp
hoelzl@59000
  1353
hoelzl@59000
  1354
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
hoelzl@59000
  1355
  using mult_mono[of a 1 b 1] by simp
hoelzl@59000
  1356
haftmann@25230
  1357
end
haftmann@25230
  1358
haftmann@35028
  1359
class linordered_idom = comm_ring_1 +
haftmann@35028
  1360
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1361
  abs_if + sgn_if
haftmann@25917
  1362
begin
haftmann@25917
  1363
hoelzl@36622
  1364
subclass linordered_semiring_1_strict ..
haftmann@35043
  1365
subclass linordered_ring_strict ..
haftmann@35028
  1366
subclass ordered_comm_ring ..
huffman@27516
  1367
subclass idom ..
haftmann@25917
  1368
haftmann@35028
  1369
subclass linordered_semidom
haftmann@28823
  1370
proof
haftmann@26193
  1371
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1372
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
  1373
qed 
haftmann@25917
  1374
haftmann@35028
  1375
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1376
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1377
  using assms by (rule neqE)
haftmann@26193
  1378
haftmann@26274
  1379
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1380
haftmann@26274
  1381
lemma mult_le_cancel_right1:
haftmann@26274
  1382
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1383
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1384
haftmann@26274
  1385
lemma mult_le_cancel_right2:
haftmann@26274
  1386
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1387
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1388
haftmann@26274
  1389
lemma mult_le_cancel_left1:
haftmann@26274
  1390
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1391
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1392
haftmann@26274
  1393
lemma mult_le_cancel_left2:
haftmann@26274
  1394
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1395
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1396
haftmann@26274
  1397
lemma mult_less_cancel_right1:
haftmann@26274
  1398
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1399
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1400
haftmann@26274
  1401
lemma mult_less_cancel_right2:
haftmann@26274
  1402
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1403
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1404
haftmann@26274
  1405
lemma mult_less_cancel_left1:
haftmann@26274
  1406
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1407
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1408
haftmann@26274
  1409
lemma mult_less_cancel_left2:
haftmann@26274
  1410
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1411
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1412
haftmann@27651
  1413
lemma sgn_sgn [simp]:
haftmann@27651
  1414
  "sgn (sgn a) = sgn a"
nipkow@29700
  1415
unfolding sgn_if by simp
haftmann@27651
  1416
haftmann@27651
  1417
lemma sgn_0_0:
haftmann@27651
  1418
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1419
unfolding sgn_if by simp
haftmann@27651
  1420
haftmann@27651
  1421
lemma sgn_1_pos:
haftmann@27651
  1422
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1423
unfolding sgn_if by simp
haftmann@27651
  1424
haftmann@27651
  1425
lemma sgn_1_neg:
haftmann@27651
  1426
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1427
unfolding sgn_if by auto
haftmann@27651
  1428
haftmann@29940
  1429
lemma sgn_pos [simp]:
haftmann@29940
  1430
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1431
unfolding sgn_1_pos .
haftmann@29940
  1432
haftmann@29940
  1433
lemma sgn_neg [simp]:
haftmann@29940
  1434
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1435
unfolding sgn_1_neg .
haftmann@29940
  1436
haftmann@27651
  1437
lemma sgn_times:
haftmann@27651
  1438
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1439
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1440
haftmann@36301
  1441
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1442
unfolding sgn_if abs_if by auto
nipkow@29700
  1443
haftmann@29940
  1444
lemma sgn_greater [simp]:
haftmann@29940
  1445
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1446
  unfolding sgn_if by auto
haftmann@29940
  1447
haftmann@29940
  1448
lemma sgn_less [simp]:
haftmann@29940
  1449
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1450
  unfolding sgn_if by auto
haftmann@29940
  1451
haftmann@36301
  1452
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1453
  by (simp add: abs_if)
huffman@29949
  1454
haftmann@36301
  1455
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1456
  by (simp add: abs_if)
haftmann@29653
  1457
nipkow@33676
  1458
lemma dvd_if_abs_eq:
haftmann@36301
  1459
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1460
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1461
huffman@55912
  1462
text {* The following lemmas can be proven in more general structures, but
haftmann@54489
  1463
are dangerous as simp rules in absence of @{thm neg_equal_zero}, 
haftmann@54489
  1464
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
haftmann@54489
  1465
haftmann@54489
  1466
lemma equation_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1467
  "1 = - a \<longleftrightarrow> a = - 1"
haftmann@54489
  1468
  by (fact equation_minus_iff)
haftmann@54489
  1469
haftmann@54489
  1470
lemma minus_equation_iff_1 [simp, no_atp]:
haftmann@54489
  1471
  "- a = 1 \<longleftrightarrow> a = - 1"
haftmann@54489
  1472
  by (subst minus_equation_iff, auto)
haftmann@54489
  1473
haftmann@54489
  1474
lemma le_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1475
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
haftmann@54489
  1476
  by (fact le_minus_iff)
haftmann@54489
  1477
haftmann@54489
  1478
lemma minus_le_iff_1 [simp, no_atp]:
haftmann@54489
  1479
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
haftmann@54489
  1480
  by (fact minus_le_iff)
haftmann@54489
  1481
haftmann@54489
  1482
lemma less_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1483
  "1 < - b \<longleftrightarrow> b < - 1"
haftmann@54489
  1484
  by (fact less_minus_iff)
haftmann@54489
  1485
haftmann@54489
  1486
lemma minus_less_iff_1 [simp, no_atp]:
haftmann@54489
  1487
  "- a < 1 \<longleftrightarrow> - 1 < a"
haftmann@54489
  1488
  by (fact minus_less_iff)
haftmann@54489
  1489
haftmann@25917
  1490
end
haftmann@25230
  1491
haftmann@26274
  1492
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1493
blanchet@54147
  1494
lemmas mult_compare_simps =
paulson@15234
  1495
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1496
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1497
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1498
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1499
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1500
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1501
    mult_cancel_right mult_cancel_left
paulson@15234
  1502
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1503
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1504
haftmann@36301
  1505
text {* Reasoning about inequalities with division *}
avigad@16775
  1506
haftmann@35028
  1507
context linordered_semidom
haftmann@25193
  1508
begin
haftmann@25193
  1509
haftmann@25193
  1510
lemma less_add_one: "a < a + 1"
paulson@14293
  1511
proof -
haftmann@25193
  1512
  have "a + 0 < a + 1"
nipkow@23482
  1513
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1514
  thus ?thesis by simp
paulson@14293
  1515
qed
paulson@14293
  1516
haftmann@25193
  1517
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1518
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1519
haftmann@25193
  1520
end
paulson@14365
  1521
haftmann@36301
  1522
context linordered_idom
haftmann@36301
  1523
begin
paulson@15234
  1524
haftmann@36301
  1525
lemma mult_right_le_one_le:
haftmann@36301
  1526
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@59833
  1527
  by (rule mult_left_le)
haftmann@36301
  1528
haftmann@36301
  1529
lemma mult_left_le_one_le:
haftmann@36301
  1530
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1531
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1532
haftmann@36301
  1533
end
haftmann@36301
  1534
haftmann@36301
  1535
text {* Absolute Value *}
paulson@14293
  1536
haftmann@35028
  1537
context linordered_idom
haftmann@25304
  1538
begin
haftmann@25304
  1539
haftmann@36301
  1540
lemma mult_sgn_abs:
haftmann@36301
  1541
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1542
  unfolding abs_if sgn_if by auto
haftmann@25304
  1543
haftmann@36301
  1544
lemma abs_one [simp]:
haftmann@36301
  1545
  "\<bar>1\<bar> = 1"
huffman@44921
  1546
  by (simp add: abs_if)
haftmann@36301
  1547
haftmann@25304
  1548
end
nipkow@24491
  1549
haftmann@35028
  1550
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1551
  assumes abs_eq_mult:
haftmann@25304
  1552
    "(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
  1553
haftmann@35028
  1554
context linordered_idom
haftmann@30961
  1555
begin
haftmann@30961
  1556
haftmann@35028
  1557
subclass ordered_ring_abs proof
huffman@35216
  1558
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1559
haftmann@30961
  1560
lemma abs_mult:
haftmann@36301
  1561
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" 
haftmann@30961
  1562
  by (rule abs_eq_mult) auto
haftmann@30961
  1563
haftmann@30961
  1564
lemma abs_mult_self:
haftmann@36301
  1565
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
haftmann@30961
  1566
  by (simp add: abs_if) 
haftmann@30961
  1567
paulson@14294
  1568
lemma abs_mult_less:
haftmann@36301
  1569
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1570
proof -
haftmann@36301
  1571
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1572
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1573
  assume "\<bar>b\<bar> < d"
paulson@14294
  1574
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1575
qed
paulson@14293
  1576
haftmann@36301
  1577
lemma abs_less_iff:
haftmann@36301
  1578
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" 
haftmann@36301
  1579
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1580
haftmann@36301
  1581
lemma abs_mult_pos:
haftmann@36301
  1582
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1583
  by (simp add: abs_mult)
haftmann@36301
  1584
hoelzl@51520
  1585
lemma abs_diff_less_iff:
hoelzl@51520
  1586
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
hoelzl@51520
  1587
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
hoelzl@51520
  1588
lp15@59865
  1589
lemma abs_diff_le_iff:
lp15@59865
  1590
   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
lp15@59865
  1591
  by (auto simp add: diff_le_eq ac_simps abs_le_iff)
lp15@59865
  1592
haftmann@36301
  1593
end
avigad@16775
  1594
haftmann@59557
  1595
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
haftmann@59557
  1596
haftmann@52435
  1597
code_identifier
haftmann@52435
  1598
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1599
paulson@14265
  1600
end