src/HOL/Rings.thy
author haftmann
Fri Jun 12 08:53:23 2015 +0200 (2015-06-12)
changeset 60429 d3d1e185cd63
parent 60353 838025c6e278
child 60516 0826b7025d07
permissions -rw-r--r--
uniform _ div _ as infix syntax for ring division
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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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   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@58952
   443
class ring_no_zero_divisors = ring + semiring_no_zero_divisors
haftmann@58952
   444
begin
haftmann@58952
   445
haftmann@26193
   446
text{*Cancellation of equalities with a common factor*}
blanchet@54147
   447
lemma mult_cancel_right [simp]:
haftmann@26193
   448
  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@26193
   449
proof -
haftmann@26193
   450
  have "(a * c = b * c) = ((a - b) * c = 0)"
huffman@35216
   451
    by (simp add: algebra_simps)
huffman@35216
   452
  thus ?thesis by (simp add: disj_commute)
haftmann@26193
   453
qed
haftmann@26193
   454
blanchet@54147
   455
lemma mult_cancel_left [simp]:
haftmann@26193
   456
  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@26193
   457
proof -
haftmann@26193
   458
  have "(c * a = c * b) = (c * (a - b) = 0)"
huffman@35216
   459
    by (simp add: algebra_simps)
huffman@35216
   460
  thus ?thesis by simp
haftmann@26193
   461
qed
haftmann@26193
   462
haftmann@58952
   463
lemma mult_left_cancel:
haftmann@58952
   464
  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
haftmann@58952
   465
  by simp 
lp15@56217
   466
haftmann@58952
   467
lemma mult_right_cancel:
haftmann@58952
   468
  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
haftmann@58952
   469
  by simp 
lp15@56217
   470
haftmann@25230
   471
end
huffman@22990
   472
huffman@23544
   473
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   474
begin
haftmann@26274
   475
huffman@36970
   476
lemma square_eq_1_iff:
huffman@36821
   477
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
huffman@36821
   478
proof -
huffman@36821
   479
  have "(x - 1) * (x + 1) = x * x - 1"
huffman@36821
   480
    by (simp add: algebra_simps)
huffman@36821
   481
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   482
    by simp
huffman@36821
   483
  thus ?thesis
huffman@36821
   484
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   485
qed
huffman@36821
   486
haftmann@26274
   487
lemma mult_cancel_right1 [simp]:
haftmann@26274
   488
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   489
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   490
haftmann@26274
   491
lemma mult_cancel_right2 [simp]:
haftmann@26274
   492
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   493
by (insert mult_cancel_right [of a c 1], simp)
haftmann@26274
   494
 
haftmann@26274
   495
lemma mult_cancel_left1 [simp]:
haftmann@26274
   496
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   497
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   498
haftmann@26274
   499
lemma mult_cancel_left2 [simp]:
haftmann@26274
   500
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   501
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   502
haftmann@26274
   503
end
huffman@22990
   504
haftmann@59910
   505
class semidom = comm_semiring_1_diff_distrib + semiring_no_zero_divisors
haftmann@59833
   506
haftmann@59833
   507
class idom = comm_ring_1 + semiring_no_zero_divisors
haftmann@25186
   508
begin
paulson@14421
   509
haftmann@59833
   510
subclass semidom ..
haftmann@59833
   511
huffman@27516
   512
subclass ring_1_no_zero_divisors ..
huffman@22990
   513
huffman@29981
   514
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   515
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   516
proof -
huffman@29981
   517
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   518
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   519
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   520
    unfolding dvd_def by simp
huffman@29981
   521
  finally show ?thesis .
huffman@29981
   522
qed
huffman@29981
   523
huffman@29981
   524
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   525
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   526
proof -
huffman@29981
   527
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   528
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   529
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   530
    unfolding dvd_def by simp
huffman@29981
   531
  finally show ?thesis .
huffman@29981
   532
qed
huffman@29981
   533
haftmann@59833
   534
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)"
haftmann@59833
   535
proof
haftmann@59833
   536
  assume "a * a = b * b"
haftmann@59833
   537
  then have "(a - b) * (a + b) = 0"
haftmann@59833
   538
    by (simp add: algebra_simps)
haftmann@59833
   539
  then show "a = b \<or> a = - b"
haftmann@59833
   540
    by (simp add: eq_neg_iff_add_eq_0)
haftmann@59833
   541
next
haftmann@59833
   542
  assume "a = b \<or> a = - b"
haftmann@59833
   543
  then show "a * a = b * b" by auto
haftmann@59833
   544
qed
haftmann@59833
   545
haftmann@25186
   546
end
haftmann@25152
   547
haftmann@35302
   548
text {*
haftmann@35302
   549
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   550
  \begin{itemize}
haftmann@35302
   551
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35302
   552
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   553
  \end{itemize}
haftmann@35302
   554
  Most of the used notions can also be looked up in 
haftmann@35302
   555
  \begin{itemize}
wenzelm@54703
   556
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35302
   557
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   558
  \end{itemize}
haftmann@35302
   559
*}
haftmann@35302
   560
haftmann@60353
   561
class divide =
haftmann@60429
   562
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
haftmann@60353
   563
haftmann@60353
   564
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60353
   565
haftmann@60353
   566
context semiring
haftmann@60353
   567
begin
haftmann@60353
   568
haftmann@60353
   569
lemma [field_simps]:
haftmann@60429
   570
  shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
haftmann@60429
   571
    and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
haftmann@60353
   572
  by (rule distrib_left distrib_right)+
haftmann@60353
   573
haftmann@60353
   574
end
haftmann@60353
   575
haftmann@60353
   576
context ring
haftmann@60353
   577
begin
haftmann@60353
   578
haftmann@60353
   579
lemma [field_simps]:
haftmann@60429
   580
  shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
haftmann@60429
   581
    and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
haftmann@60353
   582
  by (rule left_diff_distrib right_diff_distrib)+
haftmann@60353
   583
haftmann@60353
   584
end
haftmann@60353
   585
haftmann@60353
   586
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60353
   587
haftmann@60353
   588
class semidom_divide = semidom + divide +
haftmann@60429
   589
  assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
haftmann@60429
   590
  assumes divide_zero [simp]: "a div 0 = 0"
haftmann@60353
   591
begin
haftmann@60353
   592
haftmann@60353
   593
lemma nonzero_mult_divide_cancel_left [simp]:
haftmann@60429
   594
  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
haftmann@60353
   595
  using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
haftmann@60353
   596
haftmann@60353
   597
end
haftmann@60353
   598
haftmann@60353
   599
class idom_divide = idom + semidom_divide
haftmann@60353
   600
haftmann@38642
   601
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
   602
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
   603
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   604
begin
haftmann@25230
   605
haftmann@25230
   606
lemma mult_mono:
haftmann@38642
   607
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   608
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   609
apply (erule mult_left_mono, assumption)
haftmann@25230
   610
done
haftmann@25230
   611
haftmann@25230
   612
lemma mult_mono':
haftmann@38642
   613
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   614
apply (rule mult_mono)
haftmann@25230
   615
apply (fast intro: order_trans)+
haftmann@25230
   616
done
haftmann@25230
   617
haftmann@25230
   618
end
krauss@21199
   619
haftmann@38642
   620
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
   621
begin
paulson@14268
   622
huffman@27516
   623
subclass semiring_0_cancel ..
obua@23521
   624
nipkow@56536
   625
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   626
using mult_left_mono [of 0 b a] by simp
haftmann@25230
   627
haftmann@25230
   628
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   629
using mult_left_mono [of b 0 a] by simp
huffman@30692
   630
huffman@30692
   631
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   632
using mult_right_mono [of a 0 b] by simp
huffman@30692
   633
huffman@30692
   634
text {* Legacy - use @{text mult_nonpos_nonneg} *}
haftmann@25230
   635
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
haftmann@36301
   636
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
   637
haftmann@26234
   638
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
nipkow@29667
   639
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   640
haftmann@25230
   641
end
haftmann@25230
   642
haftmann@38642
   643
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
   644
begin
haftmann@25230
   645
haftmann@35028
   646
subclass ordered_cancel_semiring ..
haftmann@35028
   647
haftmann@35028
   648
subclass ordered_comm_monoid_add ..
haftmann@25304
   649
haftmann@25230
   650
lemma mult_left_less_imp_less:
haftmann@25230
   651
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   652
by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   653
 
haftmann@25230
   654
lemma mult_right_less_imp_less:
haftmann@25230
   655
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   656
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   657
haftmann@25186
   658
end
haftmann@25152
   659
haftmann@35043
   660
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
   661
begin
hoelzl@36622
   662
hoelzl@36622
   663
lemma convex_bound_le:
hoelzl@36622
   664
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   665
  shows "u * x + v * y \<le> a"
hoelzl@36622
   666
proof-
hoelzl@36622
   667
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
   668
    by (simp add: add_mono mult_left_mono)
webertj@49962
   669
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   670
qed
hoelzl@36622
   671
hoelzl@36622
   672
end
haftmann@35043
   673
haftmann@35043
   674
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
   675
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   676
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   677
begin
paulson@14341
   678
huffman@27516
   679
subclass semiring_0_cancel ..
obua@14940
   680
haftmann@35028
   681
subclass linordered_semiring
haftmann@28823
   682
proof
huffman@23550
   683
  fix a b c :: 'a
huffman@23550
   684
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   685
  from A show "c * a \<le> c * b"
haftmann@25186
   686
    unfolding le_less
haftmann@25186
   687
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   688
  from A show "a * c \<le> b * c"
haftmann@25152
   689
    unfolding le_less
haftmann@25186
   690
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   691
qed
haftmann@25152
   692
haftmann@25230
   693
lemma mult_left_le_imp_le:
haftmann@25230
   694
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   695
by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   696
 
haftmann@25230
   697
lemma mult_right_le_imp_le:
haftmann@25230
   698
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   699
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   700
nipkow@56544
   701
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
   702
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
   703
huffman@30692
   704
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
   705
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
   706
huffman@30692
   707
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
   708
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
   709
huffman@30692
   710
text {* Legacy - use @{text mult_neg_pos} *}
huffman@30692
   711
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
haftmann@36301
   712
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
   713
haftmann@25230
   714
lemma zero_less_mult_pos:
haftmann@25230
   715
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   716
apply (cases "b\<le>0")
haftmann@25230
   717
 apply (auto simp add: le_less not_less)
huffman@30692
   718
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
   719
 apply (auto dest: less_not_sym)
haftmann@25230
   720
done
haftmann@25230
   721
haftmann@25230
   722
lemma zero_less_mult_pos2:
haftmann@25230
   723
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   724
apply (cases "b\<le>0")
haftmann@25230
   725
 apply (auto simp add: le_less not_less)
huffman@30692
   726
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
   727
 apply (auto dest: less_not_sym)
haftmann@25230
   728
done
haftmann@25230
   729
haftmann@26193
   730
text{*Strict monotonicity in both arguments*}
haftmann@26193
   731
lemma mult_strict_mono:
haftmann@26193
   732
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
   733
  shows "a * c < b * d"
haftmann@26193
   734
  using assms apply (cases "c=0")
nipkow@56544
   735
  apply (simp)
haftmann@26193
   736
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
   737
  apply (force simp add: le_less)
haftmann@26193
   738
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
   739
  done
haftmann@26193
   740
haftmann@26193
   741
text{*This weaker variant has more natural premises*}
haftmann@26193
   742
lemma mult_strict_mono':
haftmann@26193
   743
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
   744
  shows "a * c < b * d"
nipkow@29667
   745
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
   746
haftmann@26193
   747
lemma mult_less_le_imp_less:
haftmann@26193
   748
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
   749
  shows "a * c < b * d"
haftmann@26193
   750
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
   751
  apply (erule less_le_trans)
haftmann@26193
   752
  apply (erule mult_left_mono)
haftmann@26193
   753
  apply simp
haftmann@26193
   754
  apply (erule mult_strict_right_mono)
haftmann@26193
   755
  apply assumption
haftmann@26193
   756
  done
haftmann@26193
   757
haftmann@26193
   758
lemma mult_le_less_imp_less:
haftmann@26193
   759
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
   760
  shows "a * c < b * d"
haftmann@26193
   761
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
   762
  apply (erule le_less_trans)
haftmann@26193
   763
  apply (erule mult_strict_left_mono)
haftmann@26193
   764
  apply simp
haftmann@26193
   765
  apply (erule mult_right_mono)
haftmann@26193
   766
  apply simp
haftmann@26193
   767
  done
haftmann@26193
   768
haftmann@25230
   769
end
haftmann@25230
   770
haftmann@35097
   771
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
   772
begin
hoelzl@36622
   773
hoelzl@36622
   774
subclass linordered_semiring_1 ..
hoelzl@36622
   775
hoelzl@36622
   776
lemma convex_bound_lt:
hoelzl@36622
   777
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   778
  shows "u * x + v * y < a"
hoelzl@36622
   779
proof -
hoelzl@36622
   780
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
   781
    by (cases "u = 0")
hoelzl@36622
   782
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
webertj@49962
   783
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   784
qed
hoelzl@36622
   785
hoelzl@36622
   786
end
haftmann@33319
   787
haftmann@38642
   788
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add + 
haftmann@38642
   789
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
   790
begin
haftmann@25152
   791
haftmann@35028
   792
subclass ordered_semiring
haftmann@28823
   793
proof
krauss@21199
   794
  fix a b c :: 'a
huffman@23550
   795
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
   796
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
haftmann@57512
   797
  thus "a * c \<le> b * c" by (simp only: mult.commute)
krauss@21199
   798
qed
paulson@14265
   799
haftmann@25267
   800
end
haftmann@25267
   801
haftmann@38642
   802
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
   803
begin
paulson@14265
   804
haftmann@38642
   805
subclass comm_semiring_0_cancel ..
haftmann@35028
   806
subclass ordered_comm_semiring ..
haftmann@35028
   807
subclass ordered_cancel_semiring ..
haftmann@25267
   808
haftmann@25267
   809
end
haftmann@25267
   810
haftmann@35028
   811
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
   812
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
   813
begin
haftmann@25267
   814
haftmann@35043
   815
subclass linordered_semiring_strict
haftmann@28823
   816
proof
huffman@23550
   817
  fix a b c :: 'a
huffman@23550
   818
  assume "a < b" "0 < c"
haftmann@38642
   819
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
haftmann@57512
   820
  thus "a * c < b * c" by (simp only: mult.commute)
huffman@23550
   821
qed
paulson@14272
   822
haftmann@35028
   823
subclass ordered_cancel_comm_semiring
haftmann@28823
   824
proof
huffman@23550
   825
  fix a b c :: 'a
huffman@23550
   826
  assume "a \<le> b" "0 \<le> c"
huffman@23550
   827
  thus "c * a \<le> c * b"
haftmann@25186
   828
    unfolding le_less
haftmann@26193
   829
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   830
qed
paulson@14272
   831
haftmann@25267
   832
end
haftmann@25230
   833
haftmann@35028
   834
class ordered_ring = ring + ordered_cancel_semiring 
haftmann@25267
   835
begin
haftmann@25230
   836
haftmann@35028
   837
subclass ordered_ab_group_add ..
paulson@14270
   838
haftmann@25230
   839
lemma less_add_iff1:
haftmann@25230
   840
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
   841
by (simp add: algebra_simps)
haftmann@25230
   842
haftmann@25230
   843
lemma less_add_iff2:
haftmann@25230
   844
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
   845
by (simp add: algebra_simps)
haftmann@25230
   846
haftmann@25230
   847
lemma le_add_iff1:
haftmann@25230
   848
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
   849
by (simp add: algebra_simps)
haftmann@25230
   850
haftmann@25230
   851
lemma le_add_iff2:
haftmann@25230
   852
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
   853
by (simp add: algebra_simps)
haftmann@25230
   854
haftmann@25230
   855
lemma mult_left_mono_neg:
haftmann@25230
   856
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
   857
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
   858
  apply simp_all
haftmann@25230
   859
  done
haftmann@25230
   860
haftmann@25230
   861
lemma mult_right_mono_neg:
haftmann@25230
   862
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
   863
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
   864
  apply simp_all
haftmann@25230
   865
  done
haftmann@25230
   866
huffman@30692
   867
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   868
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
   869
haftmann@25230
   870
lemma split_mult_pos_le:
haftmann@25230
   871
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@56536
   872
by (auto simp add: mult_nonpos_nonpos)
haftmann@25186
   873
haftmann@25186
   874
end
paulson@14270
   875
haftmann@35028
   876
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
   877
begin
haftmann@25304
   878
haftmann@35028
   879
subclass ordered_ring ..
haftmann@35028
   880
haftmann@35028
   881
subclass ordered_ab_group_add_abs
haftmann@28823
   882
proof
haftmann@25304
   883
  fix a b
haftmann@25304
   884
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@54230
   885
    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
   886
qed (auto simp add: abs_if)
haftmann@25304
   887
huffman@35631
   888
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
   889
  using linear [of 0 a]
nipkow@56536
   890
  by (auto simp add: mult_nonpos_nonpos)
huffman@35631
   891
huffman@35631
   892
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
   893
  by (simp add: not_less)
huffman@35631
   894
haftmann@25304
   895
end
obua@23521
   896
haftmann@35043
   897
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
   898
  + ordered_ab_group_add + abs_if
haftmann@25230
   899
begin
paulson@14348
   900
haftmann@35028
   901
subclass linordered_ring ..
haftmann@25304
   902
huffman@30692
   903
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
   904
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
   905
huffman@30692
   906
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
   907
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
   908
huffman@30692
   909
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
   910
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
   911
haftmann@25917
   912
subclass ring_no_zero_divisors
haftmann@28823
   913
proof
haftmann@25917
   914
  fix a b
haftmann@25917
   915
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
   916
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
   917
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
   918
  proof (cases "a < 0")
haftmann@25917
   919
    case True note A' = this
haftmann@25917
   920
    show ?thesis proof (cases "b < 0")
haftmann@25917
   921
      case True with A'
haftmann@25917
   922
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
   923
    next
haftmann@25917
   924
      case False with B have "0 < b" by auto
haftmann@25917
   925
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
   926
    qed
haftmann@25917
   927
  next
haftmann@25917
   928
    case False with A have A': "0 < a" by auto
haftmann@25917
   929
    show ?thesis proof (cases "b < 0")
haftmann@25917
   930
      case True with A'
haftmann@25917
   931
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
   932
    next
haftmann@25917
   933
      case False with B have "0 < b" by auto
nipkow@56544
   934
      with A' show ?thesis by auto
haftmann@25917
   935
    qed
haftmann@25917
   936
  qed
haftmann@25917
   937
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
   938
qed
haftmann@25304
   939
hoelzl@56480
   940
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
hoelzl@56480
   941
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
nipkow@56544
   942
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
huffman@22990
   943
hoelzl@56480
   944
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
   945
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
   946
paulson@14265
   947
lemma mult_less_0_iff:
haftmann@25917
   948
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
   949
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
   950
  apply force
haftmann@25917
   951
  done
paulson@14265
   952
paulson@14265
   953
lemma mult_le_0_iff:
haftmann@25917
   954
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
   955
  apply (insert zero_le_mult_iff [of "-a" b]) 
huffman@35216
   956
  apply force
haftmann@25917
   957
  done
haftmann@25917
   958
haftmann@26193
   959
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
   960
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
   961
haftmann@26193
   962
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
   963
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
   964
haftmann@26193
   965
lemma mult_less_cancel_right_disj:
haftmann@26193
   966
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   967
  apply (cases "c = 0")
haftmann@26193
   968
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
   969
                      mult_strict_right_mono_neg)
haftmann@26193
   970
  apply (auto simp add: not_less 
haftmann@26193
   971
                      not_le [symmetric, of "a*c"]
haftmann@26193
   972
                      not_le [symmetric, of a])
haftmann@26193
   973
  apply (erule_tac [!] notE)
haftmann@26193
   974
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
   975
                      mult_right_mono_neg)
haftmann@26193
   976
  done
haftmann@26193
   977
haftmann@26193
   978
lemma mult_less_cancel_left_disj:
haftmann@26193
   979
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   980
  apply (cases "c = 0")
haftmann@26193
   981
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
   982
                      mult_strict_left_mono_neg)
haftmann@26193
   983
  apply (auto simp add: not_less 
haftmann@26193
   984
                      not_le [symmetric, of "c*a"]
haftmann@26193
   985
                      not_le [symmetric, of a])
haftmann@26193
   986
  apply (erule_tac [!] notE)
haftmann@26193
   987
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
   988
                      mult_left_mono_neg)
haftmann@26193
   989
  done
haftmann@26193
   990
haftmann@26193
   991
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
   992
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
   993
haftmann@26193
   994
lemma mult_less_cancel_right:
haftmann@26193
   995
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   996
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
   997
haftmann@26193
   998
lemma mult_less_cancel_left:
haftmann@26193
   999
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1000
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
  1001
haftmann@26193
  1002
lemma mult_le_cancel_right:
haftmann@26193
  1003
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1004
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
  1005
haftmann@26193
  1006
lemma mult_le_cancel_left:
haftmann@26193
  1007
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1008
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
  1009
nipkow@30649
  1010
lemma mult_le_cancel_left_pos:
nipkow@30649
  1011
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
  1012
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1013
nipkow@30649
  1014
lemma mult_le_cancel_left_neg:
nipkow@30649
  1015
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
  1016
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1017
nipkow@30649
  1018
lemma mult_less_cancel_left_pos:
nipkow@30649
  1019
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
  1020
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1021
nipkow@30649
  1022
lemma mult_less_cancel_left_neg:
nipkow@30649
  1023
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
  1024
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1025
haftmann@25917
  1026
end
paulson@14265
  1027
huffman@30692
  1028
lemmas mult_sign_intros =
huffman@30692
  1029
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
  1030
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
  1031
  mult_pos_pos mult_pos_neg
huffman@30692
  1032
  mult_neg_pos mult_neg_neg
haftmann@25230
  1033
haftmann@35028
  1034
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
  1035
begin
haftmann@25230
  1036
haftmann@35028
  1037
subclass ordered_ring ..
haftmann@35028
  1038
subclass ordered_cancel_comm_semiring ..
haftmann@25230
  1039
haftmann@25267
  1040
end
haftmann@25230
  1041
haftmann@59833
  1042
class linordered_semidom = semidom + linordered_comm_semiring_strict +
haftmann@25230
  1043
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
  1044
begin
haftmann@25230
  1045
haftmann@25230
  1046
lemma pos_add_strict:
haftmann@25230
  1047
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
  1048
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
  1049
haftmann@26193
  1050
lemma zero_le_one [simp]: "0 \<le> 1"
nipkow@29667
  1051
by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
  1052
haftmann@26193
  1053
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
nipkow@29667
  1054
by (simp add: not_le) 
haftmann@26193
  1055
haftmann@26193
  1056
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
nipkow@29667
  1057
by (simp add: not_less) 
haftmann@26193
  1058
haftmann@26193
  1059
lemma less_1_mult:
haftmann@26193
  1060
  assumes "1 < m" and "1 < n"
haftmann@26193
  1061
  shows "1 < m * n"
haftmann@26193
  1062
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
  1063
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
  1064
hoelzl@59000
  1065
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
hoelzl@59000
  1066
  using mult_left_mono[of c 1 a] by simp
hoelzl@59000
  1067
hoelzl@59000
  1068
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
hoelzl@59000
  1069
  using mult_mono[of a 1 b 1] by simp
hoelzl@59000
  1070
haftmann@25230
  1071
end
haftmann@25230
  1072
haftmann@35028
  1073
class linordered_idom = comm_ring_1 +
haftmann@35028
  1074
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1075
  abs_if + sgn_if
haftmann@25917
  1076
begin
haftmann@25917
  1077
hoelzl@36622
  1078
subclass linordered_semiring_1_strict ..
haftmann@35043
  1079
subclass linordered_ring_strict ..
haftmann@35028
  1080
subclass ordered_comm_ring ..
huffman@27516
  1081
subclass idom ..
haftmann@25917
  1082
haftmann@35028
  1083
subclass linordered_semidom
haftmann@28823
  1084
proof
haftmann@26193
  1085
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1086
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
  1087
qed 
haftmann@25917
  1088
haftmann@35028
  1089
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1090
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1091
  using assms by (rule neqE)
haftmann@26193
  1092
haftmann@26274
  1093
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1094
haftmann@26274
  1095
lemma mult_le_cancel_right1:
haftmann@26274
  1096
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1097
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1098
haftmann@26274
  1099
lemma mult_le_cancel_right2:
haftmann@26274
  1100
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1101
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1102
haftmann@26274
  1103
lemma mult_le_cancel_left1:
haftmann@26274
  1104
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1105
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1106
haftmann@26274
  1107
lemma mult_le_cancel_left2:
haftmann@26274
  1108
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1109
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1110
haftmann@26274
  1111
lemma mult_less_cancel_right1:
haftmann@26274
  1112
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1113
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1114
haftmann@26274
  1115
lemma mult_less_cancel_right2:
haftmann@26274
  1116
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1117
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1118
haftmann@26274
  1119
lemma mult_less_cancel_left1:
haftmann@26274
  1120
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1121
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1122
haftmann@26274
  1123
lemma mult_less_cancel_left2:
haftmann@26274
  1124
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1125
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1126
haftmann@27651
  1127
lemma sgn_sgn [simp]:
haftmann@27651
  1128
  "sgn (sgn a) = sgn a"
nipkow@29700
  1129
unfolding sgn_if by simp
haftmann@27651
  1130
haftmann@27651
  1131
lemma sgn_0_0:
haftmann@27651
  1132
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1133
unfolding sgn_if by simp
haftmann@27651
  1134
haftmann@27651
  1135
lemma sgn_1_pos:
haftmann@27651
  1136
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1137
unfolding sgn_if by simp
haftmann@27651
  1138
haftmann@27651
  1139
lemma sgn_1_neg:
haftmann@27651
  1140
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1141
unfolding sgn_if by auto
haftmann@27651
  1142
haftmann@29940
  1143
lemma sgn_pos [simp]:
haftmann@29940
  1144
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1145
unfolding sgn_1_pos .
haftmann@29940
  1146
haftmann@29940
  1147
lemma sgn_neg [simp]:
haftmann@29940
  1148
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1149
unfolding sgn_1_neg .
haftmann@29940
  1150
haftmann@27651
  1151
lemma sgn_times:
haftmann@27651
  1152
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1153
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1154
haftmann@36301
  1155
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1156
unfolding sgn_if abs_if by auto
nipkow@29700
  1157
haftmann@29940
  1158
lemma sgn_greater [simp]:
haftmann@29940
  1159
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1160
  unfolding sgn_if by auto
haftmann@29940
  1161
haftmann@29940
  1162
lemma sgn_less [simp]:
haftmann@29940
  1163
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1164
  unfolding sgn_if by auto
haftmann@29940
  1165
haftmann@36301
  1166
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1167
  by (simp add: abs_if)
huffman@29949
  1168
haftmann@36301
  1169
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1170
  by (simp add: abs_if)
haftmann@29653
  1171
nipkow@33676
  1172
lemma dvd_if_abs_eq:
haftmann@36301
  1173
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1174
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1175
huffman@55912
  1176
text {* The following lemmas can be proven in more general structures, but
haftmann@54489
  1177
are dangerous as simp rules in absence of @{thm neg_equal_zero}, 
haftmann@54489
  1178
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
haftmann@54489
  1179
haftmann@54489
  1180
lemma equation_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1181
  "1 = - a \<longleftrightarrow> a = - 1"
haftmann@54489
  1182
  by (fact equation_minus_iff)
haftmann@54489
  1183
haftmann@54489
  1184
lemma minus_equation_iff_1 [simp, no_atp]:
haftmann@54489
  1185
  "- a = 1 \<longleftrightarrow> a = - 1"
haftmann@54489
  1186
  by (subst minus_equation_iff, auto)
haftmann@54489
  1187
haftmann@54489
  1188
lemma le_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1189
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
haftmann@54489
  1190
  by (fact le_minus_iff)
haftmann@54489
  1191
haftmann@54489
  1192
lemma minus_le_iff_1 [simp, no_atp]:
haftmann@54489
  1193
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
haftmann@54489
  1194
  by (fact minus_le_iff)
haftmann@54489
  1195
haftmann@54489
  1196
lemma less_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1197
  "1 < - b \<longleftrightarrow> b < - 1"
haftmann@54489
  1198
  by (fact less_minus_iff)
haftmann@54489
  1199
haftmann@54489
  1200
lemma minus_less_iff_1 [simp, no_atp]:
haftmann@54489
  1201
  "- a < 1 \<longleftrightarrow> - 1 < a"
haftmann@54489
  1202
  by (fact minus_less_iff)
haftmann@54489
  1203
haftmann@25917
  1204
end
haftmann@25230
  1205
haftmann@26274
  1206
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1207
blanchet@54147
  1208
lemmas mult_compare_simps =
paulson@15234
  1209
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1210
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1211
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1212
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1213
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1214
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1215
    mult_cancel_right mult_cancel_left
paulson@15234
  1216
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1217
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1218
haftmann@36301
  1219
text {* Reasoning about inequalities with division *}
avigad@16775
  1220
haftmann@35028
  1221
context linordered_semidom
haftmann@25193
  1222
begin
haftmann@25193
  1223
haftmann@25193
  1224
lemma less_add_one: "a < a + 1"
paulson@14293
  1225
proof -
haftmann@25193
  1226
  have "a + 0 < a + 1"
nipkow@23482
  1227
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1228
  thus ?thesis by simp
paulson@14293
  1229
qed
paulson@14293
  1230
haftmann@25193
  1231
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1232
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1233
haftmann@25193
  1234
end
paulson@14365
  1235
haftmann@36301
  1236
context linordered_idom
haftmann@36301
  1237
begin
paulson@15234
  1238
haftmann@36301
  1239
lemma mult_right_le_one_le:
haftmann@36301
  1240
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@59833
  1241
  by (rule mult_left_le)
haftmann@36301
  1242
haftmann@36301
  1243
lemma mult_left_le_one_le:
haftmann@36301
  1244
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1245
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1246
haftmann@36301
  1247
end
haftmann@36301
  1248
haftmann@36301
  1249
text {* Absolute Value *}
paulson@14293
  1250
haftmann@35028
  1251
context linordered_idom
haftmann@25304
  1252
begin
haftmann@25304
  1253
haftmann@36301
  1254
lemma mult_sgn_abs:
haftmann@36301
  1255
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1256
  unfolding abs_if sgn_if by auto
haftmann@25304
  1257
haftmann@36301
  1258
lemma abs_one [simp]:
haftmann@36301
  1259
  "\<bar>1\<bar> = 1"
huffman@44921
  1260
  by (simp add: abs_if)
haftmann@36301
  1261
haftmann@25304
  1262
end
nipkow@24491
  1263
haftmann@35028
  1264
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1265
  assumes abs_eq_mult:
haftmann@25304
  1266
    "(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
  1267
haftmann@35028
  1268
context linordered_idom
haftmann@30961
  1269
begin
haftmann@30961
  1270
haftmann@35028
  1271
subclass ordered_ring_abs proof
huffman@35216
  1272
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1273
haftmann@30961
  1274
lemma abs_mult:
haftmann@36301
  1275
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" 
haftmann@30961
  1276
  by (rule abs_eq_mult) auto
haftmann@30961
  1277
haftmann@30961
  1278
lemma abs_mult_self:
haftmann@36301
  1279
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
haftmann@30961
  1280
  by (simp add: abs_if) 
haftmann@30961
  1281
paulson@14294
  1282
lemma abs_mult_less:
haftmann@36301
  1283
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1284
proof -
haftmann@36301
  1285
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1286
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1287
  assume "\<bar>b\<bar> < d"
paulson@14294
  1288
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1289
qed
paulson@14293
  1290
haftmann@36301
  1291
lemma abs_less_iff:
haftmann@36301
  1292
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" 
haftmann@36301
  1293
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1294
haftmann@36301
  1295
lemma abs_mult_pos:
haftmann@36301
  1296
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1297
  by (simp add: abs_mult)
haftmann@36301
  1298
hoelzl@51520
  1299
lemma abs_diff_less_iff:
hoelzl@51520
  1300
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
hoelzl@51520
  1301
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
hoelzl@51520
  1302
lp15@59865
  1303
lemma abs_diff_le_iff:
lp15@59865
  1304
   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
lp15@59865
  1305
  by (auto simp add: diff_le_eq ac_simps abs_le_iff)
lp15@59865
  1306
haftmann@36301
  1307
end
avigad@16775
  1308
haftmann@59557
  1309
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
haftmann@59557
  1310
haftmann@52435
  1311
code_identifier
haftmann@52435
  1312
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1313
paulson@14265
  1314
end