src/HOL/Rings.thy
author haftmann
Wed Jul 08 20:19:12 2015 +0200 (2015-07-08)
changeset 60690 a9e45c9588c3
parent 60688 01488b559910
child 60758 d8d85a8172b5
permissions -rw-r--r--
tuned facts
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(*  Title:      HOL/Rings.thy
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    Author:     Gertrud Bauer
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    Author:     Steven Obua
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Markus Wenzel
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    Author:     Jeremy Avigad
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*)
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section {* Rings *}
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theory Rings
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imports Groups
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begin
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c"
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  assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
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begin
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor:
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  "a * e + (b * e + c) = (a + b) * e + c"
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by (simp add: distrib_right ac_simps)
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end
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "0 * a = 0"
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  assumes mult_zero_right [simp]: "a * 0 = 0"
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begin
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lemma mult_not_zero:
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  "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
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  by auto
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end
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
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  thus "0 * a = 0" by (simp only: add_left_cancel)
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next
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  fix a :: 'a
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  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
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  thus "a * 0 = 0" by (simp only: add_left_cancel)
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qed
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end
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a + b) * c = a * c + b * c"
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begin
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subclass semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: ac_simps)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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end
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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begin
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subclass semiring_0 ..
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end
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class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
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begin
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subclass semiring_0_cancel ..
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subclass comm_semiring_0 ..
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end
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "0 \<noteq> 1"
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begin
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lemma one_neq_zero [simp]: "1 \<noteq> 0"
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by (rule not_sym) (rule zero_neq_one)
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definition of_bool :: "bool \<Rightarrow> 'a"
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where
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  "of_bool p = (if p then 1 else 0)"
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lemma of_bool_eq [simp, code]:
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  "of_bool False = 0"
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  "of_bool True = 1"
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  by (simp_all add: of_bool_def)
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lemma of_bool_eq_iff:
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  "of_bool p = of_bool q \<longleftrightarrow> p = q"
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  by (simp add: of_bool_def)
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lemma split_of_bool [split]:
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  "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
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  by (cases p) simp_all
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lemma split_of_bool_asm:
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  "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
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  by (cases p) simp_all
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end
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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text {* Abstract divisibility *}
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class dvd = times
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begin
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definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where
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  "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
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lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
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  unfolding dvd_def ..
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding dvd_def by blast
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end
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context comm_monoid_mult
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begin
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subclass dvd .
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lemma dvd_refl [simp]:
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  "a dvd a"
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proof
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  show "a = a * 1" by simp
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qed
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lemma dvd_trans:
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  assumes "a dvd b" and "b dvd c"
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  shows "a dvd c"
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proof -
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  from assms obtain v where "b = a * v" by (auto elim!: dvdE)
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  moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
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  ultimately have "c = a * (v * w)" by (simp add: mult.assoc)
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  then show ?thesis ..
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qed
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lemma one_dvd [simp]:
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  "1 dvd a"
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  by (auto intro!: dvdI)
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lemma dvd_mult [simp]:
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  "a dvd c \<Longrightarrow> a dvd (b * c)"
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  by (auto intro!: mult.left_commute dvdI elim!: dvdE)
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lemma dvd_mult2 [simp]:
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  "a dvd b \<Longrightarrow> a dvd (b * c)"
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  using dvd_mult [of a b c] by (simp add: ac_simps)
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lemma dvd_triv_right [simp]:
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  "a dvd b * a"
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  by (rule dvd_mult) (rule dvd_refl)
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lemma dvd_triv_left [simp]:
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  "a dvd a * b"
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  by (rule dvd_mult2) (rule dvd_refl)
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lemma mult_dvd_mono:
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  assumes "a dvd b"
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    and "c dvd d"
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  shows "a * c dvd b * d"
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proof -
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  from `a dvd b` obtain b' where "b = a * b'" ..
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  moreover from `c dvd d` obtain d' where "d = c * d'" ..
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  ultimately have "b * d = (a * c) * (b' * d')" by (simp add: ac_simps)
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  then show ?thesis ..
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qed
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lemma dvd_mult_left:
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  "a * b dvd c \<Longrightarrow> a dvd c"
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  by (simp add: dvd_def mult.assoc) blast
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lemma dvd_mult_right:
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  "a * b dvd c \<Longrightarrow> b dvd c"
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  using dvd_mult_left [of b a c] by (simp add: ac_simps)
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end
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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
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begin
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subclass semiring_1 ..
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lemma dvd_0_left_iff [simp]:
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  "0 dvd a \<longleftrightarrow> a = 0"
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  by (auto intro: dvd_refl elim!: dvdE)
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lemma dvd_0_right [iff]:
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  "a dvd 0"
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proof
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  show "0 = a * 0" by simp
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qed
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lemma dvd_0_left:
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  "0 dvd a \<Longrightarrow> a = 0"
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  by simp
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lemma dvd_add [simp]:
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  assumes "a dvd b" and "a dvd c"
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  shows "a dvd (b + c)"
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proof -
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  from `a dvd b` obtain b' where "b = a * b'" ..
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  moreover from `a dvd c` obtain c' where "c = a * c'" ..
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  ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left)
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  then show ?thesis ..
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qed
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end
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class semiring_1_cancel = semiring + cancel_comm_monoid_add
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  + zero_neq_one + monoid_mult
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begin
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subclass semiring_0_cancel ..
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subclass semiring_1 ..
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end
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class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add +
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                               zero_neq_one + comm_monoid_mult +
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  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
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begin
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subclass semiring_1_cancel ..
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subclass comm_semiring_0_cancel ..
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subclass comm_semiring_1 ..
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lemma left_diff_distrib' [algebra_simps]:
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  "(b - c) * a = b * a - c * a"
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  by (simp add: algebra_simps)
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lemma dvd_add_times_triv_left_iff [simp]:
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  "a dvd c * a + b \<longleftrightarrow> a dvd b"
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proof -
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  have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
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  proof
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    assume ?Q then show ?P by simp
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  next
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    assume ?P
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    then obtain d where "a * c + b = a * d" ..
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    then have "a * c + b - a * c = a * d - a * c" by simp
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    then have "b = a * d - a * c" by simp
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    then have "b = a * (d - c)" by (simp add: algebra_simps)
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    then show ?Q ..
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  qed
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  then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
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qed
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lemma dvd_add_times_triv_right_iff [simp]:
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  "a dvd b + c * a \<longleftrightarrow> a dvd b"
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  using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)
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lemma dvd_add_triv_left_iff [simp]:
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  "a dvd a + b \<longleftrightarrow> a dvd b"
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  using dvd_add_times_triv_left_iff [of a 1 b] by simp
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lemma dvd_add_triv_right_iff [simp]:
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  "a dvd b + a \<longleftrightarrow> a dvd b"
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  using dvd_add_times_triv_right_iff [of a b 1] by simp
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lemma dvd_add_right_iff:
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  assumes "a dvd b"
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  shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P then obtain d where "b + c = a * d" ..
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  moreover from `a dvd b` obtain e where "b = a * e" ..
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  ultimately have "a * e + c = a * d" by simp
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  then have "a * e + c - a * e = a * d - a * e" by simp
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  then have "c = a * d - a * e" by simp
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  then have "c = a * (d - e)" by (simp add: algebra_simps)
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  then show ?Q ..
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next
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  assume ?Q with assms show ?P by simp
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qed
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lemma dvd_add_left_iff:
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  assumes "a dvd c"
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  shows "a dvd b + c \<longleftrightarrow> a dvd b"
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  using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps)
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end
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class ring = semiring + ab_group_add
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begin
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subclass semiring_0_cancel ..
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text {* Distribution rules *}
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lemma minus_mult_left: "- (a * b) = - a * b"
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by (rule minus_unique) (simp add: distrib_right [symmetric])
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lemma minus_mult_right: "- (a * b) = a * - b"
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by (rule minus_unique) (simp add: distrib_left [symmetric])
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text{*Extract signs from products*}
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lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
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lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
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lemma minus_mult_minus [simp]: "- a * - b = a * b"
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by simp
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lemma minus_mult_commute: "- a * b = a * - b"
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by simp
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lemma right_diff_distrib [algebra_simps]:
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  "a * (b - c) = a * b - a * c"
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  using distrib_left [of a b "-c "] by simp
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lemma left_diff_distrib [algebra_simps]:
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  "(a - b) * c = a * c - b * c"
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  using distrib_right [of a "- b" c] by simp
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lemmas ring_distribs =
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  distrib_left distrib_right left_diff_distrib right_diff_distrib
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lemma eq_add_iff1:
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  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
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by (simp add: algebra_simps)
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lemma eq_add_iff2:
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  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
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by (simp add: algebra_simps)
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end
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lemmas ring_distribs =
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  distrib_left distrib_right left_diff_distrib right_diff_distrib
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class comm_ring = comm_semiring + ab_group_add
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begin
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   353
huffman@27516
   354
subclass ring ..
huffman@28141
   355
subclass comm_semiring_0_cancel ..
haftmann@25267
   356
huffman@44350
   357
lemma square_diff_square_factored:
huffman@44350
   358
  "x * x - y * y = (x + y) * (x - y)"
huffman@44350
   359
  by (simp add: algebra_simps)
huffman@44350
   360
haftmann@25267
   361
end
obua@14738
   362
haftmann@22390
   363
class ring_1 = ring + zero_neq_one + monoid_mult
haftmann@25267
   364
begin
paulson@14265
   365
huffman@27516
   366
subclass semiring_1_cancel ..
haftmann@25267
   367
huffman@44346
   368
lemma square_diff_one_factored:
huffman@44346
   369
  "x * x - 1 = (x + 1) * (x - 1)"
huffman@44346
   370
  by (simp add: algebra_simps)
huffman@44346
   371
haftmann@25267
   372
end
haftmann@25152
   373
haftmann@22390
   374
class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
haftmann@25267
   375
begin
obua@14738
   376
huffman@27516
   377
subclass ring_1 ..
lp15@60562
   378
subclass comm_semiring_1_cancel
haftmann@59816
   379
  by unfold_locales (simp add: algebra_simps)
haftmann@58647
   380
huffman@29465
   381
lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
huffman@29408
   382
proof
huffman@29408
   383
  assume "x dvd - y"
huffman@29408
   384
  then have "x dvd - 1 * - y" by (rule dvd_mult)
huffman@29408
   385
  then show "x dvd y" by simp
huffman@29408
   386
next
huffman@29408
   387
  assume "x dvd y"
huffman@29408
   388
  then have "x dvd - 1 * y" by (rule dvd_mult)
huffman@29408
   389
  then show "x dvd - y" by simp
huffman@29408
   390
qed
huffman@29408
   391
huffman@29465
   392
lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
huffman@29408
   393
proof
huffman@29408
   394
  assume "- x dvd y"
huffman@29408
   395
  then obtain k where "y = - x * k" ..
huffman@29408
   396
  then have "y = x * - k" by simp
huffman@29408
   397
  then show "x dvd y" ..
huffman@29408
   398
next
huffman@29408
   399
  assume "x dvd y"
huffman@29408
   400
  then obtain k where "y = x * k" ..
huffman@29408
   401
  then have "y = - x * - k" by simp
huffman@29408
   402
  then show "- x dvd y" ..
huffman@29408
   403
qed
huffman@29408
   404
haftmann@54230
   405
lemma dvd_diff [simp]:
haftmann@54230
   406
  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
haftmann@54230
   407
  using dvd_add [of x y "- z"] by simp
huffman@29409
   408
haftmann@25267
   409
end
haftmann@25152
   410
haftmann@59833
   411
class semiring_no_zero_divisors = semiring_0 +
haftmann@59833
   412
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
haftmann@25230
   413
begin
haftmann@25230
   414
haftmann@59833
   415
lemma divisors_zero:
haftmann@59833
   416
  assumes "a * b = 0"
haftmann@59833
   417
  shows "a = 0 \<or> b = 0"
haftmann@59833
   418
proof (rule classical)
haftmann@59833
   419
  assume "\<not> (a = 0 \<or> b = 0)"
haftmann@59833
   420
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@59833
   421
  with no_zero_divisors have "a * b \<noteq> 0" by blast
haftmann@59833
   422
  with assms show ?thesis by simp
haftmann@59833
   423
qed
haftmann@59833
   424
haftmann@25230
   425
lemma mult_eq_0_iff [simp]:
haftmann@58952
   426
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@25230
   427
proof (cases "a = 0 \<or> b = 0")
haftmann@25230
   428
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@25230
   429
    then show ?thesis using no_zero_divisors by simp
haftmann@25230
   430
next
haftmann@25230
   431
  case True then show ?thesis by auto
haftmann@25230
   432
qed
haftmann@25230
   433
haftmann@58952
   434
end
haftmann@58952
   435
haftmann@60516
   436
class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +
haftmann@60516
   437
  assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   438
    and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@58952
   439
begin
haftmann@58952
   440
haftmann@58952
   441
lemma mult_left_cancel:
haftmann@58952
   442
  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
lp15@60562
   443
  by simp
lp15@56217
   444
haftmann@58952
   445
lemma mult_right_cancel:
haftmann@58952
   446
  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
lp15@60562
   447
  by simp
lp15@56217
   448
haftmann@25230
   449
end
huffman@22990
   450
haftmann@60516
   451
class ring_no_zero_divisors = ring + semiring_no_zero_divisors
haftmann@60516
   452
begin
haftmann@60516
   453
haftmann@60516
   454
subclass semiring_no_zero_divisors_cancel
haftmann@60516
   455
proof
haftmann@60516
   456
  fix a b c
haftmann@60516
   457
  have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"
haftmann@60516
   458
    by (simp add: algebra_simps)
haftmann@60516
   459
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   460
    by auto
haftmann@60516
   461
  finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .
haftmann@60516
   462
  have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"
haftmann@60516
   463
    by (simp add: algebra_simps)
haftmann@60516
   464
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   465
    by auto
haftmann@60516
   466
  finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .
haftmann@60516
   467
qed
haftmann@60516
   468
haftmann@60516
   469
end
haftmann@60516
   470
huffman@23544
   471
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   472
begin
haftmann@26274
   473
huffman@36970
   474
lemma square_eq_1_iff:
huffman@36821
   475
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
huffman@36821
   476
proof -
huffman@36821
   477
  have "(x - 1) * (x + 1) = x * x - 1"
huffman@36821
   478
    by (simp add: algebra_simps)
huffman@36821
   479
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   480
    by simp
huffman@36821
   481
  thus ?thesis
huffman@36821
   482
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   483
qed
huffman@36821
   484
haftmann@26274
   485
lemma mult_cancel_right1 [simp]:
haftmann@26274
   486
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   487
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   488
haftmann@26274
   489
lemma mult_cancel_right2 [simp]:
haftmann@26274
   490
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   491
by (insert mult_cancel_right [of a c 1], simp)
lp15@60562
   492
haftmann@26274
   493
lemma mult_cancel_left1 [simp]:
haftmann@26274
   494
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   495
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   496
haftmann@26274
   497
lemma mult_cancel_left2 [simp]:
haftmann@26274
   498
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   499
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   500
haftmann@26274
   501
end
huffman@22990
   502
lp15@60562
   503
class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors
haftmann@59833
   504
haftmann@59833
   505
class idom = comm_ring_1 + semiring_no_zero_divisors
haftmann@25186
   506
begin
paulson@14421
   507
haftmann@59833
   508
subclass semidom ..
haftmann@59833
   509
huffman@27516
   510
subclass ring_1_no_zero_divisors ..
huffman@22990
   511
huffman@29981
   512
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   513
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   514
proof -
huffman@29981
   515
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   516
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   517
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   518
    unfolding dvd_def by simp
huffman@29981
   519
  finally show ?thesis .
huffman@29981
   520
qed
huffman@29981
   521
huffman@29981
   522
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   523
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   524
proof -
huffman@29981
   525
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   526
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   527
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   528
    unfolding dvd_def by simp
huffman@29981
   529
  finally show ?thesis .
huffman@29981
   530
qed
huffman@29981
   531
haftmann@60516
   532
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"
haftmann@59833
   533
proof
haftmann@59833
   534
  assume "a * a = b * b"
haftmann@59833
   535
  then have "(a - b) * (a + b) = 0"
haftmann@59833
   536
    by (simp add: algebra_simps)
haftmann@59833
   537
  then show "a = b \<or> a = - b"
haftmann@59833
   538
    by (simp add: eq_neg_iff_add_eq_0)
haftmann@59833
   539
next
haftmann@59833
   540
  assume "a = b \<or> a = - b"
haftmann@59833
   541
  then show "a * a = b * b" by auto
haftmann@59833
   542
qed
haftmann@59833
   543
haftmann@25186
   544
end
haftmann@25152
   545
haftmann@35302
   546
text {*
haftmann@35302
   547
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   548
  \begin{itemize}
lp15@60562
   549
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
haftmann@35302
   550
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   551
  \end{itemize}
lp15@60562
   552
  Most of the used notions can also be looked up in
haftmann@35302
   553
  \begin{itemize}
wenzelm@54703
   554
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35302
   555
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   556
  \end{itemize}
haftmann@35302
   557
*}
haftmann@35302
   558
haftmann@60353
   559
class divide =
haftmann@60429
   560
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)
haftmann@60353
   561
haftmann@60353
   562
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60353
   563
haftmann@60353
   564
context semiring
haftmann@60353
   565
begin
haftmann@60353
   566
haftmann@60353
   567
lemma [field_simps]:
haftmann@60429
   568
  shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
haftmann@60429
   569
    and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
haftmann@60353
   570
  by (rule distrib_left distrib_right)+
haftmann@60353
   571
haftmann@60353
   572
end
haftmann@60353
   573
haftmann@60353
   574
context ring
haftmann@60353
   575
begin
haftmann@60353
   576
haftmann@60353
   577
lemma [field_simps]:
haftmann@60429
   578
  shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
haftmann@60429
   579
    and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
haftmann@60353
   580
  by (rule left_diff_distrib right_diff_distrib)+
haftmann@60353
   581
haftmann@60353
   582
end
haftmann@60353
   583
haftmann@60353
   584
setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"}) *}
haftmann@60353
   585
haftmann@60353
   586
class semidom_divide = semidom + divide +
haftmann@60429
   587
  assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
haftmann@60429
   588
  assumes divide_zero [simp]: "a div 0 = 0"
haftmann@60353
   589
begin
haftmann@60353
   590
haftmann@60353
   591
lemma nonzero_mult_divide_cancel_left [simp]:
haftmann@60429
   592
  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
haftmann@60353
   593
  using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
haftmann@60353
   594
haftmann@60516
   595
subclass semiring_no_zero_divisors_cancel
haftmann@60516
   596
proof
haftmann@60516
   597
  fix a b c
haftmann@60516
   598
  { fix a b c
haftmann@60516
   599
    show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   600
    proof (cases "c = 0")
haftmann@60516
   601
      case True then show ?thesis by simp
haftmann@60516
   602
    next
haftmann@60516
   603
      case False
haftmann@60516
   604
      { assume "a * c = b * c"
haftmann@60516
   605
        then have "a * c div c = b * c div c"
haftmann@60516
   606
          by simp
haftmann@60516
   607
        with False have "a = b"
haftmann@60516
   608
          by simp
haftmann@60516
   609
      } then show ?thesis by auto
haftmann@60516
   610
    qed
haftmann@60516
   611
  }
haftmann@60516
   612
  from this [of a c b]
haftmann@60516
   613
  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@60516
   614
    by (simp add: ac_simps)
haftmann@60516
   615
qed
haftmann@60516
   616
haftmann@60516
   617
lemma div_self [simp]:
haftmann@60516
   618
  assumes "a \<noteq> 0"
haftmann@60516
   619
  shows "a div a = 1"
haftmann@60516
   620
  using assms nonzero_mult_divide_cancel_left [of a 1] by simp
haftmann@60516
   621
haftmann@60570
   622
lemma divide_zero_left [simp]:
haftmann@60570
   623
  "0 div a = 0"
haftmann@60570
   624
proof (cases "a = 0")
haftmann@60570
   625
  case True then show ?thesis by simp
haftmann@60570
   626
next
haftmann@60570
   627
  case False then have "a * 0 div a = 0"
haftmann@60570
   628
    by (rule nonzero_mult_divide_cancel_left)
haftmann@60570
   629
  then show ?thesis by simp
haftmann@60570
   630
qed 
haftmann@60570
   631
haftmann@60690
   632
lemma divide_1 [simp]:
haftmann@60690
   633
  "a div 1 = a"
haftmann@60690
   634
  using nonzero_mult_divide_cancel_left [of 1 a] by simp
haftmann@60690
   635
haftmann@60690
   636
lemma dvd_times_left_cancel_iff [simp]:
haftmann@60690
   637
  assumes "a \<noteq> 0"
haftmann@60690
   638
  shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
haftmann@60690
   639
proof
haftmann@60690
   640
  assume ?P then obtain d where "a * c = a * b * d" ..
haftmann@60690
   641
  with assms have "c = b * d" by (simp add: ac_simps)
haftmann@60690
   642
  then show ?Q ..
haftmann@60690
   643
next
haftmann@60690
   644
  assume ?Q then obtain d where "c = b * d" .. 
haftmann@60690
   645
  then have "a * c = a * b * d" by (simp add: ac_simps)
haftmann@60690
   646
  then show ?P ..
haftmann@60690
   647
qed
haftmann@60690
   648
  
haftmann@60690
   649
lemma dvd_times_right_cancel_iff [simp]:
haftmann@60690
   650
  assumes "a \<noteq> 0"
haftmann@60690
   651
  shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
haftmann@60690
   652
using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
haftmann@60690
   653
  
haftmann@60690
   654
lemma div_dvd_iff_mult:
haftmann@60690
   655
  assumes "b \<noteq> 0" and "b dvd a"
haftmann@60690
   656
  shows "a div b dvd c \<longleftrightarrow> a dvd c * b"
haftmann@60690
   657
proof -
haftmann@60690
   658
  from \<open>b dvd a\<close> obtain d where "a = b * d" ..
haftmann@60690
   659
  with \<open>b \<noteq> 0\<close> show ?thesis by (simp add: ac_simps)
haftmann@60690
   660
qed
haftmann@60690
   661
haftmann@60690
   662
lemma dvd_div_iff_mult:
haftmann@60690
   663
  assumes "c \<noteq> 0" and "c dvd b"
haftmann@60690
   664
  shows "a dvd b div c \<longleftrightarrow> a * c dvd b"
haftmann@60690
   665
proof -
haftmann@60690
   666
  from \<open>c dvd b\<close> obtain d where "b = c * d" ..
haftmann@60690
   667
  with \<open>c \<noteq> 0\<close> show ?thesis by (simp add: mult.commute [of a])
haftmann@60690
   668
qed
haftmann@60690
   669
haftmann@60353
   670
end
haftmann@60353
   671
haftmann@60353
   672
class idom_divide = idom + semidom_divide
haftmann@60353
   673
haftmann@60517
   674
class algebraic_semidom = semidom_divide
haftmann@60517
   675
begin
haftmann@60517
   676
haftmann@60688
   677
text \<open>
haftmann@60688
   678
  Class @{class algebraic_semidom} enriches a integral domain
haftmann@60688
   679
  by notions from algebra, like units in a ring.
haftmann@60688
   680
  It is a separate class to avoid spoiling fields with notions
haftmann@60688
   681
  which are degenerated there.
haftmann@60688
   682
\<close>
haftmann@60688
   683
haftmann@60517
   684
lemma dvd_div_mult_self [simp]:
haftmann@60517
   685
  "a dvd b \<Longrightarrow> b div a * a = b"
haftmann@60517
   686
  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
haftmann@60517
   687
haftmann@60517
   688
lemma dvd_mult_div_cancel [simp]:
haftmann@60517
   689
  "a dvd b \<Longrightarrow> a * (b div a) = b"
haftmann@60517
   690
  using dvd_div_mult_self [of a b] by (simp add: ac_simps)
lp15@60562
   691
haftmann@60517
   692
lemma div_mult_swap:
haftmann@60517
   693
  assumes "c dvd b"
haftmann@60517
   694
  shows "a * (b div c) = (a * b) div c"
haftmann@60517
   695
proof (cases "c = 0")
haftmann@60517
   696
  case True then show ?thesis by simp
haftmann@60517
   697
next
haftmann@60517
   698
  case False from assms obtain d where "b = c * d" ..
haftmann@60517
   699
  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
haftmann@60517
   700
    by simp
haftmann@60517
   701
  ultimately show ?thesis by (simp add: ac_simps)
haftmann@60517
   702
qed
haftmann@60517
   703
haftmann@60517
   704
lemma dvd_div_mult:
haftmann@60517
   705
  assumes "c dvd b"
haftmann@60517
   706
  shows "b div c * a = (b * a) div c"
haftmann@60517
   707
  using assms div_mult_swap [of c b a] by (simp add: ac_simps)
haftmann@60517
   708
haftmann@60570
   709
lemma dvd_div_mult2_eq:
haftmann@60570
   710
  assumes "b * c dvd a"
haftmann@60570
   711
  shows "a div (b * c) = a div b div c"
haftmann@60570
   712
using assms proof
haftmann@60570
   713
  fix k
haftmann@60570
   714
  assume "a = b * c * k"
haftmann@60570
   715
  then show ?thesis
haftmann@60570
   716
    by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
haftmann@60570
   717
qed
haftmann@60570
   718
lp15@60562
   719
haftmann@60517
   720
text \<open>Units: invertible elements in a ring\<close>
haftmann@60517
   721
haftmann@60517
   722
abbreviation is_unit :: "'a \<Rightarrow> bool"
haftmann@60517
   723
where
haftmann@60517
   724
  "is_unit a \<equiv> a dvd 1"
haftmann@60517
   725
haftmann@60517
   726
lemma not_is_unit_0 [simp]:
haftmann@60517
   727
  "\<not> is_unit 0"
haftmann@60517
   728
  by simp
haftmann@60517
   729
lp15@60562
   730
lemma unit_imp_dvd [dest]:
haftmann@60517
   731
  "is_unit b \<Longrightarrow> b dvd a"
haftmann@60517
   732
  by (rule dvd_trans [of _ 1]) simp_all
haftmann@60517
   733
haftmann@60517
   734
lemma unit_dvdE:
haftmann@60517
   735
  assumes "is_unit a"
haftmann@60517
   736
  obtains c where "a \<noteq> 0" and "b = a * c"
haftmann@60517
   737
proof -
haftmann@60517
   738
  from assms have "a dvd b" by auto
haftmann@60517
   739
  then obtain c where "b = a * c" ..
haftmann@60517
   740
  moreover from assms have "a \<noteq> 0" by auto
haftmann@60517
   741
  ultimately show thesis using that by blast
haftmann@60517
   742
qed
haftmann@60517
   743
haftmann@60517
   744
lemma dvd_unit_imp_unit:
haftmann@60517
   745
  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
haftmann@60517
   746
  by (rule dvd_trans)
haftmann@60517
   747
haftmann@60517
   748
lemma unit_div_1_unit [simp, intro]:
haftmann@60517
   749
  assumes "is_unit a"
haftmann@60517
   750
  shows "is_unit (1 div a)"
haftmann@60517
   751
proof -
haftmann@60517
   752
  from assms have "1 = 1 div a * a" by simp
haftmann@60517
   753
  then show "is_unit (1 div a)" by (rule dvdI)
haftmann@60517
   754
qed
haftmann@60517
   755
haftmann@60517
   756
lemma is_unitE [elim?]:
haftmann@60517
   757
  assumes "is_unit a"
haftmann@60517
   758
  obtains b where "a \<noteq> 0" and "b \<noteq> 0"
haftmann@60517
   759
    and "is_unit b" and "1 div a = b" and "1 div b = a"
haftmann@60517
   760
    and "a * b = 1" and "c div a = c * b"
haftmann@60517
   761
proof (rule that)
haftmann@60517
   762
  def b \<equiv> "1 div a"
haftmann@60517
   763
  then show "1 div a = b" by simp
haftmann@60517
   764
  from b_def `is_unit a` show "is_unit b" by simp
haftmann@60517
   765
  from `is_unit a` and `is_unit b` show "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@60517
   766
  from b_def `is_unit a` show "a * b = 1" by simp
haftmann@60517
   767
  then have "1 = a * b" ..
haftmann@60517
   768
  with b_def `b \<noteq> 0` show "1 div b = a" by simp
haftmann@60517
   769
  from `is_unit a` have "a dvd c" ..
haftmann@60517
   770
  then obtain d where "c = a * d" ..
haftmann@60517
   771
  with `a \<noteq> 0` `a * b = 1` show "c div a = c * b"
haftmann@60517
   772
    by (simp add: mult.assoc mult.left_commute [of a])
haftmann@60517
   773
qed
haftmann@60517
   774
haftmann@60517
   775
lemma unit_prod [intro]:
haftmann@60517
   776
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
lp15@60562
   777
  by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
lp15@60562
   778
haftmann@60517
   779
lemma unit_div [intro]:
haftmann@60517
   780
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
haftmann@60517
   781
  by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
haftmann@60517
   782
haftmann@60517
   783
lemma mult_unit_dvd_iff:
haftmann@60517
   784
  assumes "is_unit b"
haftmann@60517
   785
  shows "a * b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   786
proof
haftmann@60517
   787
  assume "a * b dvd c"
haftmann@60517
   788
  with assms show "a dvd c"
haftmann@60517
   789
    by (simp add: dvd_mult_left)
haftmann@60517
   790
next
haftmann@60517
   791
  assume "a dvd c"
haftmann@60517
   792
  then obtain k where "c = a * k" ..
haftmann@60517
   793
  with assms have "c = (a * b) * (1 div b * k)"
haftmann@60517
   794
    by (simp add: mult_ac)
haftmann@60517
   795
  then show "a * b dvd c" by (rule dvdI)
haftmann@60517
   796
qed
haftmann@60517
   797
haftmann@60517
   798
lemma dvd_mult_unit_iff:
haftmann@60517
   799
  assumes "is_unit b"
haftmann@60517
   800
  shows "a dvd c * b \<longleftrightarrow> a dvd c"
haftmann@60517
   801
proof
haftmann@60517
   802
  assume "a dvd c * b"
haftmann@60517
   803
  with assms have "c * b dvd c * (b * (1 div b))"
haftmann@60517
   804
    by (subst mult_assoc [symmetric]) simp
haftmann@60517
   805
  also from `is_unit b` have "b * (1 div b) = 1" by (rule is_unitE) simp
haftmann@60517
   806
  finally have "c * b dvd c" by simp
haftmann@60517
   807
  with `a dvd c * b` show "a dvd c" by (rule dvd_trans)
haftmann@60517
   808
next
haftmann@60517
   809
  assume "a dvd c"
haftmann@60517
   810
  then show "a dvd c * b" by simp
haftmann@60517
   811
qed
haftmann@60517
   812
haftmann@60517
   813
lemma div_unit_dvd_iff:
haftmann@60517
   814
  "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
haftmann@60517
   815
  by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
haftmann@60517
   816
haftmann@60517
   817
lemma dvd_div_unit_iff:
haftmann@60517
   818
  "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
haftmann@60517
   819
  by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
haftmann@60517
   820
haftmann@60517
   821
lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
haftmann@60517
   822
  dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>
haftmann@60517
   823
haftmann@60517
   824
lemma unit_mult_div_div [simp]:
haftmann@60517
   825
  "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
haftmann@60517
   826
  by (erule is_unitE [of _ b]) simp
haftmann@60517
   827
haftmann@60517
   828
lemma unit_div_mult_self [simp]:
haftmann@60517
   829
  "is_unit a \<Longrightarrow> b div a * a = b"
haftmann@60517
   830
  by (rule dvd_div_mult_self) auto
haftmann@60517
   831
haftmann@60517
   832
lemma unit_div_1_div_1 [simp]:
haftmann@60517
   833
  "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
haftmann@60517
   834
  by (erule is_unitE) simp
haftmann@60517
   835
haftmann@60517
   836
lemma unit_div_mult_swap:
haftmann@60517
   837
  "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
haftmann@60517
   838
  by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
haftmann@60517
   839
haftmann@60517
   840
lemma unit_div_commute:
haftmann@60517
   841
  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
haftmann@60517
   842
  using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
haftmann@60517
   843
haftmann@60517
   844
lemma unit_eq_div1:
haftmann@60517
   845
  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
haftmann@60517
   846
  by (auto elim: is_unitE)
haftmann@60517
   847
haftmann@60517
   848
lemma unit_eq_div2:
haftmann@60517
   849
  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
haftmann@60517
   850
  using unit_eq_div1 [of b c a] by auto
haftmann@60517
   851
haftmann@60517
   852
lemma unit_mult_left_cancel:
haftmann@60517
   853
  assumes "is_unit a"
haftmann@60517
   854
  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
lp15@60562
   855
  using assms mult_cancel_left [of a b c] by auto
haftmann@60517
   856
haftmann@60517
   857
lemma unit_mult_right_cancel:
haftmann@60517
   858
  "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
haftmann@60517
   859
  using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
haftmann@60517
   860
haftmann@60517
   861
lemma unit_div_cancel:
haftmann@60517
   862
  assumes "is_unit a"
haftmann@60517
   863
  shows "b div a = c div a \<longleftrightarrow> b = c"
haftmann@60517
   864
proof -
haftmann@60517
   865
  from assms have "is_unit (1 div a)" by simp
haftmann@60517
   866
  then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
haftmann@60517
   867
    by (rule unit_mult_right_cancel)
haftmann@60517
   868
  with assms show ?thesis by simp
haftmann@60517
   869
qed
lp15@60562
   870
haftmann@60570
   871
lemma is_unit_div_mult2_eq:
haftmann@60570
   872
  assumes "is_unit b" and "is_unit c"
haftmann@60570
   873
  shows "a div (b * c) = a div b div c"
haftmann@60570
   874
proof -
haftmann@60570
   875
  from assms have "is_unit (b * c)" by (simp add: unit_prod)
haftmann@60570
   876
  then have "b * c dvd a"
haftmann@60570
   877
    by (rule unit_imp_dvd)
haftmann@60570
   878
  then show ?thesis
haftmann@60570
   879
    by (rule dvd_div_mult2_eq)
haftmann@60570
   880
qed
haftmann@60570
   881
lp15@60562
   882
lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
haftmann@60517
   883
  dvd_div_unit_iff unit_div_mult_swap unit_div_commute
lp15@60562
   884
  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
haftmann@60517
   885
  unit_eq_div1 unit_eq_div2
haftmann@60517
   886
haftmann@60685
   887
lemma is_unit_divide_mult_cancel_left:
haftmann@60685
   888
  assumes "a \<noteq> 0" and "is_unit b"
haftmann@60685
   889
  shows "a div (a * b) = 1 div b"
haftmann@60685
   890
proof -
haftmann@60685
   891
  from assms have "a div (a * b) = a div a div b"
haftmann@60685
   892
    by (simp add: mult_unit_dvd_iff dvd_div_mult2_eq)
haftmann@60685
   893
  with assms show ?thesis by simp
haftmann@60685
   894
qed
haftmann@60685
   895
haftmann@60685
   896
lemma is_unit_divide_mult_cancel_right:
haftmann@60685
   897
  assumes "a \<noteq> 0" and "is_unit b"
haftmann@60685
   898
  shows "a div (b * a) = 1 div b"
haftmann@60685
   899
  using assms is_unit_divide_mult_cancel_left [of a b] by (simp add: ac_simps)
haftmann@60685
   900
haftmann@60685
   901
end
haftmann@60685
   902
haftmann@60685
   903
class normalization_semidom = algebraic_semidom +
haftmann@60685
   904
  fixes normalize :: "'a \<Rightarrow> 'a"
haftmann@60685
   905
    and unit_factor :: "'a \<Rightarrow> 'a"
haftmann@60685
   906
  assumes unit_factor_mult_normalize [simp]: "unit_factor a * normalize a = a"
haftmann@60685
   907
  assumes normalize_0 [simp]: "normalize 0 = 0"
haftmann@60685
   908
    and unit_factor_0 [simp]: "unit_factor 0 = 0"
haftmann@60685
   909
  assumes is_unit_normalize:
haftmann@60685
   910
    "is_unit a  \<Longrightarrow> normalize a = 1"
haftmann@60685
   911
  assumes unit_factor_is_unit [iff]: 
haftmann@60685
   912
    "a \<noteq> 0 \<Longrightarrow> is_unit (unit_factor a)"
haftmann@60685
   913
  assumes unit_factor_mult: "unit_factor (a * b) = unit_factor a * unit_factor b"
haftmann@60685
   914
begin
haftmann@60685
   915
haftmann@60688
   916
text \<open>
haftmann@60688
   917
  Class @{class normalization_semidom} cultivates the idea that
haftmann@60688
   918
  each integral domain can be split into equivalence classes
haftmann@60688
   919
  whose representants are associated, i.e. divide each other.
haftmann@60688
   920
  @{const normalize} specifies a canonical representant for each equivalence
haftmann@60688
   921
  class.  The rationale behind this is that it is easier to reason about equality
haftmann@60688
   922
  than equivalences, hence we prefer to think about equality of normalized
haftmann@60688
   923
  values rather than associated elements.
haftmann@60688
   924
\<close>
haftmann@60688
   925
haftmann@60685
   926
lemma unit_factor_dvd [simp]:
haftmann@60685
   927
  "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
haftmann@60685
   928
  by (rule unit_imp_dvd) simp
haftmann@60685
   929
haftmann@60685
   930
lemma unit_factor_self [simp]:
haftmann@60685
   931
  "unit_factor a dvd a"
haftmann@60685
   932
  by (cases "a = 0") simp_all 
haftmann@60685
   933
  
haftmann@60685
   934
lemma normalize_mult_unit_factor [simp]:
haftmann@60685
   935
  "normalize a * unit_factor a = a"
haftmann@60685
   936
  using unit_factor_mult_normalize [of a] by (simp add: ac_simps)
haftmann@60685
   937
haftmann@60685
   938
lemma normalize_eq_0_iff [simp]:
haftmann@60685
   939
  "normalize a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
haftmann@60685
   940
proof
haftmann@60685
   941
  assume ?P
haftmann@60685
   942
  moreover have "unit_factor a * normalize a = a" by simp
haftmann@60685
   943
  ultimately show ?Q by simp 
haftmann@60685
   944
next
haftmann@60685
   945
  assume ?Q then show ?P by simp
haftmann@60685
   946
qed
haftmann@60685
   947
haftmann@60685
   948
lemma unit_factor_eq_0_iff [simp]:
haftmann@60685
   949
  "unit_factor a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
haftmann@60685
   950
proof
haftmann@60685
   951
  assume ?P
haftmann@60685
   952
  moreover have "unit_factor a * normalize a = a" by simp
haftmann@60685
   953
  ultimately show ?Q by simp 
haftmann@60685
   954
next
haftmann@60685
   955
  assume ?Q then show ?P by simp
haftmann@60685
   956
qed
haftmann@60685
   957
haftmann@60685
   958
lemma is_unit_unit_factor:
haftmann@60685
   959
  assumes "is_unit a" shows "unit_factor a = a"
haftmann@60685
   960
proof - 
haftmann@60685
   961
  from assms have "normalize a = 1" by (rule is_unit_normalize)
haftmann@60685
   962
  moreover from unit_factor_mult_normalize have "unit_factor a * normalize a = a" .
haftmann@60685
   963
  ultimately show ?thesis by simp
haftmann@60685
   964
qed
haftmann@60685
   965
haftmann@60685
   966
lemma unit_factor_1 [simp]:
haftmann@60685
   967
  "unit_factor 1 = 1"
haftmann@60685
   968
  by (rule is_unit_unit_factor) simp
haftmann@60685
   969
haftmann@60685
   970
lemma normalize_1 [simp]:
haftmann@60685
   971
  "normalize 1 = 1"
haftmann@60685
   972
  by (rule is_unit_normalize) simp
haftmann@60685
   973
haftmann@60685
   974
lemma normalize_1_iff:
haftmann@60685
   975
  "normalize a = 1 \<longleftrightarrow> is_unit a" (is "?P \<longleftrightarrow> ?Q")
haftmann@60685
   976
proof
haftmann@60685
   977
  assume ?Q then show ?P by (rule is_unit_normalize)
haftmann@60685
   978
next
haftmann@60685
   979
  assume ?P
haftmann@60685
   980
  then have "a \<noteq> 0" by auto
haftmann@60685
   981
  from \<open>?P\<close> have "unit_factor a * normalize a = unit_factor a * 1"
haftmann@60685
   982
    by simp
haftmann@60685
   983
  then have "unit_factor a = a"
haftmann@60685
   984
    by simp
haftmann@60685
   985
  moreover have "is_unit (unit_factor a)"
haftmann@60685
   986
    using \<open>a \<noteq> 0\<close> by simp
haftmann@60685
   987
  ultimately show ?Q by simp
haftmann@60685
   988
qed
haftmann@60685
   989
  
haftmann@60685
   990
lemma div_normalize [simp]:
haftmann@60685
   991
  "a div normalize a = unit_factor a"
haftmann@60685
   992
proof (cases "a = 0")
haftmann@60685
   993
  case True then show ?thesis by simp
haftmann@60685
   994
next
haftmann@60685
   995
  case False then have "normalize a \<noteq> 0" by simp 
haftmann@60685
   996
  with nonzero_mult_divide_cancel_right
haftmann@60685
   997
  have "unit_factor a * normalize a div normalize a = unit_factor a" by blast
haftmann@60685
   998
  then show ?thesis by simp
haftmann@60685
   999
qed
haftmann@60685
  1000
haftmann@60685
  1001
lemma div_unit_factor [simp]:
haftmann@60685
  1002
  "a div unit_factor a = normalize a"
haftmann@60685
  1003
proof (cases "a = 0")
haftmann@60685
  1004
  case True then show ?thesis by simp
haftmann@60685
  1005
next
haftmann@60685
  1006
  case False then have "unit_factor a \<noteq> 0" by simp 
haftmann@60685
  1007
  with nonzero_mult_divide_cancel_left
haftmann@60685
  1008
  have "unit_factor a * normalize a div unit_factor a = normalize a" by blast
haftmann@60685
  1009
  then show ?thesis by simp
haftmann@60685
  1010
qed
haftmann@60685
  1011
haftmann@60685
  1012
lemma normalize_div [simp]:
haftmann@60685
  1013
  "normalize a div a = 1 div unit_factor a"
haftmann@60685
  1014
proof (cases "a = 0")
haftmann@60685
  1015
  case True then show ?thesis by simp
haftmann@60685
  1016
next
haftmann@60685
  1017
  case False
haftmann@60685
  1018
  have "normalize a div a = normalize a div (unit_factor a * normalize a)"
haftmann@60685
  1019
    by simp
haftmann@60685
  1020
  also have "\<dots> = 1 div unit_factor a"
haftmann@60685
  1021
    using False by (subst is_unit_divide_mult_cancel_right) simp_all
haftmann@60685
  1022
  finally show ?thesis .
haftmann@60685
  1023
qed
haftmann@60685
  1024
haftmann@60685
  1025
lemma mult_one_div_unit_factor [simp]:
haftmann@60685
  1026
  "a * (1 div unit_factor b) = a div unit_factor b"
haftmann@60685
  1027
  by (cases "b = 0") simp_all
haftmann@60685
  1028
haftmann@60685
  1029
lemma normalize_mult:
haftmann@60685
  1030
  "normalize (a * b) = normalize a * normalize b"
haftmann@60685
  1031
proof (cases "a = 0 \<or> b = 0")
haftmann@60685
  1032
  case True then show ?thesis by auto
haftmann@60685
  1033
next
haftmann@60685
  1034
  case False
haftmann@60685
  1035
  from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" .
haftmann@60685
  1036
  then have "normalize (a * b) = a * b div unit_factor (a * b)" by simp
haftmann@60685
  1037
  also have "\<dots> = a * b div unit_factor (b * a)" by (simp add: ac_simps)
haftmann@60685
  1038
  also have "\<dots> = a * b div unit_factor b div unit_factor a"
haftmann@60685
  1039
    using False by (simp add: unit_factor_mult is_unit_div_mult2_eq [symmetric])
haftmann@60685
  1040
  also have "\<dots> = a * (b div unit_factor b) div unit_factor a"
haftmann@60685
  1041
    using False by (subst unit_div_mult_swap) simp_all
haftmann@60685
  1042
  also have "\<dots> = normalize a * normalize b"
haftmann@60685
  1043
    using False by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
haftmann@60685
  1044
  finally show ?thesis .
haftmann@60685
  1045
qed
haftmann@60685
  1046
 
haftmann@60685
  1047
lemma unit_factor_idem [simp]:
haftmann@60685
  1048
  "unit_factor (unit_factor a) = unit_factor a"
haftmann@60685
  1049
  by (cases "a = 0") (auto intro: is_unit_unit_factor)
haftmann@60685
  1050
haftmann@60685
  1051
lemma normalize_unit_factor [simp]:
haftmann@60685
  1052
  "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
haftmann@60685
  1053
  by (rule is_unit_normalize) simp
haftmann@60685
  1054
  
haftmann@60685
  1055
lemma normalize_idem [simp]:
haftmann@60685
  1056
  "normalize (normalize a) = normalize a"
haftmann@60685
  1057
proof (cases "a = 0")
haftmann@60685
  1058
  case True then show ?thesis by simp
haftmann@60685
  1059
next
haftmann@60685
  1060
  case False
haftmann@60685
  1061
  have "normalize a = normalize (unit_factor a * normalize a)" by simp
haftmann@60685
  1062
  also have "\<dots> = normalize (unit_factor a) * normalize (normalize a)"
haftmann@60685
  1063
    by (simp only: normalize_mult)
haftmann@60685
  1064
  finally show ?thesis using False by simp_all
haftmann@60685
  1065
qed
haftmann@60685
  1066
haftmann@60685
  1067
lemma unit_factor_normalize [simp]:
haftmann@60685
  1068
  assumes "a \<noteq> 0"
haftmann@60685
  1069
  shows "unit_factor (normalize a) = 1"
haftmann@60685
  1070
proof -
haftmann@60685
  1071
  from assms have "normalize a \<noteq> 0" by simp
haftmann@60685
  1072
  have "unit_factor (normalize a) * normalize (normalize a) = normalize a"
haftmann@60685
  1073
    by (simp only: unit_factor_mult_normalize)
haftmann@60685
  1074
  then have "unit_factor (normalize a) * normalize a = normalize a"
haftmann@60685
  1075
    by simp
haftmann@60685
  1076
  with \<open>normalize a \<noteq> 0\<close>
haftmann@60685
  1077
  have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
haftmann@60685
  1078
    by simp
haftmann@60685
  1079
  with \<open>normalize a \<noteq> 0\<close>
haftmann@60685
  1080
  show ?thesis by simp
haftmann@60685
  1081
qed
haftmann@60685
  1082
haftmann@60685
  1083
lemma dvd_unit_factor_div:
haftmann@60685
  1084
  assumes "b dvd a"
haftmann@60685
  1085
  shows "unit_factor (a div b) = unit_factor a div unit_factor b"
haftmann@60685
  1086
proof -
haftmann@60685
  1087
  from assms have "a = a div b * b"
haftmann@60685
  1088
    by simp
haftmann@60685
  1089
  then have "unit_factor a = unit_factor (a div b * b)"
haftmann@60685
  1090
    by simp
haftmann@60685
  1091
  then show ?thesis
haftmann@60685
  1092
    by (cases "b = 0") (simp_all add: unit_factor_mult)
haftmann@60685
  1093
qed
haftmann@60685
  1094
haftmann@60685
  1095
lemma dvd_normalize_div:
haftmann@60685
  1096
  assumes "b dvd a"
haftmann@60685
  1097
  shows "normalize (a div b) = normalize a div normalize b"
haftmann@60685
  1098
proof -
haftmann@60685
  1099
  from assms have "a = a div b * b"
haftmann@60685
  1100
    by simp
haftmann@60685
  1101
  then have "normalize a = normalize (a div b * b)"
haftmann@60685
  1102
    by simp
haftmann@60685
  1103
  then show ?thesis
haftmann@60685
  1104
    by (cases "b = 0") (simp_all add: normalize_mult)
haftmann@60685
  1105
qed
haftmann@60685
  1106
haftmann@60685
  1107
lemma normalize_dvd_iff [simp]:
haftmann@60685
  1108
  "normalize a dvd b \<longleftrightarrow> a dvd b"
haftmann@60685
  1109
proof -
haftmann@60685
  1110
  have "normalize a dvd b \<longleftrightarrow> unit_factor a * normalize a dvd b"
haftmann@60685
  1111
    using mult_unit_dvd_iff [of "unit_factor a" "normalize a" b]
haftmann@60685
  1112
      by (cases "a = 0") simp_all
haftmann@60685
  1113
  then show ?thesis by simp
haftmann@60685
  1114
qed
haftmann@60685
  1115
haftmann@60685
  1116
lemma dvd_normalize_iff [simp]:
haftmann@60685
  1117
  "a dvd normalize b \<longleftrightarrow> a dvd b"
haftmann@60685
  1118
proof -
haftmann@60685
  1119
  have "a dvd normalize  b \<longleftrightarrow> a dvd normalize b * unit_factor b"
haftmann@60685
  1120
    using dvd_mult_unit_iff [of "unit_factor b" a "normalize b"]
haftmann@60685
  1121
      by (cases "b = 0") simp_all
haftmann@60685
  1122
  then show ?thesis by simp
haftmann@60685
  1123
qed
haftmann@60685
  1124
haftmann@60688
  1125
text \<open>
haftmann@60688
  1126
  We avoid an explicit definition of associated elements but prefer
haftmann@60688
  1127
  explicit normalisation instead.  In theory we could define an abbreviation
haftmann@60688
  1128
  like @{prop "associated a b \<longleftrightarrow> normalize a = normalize b"} but this is
haftmann@60688
  1129
  counterproductive without suggestive infix syntax, which we do not want
haftmann@60688
  1130
  to sacrifice for this purpose here.
haftmann@60688
  1131
\<close>
haftmann@60685
  1132
haftmann@60688
  1133
lemma associatedI:
haftmann@60688
  1134
  assumes "a dvd b" and "b dvd a"
haftmann@60688
  1135
  shows "normalize a = normalize b"
haftmann@60685
  1136
proof (cases "a = 0 \<or> b = 0")
haftmann@60688
  1137
  case True with assms show ?thesis by auto
haftmann@60685
  1138
next
haftmann@60685
  1139
  case False
haftmann@60688
  1140
  from \<open>a dvd b\<close> obtain c where b: "b = a * c" ..
haftmann@60688
  1141
  moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
haftmann@60688
  1142
  ultimately have "b * 1 = b * (c * d)" by (simp add: ac_simps)
haftmann@60688
  1143
  with False have "1 = c * d"
haftmann@60688
  1144
    unfolding mult_cancel_left by simp
haftmann@60688
  1145
  then have "is_unit c" and "is_unit d" by auto
haftmann@60688
  1146
  with a b show ?thesis by (simp add: normalize_mult is_unit_normalize)
haftmann@60688
  1147
qed
haftmann@60688
  1148
haftmann@60688
  1149
lemma associatedD1:
haftmann@60688
  1150
  "normalize a = normalize b \<Longrightarrow> a dvd b"
haftmann@60688
  1151
  using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric]
haftmann@60688
  1152
  by simp
haftmann@60688
  1153
haftmann@60688
  1154
lemma associatedD2:
haftmann@60688
  1155
  "normalize a = normalize b \<Longrightarrow> b dvd a"
haftmann@60688
  1156
  using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric]
haftmann@60688
  1157
  by simp
haftmann@60688
  1158
haftmann@60688
  1159
lemma associated_unit:
haftmann@60688
  1160
  "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
haftmann@60688
  1161
  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
haftmann@60688
  1162
haftmann@60688
  1163
lemma associated_iff_dvd:
haftmann@60688
  1164
  "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a" (is "?P \<longleftrightarrow> ?Q")
haftmann@60688
  1165
proof
haftmann@60688
  1166
  assume ?Q then show ?P by (auto intro!: associatedI)
haftmann@60688
  1167
next
haftmann@60688
  1168
  assume ?P
haftmann@60688
  1169
  then have "unit_factor a * normalize a = unit_factor a * normalize b"
haftmann@60688
  1170
    by simp
haftmann@60688
  1171
  then have *: "normalize b * unit_factor a = a"
haftmann@60688
  1172
    by (simp add: ac_simps)
haftmann@60688
  1173
  show ?Q
haftmann@60688
  1174
  proof (cases "a = 0 \<or> b = 0")
haftmann@60688
  1175
    case True with \<open>?P\<close> show ?thesis by auto
haftmann@60685
  1176
  next
haftmann@60688
  1177
    case False 
haftmann@60688
  1178
    then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
haftmann@60688
  1179
      by (simp_all add: mult_unit_dvd_iff dvd_mult_unit_iff)
haftmann@60688
  1180
    with * show ?thesis by simp
haftmann@60685
  1181
  qed
haftmann@60685
  1182
qed
haftmann@60685
  1183
haftmann@60685
  1184
lemma associated_eqI:
haftmann@60688
  1185
  assumes "a dvd b" and "b dvd a"
haftmann@60688
  1186
  assumes "normalize a = a" and "normalize b = b"
haftmann@60685
  1187
  shows "a = b"
haftmann@60688
  1188
proof -
haftmann@60688
  1189
  from assms have "normalize a = normalize b"
haftmann@60688
  1190
    unfolding associated_iff_dvd by simp
haftmann@60688
  1191
  with \<open>normalize a = a\<close> have "a = normalize b" by simp
haftmann@60688
  1192
  with \<open>normalize b = b\<close> show "a = b" by simp
haftmann@60685
  1193
qed
haftmann@60685
  1194
haftmann@60685
  1195
end
haftmann@60685
  1196
haftmann@38642
  1197
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
  1198
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
  1199
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
  1200
begin
haftmann@25230
  1201
haftmann@25230
  1202
lemma mult_mono:
haftmann@38642
  1203
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
  1204
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
  1205
apply (erule mult_left_mono, assumption)
haftmann@25230
  1206
done
haftmann@25230
  1207
haftmann@25230
  1208
lemma mult_mono':
haftmann@38642
  1209
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
  1210
apply (rule mult_mono)
haftmann@25230
  1211
apply (fast intro: order_trans)+
haftmann@25230
  1212
done
haftmann@25230
  1213
haftmann@25230
  1214
end
krauss@21199
  1215
haftmann@38642
  1216
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
  1217
begin
paulson@14268
  1218
huffman@27516
  1219
subclass semiring_0_cancel ..
obua@23521
  1220
nipkow@56536
  1221
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
  1222
using mult_left_mono [of 0 b a] by simp
haftmann@25230
  1223
haftmann@25230
  1224
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
  1225
using mult_left_mono [of b 0 a] by simp
huffman@30692
  1226
huffman@30692
  1227
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
  1228
using mult_right_mono [of a 0 b] by simp
huffman@30692
  1229
huffman@30692
  1230
text {* Legacy - use @{text mult_nonpos_nonneg} *}
lp15@60562
  1231
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
haftmann@36301
  1232
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
  1233
lp15@60562
  1234
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
nipkow@29667
  1235
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
  1236
haftmann@25230
  1237
end
haftmann@25230
  1238
haftmann@38642
  1239
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
  1240
begin
haftmann@25230
  1241
haftmann@35028
  1242
subclass ordered_cancel_semiring ..
haftmann@35028
  1243
haftmann@35028
  1244
subclass ordered_comm_monoid_add ..
haftmann@25304
  1245
haftmann@25230
  1246
lemma mult_left_less_imp_less:
haftmann@25230
  1247
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
  1248
by (force simp add: mult_left_mono not_le [symmetric])
lp15@60562
  1249
haftmann@25230
  1250
lemma mult_right_less_imp_less:
haftmann@25230
  1251
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
  1252
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
  1253
haftmann@25186
  1254
end
haftmann@25152
  1255
haftmann@35043
  1256
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
  1257
begin
hoelzl@36622
  1258
hoelzl@36622
  1259
lemma convex_bound_le:
hoelzl@36622
  1260
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
  1261
  shows "u * x + v * y \<le> a"
hoelzl@36622
  1262
proof-
hoelzl@36622
  1263
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
  1264
    by (simp add: add_mono mult_left_mono)
webertj@49962
  1265
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
  1266
qed
hoelzl@36622
  1267
hoelzl@36622
  1268
end
haftmann@35043
  1269
haftmann@35043
  1270
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
  1271
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
  1272
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
  1273
begin
paulson@14341
  1274
huffman@27516
  1275
subclass semiring_0_cancel ..
obua@14940
  1276
haftmann@35028
  1277
subclass linordered_semiring
haftmann@28823
  1278
proof
huffman@23550
  1279
  fix a b c :: 'a
huffman@23550
  1280
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
  1281
  from A show "c * a \<le> c * b"
haftmann@25186
  1282
    unfolding le_less
haftmann@25186
  1283
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
  1284
  from A show "a * c \<le> b * c"
haftmann@25152
  1285
    unfolding le_less
haftmann@25186
  1286
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
  1287
qed
haftmann@25152
  1288
haftmann@25230
  1289
lemma mult_left_le_imp_le:
haftmann@25230
  1290
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
  1291
by (force simp add: mult_strict_left_mono _not_less [symmetric])
lp15@60562
  1292
haftmann@25230
  1293
lemma mult_right_le_imp_le:
haftmann@25230
  1294
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
  1295
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
  1296
nipkow@56544
  1297
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
  1298
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
  1299
huffman@30692
  1300
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
  1301
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
  1302
huffman@30692
  1303
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
  1304
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
  1305
huffman@30692
  1306
text {* Legacy - use @{text mult_neg_pos} *}
lp15@60562
  1307
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
haftmann@36301
  1308
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
  1309
haftmann@25230
  1310
lemma zero_less_mult_pos:
haftmann@25230
  1311
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1312
apply (cases "b\<le>0")
haftmann@25230
  1313
 apply (auto simp add: le_less not_less)
huffman@30692
  1314
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
  1315
 apply (auto dest: less_not_sym)
haftmann@25230
  1316
done
haftmann@25230
  1317
haftmann@25230
  1318
lemma zero_less_mult_pos2:
haftmann@25230
  1319
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
  1320
apply (cases "b\<le>0")
haftmann@25230
  1321
 apply (auto simp add: le_less not_less)
huffman@30692
  1322
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
  1323
 apply (auto dest: less_not_sym)
haftmann@25230
  1324
done
haftmann@25230
  1325
haftmann@26193
  1326
text{*Strict monotonicity in both arguments*}
haftmann@26193
  1327
lemma mult_strict_mono:
haftmann@26193
  1328
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
  1329
  shows "a * c < b * d"
haftmann@26193
  1330
  using assms apply (cases "c=0")
nipkow@56544
  1331
  apply (simp)
haftmann@26193
  1332
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
  1333
  apply (force simp add: le_less)
haftmann@26193
  1334
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
  1335
  done
haftmann@26193
  1336
haftmann@26193
  1337
text{*This weaker variant has more natural premises*}
haftmann@26193
  1338
lemma mult_strict_mono':
haftmann@26193
  1339
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
  1340
  shows "a * c < b * d"
nipkow@29667
  1341
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
  1342
haftmann@26193
  1343
lemma mult_less_le_imp_less:
haftmann@26193
  1344
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
  1345
  shows "a * c < b * d"
haftmann@26193
  1346
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
  1347
  apply (erule less_le_trans)
haftmann@26193
  1348
  apply (erule mult_left_mono)
haftmann@26193
  1349
  apply simp
haftmann@26193
  1350
  apply (erule mult_strict_right_mono)
haftmann@26193
  1351
  apply assumption
haftmann@26193
  1352
  done
haftmann@26193
  1353
haftmann@26193
  1354
lemma mult_le_less_imp_less:
haftmann@26193
  1355
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
  1356
  shows "a * c < b * d"
haftmann@26193
  1357
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
  1358
  apply (erule le_less_trans)
haftmann@26193
  1359
  apply (erule mult_strict_left_mono)
haftmann@26193
  1360
  apply simp
haftmann@26193
  1361
  apply (erule mult_right_mono)
haftmann@26193
  1362
  apply simp
haftmann@26193
  1363
  done
haftmann@26193
  1364
haftmann@25230
  1365
end
haftmann@25230
  1366
haftmann@35097
  1367
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
  1368
begin
hoelzl@36622
  1369
hoelzl@36622
  1370
subclass linordered_semiring_1 ..
hoelzl@36622
  1371
hoelzl@36622
  1372
lemma convex_bound_lt:
hoelzl@36622
  1373
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
  1374
  shows "u * x + v * y < a"
hoelzl@36622
  1375
proof -
hoelzl@36622
  1376
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
  1377
    by (cases "u = 0")
hoelzl@36622
  1378
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
webertj@49962
  1379
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
  1380
qed
hoelzl@36622
  1381
hoelzl@36622
  1382
end
haftmann@33319
  1383
lp15@60562
  1384
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +
haftmann@38642
  1385
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
  1386
begin
haftmann@25152
  1387
haftmann@35028
  1388
subclass ordered_semiring
haftmann@28823
  1389
proof
krauss@21199
  1390
  fix a b c :: 'a
huffman@23550
  1391
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
  1392
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
haftmann@57512
  1393
  thus "a * c \<le> b * c" by (simp only: mult.commute)
krauss@21199
  1394
qed
paulson@14265
  1395
haftmann@25267
  1396
end
haftmann@25267
  1397
haftmann@38642
  1398
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
  1399
begin
paulson@14265
  1400
haftmann@38642
  1401
subclass comm_semiring_0_cancel ..
haftmann@35028
  1402
subclass ordered_comm_semiring ..
haftmann@35028
  1403
subclass ordered_cancel_semiring ..
haftmann@25267
  1404
haftmann@25267
  1405
end
haftmann@25267
  1406
haftmann@35028
  1407
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
  1408
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
  1409
begin
haftmann@25267
  1410
haftmann@35043
  1411
subclass linordered_semiring_strict
haftmann@28823
  1412
proof
huffman@23550
  1413
  fix a b c :: 'a
huffman@23550
  1414
  assume "a < b" "0 < c"
haftmann@38642
  1415
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
haftmann@57512
  1416
  thus "a * c < b * c" by (simp only: mult.commute)
huffman@23550
  1417
qed
paulson@14272
  1418
haftmann@35028
  1419
subclass ordered_cancel_comm_semiring
haftmann@28823
  1420
proof
huffman@23550
  1421
  fix a b c :: 'a
huffman@23550
  1422
  assume "a \<le> b" "0 \<le> c"
huffman@23550
  1423
  thus "c * a \<le> c * b"
haftmann@25186
  1424
    unfolding le_less
haftmann@26193
  1425
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
  1426
qed
paulson@14272
  1427
haftmann@25267
  1428
end
haftmann@25230
  1429
lp15@60562
  1430
class ordered_ring = ring + ordered_cancel_semiring
haftmann@25267
  1431
begin
haftmann@25230
  1432
haftmann@35028
  1433
subclass ordered_ab_group_add ..
paulson@14270
  1434
haftmann@25230
  1435
lemma less_add_iff1:
haftmann@25230
  1436
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
  1437
by (simp add: algebra_simps)
haftmann@25230
  1438
haftmann@25230
  1439
lemma less_add_iff2:
haftmann@25230
  1440
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
  1441
by (simp add: algebra_simps)
haftmann@25230
  1442
haftmann@25230
  1443
lemma le_add_iff1:
haftmann@25230
  1444
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
  1445
by (simp add: algebra_simps)
haftmann@25230
  1446
haftmann@25230
  1447
lemma le_add_iff2:
haftmann@25230
  1448
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
  1449
by (simp add: algebra_simps)
haftmann@25230
  1450
haftmann@25230
  1451
lemma mult_left_mono_neg:
haftmann@25230
  1452
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
  1453
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
  1454
  apply simp_all
haftmann@25230
  1455
  done
haftmann@25230
  1456
haftmann@25230
  1457
lemma mult_right_mono_neg:
haftmann@25230
  1458
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
  1459
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
  1460
  apply simp_all
haftmann@25230
  1461
  done
haftmann@25230
  1462
huffman@30692
  1463
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
  1464
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
  1465
haftmann@25230
  1466
lemma split_mult_pos_le:
haftmann@25230
  1467
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@56536
  1468
by (auto simp add: mult_nonpos_nonpos)
haftmann@25186
  1469
haftmann@25186
  1470
end
paulson@14270
  1471
haftmann@35028
  1472
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
  1473
begin
haftmann@25304
  1474
haftmann@35028
  1475
subclass ordered_ring ..
haftmann@35028
  1476
haftmann@35028
  1477
subclass ordered_ab_group_add_abs
haftmann@28823
  1478
proof
haftmann@25304
  1479
  fix a b
haftmann@25304
  1480
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@54230
  1481
    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
  1482
qed (auto simp add: abs_if)
haftmann@25304
  1483
huffman@35631
  1484
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
  1485
  using linear [of 0 a]
nipkow@56536
  1486
  by (auto simp add: mult_nonpos_nonpos)
huffman@35631
  1487
huffman@35631
  1488
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
  1489
  by (simp add: not_less)
huffman@35631
  1490
haftmann@25304
  1491
end
obua@23521
  1492
haftmann@35043
  1493
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
  1494
  + ordered_ab_group_add + abs_if
haftmann@25230
  1495
begin
paulson@14348
  1496
haftmann@35028
  1497
subclass linordered_ring ..
haftmann@25304
  1498
huffman@30692
  1499
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
  1500
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
  1501
huffman@30692
  1502
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
  1503
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
  1504
huffman@30692
  1505
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
  1506
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
  1507
haftmann@25917
  1508
subclass ring_no_zero_divisors
haftmann@28823
  1509
proof
haftmann@25917
  1510
  fix a b
haftmann@25917
  1511
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
  1512
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
  1513
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
  1514
  proof (cases "a < 0")
haftmann@25917
  1515
    case True note A' = this
haftmann@25917
  1516
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1517
      case True with A'
haftmann@25917
  1518
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
  1519
    next
haftmann@25917
  1520
      case False with B have "0 < b" by auto
haftmann@25917
  1521
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
  1522
    qed
haftmann@25917
  1523
  next
haftmann@25917
  1524
    case False with A have A': "0 < a" by auto
haftmann@25917
  1525
    show ?thesis proof (cases "b < 0")
haftmann@25917
  1526
      case True with A'
haftmann@25917
  1527
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
  1528
    next
haftmann@25917
  1529
      case False with B have "0 < b" by auto
nipkow@56544
  1530
      with A' show ?thesis by auto
haftmann@25917
  1531
    qed
haftmann@25917
  1532
  qed
haftmann@25917
  1533
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
  1534
qed
haftmann@25304
  1535
hoelzl@56480
  1536
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
hoelzl@56480
  1537
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
nipkow@56544
  1538
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
huffman@22990
  1539
hoelzl@56480
  1540
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
  1541
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
  1542
paulson@14265
  1543
lemma mult_less_0_iff:
haftmann@25917
  1544
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
  1545
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
  1546
  apply force
haftmann@25917
  1547
  done
paulson@14265
  1548
paulson@14265
  1549
lemma mult_le_0_iff:
haftmann@25917
  1550
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
lp15@60562
  1551
  apply (insert zero_le_mult_iff [of "-a" b])
huffman@35216
  1552
  apply force
haftmann@25917
  1553
  done
haftmann@25917
  1554
haftmann@26193
  1555
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
  1556
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
  1557
haftmann@26193
  1558
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
  1559
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
  1560
haftmann@26193
  1561
lemma mult_less_cancel_right_disj:
haftmann@26193
  1562
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1563
  apply (cases "c = 0")
lp15@60562
  1564
  apply (auto simp add: neq_iff mult_strict_right_mono
haftmann@26193
  1565
                      mult_strict_right_mono_neg)
lp15@60562
  1566
  apply (auto simp add: not_less
haftmann@26193
  1567
                      not_le [symmetric, of "a*c"]
haftmann@26193
  1568
                      not_le [symmetric, of a])
haftmann@26193
  1569
  apply (erule_tac [!] notE)
lp15@60562
  1570
  apply (auto simp add: less_imp_le mult_right_mono
haftmann@26193
  1571
                      mult_right_mono_neg)
haftmann@26193
  1572
  done
haftmann@26193
  1573
haftmann@26193
  1574
lemma mult_less_cancel_left_disj:
haftmann@26193
  1575
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
  1576
  apply (cases "c = 0")
lp15@60562
  1577
  apply (auto simp add: neq_iff mult_strict_left_mono
haftmann@26193
  1578
                      mult_strict_left_mono_neg)
lp15@60562
  1579
  apply (auto simp add: not_less
haftmann@26193
  1580
                      not_le [symmetric, of "c*a"]
haftmann@26193
  1581
                      not_le [symmetric, of a])
haftmann@26193
  1582
  apply (erule_tac [!] notE)
lp15@60562
  1583
  apply (auto simp add: less_imp_le mult_left_mono
haftmann@26193
  1584
                      mult_left_mono_neg)
haftmann@26193
  1585
  done
haftmann@26193
  1586
haftmann@26193
  1587
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
  1588
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
  1589
haftmann@26193
  1590
lemma mult_less_cancel_right:
haftmann@26193
  1591
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1592
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
  1593
haftmann@26193
  1594
lemma mult_less_cancel_left:
haftmann@26193
  1595
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
  1596
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
  1597
haftmann@26193
  1598
lemma mult_le_cancel_right:
haftmann@26193
  1599
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1600
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
  1601
haftmann@26193
  1602
lemma mult_le_cancel_left:
haftmann@26193
  1603
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
  1604
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
  1605
nipkow@30649
  1606
lemma mult_le_cancel_left_pos:
nipkow@30649
  1607
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
  1608
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1609
nipkow@30649
  1610
lemma mult_le_cancel_left_neg:
nipkow@30649
  1611
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
  1612
by (auto simp: mult_le_cancel_left)
nipkow@30649
  1613
nipkow@30649
  1614
lemma mult_less_cancel_left_pos:
nipkow@30649
  1615
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
  1616
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1617
nipkow@30649
  1618
lemma mult_less_cancel_left_neg:
nipkow@30649
  1619
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
  1620
by (auto simp: mult_less_cancel_left)
nipkow@30649
  1621
haftmann@25917
  1622
end
paulson@14265
  1623
huffman@30692
  1624
lemmas mult_sign_intros =
huffman@30692
  1625
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
  1626
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
  1627
  mult_pos_pos mult_pos_neg
huffman@30692
  1628
  mult_neg_pos mult_neg_neg
haftmann@25230
  1629
haftmann@35028
  1630
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
  1631
begin
haftmann@25230
  1632
haftmann@35028
  1633
subclass ordered_ring ..
haftmann@35028
  1634
subclass ordered_cancel_comm_semiring ..
haftmann@25230
  1635
haftmann@25267
  1636
end
haftmann@25230
  1637
haftmann@59833
  1638
class linordered_semidom = semidom + linordered_comm_semiring_strict +
haftmann@25230
  1639
  assumes zero_less_one [simp]: "0 < 1"
lp15@60562
  1640
  assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
haftmann@25230
  1641
begin
haftmann@25230
  1642
lp15@60562
  1643
text {* Addition is the inverse of subtraction. *}
lp15@60562
  1644
lp15@60562
  1645
lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"
lp15@60562
  1646
  by (frule le_add_diff_inverse2) (simp add: add.commute)
lp15@60562
  1647
lp15@60562
  1648
lemma add_diff_inverse: "~ a<b \<Longrightarrow> b + (a - b) = a"
lp15@60562
  1649
  by simp
lp15@60615
  1650
lp15@60615
  1651
lemma add_le_imp_le_diff: 
lp15@60615
  1652
  shows "i + k \<le> n \<Longrightarrow> i \<le> n - k"
lp15@60615
  1653
  apply (subst add_le_cancel_right [where c=k, symmetric])
lp15@60615
  1654
  apply (frule le_add_diff_inverse2)
lp15@60615
  1655
  apply (simp only: add.assoc [symmetric])
lp15@60615
  1656
  using add_implies_diff by fastforce
lp15@60615
  1657
lp15@60615
  1658
lemma add_le_add_imp_diff_le: 
lp15@60615
  1659
  assumes a1: "i + k \<le> n"
lp15@60615
  1660
      and a2: "n \<le> j + k"
lp15@60615
  1661
  shows "\<lbrakk>i + k \<le> n; n \<le> j + k\<rbrakk> \<Longrightarrow> n - k \<le> j"
lp15@60615
  1662
proof -
lp15@60615
  1663
  have "n - (i + k) + (i + k) = n"
lp15@60615
  1664
    using a1 by simp
lp15@60615
  1665
  moreover have "n - k = n - k - i + i"
lp15@60615
  1666
    using a1 by (simp add: add_le_imp_le_diff)
lp15@60615
  1667
  ultimately show ?thesis
lp15@60615
  1668
    using a2
lp15@60615
  1669
    apply (simp add: add.assoc [symmetric])
lp15@60615
  1670
    apply (rule add_le_imp_le_diff [of _ k "j+k", simplified add_diff_cancel_right'])
lp15@60615
  1671
    by (simp add: add.commute diff_diff_add)
lp15@60615
  1672
qed
lp15@60615
  1673
haftmann@25230
  1674
lemma pos_add_strict:
haftmann@25230
  1675
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
  1676
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
  1677
haftmann@26193
  1678
lemma zero_le_one [simp]: "0 \<le> 1"
lp15@60562
  1679
by (rule zero_less_one [THEN less_imp_le])
haftmann@26193
  1680
haftmann@26193
  1681
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
lp15@60562
  1682
by (simp add: not_le)
haftmann@26193
  1683
haftmann@26193
  1684
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
lp15@60562
  1685
by (simp add: not_less)
haftmann@26193
  1686
haftmann@26193
  1687
lemma less_1_mult:
haftmann@26193
  1688
  assumes "1 < m" and "1 < n"
haftmann@26193
  1689
  shows "1 < m * n"
haftmann@26193
  1690
  using assms mult_strict_mono [of 1 m 1 n]
lp15@60562
  1691
    by (simp add:  less_trans [OF zero_less_one])
haftmann@26193
  1692
hoelzl@59000
  1693
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
hoelzl@59000
  1694
  using mult_left_mono[of c 1 a] by simp
hoelzl@59000
  1695
hoelzl@59000
  1696
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
hoelzl@59000
  1697
  using mult_mono[of a 1 b 1] by simp
hoelzl@59000
  1698
haftmann@25230
  1699
end
haftmann@25230
  1700
haftmann@35028
  1701
class linordered_idom = comm_ring_1 +
haftmann@35028
  1702
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1703
  abs_if + sgn_if
haftmann@25917
  1704
begin
haftmann@25917
  1705
hoelzl@36622
  1706
subclass linordered_semiring_1_strict ..
haftmann@35043
  1707
subclass linordered_ring_strict ..
haftmann@35028
  1708
subclass ordered_comm_ring ..
huffman@27516
  1709
subclass idom ..
haftmann@25917
  1710
haftmann@35028
  1711
subclass linordered_semidom
haftmann@28823
  1712
proof
haftmann@26193
  1713
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1714
  thus "0 < 1" by (simp add: le_less)
lp15@60562
  1715
  show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"
lp15@60562
  1716
    by simp
lp15@60562
  1717
qed
haftmann@25917
  1718
haftmann@35028
  1719
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1720
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1721
  using assms by (rule neqE)
haftmann@26193
  1722
haftmann@26274
  1723
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1724
haftmann@26274
  1725
lemma mult_le_cancel_right1:
haftmann@26274
  1726
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1727
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1728
haftmann@26274
  1729
lemma mult_le_cancel_right2:
haftmann@26274
  1730
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1731
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1732
haftmann@26274
  1733
lemma mult_le_cancel_left1:
haftmann@26274
  1734
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1735
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1736
haftmann@26274
  1737
lemma mult_le_cancel_left2:
haftmann@26274
  1738
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1739
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1740
haftmann@26274
  1741
lemma mult_less_cancel_right1:
haftmann@26274
  1742
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1743
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1744
haftmann@26274
  1745
lemma mult_less_cancel_right2:
haftmann@26274
  1746
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1747
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1748
haftmann@26274
  1749
lemma mult_less_cancel_left1:
haftmann@26274
  1750
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1751
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1752
haftmann@26274
  1753
lemma mult_less_cancel_left2:
haftmann@26274
  1754
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1755
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1756
haftmann@27651
  1757
lemma sgn_sgn [simp]:
haftmann@27651
  1758
  "sgn (sgn a) = sgn a"
nipkow@29700
  1759
unfolding sgn_if by simp
haftmann@27651
  1760
haftmann@27651
  1761
lemma sgn_0_0:
haftmann@27651
  1762
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1763
unfolding sgn_if by simp
haftmann@27651
  1764
haftmann@27651
  1765
lemma sgn_1_pos:
haftmann@27651
  1766
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1767
unfolding sgn_if by simp
haftmann@27651
  1768
haftmann@27651
  1769
lemma sgn_1_neg:
haftmann@27651
  1770
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1771
unfolding sgn_if by auto
haftmann@27651
  1772
haftmann@29940
  1773
lemma sgn_pos [simp]:
haftmann@29940
  1774
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1775
unfolding sgn_1_pos .
haftmann@29940
  1776
haftmann@29940
  1777
lemma sgn_neg [simp]:
haftmann@29940
  1778
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1779
unfolding sgn_1_neg .
haftmann@29940
  1780
haftmann@27651
  1781
lemma sgn_times:
haftmann@27651
  1782
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1783
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1784
haftmann@36301
  1785
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1786
unfolding sgn_if abs_if by auto
nipkow@29700
  1787
haftmann@29940
  1788
lemma sgn_greater [simp]:
haftmann@29940
  1789
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1790
  unfolding sgn_if by auto
haftmann@29940
  1791
haftmann@29940
  1792
lemma sgn_less [simp]:
haftmann@29940
  1793
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1794
  unfolding sgn_if by auto
haftmann@29940
  1795
haftmann@36301
  1796
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1797
  by (simp add: abs_if)
huffman@29949
  1798
haftmann@36301
  1799
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1800
  by (simp add: abs_if)
haftmann@29653
  1801
nipkow@33676
  1802
lemma dvd_if_abs_eq:
haftmann@36301
  1803
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1804
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1805
huffman@55912
  1806
text {* The following lemmas can be proven in more general structures, but
lp15@60562
  1807
are dangerous as simp rules in absence of @{thm neg_equal_zero},
haftmann@54489
  1808
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
haftmann@54489
  1809
haftmann@54489
  1810
lemma equation_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1811
  "1 = - a \<longleftrightarrow> a = - 1"
haftmann@54489
  1812
  by (fact equation_minus_iff)
haftmann@54489
  1813
haftmann@54489
  1814
lemma minus_equation_iff_1 [simp, no_atp]:
haftmann@54489
  1815
  "- a = 1 \<longleftrightarrow> a = - 1"
haftmann@54489
  1816
  by (subst minus_equation_iff, auto)
haftmann@54489
  1817
haftmann@54489
  1818
lemma le_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1819
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
haftmann@54489
  1820
  by (fact le_minus_iff)
haftmann@54489
  1821
haftmann@54489
  1822
lemma minus_le_iff_1 [simp, no_atp]:
haftmann@54489
  1823
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
haftmann@54489
  1824
  by (fact minus_le_iff)
haftmann@54489
  1825
haftmann@54489
  1826
lemma less_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1827
  "1 < - b \<longleftrightarrow> b < - 1"
haftmann@54489
  1828
  by (fact less_minus_iff)
haftmann@54489
  1829
haftmann@54489
  1830
lemma minus_less_iff_1 [simp, no_atp]:
haftmann@54489
  1831
  "- a < 1 \<longleftrightarrow> - 1 < a"
haftmann@54489
  1832
  by (fact minus_less_iff)
haftmann@54489
  1833
haftmann@25917
  1834
end
haftmann@25230
  1835
haftmann@26274
  1836
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1837
blanchet@54147
  1838
lemmas mult_compare_simps =
paulson@15234
  1839
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1840
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1841
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1842
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1843
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1844
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1845
    mult_cancel_right mult_cancel_left
paulson@15234
  1846
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1847
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1848
haftmann@36301
  1849
text {* Reasoning about inequalities with division *}
avigad@16775
  1850
haftmann@35028
  1851
context linordered_semidom
haftmann@25193
  1852
begin
haftmann@25193
  1853
haftmann@25193
  1854
lemma less_add_one: "a < a + 1"
paulson@14293
  1855
proof -
haftmann@25193
  1856
  have "a + 0 < a + 1"
nipkow@23482
  1857
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1858
  thus ?thesis by simp
paulson@14293
  1859
qed
paulson@14293
  1860
haftmann@25193
  1861
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1862
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1863
haftmann@25193
  1864
end
paulson@14365
  1865
haftmann@36301
  1866
context linordered_idom
haftmann@36301
  1867
begin
paulson@15234
  1868
haftmann@36301
  1869
lemma mult_right_le_one_le:
haftmann@36301
  1870
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@59833
  1871
  by (rule mult_left_le)
haftmann@36301
  1872
haftmann@36301
  1873
lemma mult_left_le_one_le:
haftmann@36301
  1874
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1875
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1876
haftmann@36301
  1877
end
haftmann@36301
  1878
haftmann@36301
  1879
text {* Absolute Value *}
paulson@14293
  1880
haftmann@35028
  1881
context linordered_idom
haftmann@25304
  1882
begin
haftmann@25304
  1883
haftmann@36301
  1884
lemma mult_sgn_abs:
haftmann@36301
  1885
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1886
  unfolding abs_if sgn_if by auto
haftmann@25304
  1887
haftmann@36301
  1888
lemma abs_one [simp]:
haftmann@36301
  1889
  "\<bar>1\<bar> = 1"
huffman@44921
  1890
  by (simp add: abs_if)
haftmann@36301
  1891
haftmann@25304
  1892
end
nipkow@24491
  1893
haftmann@35028
  1894
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1895
  assumes abs_eq_mult:
haftmann@25304
  1896
    "(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
  1897
haftmann@35028
  1898
context linordered_idom
haftmann@30961
  1899
begin
haftmann@30961
  1900
haftmann@35028
  1901
subclass ordered_ring_abs proof
huffman@35216
  1902
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1903
haftmann@30961
  1904
lemma abs_mult:
lp15@60562
  1905
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
haftmann@30961
  1906
  by (rule abs_eq_mult) auto
haftmann@30961
  1907
haftmann@30961
  1908
lemma abs_mult_self:
haftmann@36301
  1909
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
lp15@60562
  1910
  by (simp add: abs_if)
haftmann@30961
  1911
paulson@14294
  1912
lemma abs_mult_less:
haftmann@36301
  1913
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1914
proof -
haftmann@36301
  1915
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1916
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1917
  assume "\<bar>b\<bar> < d"
lp15@60562
  1918
  thus ?thesis by (simp add: ac cpos mult_strict_mono)
paulson@14294
  1919
qed
paulson@14293
  1920
haftmann@36301
  1921
lemma abs_less_iff:
lp15@60562
  1922
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
haftmann@36301
  1923
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1924
haftmann@36301
  1925
lemma abs_mult_pos:
haftmann@36301
  1926
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1927
  by (simp add: abs_mult)
haftmann@36301
  1928
hoelzl@51520
  1929
lemma abs_diff_less_iff:
hoelzl@51520
  1930
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
hoelzl@51520
  1931
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
hoelzl@51520
  1932
lp15@59865
  1933
lemma abs_diff_le_iff:
lp15@59865
  1934
   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
lp15@59865
  1935
  by (auto simp add: diff_le_eq ac_simps abs_le_iff)
lp15@59865
  1936
haftmann@36301
  1937
end
avigad@16775
  1938
haftmann@59557
  1939
hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
haftmann@59557
  1940
haftmann@52435
  1941
code_identifier
haftmann@52435
  1942
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1943
paulson@14265
  1944
end