src/HOL/Rings.thy
author hoelzl
Thu Nov 13 17:19:52 2014 +0100 (2014-11-13)
changeset 59000 6eb0725503fc
parent 58952 5d82cdef6c1b
child 59009 348561aa3869
permissions -rw-r--r--
import general theorems from AFP/Markov_Models
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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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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult + dvd
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  (*previously almost_semiring*)
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begin
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subclass semiring_1 ..
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lemma dvd_refl[simp]: "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 dvd_0_left_iff [simp]: "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]: "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 one_dvd [simp]: "1 dvd a"
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by (auto intro!: dvdI)
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lemma dvd_mult[simp]: "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]: "a dvd b \<Longrightarrow> a dvd (b * c)"
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  apply (subst mult.commute)
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  apply (erule dvd_mult)
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  done
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lemma dvd_triv_right [simp]: "a dvd b * a"
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by (rule dvd_mult) (rule dvd_refl)
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lemma dvd_triv_left [simp]: "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: "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: "a * b dvd c \<Longrightarrow> b dvd c"
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  unfolding mult.commute [of a] by (rule dvd_mult_left)
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lemma dvd_0_left: "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" 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_dvd = comm_semiring_1 +
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  assumes dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b"
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  assumes dvd_addD: "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"
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begin
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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"
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  using assms by (auto dest: dvd_addD)
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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 no_zero_divisors = zero + times +
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  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
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begin
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lemma divisors_zero:
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  assumes "a * b = 0"
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  shows "a = 0 \<or> b = 0"
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proof (rule classical)
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  assume "\<not> (a = 0 \<or> b = 0)"
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  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
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  with no_zero_divisors have "a * b \<noteq> 0" by blast
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  with assms show ?thesis by simp
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qed
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end
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class semiring_1_cancel = semiring + cancel_comm_monoid_add
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  + zero_neq_one + monoid_mult
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begin
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subclass semiring_0_cancel ..
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subclass semiring_1 ..
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end
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class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add
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  + zero_neq_one + comm_monoid_mult
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begin
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subclass semiring_1_cancel ..
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subclass comm_semiring_0_cancel ..
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subclass comm_semiring_1 ..
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end
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class 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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subclass ring ..
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subclass comm_semiring_0_cancel ..
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lemma square_diff_square_factored:
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  "x * x - y * y = (x + y) * (x - y)"
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  by (simp add: algebra_simps)
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end
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class ring_1 = ring + zero_neq_one + monoid_mult
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begin
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subclass semiring_1_cancel ..
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lemma square_diff_one_factored:
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  "x * x - 1 = (x + 1) * (x - 1)"
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  by (simp add: algebra_simps)
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end
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class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
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  (*previously ring*)
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begin
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subclass ring_1 ..
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subclass comm_semiring_1_cancel ..
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subclass semiring_dvd
haftmann@58647
   357
proof
haftmann@58647
   358
  fix a b c
haftmann@58647
   359
  show "a dvd c * a + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
haftmann@58647
   360
  proof
haftmann@58647
   361
    assume ?Q then show ?P by simp
haftmann@58647
   362
  next
haftmann@58647
   363
    assume ?P then obtain d where "c * a + b = a * d" ..
haftmann@58647
   364
    then have "b = a * (d - c)" by (simp add: algebra_simps)
haftmann@58647
   365
    then show ?Q ..
haftmann@58647
   366
  qed
haftmann@58647
   367
  assume "a dvd b + c" and "a dvd b"
haftmann@58647
   368
  show "a dvd c"
haftmann@58647
   369
  proof -
haftmann@58647
   370
    from `a dvd b` obtain d where "b = a * d" ..
haftmann@58647
   371
    moreover from `a dvd b + c` obtain e where "b + c = a * e" ..
haftmann@58647
   372
    ultimately have "a * d + c = a * e" by simp
haftmann@58647
   373
    then have "c = a * (e - d)" by (simp add: algebra_simps)
haftmann@58647
   374
    then show "a dvd c" ..
haftmann@58647
   375
  qed
haftmann@58647
   376
qed
haftmann@58647
   377
huffman@29465
   378
lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
huffman@29408
   379
proof
huffman@29408
   380
  assume "x dvd - y"
huffman@29408
   381
  then have "x dvd - 1 * - y" by (rule dvd_mult)
huffman@29408
   382
  then show "x dvd y" by simp
huffman@29408
   383
next
huffman@29408
   384
  assume "x dvd y"
huffman@29408
   385
  then have "x dvd - 1 * y" by (rule dvd_mult)
huffman@29408
   386
  then show "x dvd - y" by simp
huffman@29408
   387
qed
huffman@29408
   388
huffman@29465
   389
lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
huffman@29408
   390
proof
huffman@29408
   391
  assume "- x dvd y"
huffman@29408
   392
  then obtain k where "y = - x * k" ..
huffman@29408
   393
  then have "y = x * - k" by simp
huffman@29408
   394
  then show "x dvd y" ..
huffman@29408
   395
next
huffman@29408
   396
  assume "x dvd y"
huffman@29408
   397
  then obtain k where "y = x * k" ..
huffman@29408
   398
  then have "y = - x * - k" by simp
huffman@29408
   399
  then show "- x dvd y" ..
huffman@29408
   400
qed
huffman@29408
   401
haftmann@54230
   402
lemma dvd_diff [simp]:
haftmann@54230
   403
  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
haftmann@54230
   404
  using dvd_add [of x y "- z"] by simp
huffman@29409
   405
haftmann@25267
   406
end
haftmann@25152
   407
haftmann@58952
   408
class semiring_no_zero_divisors = semiring_0 + no_zero_divisors
haftmann@25230
   409
begin
haftmann@25230
   410
haftmann@25230
   411
lemma mult_eq_0_iff [simp]:
haftmann@58952
   412
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@25230
   413
proof (cases "a = 0 \<or> b = 0")
haftmann@25230
   414
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
haftmann@25230
   415
    then show ?thesis using no_zero_divisors by simp
haftmann@25230
   416
next
haftmann@25230
   417
  case True then show ?thesis by auto
haftmann@25230
   418
qed
haftmann@25230
   419
haftmann@58952
   420
end
haftmann@58952
   421
haftmann@58952
   422
class ring_no_zero_divisors = ring + semiring_no_zero_divisors
haftmann@58952
   423
begin
haftmann@58952
   424
haftmann@26193
   425
text{*Cancellation of equalities with a common factor*}
blanchet@54147
   426
lemma mult_cancel_right [simp]:
haftmann@26193
   427
  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@26193
   428
proof -
haftmann@26193
   429
  have "(a * c = b * c) = ((a - b) * c = 0)"
huffman@35216
   430
    by (simp add: algebra_simps)
huffman@35216
   431
  thus ?thesis by (simp add: disj_commute)
haftmann@26193
   432
qed
haftmann@26193
   433
blanchet@54147
   434
lemma mult_cancel_left [simp]:
haftmann@26193
   435
  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@26193
   436
proof -
haftmann@26193
   437
  have "(c * a = c * b) = (c * (a - b) = 0)"
huffman@35216
   438
    by (simp add: algebra_simps)
huffman@35216
   439
  thus ?thesis by simp
haftmann@26193
   440
qed
haftmann@26193
   441
haftmann@58952
   442
lemma mult_left_cancel:
haftmann@58952
   443
  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
haftmann@58952
   444
  by simp 
lp15@56217
   445
haftmann@58952
   446
lemma mult_right_cancel:
haftmann@58952
   447
  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
haftmann@58952
   448
  by simp 
lp15@56217
   449
haftmann@25230
   450
end
huffman@22990
   451
huffman@23544
   452
class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
haftmann@26274
   453
begin
haftmann@26274
   454
huffman@36970
   455
lemma square_eq_1_iff:
huffman@36821
   456
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
huffman@36821
   457
proof -
huffman@36821
   458
  have "(x - 1) * (x + 1) = x * x - 1"
huffman@36821
   459
    by (simp add: algebra_simps)
huffman@36821
   460
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
huffman@36821
   461
    by simp
huffman@36821
   462
  thus ?thesis
huffman@36821
   463
    by (simp add: eq_neg_iff_add_eq_0)
huffman@36821
   464
qed
huffman@36821
   465
haftmann@26274
   466
lemma mult_cancel_right1 [simp]:
haftmann@26274
   467
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   468
by (insert mult_cancel_right [of 1 c b], force)
haftmann@26274
   469
haftmann@26274
   470
lemma mult_cancel_right2 [simp]:
haftmann@26274
   471
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   472
by (insert mult_cancel_right [of a c 1], simp)
haftmann@26274
   473
 
haftmann@26274
   474
lemma mult_cancel_left1 [simp]:
haftmann@26274
   475
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
nipkow@29667
   476
by (insert mult_cancel_left [of c 1 b], force)
haftmann@26274
   477
haftmann@26274
   478
lemma mult_cancel_left2 [simp]:
haftmann@26274
   479
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
nipkow@29667
   480
by (insert mult_cancel_left [of c a 1], simp)
haftmann@26274
   481
haftmann@26274
   482
end
huffman@22990
   483
haftmann@22390
   484
class idom = comm_ring_1 + no_zero_divisors
haftmann@25186
   485
begin
paulson@14421
   486
huffman@27516
   487
subclass ring_1_no_zero_divisors ..
huffman@22990
   488
huffman@29915
   489
lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> (a = b \<or> a = - b)"
huffman@29915
   490
proof
huffman@29915
   491
  assume "a * a = b * b"
huffman@29915
   492
  then have "(a - b) * (a + b) = 0"
huffman@29915
   493
    by (simp add: algebra_simps)
huffman@29915
   494
  then show "a = b \<or> a = - b"
huffman@35216
   495
    by (simp add: eq_neg_iff_add_eq_0)
huffman@29915
   496
next
huffman@29915
   497
  assume "a = b \<or> a = - b"
huffman@29915
   498
  then show "a * a = b * b" by auto
huffman@29915
   499
qed
huffman@29915
   500
huffman@29981
   501
lemma dvd_mult_cancel_right [simp]:
huffman@29981
   502
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   503
proof -
huffman@29981
   504
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   505
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   506
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   507
    unfolding dvd_def by simp
huffman@29981
   508
  finally show ?thesis .
huffman@29981
   509
qed
huffman@29981
   510
huffman@29981
   511
lemma dvd_mult_cancel_left [simp]:
huffman@29981
   512
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   513
proof -
huffman@29981
   514
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
haftmann@57514
   515
    unfolding dvd_def by (simp add: ac_simps)
huffman@29981
   516
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
huffman@29981
   517
    unfolding dvd_def by simp
huffman@29981
   518
  finally show ?thesis .
huffman@29981
   519
qed
huffman@29981
   520
haftmann@25186
   521
end
haftmann@25152
   522
haftmann@35302
   523
text {*
haftmann@35302
   524
  The theory of partially ordered rings is taken from the books:
haftmann@35302
   525
  \begin{itemize}
haftmann@35302
   526
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35302
   527
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35302
   528
  \end{itemize}
haftmann@35302
   529
  Most of the used notions can also be looked up in 
haftmann@35302
   530
  \begin{itemize}
wenzelm@54703
   531
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
haftmann@35302
   532
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35302
   533
  \end{itemize}
haftmann@35302
   534
*}
haftmann@35302
   535
haftmann@38642
   536
class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
haftmann@38642
   537
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@38642
   538
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
haftmann@25230
   539
begin
haftmann@25230
   540
haftmann@25230
   541
lemma mult_mono:
haftmann@38642
   542
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   543
apply (erule mult_right_mono [THEN order_trans], assumption)
haftmann@25230
   544
apply (erule mult_left_mono, assumption)
haftmann@25230
   545
done
haftmann@25230
   546
haftmann@25230
   547
lemma mult_mono':
haftmann@38642
   548
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
haftmann@25230
   549
apply (rule mult_mono)
haftmann@25230
   550
apply (fast intro: order_trans)+
haftmann@25230
   551
done
haftmann@25230
   552
haftmann@25230
   553
end
krauss@21199
   554
haftmann@38642
   555
class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
haftmann@25267
   556
begin
paulson@14268
   557
huffman@27516
   558
subclass semiring_0_cancel ..
obua@23521
   559
nipkow@56536
   560
lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   561
using mult_left_mono [of 0 b a] by simp
haftmann@25230
   562
haftmann@25230
   563
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   564
using mult_left_mono [of b 0 a] by simp
huffman@30692
   565
huffman@30692
   566
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
haftmann@36301
   567
using mult_right_mono [of a 0 b] by simp
huffman@30692
   568
huffman@30692
   569
text {* Legacy - use @{text mult_nonpos_nonneg} *}
haftmann@25230
   570
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
haftmann@36301
   571
by (drule mult_right_mono [of b 0], auto)
haftmann@25230
   572
haftmann@26234
   573
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
nipkow@29667
   574
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
haftmann@25230
   575
haftmann@25230
   576
end
haftmann@25230
   577
haftmann@38642
   578
class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
haftmann@25267
   579
begin
haftmann@25230
   580
haftmann@35028
   581
subclass ordered_cancel_semiring ..
haftmann@35028
   582
haftmann@35028
   583
subclass ordered_comm_monoid_add ..
haftmann@25304
   584
haftmann@25230
   585
lemma mult_left_less_imp_less:
haftmann@25230
   586
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   587
by (force simp add: mult_left_mono not_le [symmetric])
haftmann@25230
   588
 
haftmann@25230
   589
lemma mult_right_less_imp_less:
haftmann@25230
   590
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
nipkow@29667
   591
by (force simp add: mult_right_mono not_le [symmetric])
obua@23521
   592
haftmann@25186
   593
end
haftmann@25152
   594
haftmann@35043
   595
class linordered_semiring_1 = linordered_semiring + semiring_1
hoelzl@36622
   596
begin
hoelzl@36622
   597
hoelzl@36622
   598
lemma convex_bound_le:
hoelzl@36622
   599
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   600
  shows "u * x + v * y \<le> a"
hoelzl@36622
   601
proof-
hoelzl@36622
   602
  from assms have "u * x + v * y \<le> u * a + v * a"
hoelzl@36622
   603
    by (simp add: add_mono mult_left_mono)
webertj@49962
   604
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   605
qed
hoelzl@36622
   606
hoelzl@36622
   607
end
haftmann@35043
   608
haftmann@35043
   609
class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
haftmann@25062
   610
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25062
   611
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
haftmann@25267
   612
begin
paulson@14341
   613
huffman@27516
   614
subclass semiring_0_cancel ..
obua@14940
   615
haftmann@35028
   616
subclass linordered_semiring
haftmann@28823
   617
proof
huffman@23550
   618
  fix a b c :: 'a
huffman@23550
   619
  assume A: "a \<le> b" "0 \<le> c"
huffman@23550
   620
  from A show "c * a \<le> c * b"
haftmann@25186
   621
    unfolding le_less
haftmann@25186
   622
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   623
  from A show "a * c \<le> b * c"
haftmann@25152
   624
    unfolding le_less
haftmann@25186
   625
    using mult_strict_right_mono by (cases "c = 0") auto
haftmann@25152
   626
qed
haftmann@25152
   627
haftmann@25230
   628
lemma mult_left_le_imp_le:
haftmann@25230
   629
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   630
by (force simp add: mult_strict_left_mono _not_less [symmetric])
haftmann@25230
   631
 
haftmann@25230
   632
lemma mult_right_le_imp_le:
haftmann@25230
   633
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
nipkow@29667
   634
by (force simp add: mult_strict_right_mono not_less [symmetric])
haftmann@25230
   635
nipkow@56544
   636
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
haftmann@36301
   637
using mult_strict_left_mono [of 0 b a] by simp
huffman@30692
   638
huffman@30692
   639
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
haftmann@36301
   640
using mult_strict_left_mono [of b 0 a] by simp
huffman@30692
   641
huffman@30692
   642
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
haftmann@36301
   643
using mult_strict_right_mono [of a 0 b] by simp
huffman@30692
   644
huffman@30692
   645
text {* Legacy - use @{text mult_neg_pos} *}
huffman@30692
   646
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
haftmann@36301
   647
by (drule mult_strict_right_mono [of b 0], auto)
haftmann@25230
   648
haftmann@25230
   649
lemma zero_less_mult_pos:
haftmann@25230
   650
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   651
apply (cases "b\<le>0")
haftmann@25230
   652
 apply (auto simp add: le_less not_less)
huffman@30692
   653
apply (drule_tac mult_pos_neg [of a b])
haftmann@25230
   654
 apply (auto dest: less_not_sym)
haftmann@25230
   655
done
haftmann@25230
   656
haftmann@25230
   657
lemma zero_less_mult_pos2:
haftmann@25230
   658
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
huffman@30692
   659
apply (cases "b\<le>0")
haftmann@25230
   660
 apply (auto simp add: le_less not_less)
huffman@30692
   661
apply (drule_tac mult_pos_neg2 [of a b])
haftmann@25230
   662
 apply (auto dest: less_not_sym)
haftmann@25230
   663
done
haftmann@25230
   664
haftmann@26193
   665
text{*Strict monotonicity in both arguments*}
haftmann@26193
   666
lemma mult_strict_mono:
haftmann@26193
   667
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
haftmann@26193
   668
  shows "a * c < b * d"
haftmann@26193
   669
  using assms apply (cases "c=0")
nipkow@56544
   670
  apply (simp)
haftmann@26193
   671
  apply (erule mult_strict_right_mono [THEN less_trans])
huffman@30692
   672
  apply (force simp add: le_less)
haftmann@26193
   673
  apply (erule mult_strict_left_mono, assumption)
haftmann@26193
   674
  done
haftmann@26193
   675
haftmann@26193
   676
text{*This weaker variant has more natural premises*}
haftmann@26193
   677
lemma mult_strict_mono':
haftmann@26193
   678
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
haftmann@26193
   679
  shows "a * c < b * d"
nipkow@29667
   680
by (rule mult_strict_mono) (insert assms, auto)
haftmann@26193
   681
haftmann@26193
   682
lemma mult_less_le_imp_less:
haftmann@26193
   683
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
haftmann@26193
   684
  shows "a * c < b * d"
haftmann@26193
   685
  using assms apply (subgoal_tac "a * c < b * c")
haftmann@26193
   686
  apply (erule less_le_trans)
haftmann@26193
   687
  apply (erule mult_left_mono)
haftmann@26193
   688
  apply simp
haftmann@26193
   689
  apply (erule mult_strict_right_mono)
haftmann@26193
   690
  apply assumption
haftmann@26193
   691
  done
haftmann@26193
   692
haftmann@26193
   693
lemma mult_le_less_imp_less:
haftmann@26193
   694
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
haftmann@26193
   695
  shows "a * c < b * d"
haftmann@26193
   696
  using assms apply (subgoal_tac "a * c \<le> b * c")
haftmann@26193
   697
  apply (erule le_less_trans)
haftmann@26193
   698
  apply (erule mult_strict_left_mono)
haftmann@26193
   699
  apply simp
haftmann@26193
   700
  apply (erule mult_right_mono)
haftmann@26193
   701
  apply simp
haftmann@26193
   702
  done
haftmann@26193
   703
haftmann@26193
   704
lemma mult_less_imp_less_left:
haftmann@26193
   705
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
haftmann@26193
   706
  shows "a < b"
haftmann@26193
   707
proof (rule ccontr)
haftmann@26193
   708
  assume "\<not>  a < b"
haftmann@26193
   709
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   710
  hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)
nipkow@29667
   711
  with this and less show False by (simp add: not_less [symmetric])
haftmann@26193
   712
qed
haftmann@26193
   713
haftmann@26193
   714
lemma mult_less_imp_less_right:
haftmann@26193
   715
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
haftmann@26193
   716
  shows "a < b"
haftmann@26193
   717
proof (rule ccontr)
haftmann@26193
   718
  assume "\<not> a < b"
haftmann@26193
   719
  hence "b \<le> a" by (simp add: linorder_not_less)
haftmann@26193
   720
  hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)
nipkow@29667
   721
  with this and less show False by (simp add: not_less [symmetric])
haftmann@26193
   722
qed  
haftmann@26193
   723
haftmann@25230
   724
end
haftmann@25230
   725
haftmann@35097
   726
class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
hoelzl@36622
   727
begin
hoelzl@36622
   728
hoelzl@36622
   729
subclass linordered_semiring_1 ..
hoelzl@36622
   730
hoelzl@36622
   731
lemma convex_bound_lt:
hoelzl@36622
   732
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
hoelzl@36622
   733
  shows "u * x + v * y < a"
hoelzl@36622
   734
proof -
hoelzl@36622
   735
  from assms have "u * x + v * y < u * a + v * a"
hoelzl@36622
   736
    by (cases "u = 0")
hoelzl@36622
   737
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
webertj@49962
   738
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
hoelzl@36622
   739
qed
hoelzl@36622
   740
hoelzl@36622
   741
end
haftmann@33319
   742
haftmann@38642
   743
class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add + 
haftmann@38642
   744
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
haftmann@25186
   745
begin
haftmann@25152
   746
haftmann@35028
   747
subclass ordered_semiring
haftmann@28823
   748
proof
krauss@21199
   749
  fix a b c :: 'a
huffman@23550
   750
  assume "a \<le> b" "0 \<le> c"
haftmann@38642
   751
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
haftmann@57512
   752
  thus "a * c \<le> b * c" by (simp only: mult.commute)
krauss@21199
   753
qed
paulson@14265
   754
haftmann@25267
   755
end
haftmann@25267
   756
haftmann@38642
   757
class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
haftmann@25267
   758
begin
paulson@14265
   759
haftmann@38642
   760
subclass comm_semiring_0_cancel ..
haftmann@35028
   761
subclass ordered_comm_semiring ..
haftmann@35028
   762
subclass ordered_cancel_semiring ..
haftmann@25267
   763
haftmann@25267
   764
end
haftmann@25267
   765
haftmann@35028
   766
class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
haftmann@38642
   767
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
haftmann@25267
   768
begin
haftmann@25267
   769
haftmann@35043
   770
subclass linordered_semiring_strict
haftmann@28823
   771
proof
huffman@23550
   772
  fix a b c :: 'a
huffman@23550
   773
  assume "a < b" "0 < c"
haftmann@38642
   774
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
haftmann@57512
   775
  thus "a * c < b * c" by (simp only: mult.commute)
huffman@23550
   776
qed
paulson@14272
   777
haftmann@35028
   778
subclass ordered_cancel_comm_semiring
haftmann@28823
   779
proof
huffman@23550
   780
  fix a b c :: 'a
huffman@23550
   781
  assume "a \<le> b" "0 \<le> c"
huffman@23550
   782
  thus "c * a \<le> c * b"
haftmann@25186
   783
    unfolding le_less
haftmann@26193
   784
    using mult_strict_left_mono by (cases "c = 0") auto
huffman@23550
   785
qed
paulson@14272
   786
haftmann@25267
   787
end
haftmann@25230
   788
haftmann@35028
   789
class ordered_ring = ring + ordered_cancel_semiring 
haftmann@25267
   790
begin
haftmann@25230
   791
haftmann@35028
   792
subclass ordered_ab_group_add ..
paulson@14270
   793
haftmann@25230
   794
lemma less_add_iff1:
haftmann@25230
   795
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
nipkow@29667
   796
by (simp add: algebra_simps)
haftmann@25230
   797
haftmann@25230
   798
lemma less_add_iff2:
haftmann@25230
   799
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
nipkow@29667
   800
by (simp add: algebra_simps)
haftmann@25230
   801
haftmann@25230
   802
lemma le_add_iff1:
haftmann@25230
   803
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
nipkow@29667
   804
by (simp add: algebra_simps)
haftmann@25230
   805
haftmann@25230
   806
lemma le_add_iff2:
haftmann@25230
   807
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
nipkow@29667
   808
by (simp add: algebra_simps)
haftmann@25230
   809
haftmann@25230
   810
lemma mult_left_mono_neg:
haftmann@25230
   811
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
haftmann@36301
   812
  apply (drule mult_left_mono [of _ _ "- c"])
huffman@35216
   813
  apply simp_all
haftmann@25230
   814
  done
haftmann@25230
   815
haftmann@25230
   816
lemma mult_right_mono_neg:
haftmann@25230
   817
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
haftmann@36301
   818
  apply (drule mult_right_mono [of _ _ "- c"])
huffman@35216
   819
  apply simp_all
haftmann@25230
   820
  done
haftmann@25230
   821
huffman@30692
   822
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
haftmann@36301
   823
using mult_right_mono_neg [of a 0 b] by simp
haftmann@25230
   824
haftmann@25230
   825
lemma split_mult_pos_le:
haftmann@25230
   826
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
nipkow@56536
   827
by (auto simp add: mult_nonpos_nonpos)
haftmann@25186
   828
haftmann@25186
   829
end
paulson@14270
   830
haftmann@35028
   831
class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
haftmann@25304
   832
begin
haftmann@25304
   833
haftmann@35028
   834
subclass ordered_ring ..
haftmann@35028
   835
haftmann@35028
   836
subclass ordered_ab_group_add_abs
haftmann@28823
   837
proof
haftmann@25304
   838
  fix a b
haftmann@25304
   839
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@54230
   840
    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
   841
qed (auto simp add: abs_if)
haftmann@25304
   842
huffman@35631
   843
lemma zero_le_square [simp]: "0 \<le> a * a"
huffman@35631
   844
  using linear [of 0 a]
nipkow@56536
   845
  by (auto simp add: mult_nonpos_nonpos)
huffman@35631
   846
huffman@35631
   847
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
huffman@35631
   848
  by (simp add: not_less)
huffman@35631
   849
haftmann@25304
   850
end
obua@23521
   851
haftmann@35028
   852
(* The "strict" suffix can be seen as describing the combination of linordered_ring and no_zero_divisors.
haftmann@35043
   853
   Basically, linordered_ring + no_zero_divisors = linordered_ring_strict.
haftmann@25230
   854
 *)
haftmann@35043
   855
class linordered_ring_strict = ring + linordered_semiring_strict
haftmann@25304
   856
  + ordered_ab_group_add + abs_if
haftmann@25230
   857
begin
paulson@14348
   858
haftmann@35028
   859
subclass linordered_ring ..
haftmann@25304
   860
huffman@30692
   861
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
huffman@30692
   862
using mult_strict_left_mono [of b a "- c"] by simp
huffman@30692
   863
huffman@30692
   864
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
huffman@30692
   865
using mult_strict_right_mono [of b a "- c"] by simp
huffman@30692
   866
huffman@30692
   867
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
haftmann@36301
   868
using mult_strict_right_mono_neg [of a 0 b] by simp
obua@14738
   869
haftmann@25917
   870
subclass ring_no_zero_divisors
haftmann@28823
   871
proof
haftmann@25917
   872
  fix a b
haftmann@25917
   873
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
haftmann@25917
   874
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
haftmann@25917
   875
  have "a * b < 0 \<or> 0 < a * b"
haftmann@25917
   876
  proof (cases "a < 0")
haftmann@25917
   877
    case True note A' = this
haftmann@25917
   878
    show ?thesis proof (cases "b < 0")
haftmann@25917
   879
      case True with A'
haftmann@25917
   880
      show ?thesis by (auto dest: mult_neg_neg)
haftmann@25917
   881
    next
haftmann@25917
   882
      case False with B have "0 < b" by auto
haftmann@25917
   883
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
haftmann@25917
   884
    qed
haftmann@25917
   885
  next
haftmann@25917
   886
    case False with A have A': "0 < a" by auto
haftmann@25917
   887
    show ?thesis proof (cases "b < 0")
haftmann@25917
   888
      case True with A'
haftmann@25917
   889
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
haftmann@25917
   890
    next
haftmann@25917
   891
      case False with B have "0 < b" by auto
nipkow@56544
   892
      with A' show ?thesis by auto
haftmann@25917
   893
    qed
haftmann@25917
   894
  qed
haftmann@25917
   895
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
haftmann@25917
   896
qed
haftmann@25304
   897
hoelzl@56480
   898
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
hoelzl@56480
   899
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
nipkow@56544
   900
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
huffman@22990
   901
hoelzl@56480
   902
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
   903
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
paulson@14265
   904
paulson@14265
   905
lemma mult_less_0_iff:
haftmann@25917
   906
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
huffman@35216
   907
  apply (insert zero_less_mult_iff [of "-a" b])
huffman@35216
   908
  apply force
haftmann@25917
   909
  done
paulson@14265
   910
paulson@14265
   911
lemma mult_le_0_iff:
haftmann@25917
   912
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
haftmann@25917
   913
  apply (insert zero_le_mult_iff [of "-a" b]) 
huffman@35216
   914
  apply force
haftmann@25917
   915
  done
haftmann@25917
   916
haftmann@26193
   917
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
haftmann@26193
   918
   also with the relations @{text "\<le>"} and equality.*}
haftmann@26193
   919
haftmann@26193
   920
text{*These ``disjunction'' versions produce two cases when the comparison is
haftmann@26193
   921
 an assumption, but effectively four when the comparison is a goal.*}
haftmann@26193
   922
haftmann@26193
   923
lemma mult_less_cancel_right_disj:
haftmann@26193
   924
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   925
  apply (cases "c = 0")
haftmann@26193
   926
  apply (auto simp add: neq_iff mult_strict_right_mono 
haftmann@26193
   927
                      mult_strict_right_mono_neg)
haftmann@26193
   928
  apply (auto simp add: not_less 
haftmann@26193
   929
                      not_le [symmetric, of "a*c"]
haftmann@26193
   930
                      not_le [symmetric, of a])
haftmann@26193
   931
  apply (erule_tac [!] notE)
haftmann@26193
   932
  apply (auto simp add: less_imp_le mult_right_mono 
haftmann@26193
   933
                      mult_right_mono_neg)
haftmann@26193
   934
  done
haftmann@26193
   935
haftmann@26193
   936
lemma mult_less_cancel_left_disj:
haftmann@26193
   937
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
haftmann@26193
   938
  apply (cases "c = 0")
haftmann@26193
   939
  apply (auto simp add: neq_iff mult_strict_left_mono 
haftmann@26193
   940
                      mult_strict_left_mono_neg)
haftmann@26193
   941
  apply (auto simp add: not_less 
haftmann@26193
   942
                      not_le [symmetric, of "c*a"]
haftmann@26193
   943
                      not_le [symmetric, of a])
haftmann@26193
   944
  apply (erule_tac [!] notE)
haftmann@26193
   945
  apply (auto simp add: less_imp_le mult_left_mono 
haftmann@26193
   946
                      mult_left_mono_neg)
haftmann@26193
   947
  done
haftmann@26193
   948
haftmann@26193
   949
text{*The ``conjunction of implication'' lemmas produce two cases when the
haftmann@26193
   950
comparison is a goal, but give four when the comparison is an assumption.*}
haftmann@26193
   951
haftmann@26193
   952
lemma mult_less_cancel_right:
haftmann@26193
   953
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   954
  using mult_less_cancel_right_disj [of a c b] by auto
haftmann@26193
   955
haftmann@26193
   956
lemma mult_less_cancel_left:
haftmann@26193
   957
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
haftmann@26193
   958
  using mult_less_cancel_left_disj [of c a b] by auto
haftmann@26193
   959
haftmann@26193
   960
lemma mult_le_cancel_right:
haftmann@26193
   961
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
   962
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
haftmann@26193
   963
haftmann@26193
   964
lemma mult_le_cancel_left:
haftmann@26193
   965
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
nipkow@29667
   966
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
haftmann@26193
   967
nipkow@30649
   968
lemma mult_le_cancel_left_pos:
nipkow@30649
   969
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
nipkow@30649
   970
by (auto simp: mult_le_cancel_left)
nipkow@30649
   971
nipkow@30649
   972
lemma mult_le_cancel_left_neg:
nipkow@30649
   973
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
nipkow@30649
   974
by (auto simp: mult_le_cancel_left)
nipkow@30649
   975
nipkow@30649
   976
lemma mult_less_cancel_left_pos:
nipkow@30649
   977
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
nipkow@30649
   978
by (auto simp: mult_less_cancel_left)
nipkow@30649
   979
nipkow@30649
   980
lemma mult_less_cancel_left_neg:
nipkow@30649
   981
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
nipkow@30649
   982
by (auto simp: mult_less_cancel_left)
nipkow@30649
   983
haftmann@25917
   984
end
paulson@14265
   985
huffman@30692
   986
lemmas mult_sign_intros =
huffman@30692
   987
  mult_nonneg_nonneg mult_nonneg_nonpos
huffman@30692
   988
  mult_nonpos_nonneg mult_nonpos_nonpos
huffman@30692
   989
  mult_pos_pos mult_pos_neg
huffman@30692
   990
  mult_neg_pos mult_neg_neg
haftmann@25230
   991
haftmann@35028
   992
class ordered_comm_ring = comm_ring + ordered_comm_semiring
haftmann@25267
   993
begin
haftmann@25230
   994
haftmann@35028
   995
subclass ordered_ring ..
haftmann@35028
   996
subclass ordered_cancel_comm_semiring ..
haftmann@25230
   997
haftmann@25267
   998
end
haftmann@25230
   999
haftmann@35028
  1000
class linordered_semidom = comm_semiring_1_cancel + linordered_comm_semiring_strict +
haftmann@35028
  1001
  (*previously linordered_semiring*)
haftmann@25230
  1002
  assumes zero_less_one [simp]: "0 < 1"
haftmann@25230
  1003
begin
haftmann@25230
  1004
haftmann@25230
  1005
lemma pos_add_strict:
haftmann@25230
  1006
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36301
  1007
  using add_strict_mono [of 0 a b c] by simp
haftmann@25230
  1008
haftmann@26193
  1009
lemma zero_le_one [simp]: "0 \<le> 1"
nipkow@29667
  1010
by (rule zero_less_one [THEN less_imp_le]) 
haftmann@26193
  1011
haftmann@26193
  1012
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
nipkow@29667
  1013
by (simp add: not_le) 
haftmann@26193
  1014
haftmann@26193
  1015
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
nipkow@29667
  1016
by (simp add: not_less) 
haftmann@26193
  1017
haftmann@26193
  1018
lemma less_1_mult:
haftmann@26193
  1019
  assumes "1 < m" and "1 < n"
haftmann@26193
  1020
  shows "1 < m * n"
haftmann@26193
  1021
  using assms mult_strict_mono [of 1 m 1 n]
haftmann@26193
  1022
    by (simp add:  less_trans [OF zero_less_one]) 
haftmann@26193
  1023
hoelzl@59000
  1024
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
hoelzl@59000
  1025
  using mult_left_mono[of c 1 a] by simp
hoelzl@59000
  1026
hoelzl@59000
  1027
lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
hoelzl@59000
  1028
  using mult_mono[of a 1 b 1] by simp
hoelzl@59000
  1029
haftmann@25230
  1030
end
haftmann@25230
  1031
haftmann@35028
  1032
class linordered_idom = comm_ring_1 +
haftmann@35028
  1033
  linordered_comm_semiring_strict + ordered_ab_group_add +
haftmann@25230
  1034
  abs_if + sgn_if
haftmann@35028
  1035
  (*previously linordered_ring*)
haftmann@25917
  1036
begin
haftmann@25917
  1037
hoelzl@36622
  1038
subclass linordered_semiring_1_strict ..
haftmann@35043
  1039
subclass linordered_ring_strict ..
haftmann@35028
  1040
subclass ordered_comm_ring ..
huffman@27516
  1041
subclass idom ..
haftmann@25917
  1042
haftmann@35028
  1043
subclass linordered_semidom
haftmann@28823
  1044
proof
haftmann@26193
  1045
  have "0 \<le> 1 * 1" by (rule zero_le_square)
haftmann@26193
  1046
  thus "0 < 1" by (simp add: le_less)
haftmann@25917
  1047
qed 
haftmann@25917
  1048
haftmann@35028
  1049
lemma linorder_neqE_linordered_idom:
haftmann@26193
  1050
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
haftmann@26193
  1051
  using assms by (rule neqE)
haftmann@26193
  1052
haftmann@26274
  1053
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
haftmann@26274
  1054
haftmann@26274
  1055
lemma mult_le_cancel_right1:
haftmann@26274
  1056
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1057
by (insert mult_le_cancel_right [of 1 c b], simp)
haftmann@26274
  1058
haftmann@26274
  1059
lemma mult_le_cancel_right2:
haftmann@26274
  1060
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1061
by (insert mult_le_cancel_right [of a c 1], simp)
haftmann@26274
  1062
haftmann@26274
  1063
lemma mult_le_cancel_left1:
haftmann@26274
  1064
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
nipkow@29667
  1065
by (insert mult_le_cancel_left [of c 1 b], simp)
haftmann@26274
  1066
haftmann@26274
  1067
lemma mult_le_cancel_left2:
haftmann@26274
  1068
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
nipkow@29667
  1069
by (insert mult_le_cancel_left [of c a 1], simp)
haftmann@26274
  1070
haftmann@26274
  1071
lemma mult_less_cancel_right1:
haftmann@26274
  1072
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1073
by (insert mult_less_cancel_right [of 1 c b], simp)
haftmann@26274
  1074
haftmann@26274
  1075
lemma mult_less_cancel_right2:
haftmann@26274
  1076
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1077
by (insert mult_less_cancel_right [of a c 1], simp)
haftmann@26274
  1078
haftmann@26274
  1079
lemma mult_less_cancel_left1:
haftmann@26274
  1080
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
nipkow@29667
  1081
by (insert mult_less_cancel_left [of c 1 b], simp)
haftmann@26274
  1082
haftmann@26274
  1083
lemma mult_less_cancel_left2:
haftmann@26274
  1084
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
nipkow@29667
  1085
by (insert mult_less_cancel_left [of c a 1], simp)
haftmann@26274
  1086
haftmann@27651
  1087
lemma sgn_sgn [simp]:
haftmann@27651
  1088
  "sgn (sgn a) = sgn a"
nipkow@29700
  1089
unfolding sgn_if by simp
haftmann@27651
  1090
haftmann@27651
  1091
lemma sgn_0_0:
haftmann@27651
  1092
  "sgn a = 0 \<longleftrightarrow> a = 0"
nipkow@29700
  1093
unfolding sgn_if by simp
haftmann@27651
  1094
haftmann@27651
  1095
lemma sgn_1_pos:
haftmann@27651
  1096
  "sgn a = 1 \<longleftrightarrow> a > 0"
huffman@35216
  1097
unfolding sgn_if by simp
haftmann@27651
  1098
haftmann@27651
  1099
lemma sgn_1_neg:
haftmann@27651
  1100
  "sgn a = - 1 \<longleftrightarrow> a < 0"
huffman@35216
  1101
unfolding sgn_if by auto
haftmann@27651
  1102
haftmann@29940
  1103
lemma sgn_pos [simp]:
haftmann@29940
  1104
  "0 < a \<Longrightarrow> sgn a = 1"
haftmann@29940
  1105
unfolding sgn_1_pos .
haftmann@29940
  1106
haftmann@29940
  1107
lemma sgn_neg [simp]:
haftmann@29940
  1108
  "a < 0 \<Longrightarrow> sgn a = - 1"
haftmann@29940
  1109
unfolding sgn_1_neg .
haftmann@29940
  1110
haftmann@27651
  1111
lemma sgn_times:
haftmann@27651
  1112
  "sgn (a * b) = sgn a * sgn b"
nipkow@29667
  1113
by (auto simp add: sgn_if zero_less_mult_iff)
haftmann@27651
  1114
haftmann@36301
  1115
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
nipkow@29700
  1116
unfolding sgn_if abs_if by auto
nipkow@29700
  1117
haftmann@29940
  1118
lemma sgn_greater [simp]:
haftmann@29940
  1119
  "0 < sgn a \<longleftrightarrow> 0 < a"
haftmann@29940
  1120
  unfolding sgn_if by auto
haftmann@29940
  1121
haftmann@29940
  1122
lemma sgn_less [simp]:
haftmann@29940
  1123
  "sgn a < 0 \<longleftrightarrow> a < 0"
haftmann@29940
  1124
  unfolding sgn_if by auto
haftmann@29940
  1125
haftmann@36301
  1126
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
huffman@29949
  1127
  by (simp add: abs_if)
huffman@29949
  1128
haftmann@36301
  1129
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
huffman@29949
  1130
  by (simp add: abs_if)
haftmann@29653
  1131
nipkow@33676
  1132
lemma dvd_if_abs_eq:
haftmann@36301
  1133
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
nipkow@33676
  1134
by(subst abs_dvd_iff[symmetric]) simp
nipkow@33676
  1135
huffman@55912
  1136
text {* The following lemmas can be proven in more general structures, but
haftmann@54489
  1137
are dangerous as simp rules in absence of @{thm neg_equal_zero}, 
haftmann@54489
  1138
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}
haftmann@54489
  1139
haftmann@54489
  1140
lemma equation_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1141
  "1 = - a \<longleftrightarrow> a = - 1"
haftmann@54489
  1142
  by (fact equation_minus_iff)
haftmann@54489
  1143
haftmann@54489
  1144
lemma minus_equation_iff_1 [simp, no_atp]:
haftmann@54489
  1145
  "- a = 1 \<longleftrightarrow> a = - 1"
haftmann@54489
  1146
  by (subst minus_equation_iff, auto)
haftmann@54489
  1147
haftmann@54489
  1148
lemma le_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1149
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
haftmann@54489
  1150
  by (fact le_minus_iff)
haftmann@54489
  1151
haftmann@54489
  1152
lemma minus_le_iff_1 [simp, no_atp]:
haftmann@54489
  1153
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
haftmann@54489
  1154
  by (fact minus_le_iff)
haftmann@54489
  1155
haftmann@54489
  1156
lemma less_minus_iff_1 [simp, no_atp]:
haftmann@54489
  1157
  "1 < - b \<longleftrightarrow> b < - 1"
haftmann@54489
  1158
  by (fact less_minus_iff)
haftmann@54489
  1159
haftmann@54489
  1160
lemma minus_less_iff_1 [simp, no_atp]:
haftmann@54489
  1161
  "- a < 1 \<longleftrightarrow> - 1 < a"
haftmann@54489
  1162
  by (fact minus_less_iff)
haftmann@54489
  1163
haftmann@25917
  1164
end
haftmann@25230
  1165
haftmann@26274
  1166
text {* Simprules for comparisons where common factors can be cancelled. *}
paulson@15234
  1167
blanchet@54147
  1168
lemmas mult_compare_simps =
paulson@15234
  1169
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
  1170
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
  1171
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
  1172
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
  1173
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
  1174
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
  1175
    mult_cancel_right mult_cancel_left
paulson@15234
  1176
    mult_cancel_right1 mult_cancel_right2
paulson@15234
  1177
    mult_cancel_left1 mult_cancel_left2
paulson@15234
  1178
haftmann@36301
  1179
text {* Reasoning about inequalities with division *}
avigad@16775
  1180
haftmann@35028
  1181
context linordered_semidom
haftmann@25193
  1182
begin
haftmann@25193
  1183
haftmann@25193
  1184
lemma less_add_one: "a < a + 1"
paulson@14293
  1185
proof -
haftmann@25193
  1186
  have "a + 0 < a + 1"
nipkow@23482
  1187
    by (blast intro: zero_less_one add_strict_left_mono)
paulson@14293
  1188
  thus ?thesis by simp
paulson@14293
  1189
qed
paulson@14293
  1190
haftmann@25193
  1191
lemma zero_less_two: "0 < 1 + 1"
nipkow@29667
  1192
by (blast intro: less_trans zero_less_one less_add_one)
haftmann@25193
  1193
haftmann@25193
  1194
end
paulson@14365
  1195
haftmann@36301
  1196
context linordered_idom
haftmann@36301
  1197
begin
paulson@15234
  1198
haftmann@36301
  1199
lemma mult_right_le_one_le:
haftmann@36301
  1200
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
haftmann@36301
  1201
  by (auto simp add: mult_le_cancel_left2)
haftmann@36301
  1202
haftmann@36301
  1203
lemma mult_left_le_one_le:
haftmann@36301
  1204
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
haftmann@36301
  1205
  by (auto simp add: mult_le_cancel_right2)
haftmann@36301
  1206
haftmann@36301
  1207
end
haftmann@36301
  1208
haftmann@36301
  1209
text {* Absolute Value *}
paulson@14293
  1210
haftmann@35028
  1211
context linordered_idom
haftmann@25304
  1212
begin
haftmann@25304
  1213
haftmann@36301
  1214
lemma mult_sgn_abs:
haftmann@36301
  1215
  "sgn x * \<bar>x\<bar> = x"
haftmann@25304
  1216
  unfolding abs_if sgn_if by auto
haftmann@25304
  1217
haftmann@36301
  1218
lemma abs_one [simp]:
haftmann@36301
  1219
  "\<bar>1\<bar> = 1"
huffman@44921
  1220
  by (simp add: abs_if)
haftmann@36301
  1221
haftmann@25304
  1222
end
nipkow@24491
  1223
haftmann@35028
  1224
class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
haftmann@25304
  1225
  assumes abs_eq_mult:
haftmann@25304
  1226
    "(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
  1227
haftmann@35028
  1228
context linordered_idom
haftmann@30961
  1229
begin
haftmann@30961
  1230
haftmann@35028
  1231
subclass ordered_ring_abs proof
huffman@35216
  1232
qed (auto simp add: abs_if not_less mult_less_0_iff)
haftmann@30961
  1233
haftmann@30961
  1234
lemma abs_mult:
haftmann@36301
  1235
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>" 
haftmann@30961
  1236
  by (rule abs_eq_mult) auto
haftmann@30961
  1237
haftmann@30961
  1238
lemma abs_mult_self:
haftmann@36301
  1239
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
haftmann@30961
  1240
  by (simp add: abs_if) 
haftmann@30961
  1241
paulson@14294
  1242
lemma abs_mult_less:
haftmann@36301
  1243
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
paulson@14294
  1244
proof -
haftmann@36301
  1245
  assume ac: "\<bar>a\<bar> < c"
haftmann@36301
  1246
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
haftmann@36301
  1247
  assume "\<bar>b\<bar> < d"
paulson@14294
  1248
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1249
qed
paulson@14293
  1250
haftmann@36301
  1251
lemma abs_less_iff:
haftmann@36301
  1252
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b" 
haftmann@36301
  1253
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
obua@14738
  1254
haftmann@36301
  1255
lemma abs_mult_pos:
haftmann@36301
  1256
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
haftmann@36301
  1257
  by (simp add: abs_mult)
haftmann@36301
  1258
hoelzl@51520
  1259
lemma abs_diff_less_iff:
hoelzl@51520
  1260
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
hoelzl@51520
  1261
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)
hoelzl@51520
  1262
haftmann@36301
  1263
end
avigad@16775
  1264
haftmann@52435
  1265
code_identifier
haftmann@52435
  1266
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1267
paulson@14265
  1268
end
haftmann@52435
  1269