src/HOL/Rings.thy

Useful lemmas. The theorem concerning swapping the variables in a double integral.

(*  Title:      HOL/Rings.thy
    Author:     Gertrud Bauer
    Author:     Steven Obua
    Author:     Tobias Nipkow
    Author:     Lawrence C Paulson
    Author:     Markus Wenzel
    Author:     Jeremy Avigad
*)

section {* Rings *}

theory Rings
imports Groups
begin

class semiring = ab_semigroup_add + semigroup_mult +
  assumes distrib_right[algebra_simps]: "(a + b) * c = a * c + b * c"
  assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
begin

text{*For the @{text combine_numerals} simproc*}
lemma combine_common_factor:
  "a * e + (b * e + c) = (a + b) * e + c"
by (simp add: distrib_right ac_simps)

end

class mult_zero = times + zero +
  assumes mult_zero_left [simp]: "0 * a = 0"
  assumes mult_zero_right [simp]: "a * 0 = 0"
begin

lemma mult_not_zero:
  "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
  by auto

end

class semiring_0 = semiring + comm_monoid_add + mult_zero

class semiring_0_cancel = semiring + cancel_comm_monoid_add
begin

subclass semiring_0
proof
  fix a :: 'a
  have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
  thus "0 * a = 0" by (simp only: add_left_cancel)
next
  fix a :: 'a
  have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
  thus "a * 0 = 0" by (simp only: add_left_cancel)
qed

end

class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
  assumes distrib: "(a + b) * c = a * c + b * c"
begin

subclass semiring
proof
  fix a b c :: 'a
  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
  have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
  also have "... = b * a + c * a" by (simp only: distrib)
  also have "... = a * b + a * c" by (simp add: ac_simps)
  finally show "a * (b + c) = a * b + a * c" by blast
qed

end

class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
begin

subclass semiring_0 ..

end

class comm_semiring_0_cancel = comm_semiring + cancel_comm_monoid_add
begin

subclass semiring_0_cancel ..

subclass comm_semiring_0 ..

end

class zero_neq_one = zero + one +
  assumes zero_neq_one [simp]: "0 \<noteq> 1"
begin

lemma one_neq_zero [simp]: "1 \<noteq> 0"
by (rule not_sym) (rule zero_neq_one)

definition of_bool :: "bool \<Rightarrow> 'a"
where
  "of_bool p = (if p then 1 else 0)"

lemma of_bool_eq [simp, code]:
  "of_bool False = 0"
  "of_bool True = 1"
  by (simp_all add: of_bool_def)

lemma of_bool_eq_iff:
  "of_bool p = of_bool q \<longleftrightarrow> p = q"
  by (simp add: of_bool_def)

lemma split_of_bool [split]:
  "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
  by (cases p) simp_all

lemma split_of_bool_asm:
  "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
  by (cases p) simp_all

end

class semiring_1 = zero_neq_one + semiring_0 + monoid_mult

text {* Abstract divisibility *}

class dvd = times
begin

definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where
  "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"

lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
  unfolding dvd_def ..

lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>k. a = b * k \<Longrightarrow> P) \<Longrightarrow> P"
  unfolding dvd_def by blast

end

context comm_monoid_mult
begin

subclass dvd .

lemma dvd_refl [simp]:
  "a dvd a"
proof
  show "a = a * 1" by simp
qed

lemma dvd_trans:
  assumes "a dvd b" and "b dvd c"
  shows "a dvd c"
proof -
  from assms obtain v where "b = a * v" by (auto elim!: dvdE)
  moreover from assms obtain w where "c = b * w" by (auto elim!: dvdE)
  ultimately have "c = a * (v * w)" by (simp add: mult.assoc)
  then show ?thesis ..
qed

lemma one_dvd [simp]:
  "1 dvd a"
  by (auto intro!: dvdI)

lemma dvd_mult [simp]:
  "a dvd c \<Longrightarrow> a dvd (b * c)"
  by (auto intro!: mult.left_commute dvdI elim!: dvdE)

lemma dvd_mult2 [simp]:
  "a dvd b \<Longrightarrow> a dvd (b * c)"
  using dvd_mult [of a b c] by (simp add: ac_simps)

lemma dvd_triv_right [simp]:
  "a dvd b * a"
  by (rule dvd_mult) (rule dvd_refl)

lemma dvd_triv_left [simp]:
  "a dvd a * b"
  by (rule dvd_mult2) (rule dvd_refl)

lemma mult_dvd_mono:
  assumes "a dvd b"
    and "c dvd d"
  shows "a * c dvd b * d"
proof -
  from `a dvd b` obtain b' where "b = a * b'" ..
  moreover from `c dvd d` obtain d' where "d = c * d'" ..
  ultimately have "b * d = (a * c) * (b' * d')" by (simp add: ac_simps)
  then show ?thesis ..
qed

lemma dvd_mult_left:
  "a * b dvd c \<Longrightarrow> a dvd c"
  by (simp add: dvd_def mult.assoc) blast

lemma dvd_mult_right:
  "a * b dvd c \<Longrightarrow> b dvd c"
  using dvd_mult_left [of b a c] by (simp add: ac_simps)

end

class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
begin

subclass semiring_1 ..

lemma dvd_0_left_iff [simp]:
  "0 dvd a \<longleftrightarrow> a = 0"
  by (auto intro: dvd_refl elim!: dvdE)

lemma dvd_0_right [iff]:
  "a dvd 0"
proof
  show "0 = a * 0" by simp
qed

lemma dvd_0_left:
  "0 dvd a \<Longrightarrow> a = 0"
  by simp

lemma dvd_add [simp]:
  assumes "a dvd b" and "a dvd c"
  shows "a dvd (b + c)"
proof -
  from `a dvd b` obtain b' where "b = a * b'" ..
  moreover from `a dvd c` obtain c' where "c = a * c'" ..
  ultimately have "b + c = a * (b' + c')" by (simp add: distrib_left)
  then show ?thesis ..
qed

end

class semiring_1_cancel = semiring + cancel_comm_monoid_add
  + zero_neq_one + monoid_mult
begin

subclass semiring_0_cancel ..

subclass semiring_1 ..

end

class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add +
                               zero_neq_one + comm_monoid_mult +
  assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
begin

subclass semiring_1_cancel ..
subclass comm_semiring_0_cancel ..
subclass comm_semiring_1 ..

lemma left_diff_distrib' [algebra_simps]:
  "(b - c) * a = b * a - c * a"
  by (simp add: algebra_simps)

lemma dvd_add_times_triv_left_iff [simp]:
  "a dvd c * a + b \<longleftrightarrow> a dvd b"
proof -
  have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
  proof
    assume ?Q then show ?P by simp
  next
    assume ?P
    then obtain d where "a * c + b = a * d" ..
    then have "a * c + b - a * c = a * d - a * c" by simp
    then have "b = a * d - a * c" by simp
    then have "b = a * (d - c)" by (simp add: algebra_simps)
    then show ?Q ..
  qed
  then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
qed

lemma dvd_add_times_triv_right_iff [simp]:
  "a dvd b + c * a \<longleftrightarrow> a dvd b"
  using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)

lemma dvd_add_triv_left_iff [simp]:
  "a dvd a + b \<longleftrightarrow> a dvd b"
  using dvd_add_times_triv_left_iff [of a 1 b] by simp

lemma dvd_add_triv_right_iff [simp]:
  "a dvd b + a \<longleftrightarrow> a dvd b"
  using dvd_add_times_triv_right_iff [of a b 1] by simp

lemma dvd_add_right_iff:
  assumes "a dvd b"
  shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
proof
  assume ?P then obtain d where "b + c = a * d" ..
  moreover from `a dvd b` obtain e where "b = a * e" ..
  ultimately have "a * e + c = a * d" by simp
  then have "a * e + c - a * e = a * d - a * e" by simp
  then have "c = a * d - a * e" by simp
  then have "c = a * (d - e)" by (simp add: algebra_simps)
  then show ?Q ..
next
  assume ?Q with assms show ?P by simp
qed

lemma dvd_add_left_iff:
  assumes "a dvd c"
  shows "a dvd b + c \<longleftrightarrow> a dvd b"
  using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps)

end

class ring = semiring + ab_group_add
begin

subclass semiring_0_cancel ..

text {* Distribution rules *}

lemma minus_mult_left: "- (a * b) = - a * b"
by (rule minus_unique) (simp add: distrib_right [symmetric])

lemma minus_mult_right: "- (a * b) = a * - b"
by (rule minus_unique) (simp add: distrib_left [symmetric])

text{*Extract signs from products*}
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
lemmas mult_minus_right [simp] = minus_mult_right [symmetric]

lemma minus_mult_minus [simp]: "- a * - b = a * b"
by simp

lemma minus_mult_commute: "- a * b = a * - b"
by simp

lemma right_diff_distrib [algebra_simps]:
  "a * (b - c) = a * b - a * c"
  using distrib_left [of a b "-c "] by simp

lemma left_diff_distrib [algebra_simps]:
  "(a - b) * c = a * c - b * c"
  using distrib_right [of a "- b" c] by simp

lemmas ring_distribs =
  distrib_left distrib_right left_diff_distrib right_diff_distrib

lemma eq_add_iff1:
  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
by (simp add: algebra_simps)

lemma eq_add_iff2:
  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
by (simp add: algebra_simps)

end

lemmas ring_distribs =
  distrib_left distrib_right left_diff_distrib right_diff_distrib

class comm_ring = comm_semiring + ab_group_add
begin

subclass ring ..
subclass comm_semiring_0_cancel ..

lemma square_diff_square_factored:
  "x * x - y * y = (x + y) * (x - y)"
  by (simp add: algebra_simps)

end

class ring_1 = ring + zero_neq_one + monoid_mult
begin

subclass semiring_1_cancel ..

lemma square_diff_one_factored:
  "x * x - 1 = (x + 1) * (x - 1)"
  by (simp add: algebra_simps)

end

class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
begin

subclass ring_1 ..
subclass comm_semiring_1_cancel
  by unfold_locales (simp add: algebra_simps)

lemma dvd_minus_iff [simp]: "x dvd - y \<longleftrightarrow> x dvd y"
proof
  assume "x dvd - y"
  then have "x dvd - 1 * - y" by (rule dvd_mult)
  then show "x dvd y" by simp
next
  assume "x dvd y"
  then have "x dvd - 1 * y" by (rule dvd_mult)
  then show "x dvd - y" by simp
qed

lemma minus_dvd_iff [simp]: "- x dvd y \<longleftrightarrow> x dvd y"
proof
  assume "- x dvd y"
  then obtain k where "y = - x * k" ..
  then have "y = x * - k" by simp
  then show "x dvd y" ..
next
  assume "x dvd y"
  then obtain k where "y = x * k" ..
  then have "y = - x * - k" by simp
  then show "- x dvd y" ..
qed

lemma dvd_diff [simp]:
  "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
  using dvd_add [of x y "- z"] by simp

end

class semiring_no_zero_divisors = semiring_0 +
  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
begin

lemma divisors_zero:
  assumes "a * b = 0"
  shows "a = 0 \<or> b = 0"
proof (rule classical)
  assume "\<not> (a = 0 \<or> b = 0)"
  then have "a \<noteq> 0" and "b \<noteq> 0" by auto
  with no_zero_divisors have "a * b \<noteq> 0" by blast
  with assms show ?thesis by simp
qed

lemma mult_eq_0_iff [simp]:
  shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
proof (cases "a = 0 \<or> b = 0")
  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
    then show ?thesis using no_zero_divisors by simp
next
  case True then show ?thesis by auto
qed

end

class semiring_no_zero_divisors_cancel = semiring_no_zero_divisors +
  assumes mult_cancel_right [simp]: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
    and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
begin

lemma mult_left_cancel:
  "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
  by simp

lemma mult_right_cancel:
  "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
  by simp

end

class ring_no_zero_divisors = ring + semiring_no_zero_divisors
begin

subclass semiring_no_zero_divisors_cancel
proof
  fix a b c
  have "a * c = b * c \<longleftrightarrow> (a - b) * c = 0"
    by (simp add: algebra_simps)
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
    by auto
  finally show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" .
  have "c * a = c * b \<longleftrightarrow> c * (a - b) = 0"
    by (simp add: algebra_simps)
  also have "\<dots> \<longleftrightarrow> c = 0 \<or> a = b"
    by auto
  finally show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" .
qed

end

class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
begin

lemma square_eq_1_iff:
  "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
proof -
  have "(x - 1) * (x + 1) = x * x - 1"
    by (simp add: algebra_simps)
  hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
    by simp
  thus ?thesis
    by (simp add: eq_neg_iff_add_eq_0)
qed

lemma mult_cancel_right1 [simp]:
  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
by (insert mult_cancel_right [of 1 c b], force)

lemma mult_cancel_right2 [simp]:
  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
by (insert mult_cancel_right [of a c 1], simp)

lemma mult_cancel_left1 [simp]:
  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
by (insert mult_cancel_left [of c 1 b], force)

lemma mult_cancel_left2 [simp]:
  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
by (insert mult_cancel_left [of c a 1], simp)

end

class semidom = comm_semiring_1_cancel + semiring_no_zero_divisors

class idom = comm_ring_1 + semiring_no_zero_divisors
begin

subclass semidom ..

subclass ring_1_no_zero_divisors ..

lemma dvd_mult_cancel_right [simp]:
  "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
proof -
  have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
    unfolding dvd_def by (simp add: ac_simps)
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
    unfolding dvd_def by simp
  finally show ?thesis .
qed

lemma dvd_mult_cancel_left [simp]:
  "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
proof -
  have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
    unfolding dvd_def by (simp add: ac_simps)
  also have "(\<exists>k. b * c = (a * k) * c) \<longleftrightarrow> c = 0 \<or> a dvd b"
    unfolding dvd_def by simp
  finally show ?thesis .
qed

lemma square_eq_iff: "a * a = b * b \<longleftrightarrow> a = b \<or> a = - b"
proof
  assume "a * a = b * b"
  then have "(a - b) * (a + b) = 0"
    by (simp add: algebra_simps)
  then show "a = b \<or> a = - b"
    by (simp add: eq_neg_iff_add_eq_0)
next
  assume "a = b \<or> a = - b"
  then show "a * a = b * b" by auto
qed

end

text {*
  The theory of partially ordered rings is taken from the books:
  \begin{itemize}
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
  \end{itemize}
  Most of the used notions can also be looked up in
  \begin{itemize}
  \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
  \item \emph{Algebra I} by van der Waerden, Springer.
  \end{itemize}
*}

class divide =
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "div" 70)

setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a \<Rightarrow> 'a \<Rightarrow> 'a"}) *}

context semiring
begin

lemma [field_simps]:
  shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
    and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
  by (rule distrib_left distrib_right)+

end

context ring
begin

lemma [field_simps]:
  shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
    and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
  by (rule left_diff_distrib right_diff_distrib)+

end

setup {* Sign.add_const_constraint (@{const_name "divide"}, SOME @{typ "'a::divide \<Rightarrow> 'a \<Rightarrow> 'a"}) *}

class semidom_divide = semidom + divide +
  assumes nonzero_mult_divide_cancel_right [simp]: "b \<noteq> 0 \<Longrightarrow> (a * b) div b = a"
  assumes divide_zero [simp]: "a div 0 = 0"
begin

lemma nonzero_mult_divide_cancel_left [simp]:
  "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
  using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)

subclass semiring_no_zero_divisors_cancel
proof
  fix a b c
  { fix a b c
    show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
    proof (cases "c = 0")
      case True then show ?thesis by simp
    next
      case False
      { assume "a * c = b * c"
        then have "a * c div c = b * c div c"
          by simp
        with False have "a = b"
          by simp
      } then show ?thesis by auto
    qed
  }
  from this [of a c b]
  show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
    by (simp add: ac_simps)
qed

lemma div_self [simp]:
  assumes "a \<noteq> 0"
  shows "a div a = 1"
  using assms nonzero_mult_divide_cancel_left [of a 1] by simp

lemma divide_zero_left [simp]:
  "0 div a = 0"
proof (cases "a = 0")
  case True then show ?thesis by simp
next
  case False then have "a * 0 div a = 0"
    by (rule nonzero_mult_divide_cancel_left)
  then show ?thesis by simp
qed 

end

class idom_divide = idom + semidom_divide

class algebraic_semidom = semidom_divide
begin

lemma dvd_div_mult_self [simp]:
  "a dvd b \<Longrightarrow> b div a * a = b"
  by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)

lemma dvd_mult_div_cancel [simp]:
  "a dvd b \<Longrightarrow> a * (b div a) = b"
  using dvd_div_mult_self [of a b] by (simp add: ac_simps)

lemma div_mult_swap:
  assumes "c dvd b"
  shows "a * (b div c) = (a * b) div c"
proof (cases "c = 0")
  case True then show ?thesis by simp
next
  case False from assms obtain d where "b = c * d" ..
  moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
    by simp
  ultimately show ?thesis by (simp add: ac_simps)
qed

lemma dvd_div_mult:
  assumes "c dvd b"
  shows "b div c * a = (b * a) div c"
  using assms div_mult_swap [of c b a] by (simp add: ac_simps)

lemma dvd_div_mult2_eq:
  assumes "b * c dvd a"
  shows "a div (b * c) = a div b div c"
using assms proof
  fix k
  assume "a = b * c * k"
  then show ?thesis
    by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
qed


text \<open>Units: invertible elements in a ring\<close>

abbreviation is_unit :: "'a \<Rightarrow> bool"
where
  "is_unit a \<equiv> a dvd 1"

lemma not_is_unit_0 [simp]:
  "\<not> is_unit 0"
  by simp

lemma unit_imp_dvd [dest]:
  "is_unit b \<Longrightarrow> b dvd a"
  by (rule dvd_trans [of _ 1]) simp_all

lemma unit_dvdE:
  assumes "is_unit a"
  obtains c where "a \<noteq> 0" and "b = a * c"
proof -
  from assms have "a dvd b" by auto
  then obtain c where "b = a * c" ..
  moreover from assms have "a \<noteq> 0" by auto
  ultimately show thesis using that by blast
qed

lemma dvd_unit_imp_unit:
  "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
  by (rule dvd_trans)

lemma unit_div_1_unit [simp, intro]:
  assumes "is_unit a"
  shows "is_unit (1 div a)"
proof -
  from assms have "1 = 1 div a * a" by simp
  then show "is_unit (1 div a)" by (rule dvdI)
qed

lemma is_unitE [elim?]:
  assumes "is_unit a"
  obtains b where "a \<noteq> 0" and "b \<noteq> 0"
    and "is_unit b" and "1 div a = b" and "1 div b = a"
    and "a * b = 1" and "c div a = c * b"
proof (rule that)
  def b \<equiv> "1 div a"
  then show "1 div a = b" by simp
  from b_def `is_unit a` show "is_unit b" by simp
  from `is_unit a` and `is_unit b` show "a \<noteq> 0" and "b \<noteq> 0" by auto
  from b_def `is_unit a` show "a * b = 1" by simp
  then have "1 = a * b" ..
  with b_def `b \<noteq> 0` show "1 div b = a" by simp
  from `is_unit a` have "a dvd c" ..
  then obtain d where "c = a * d" ..
  with `a \<noteq> 0` `a * b = 1` show "c div a = c * b"
    by (simp add: mult.assoc mult.left_commute [of a])
qed

lemma unit_prod [intro]:
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
  by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)

lemma unit_div [intro]:
  "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
  by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)

lemma mult_unit_dvd_iff:
  assumes "is_unit b"
  shows "a * b dvd c \<longleftrightarrow> a dvd c"
proof
  assume "a * b dvd c"
  with assms show "a dvd c"
    by (simp add: dvd_mult_left)
next
  assume "a dvd c"
  then obtain k where "c = a * k" ..
  with assms have "c = (a * b) * (1 div b * k)"
    by (simp add: mult_ac)
  then show "a * b dvd c" by (rule dvdI)
qed

lemma dvd_mult_unit_iff:
  assumes "is_unit b"
  shows "a dvd c * b \<longleftrightarrow> a dvd c"
proof
  assume "a dvd c * b"
  with assms have "c * b dvd c * (b * (1 div b))"
    by (subst mult_assoc [symmetric]) simp
  also from `is_unit b` have "b * (1 div b) = 1" by (rule is_unitE) simp
  finally have "c * b dvd c" by simp
  with `a dvd c * b` show "a dvd c" by (rule dvd_trans)
next
  assume "a dvd c"
  then show "a dvd c * b" by simp
qed

lemma div_unit_dvd_iff:
  "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
  by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)

lemma dvd_div_unit_iff:
  "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
  by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)

lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
  dvd_mult_unit_iff dvd_div_unit_iff -- \<open>FIXME consider fact collection\<close>

lemma unit_mult_div_div [simp]:
  "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
  by (erule is_unitE [of _ b]) simp

lemma unit_div_mult_self [simp]:
  "is_unit a \<Longrightarrow> b div a * a = b"
  by (rule dvd_div_mult_self) auto

lemma unit_div_1_div_1 [simp]:
  "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
  by (erule is_unitE) simp

lemma unit_div_mult_swap:
  "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
  by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])

lemma unit_div_commute:
  "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
  using unit_div_mult_swap [of b c a] by (simp add: ac_simps)

lemma unit_eq_div1:
  "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
  by (auto elim: is_unitE)

lemma unit_eq_div2:
  "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
  using unit_eq_div1 [of b c a] by auto

lemma unit_mult_left_cancel:
  assumes "is_unit a"
  shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
  using assms mult_cancel_left [of a b c] by auto

lemma unit_mult_right_cancel:
  "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
  using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)

lemma unit_div_cancel:
  assumes "is_unit a"
  shows "b div a = c div a \<longleftrightarrow> b = c"
proof -
  from assms have "is_unit (1 div a)" by simp
  then have "b * (1 div a) = c * (1 div a) \<longleftrightarrow> b = c"
    by (rule unit_mult_right_cancel)
  with assms show ?thesis by simp
qed

lemma is_unit_div_mult2_eq:
  assumes "is_unit b" and "is_unit c"
  shows "a div (b * c) = a div b div c"
proof -
  from assms have "is_unit (b * c)" by (simp add: unit_prod)
  then have "b * c dvd a"
    by (rule unit_imp_dvd)
  then show ?thesis
    by (rule dvd_div_mult2_eq)
qed


text \<open>Associated elements in a ring --- an equivalence relation induced
  by the quasi-order divisibility.\<close>

definition associated :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
where
  "associated a b \<longleftrightarrow> a dvd b \<and> b dvd a"

lemma associatedI:
  "a dvd b \<Longrightarrow> b dvd a \<Longrightarrow> associated a b"
  by (simp add: associated_def)

lemma associatedD1:
  "associated a b \<Longrightarrow> a dvd b"
  by (simp add: associated_def)

lemma associatedD2:
  "associated a b \<Longrightarrow> b dvd a"
  by (simp add: associated_def)

lemma associated_refl [simp]:
  "associated a a"
  by (auto intro: associatedI)

lemma associated_sym:
  "associated b a \<longleftrightarrow> associated a b"
  by (auto intro: associatedI dest: associatedD1 associatedD2)

lemma associated_trans:
  "associated a b \<Longrightarrow> associated b c \<Longrightarrow> associated a c"
  by (auto intro: associatedI dvd_trans dest: associatedD1 associatedD2)

lemma associated_0 [simp]:
  "associated 0 b \<longleftrightarrow> b = 0"
  "associated a 0 \<longleftrightarrow> a = 0"
  by (auto dest: associatedD1 associatedD2)

lemma associated_unit:
  "associated a b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
  using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)

lemma is_unit_associatedI:
  assumes "is_unit c" and "a = c * b"
  shows "associated a b"
proof (rule associatedI)
  from `a = c * b` show "b dvd a" by auto
  from `is_unit c` obtain d where "c * d = 1" by (rule is_unitE)
  moreover from `a = c * b` have "d * a = d * (c * b)" by simp
  ultimately have "b = a * d" by (simp add: ac_simps)
  then show "a dvd b" ..
qed

lemma associated_is_unitE:
  assumes "associated a b"
  obtains c where "is_unit c" and "a = c * b"
proof (cases "b = 0")
  case True with assms have "is_unit 1" and "a = 1 * b" by simp_all
  with that show thesis .
next
  case False
  from assms have "a dvd b" and "b dvd a" by (auto dest: associatedD1 associatedD2)
  then obtain c d where "b = a * d" and "a = b * c" by (blast elim: dvdE)
  then have "a = c * b" and "(c * d) * b = 1 * b" by (simp_all add: ac_simps)
  with False have "c * d = 1" using mult_cancel_right [of "c * d" b 1] by simp
  then have "is_unit c" by auto
  with `a = c * b` that show thesis by blast
qed

lemmas unit_simps = mult_unit_dvd_iff div_unit_dvd_iff dvd_mult_unit_iff
  dvd_div_unit_iff unit_div_mult_swap unit_div_commute
  unit_mult_left_cancel unit_mult_right_cancel unit_div_cancel
  unit_eq_div1 unit_eq_div2

end

class ordered_semiring = semiring + comm_monoid_add + ordered_ab_semigroup_add +
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
begin

lemma mult_mono:
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
apply (erule mult_right_mono [THEN order_trans], assumption)
apply (erule mult_left_mono, assumption)
done

lemma mult_mono':
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
apply (rule mult_mono)
apply (fast intro: order_trans)+
done

end

class ordered_cancel_semiring = ordered_semiring + cancel_comm_monoid_add
begin

subclass semiring_0_cancel ..

lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
using mult_left_mono [of 0 b a] by simp

lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
using mult_left_mono [of b 0 a] by simp

lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
using mult_right_mono [of a 0 b] by simp

text {* Legacy - use @{text mult_nonpos_nonneg} *}
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
by (drule mult_right_mono [of b 0], auto)

lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

end

class linordered_semiring = ordered_semiring + linordered_cancel_ab_semigroup_add
begin

subclass ordered_cancel_semiring ..

subclass ordered_comm_monoid_add ..

lemma mult_left_less_imp_less:
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
by (force simp add: mult_left_mono not_le [symmetric])

lemma mult_right_less_imp_less:
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
by (force simp add: mult_right_mono not_le [symmetric])

end

class linordered_semiring_1 = linordered_semiring + semiring_1
begin

lemma convex_bound_le:
  assumes "x \<le> a" "y \<le> a" "0 \<le> u" "0 \<le> v" "u + v = 1"
  shows "u * x + v * y \<le> a"
proof-
  from assms have "u * x + v * y \<le> u * a + v * a"
    by (simp add: add_mono mult_left_mono)
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
qed

end

class linordered_semiring_strict = semiring + comm_monoid_add + linordered_cancel_ab_semigroup_add +
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
begin

subclass semiring_0_cancel ..

subclass linordered_semiring
proof
  fix a b c :: 'a
  assume A: "a \<le> b" "0 \<le> c"
  from A show "c * a \<le> c * b"
    unfolding le_less
    using mult_strict_left_mono by (cases "c = 0") auto
  from A show "a * c \<le> b * c"
    unfolding le_less
    using mult_strict_right_mono by (cases "c = 0") auto
qed

lemma mult_left_le_imp_le:
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
by (force simp add: mult_strict_left_mono _not_less [symmetric])

lemma mult_right_le_imp_le:
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
by (force simp add: mult_strict_right_mono not_less [symmetric])

lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
using mult_strict_left_mono [of 0 b a] by simp

lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
using mult_strict_left_mono [of b 0 a] by simp

lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
using mult_strict_right_mono [of a 0 b] by simp

text {* Legacy - use @{text mult_neg_pos} *}
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
by (drule mult_strict_right_mono [of b 0], auto)

lemma zero_less_mult_pos:
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
apply (cases "b\<le>0")
 apply (auto simp add: le_less not_less)
apply (drule_tac mult_pos_neg [of a b])
 apply (auto dest: less_not_sym)
done

lemma zero_less_mult_pos2:
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
apply (cases "b\<le>0")
 apply (auto simp add: le_less not_less)
apply (drule_tac mult_pos_neg2 [of a b])
 apply (auto dest: less_not_sym)
done

text{*Strict monotonicity in both arguments*}
lemma mult_strict_mono:
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
  shows "a * c < b * d"
  using assms apply (cases "c=0")
  apply (simp)
  apply (erule mult_strict_right_mono [THEN less_trans])
  apply (force simp add: le_less)
  apply (erule mult_strict_left_mono, assumption)
  done

text{*This weaker variant has more natural premises*}
lemma mult_strict_mono':
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
  shows "a * c < b * d"
by (rule mult_strict_mono) (insert assms, auto)

lemma mult_less_le_imp_less:
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
  shows "a * c < b * d"
  using assms apply (subgoal_tac "a * c < b * c")
  apply (erule less_le_trans)
  apply (erule mult_left_mono)
  apply simp
  apply (erule mult_strict_right_mono)
  apply assumption
  done

lemma mult_le_less_imp_less:
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
  shows "a * c < b * d"
  using assms apply (subgoal_tac "a * c \<le> b * c")
  apply (erule le_less_trans)
  apply (erule mult_strict_left_mono)
  apply simp
  apply (erule mult_right_mono)
  apply simp
  done

end

class linordered_semiring_1_strict = linordered_semiring_strict + semiring_1
begin

subclass linordered_semiring_1 ..

lemma convex_bound_lt:
  assumes "x < a" "y < a" "0 \<le> u" "0 \<le> v" "u + v = 1"
  shows "u * x + v * y < a"
proof -
  from assms have "u * x + v * y < u * a + v * a"
    by (cases "u = 0")
       (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
  thus ?thesis using assms unfolding distrib_right[symmetric] by simp
qed

end

class ordered_comm_semiring = comm_semiring_0 + ordered_ab_semigroup_add +
  assumes comm_mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
begin

subclass ordered_semiring
proof
  fix a b c :: 'a
  assume "a \<le> b" "0 \<le> c"
  thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
  thus "a * c \<le> b * c" by (simp only: mult.commute)
qed

end

class ordered_cancel_comm_semiring = ordered_comm_semiring + cancel_comm_monoid_add
begin

subclass comm_semiring_0_cancel ..
subclass ordered_comm_semiring ..
subclass ordered_cancel_semiring ..

end

class linordered_comm_semiring_strict = comm_semiring_0 + linordered_cancel_ab_semigroup_add +
  assumes comm_mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
begin

subclass linordered_semiring_strict
proof
  fix a b c :: 'a
  assume "a < b" "0 < c"
  thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
  thus "a * c < b * c" by (simp only: mult.commute)
qed

subclass ordered_cancel_comm_semiring
proof
  fix a b c :: 'a
  assume "a \<le> b" "0 \<le> c"
  thus "c * a \<le> c * b"
    unfolding le_less
    using mult_strict_left_mono by (cases "c = 0") auto
qed

end

class ordered_ring = ring + ordered_cancel_semiring
begin

subclass ordered_ab_group_add ..

lemma less_add_iff1:
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
by (simp add: algebra_simps)

lemma less_add_iff2:
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
by (simp add: algebra_simps)

lemma le_add_iff1:
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
by (simp add: algebra_simps)

lemma le_add_iff2:
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
by (simp add: algebra_simps)

lemma mult_left_mono_neg:
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
  apply (drule mult_left_mono [of _ _ "- c"])
  apply simp_all
  done

lemma mult_right_mono_neg:
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
  apply (drule mult_right_mono [of _ _ "- c"])
  apply simp_all
  done

lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
using mult_right_mono_neg [of a 0 b] by simp

lemma split_mult_pos_le:
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
by (auto simp add: mult_nonpos_nonpos)

end

class linordered_ring = ring + linordered_semiring + linordered_ab_group_add + abs_if
begin

subclass ordered_ring ..

subclass ordered_ab_group_add_abs
proof
  fix a b
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
    by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
qed (auto simp add: abs_if)

lemma zero_le_square [simp]: "0 \<le> a * a"
  using linear [of 0 a]
  by (auto simp add: mult_nonpos_nonpos)

lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
  by (simp add: not_less)

end

class linordered_ring_strict = ring + linordered_semiring_strict
  + ordered_ab_group_add + abs_if
begin

subclass linordered_ring ..

lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
using mult_strict_left_mono [of b a "- c"] by simp

lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
using mult_strict_right_mono [of b a "- c"] by simp

lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
using mult_strict_right_mono_neg [of a 0 b] by simp

subclass ring_no_zero_divisors
proof
  fix a b
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
  have "a * b < 0 \<or> 0 < a * b"
  proof (cases "a < 0")
    case True note A' = this
    show ?thesis proof (cases "b < 0")
      case True with A'
      show ?thesis by (auto dest: mult_neg_neg)
    next
      case False with B have "0 < b" by auto
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
    qed
  next
    case False with A have A': "0 < a" by auto
    show ?thesis proof (cases "b < 0")
      case True with A'
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
    next
      case False with B have "0 < b" by auto
      with A' show ?thesis by auto
    qed
  qed
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
qed

lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
  by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
     (auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)

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"
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)

lemma mult_less_0_iff:
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
  apply (insert zero_less_mult_iff [of "-a" b])
  apply force
  done

lemma mult_le_0_iff:
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
  apply (insert zero_le_mult_iff [of "-a" b])
  apply force
  done

text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
   also with the relations @{text "\<le>"} and equality.*}

text{*These ``disjunction'' versions produce two cases when the comparison is
 an assumption, but effectively four when the comparison is a goal.*}

lemma mult_less_cancel_right_disj:
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
  apply (cases "c = 0")
  apply (auto simp add: neq_iff mult_strict_right_mono
                      mult_strict_right_mono_neg)
  apply (auto simp add: not_less
                      not_le [symmetric, of "a*c"]
                      not_le [symmetric, of a])
  apply (erule_tac [!] notE)
  apply (auto simp add: less_imp_le mult_right_mono
                      mult_right_mono_neg)
  done

lemma mult_less_cancel_left_disj:
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
  apply (cases "c = 0")
  apply (auto simp add: neq_iff mult_strict_left_mono
                      mult_strict_left_mono_neg)
  apply (auto simp add: not_less
                      not_le [symmetric, of "c*a"]
                      not_le [symmetric, of a])
  apply (erule_tac [!] notE)
  apply (auto simp add: less_imp_le mult_left_mono
                      mult_left_mono_neg)
  done

text{*The ``conjunction of implication'' lemmas produce two cases when the
comparison is a goal, but give four when the comparison is an assumption.*}

lemma mult_less_cancel_right:
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
  using mult_less_cancel_right_disj [of a c b] by auto

lemma mult_less_cancel_left:
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
  using mult_less_cancel_left_disj [of c a b] by auto

lemma mult_le_cancel_right:
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
by (simp add: not_less [symmetric] mult_less_cancel_right_disj)

lemma mult_le_cancel_left:
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
by (simp add: not_less [symmetric] mult_less_cancel_left_disj)

lemma mult_le_cancel_left_pos:
  "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
by (auto simp: mult_le_cancel_left)

lemma mult_le_cancel_left_neg:
  "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
by (auto simp: mult_le_cancel_left)

lemma mult_less_cancel_left_pos:
  "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
by (auto simp: mult_less_cancel_left)

lemma mult_less_cancel_left_neg:
  "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
by (auto simp: mult_less_cancel_left)

end

lemmas mult_sign_intros =
  mult_nonneg_nonneg mult_nonneg_nonpos
  mult_nonpos_nonneg mult_nonpos_nonpos
  mult_pos_pos mult_pos_neg
  mult_neg_pos mult_neg_neg

class ordered_comm_ring = comm_ring + ordered_comm_semiring
begin

subclass ordered_ring ..
subclass ordered_cancel_comm_semiring ..

end

class linordered_semidom = semidom + linordered_comm_semiring_strict +
  assumes zero_less_one [simp]: "0 < 1"
  assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
begin

text {* Addition is the inverse of subtraction. *}

lemma le_add_diff_inverse [simp]: "b \<le> a \<Longrightarrow> b + (a - b) = a"
  by (frule le_add_diff_inverse2) (simp add: add.commute)

lemma add_diff_inverse: "~ a<b \<Longrightarrow> b + (a - b) = a"
  by simp

lemma add_le_imp_le_diff: 
  shows "i + k \<le> n \<Longrightarrow> i \<le> n - k"
  apply (subst add_le_cancel_right [where c=k, symmetric])
  apply (frule le_add_diff_inverse2)
  apply (simp only: add.assoc [symmetric])
  using add_implies_diff by fastforce

lemma add_le_add_imp_diff_le: 
  assumes a1: "i + k \<le> n"
      and a2: "n \<le> j + k"
  shows "\<lbrakk>i + k \<le> n; n \<le> j + k\<rbrakk> \<Longrightarrow> n - k \<le> j"
proof -
  have "n - (i + k) + (i + k) = n"
    using a1 by simp
  moreover have "n - k = n - k - i + i"
    using a1 by (simp add: add_le_imp_le_diff)
  ultimately show ?thesis
    using a2
    apply (simp add: add.assoc [symmetric])
    apply (rule add_le_imp_le_diff [of _ k "j+k", simplified add_diff_cancel_right'])
    by (simp add: add.commute diff_diff_add)
qed

lemma pos_add_strict:
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
  using add_strict_mono [of 0 a b c] by simp

lemma zero_le_one [simp]: "0 \<le> 1"
by (rule zero_less_one [THEN less_imp_le])

lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
by (simp add: not_le)

lemma not_one_less_zero [simp]: "\<not> 1 < 0"
by (simp add: not_less)

lemma less_1_mult:
  assumes "1 < m" and "1 < n"
  shows "1 < m * n"
  using assms mult_strict_mono [of 1 m 1 n]
    by (simp add:  less_trans [OF zero_less_one])

lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
  using mult_left_mono[of c 1 a] by simp

lemma mult_le_one: "a \<le> 1 \<Longrightarrow> 0 \<le> b \<Longrightarrow> b \<le> 1 \<Longrightarrow> a * b \<le> 1"
  using mult_mono[of a 1 b 1] by simp

end

class linordered_idom = comm_ring_1 +
  linordered_comm_semiring_strict + ordered_ab_group_add +
  abs_if + sgn_if
begin

subclass linordered_semiring_1_strict ..
subclass linordered_ring_strict ..
subclass ordered_comm_ring ..
subclass idom ..

subclass linordered_semidom
proof
  have "0 \<le> 1 * 1" by (rule zero_le_square)
  thus "0 < 1" by (simp add: le_less)
  show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"
    by simp
qed

lemma linorder_neqE_linordered_idom:
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
  using assms by (rule neqE)

text {* These cancellation simprules also produce two cases when the comparison is a goal. *}

lemma mult_le_cancel_right1:
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
by (insert mult_le_cancel_right [of 1 c b], simp)

lemma mult_le_cancel_right2:
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
by (insert mult_le_cancel_right [of a c 1], simp)

lemma mult_le_cancel_left1:
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
by (insert mult_le_cancel_left [of c 1 b], simp)

lemma mult_le_cancel_left2:
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
by (insert mult_le_cancel_left [of c a 1], simp)

lemma mult_less_cancel_right1:
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
by (insert mult_less_cancel_right [of 1 c b], simp)

lemma mult_less_cancel_right2:
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
by (insert mult_less_cancel_right [of a c 1], simp)

lemma mult_less_cancel_left1:
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
by (insert mult_less_cancel_left [of c 1 b], simp)

lemma mult_less_cancel_left2:
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
by (insert mult_less_cancel_left [of c a 1], simp)

lemma sgn_sgn [simp]:
  "sgn (sgn a) = sgn a"
unfolding sgn_if by simp

lemma sgn_0_0:
  "sgn a = 0 \<longleftrightarrow> a = 0"
unfolding sgn_if by simp

lemma sgn_1_pos:
  "sgn a = 1 \<longleftrightarrow> a > 0"
unfolding sgn_if by simp

lemma sgn_1_neg:
  "sgn a = - 1 \<longleftrightarrow> a < 0"
unfolding sgn_if by auto

lemma sgn_pos [simp]:
  "0 < a \<Longrightarrow> sgn a = 1"
unfolding sgn_1_pos .

lemma sgn_neg [simp]:
  "a < 0 \<Longrightarrow> sgn a = - 1"
unfolding sgn_1_neg .

lemma sgn_times:
  "sgn (a * b) = sgn a * sgn b"
by (auto simp add: sgn_if zero_less_mult_iff)

lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
unfolding sgn_if abs_if by auto

lemma sgn_greater [simp]:
  "0 < sgn a \<longleftrightarrow> 0 < a"
  unfolding sgn_if by auto

lemma sgn_less [simp]:
  "sgn a < 0 \<longleftrightarrow> a < 0"
  unfolding sgn_if by auto

lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
  by (simp add: abs_if)

lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
  by (simp add: abs_if)

lemma dvd_if_abs_eq:
  "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
by(subst abs_dvd_iff[symmetric]) simp

text {* The following lemmas can be proven in more general structures, but
are dangerous as simp rules in absence of @{thm neg_equal_zero},
@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}. *}

lemma equation_minus_iff_1 [simp, no_atp]:
  "1 = - a \<longleftrightarrow> a = - 1"
  by (fact equation_minus_iff)

lemma minus_equation_iff_1 [simp, no_atp]:
  "- a = 1 \<longleftrightarrow> a = - 1"
  by (subst minus_equation_iff, auto)

lemma le_minus_iff_1 [simp, no_atp]:
  "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
  by (fact le_minus_iff)

lemma minus_le_iff_1 [simp, no_atp]:
  "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
  by (fact minus_le_iff)

lemma less_minus_iff_1 [simp, no_atp]:
  "1 < - b \<longleftrightarrow> b < - 1"
  by (fact less_minus_iff)

lemma minus_less_iff_1 [simp, no_atp]:
  "- a < 1 \<longleftrightarrow> - 1 < a"
  by (fact minus_less_iff)

end

text {* Simprules for comparisons where common factors can be cancelled. *}

lemmas mult_compare_simps =
    mult_le_cancel_right mult_le_cancel_left
    mult_le_cancel_right1 mult_le_cancel_right2
    mult_le_cancel_left1 mult_le_cancel_left2
    mult_less_cancel_right mult_less_cancel_left
    mult_less_cancel_right1 mult_less_cancel_right2
    mult_less_cancel_left1 mult_less_cancel_left2
    mult_cancel_right mult_cancel_left
    mult_cancel_right1 mult_cancel_right2
    mult_cancel_left1 mult_cancel_left2

text {* Reasoning about inequalities with division *}

context linordered_semidom
begin

lemma less_add_one: "a < a + 1"
proof -
  have "a + 0 < a + 1"
    by (blast intro: zero_less_one add_strict_left_mono)
  thus ?thesis by simp
qed

lemma zero_less_two: "0 < 1 + 1"
by (blast intro: less_trans zero_less_one less_add_one)

end

context linordered_idom
begin

lemma mult_right_le_one_le:
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
  by (rule mult_left_le)

lemma mult_left_le_one_le:
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
  by (auto simp add: mult_le_cancel_right2)

end

text {* Absolute Value *}

context linordered_idom
begin

lemma mult_sgn_abs:
  "sgn x * \<bar>x\<bar> = x"
  unfolding abs_if sgn_if by auto

lemma abs_one [simp]:
  "\<bar>1\<bar> = 1"
  by (simp add: abs_if)

end

class ordered_ring_abs = ordered_ring + ordered_ab_group_add_abs +
  assumes abs_eq_mult:
    "(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>"

context linordered_idom
begin

subclass ordered_ring_abs proof
qed (auto simp add: abs_if not_less mult_less_0_iff)

lemma abs_mult:
  "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
  by (rule abs_eq_mult) auto

lemma abs_mult_self:
  "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
  by (simp add: abs_if)

lemma abs_mult_less:
  "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
proof -
  assume ac: "\<bar>a\<bar> < c"
  hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
  assume "\<bar>b\<bar> < d"
  thus ?thesis by (simp add: ac cpos mult_strict_mono)
qed

lemma abs_less_iff:
  "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
  by (simp add: less_le abs_le_iff) (auto simp add: abs_if)

lemma abs_mult_pos:
  "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
  by (simp add: abs_mult)

lemma abs_diff_less_iff:
  "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
  by (auto simp add: diff_less_eq ac_simps abs_less_iff)

lemma abs_diff_le_iff:
   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
  by (auto simp add: diff_le_eq ac_simps abs_le_iff)

end

hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib

code_identifier
  code_module Rings \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

end