(*  Title:      HOL/Fields.thy

    Author:     Gertrud Bauer

    Author:     Steven Obua

    Author:     Tobias Nipkow

    Author:     Lawrence C Paulson

    Author:     Markus Wenzel

    Author:     Jeremy Avigad

*)

header {* Fields *}

theory Fields

imports Rings

begin

class field = comm_ring_1 + inverse +

  assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"

  assumes divide_inverse: "a / b = a * inverse b"

begin

subclass division_ring

proof

  fix a :: 'a

  assume "a \<noteq> 0"

  thus "inverse a * a = 1" by (rule field_inverse)

  thus "a * inverse a = 1" by (simp only: mult_commute)

qed

subclass idom ..

lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"

proof

  assume neq: "b \<noteq> 0"

  {

    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)

    also assume "a / b = 1"

    finally show "a = b" by simp

  next

    assume "a = b"

    with neq show "a / b = 1" by (simp add: divide_inverse)

  }

qed

lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"

by (simp add: divide_inverse)

lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"

by (simp add: divide_inverse)

lemma divide_zero_left [simp]: "0 / a = 0"

by (simp add: divide_inverse)

lemma inverse_eq_divide: "inverse a = 1 / a"

by (simp add: divide_inverse)

lemma add_divide_distrib: "(a+b) / c = a/c + b/c"

by (simp add: divide_inverse algebra_simps)

text{*There is no slick version using division by zero.*}

lemma inverse_add:

  "[| a \<noteq> 0;  b \<noteq> 0 |]

   ==> inverse a + inverse b = (a + b) * inverse a * inverse b"

by (simp add: division_ring_inverse_add mult_ac)

lemma nonzero_mult_divide_mult_cancel_left [simp, noatp]:

assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b"

proof -

  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"

    by (simp add: divide_inverse nonzero_inverse_mult_distrib)

  also have "... =  a * inverse b * (inverse c * c)"

    by (simp only: mult_ac)

  also have "... =  a * inverse b" by simp

    finally show ?thesis by (simp add: divide_inverse)

qed

lemma nonzero_mult_divide_mult_cancel_right [simp, noatp]:

  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b"

by (simp add: mult_commute [of _ c])

lemma divide_1 [simp]: "a / 1 = a"

by (simp add: divide_inverse)

lemma times_divide_eq_right: "a * (b / c) = (a * b) / c"

by (simp add: divide_inverse mult_assoc)

lemma times_divide_eq_left: "(b / c) * a = (b * a) / c"

by (simp add: divide_inverse mult_ac)

text {* These are later declared as simp rules. *}

lemmas times_divide_eq [noatp] = times_divide_eq_right times_divide_eq_left

lemma add_frac_eq:

  assumes "y \<noteq> 0" and "z \<noteq> 0"

  shows "x / y + w / z = (x * z + w * y) / (y * z)"

proof -

  have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"

    using assms by simp

  also have "\<dots> = (x * z + y * w) / (y * z)"

    by (simp only: add_divide_distrib)

  finally show ?thesis

    by (simp only: mult_commute)

qed

text{*Special Cancellation Simprules for Division*}

lemma nonzero_mult_divide_cancel_right [simp, noatp]:

  "b \<noteq> 0 \<Longrightarrow> a * b / b = a"

using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp

lemma nonzero_mult_divide_cancel_left [simp, noatp]:

  "a \<noteq> 0 \<Longrightarrow> a * b / a = b"

using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp

lemma nonzero_divide_mult_cancel_right [simp, noatp]:

  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a"

using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp

lemma nonzero_divide_mult_cancel_left [simp, noatp]:

  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b"

using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp

lemma nonzero_mult_divide_mult_cancel_left2 [simp, noatp]:

  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b"

using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac)

lemma nonzero_mult_divide_mult_cancel_right2 [simp, noatp]:

  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b"

using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac)

lemma minus_divide_left: "- (a / b) = (-a) / b"

by (simp add: divide_inverse)

lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"

by (simp add: divide_inverse nonzero_inverse_minus_eq)

lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"

by (simp add: divide_inverse nonzero_inverse_minus_eq)

lemma divide_minus_left [simp, noatp]: "(-a) / b = - (a / b)"

by (simp add: divide_inverse)

lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"

by (simp add: diff_minus add_divide_distrib)

lemma add_divide_eq_iff:

  "z \<noteq> 0 \<Longrightarrow> x + y / z = (z * x + y) / z"

by (simp add: add_divide_distrib)

lemma divide_add_eq_iff:

  "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + z * y) / z"

by (simp add: add_divide_distrib)

lemma diff_divide_eq_iff:

  "z \<noteq> 0 \<Longrightarrow> x - y / z = (z * x - y) / z"

by (simp add: diff_divide_distrib)

lemma divide_diff_eq_iff:

  "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - z * y) / z"

by (simp add: diff_divide_distrib)

lemma nonzero_eq_divide_eq: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"

proof -

  assume [simp]: "c \<noteq> 0"

  have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp

  also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc)

  finally show ?thesis .

qed

lemma nonzero_divide_eq_eq: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"

proof -

  assume [simp]: "c \<noteq> 0"

  have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp

  also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc) 

  finally show ?thesis .

qed

lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"

by simp

lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"

by (erule subst, simp)

lemmas field_eq_simps[noatp] = algebra_simps

  (* pull / out*)

  add_divide_eq_iff divide_add_eq_iff

  diff_divide_eq_iff divide_diff_eq_iff

  (* multiply eqn *)

  nonzero_eq_divide_eq nonzero_divide_eq_eq

(* is added later:

  times_divide_eq_left times_divide_eq_right

*)

text{*An example:*}

lemma "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f\<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1"

apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")

 apply(simp add:field_eq_simps)

apply(simp)

done

lemma diff_frac_eq:

  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"

by (simp add: field_eq_simps times_divide_eq)

lemma frac_eq_eq:

  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"

by (simp add: field_eq_simps times_divide_eq)

end

class division_by_zero = zero + inverse +

  assumes inverse_zero [simp]: "inverse 0 = 0"

lemma divide_zero [simp]:

  "a / 0 = (0::'a::{field,division_by_zero})"

by (simp add: divide_inverse)

lemma divide_self_if [simp]:

  "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"

by simp

class linordered_field = field + linordered_idom

lemma inverse_nonzero_iff_nonzero [simp]:

   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"

by (force dest: inverse_zero_imp_zero) 

lemma inverse_minus_eq [simp]:

   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"

proof cases

  assume "a=0" thus ?thesis by (simp add: inverse_zero)

next

  assume "a\<noteq>0" 

  thus ?thesis by (simp add: nonzero_inverse_minus_eq)

qed

lemma inverse_eq_imp_eq:

  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"

apply (cases "a=0 | b=0") 

 apply (force dest!: inverse_zero_imp_zero

              simp add: eq_commute [of "0::'a"])

apply (force dest!: nonzero_inverse_eq_imp_eq) 

done

lemma inverse_eq_iff_eq [simp]:

  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"

by (force dest!: inverse_eq_imp_eq)

lemma inverse_inverse_eq [simp]:

     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"

  proof cases

    assume "a=0" thus ?thesis by simp

  next

    assume "a\<noteq>0" 

    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)

  qed

text{*This version builds in division by zero while also re-orienting

      the right-hand side.*}

lemma inverse_mult_distrib [simp]:

     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"

  proof cases

    assume "a \<noteq> 0 & b \<noteq> 0" 

    thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute)

  next

    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

    thus ?thesis by force

  qed

lemma inverse_divide [simp]:

  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"

by (simp add: divide_inverse mult_commute)

subsection {* Calculations with fractions *}

text{* There is a whole bunch of simp-rules just for class @{text

field} but none for class @{text field} and @{text nonzero_divides}

because the latter are covered by a simproc. *}

lemma mult_divide_mult_cancel_left:

  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"

apply (cases "b = 0")

apply (simp_all add: nonzero_mult_divide_mult_cancel_left)

done

lemma mult_divide_mult_cancel_right:

  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"

apply (cases "b = 0")

apply (simp_all add: nonzero_mult_divide_mult_cancel_right)

done

lemma divide_divide_eq_right [simp,noatp]:

  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"

by (simp add: divide_inverse mult_ac)

lemma divide_divide_eq_left [simp,noatp]:

  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"

by (simp add: divide_inverse mult_assoc)

subsubsection{*Special Cancellation Simprules for Division*}

lemma mult_divide_mult_cancel_left_if[simp,noatp]:

fixes c :: "'a :: {field,division_by_zero}"

shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"

by (simp add: mult_divide_mult_cancel_left)

subsection {* Division and Unary Minus *}

lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"

by (simp add: divide_inverse)

lemma divide_minus_right [simp, noatp]:

  "a / -(b::'a::{field,division_by_zero}) = -(a / b)"

by (simp add: divide_inverse)

lemma minus_divide_divide:

  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"

apply (cases "b=0", simp) 

apply (simp add: nonzero_minus_divide_divide) 

done

lemma eq_divide_eq:

  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"

by (simp add: nonzero_eq_divide_eq)

lemma divide_eq_eq:

  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"

by (force simp add: nonzero_divide_eq_eq)

subsection {* Ordered Fields *}

lemma positive_imp_inverse_positive: 

assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::linordered_field)"

proof -

  have "0 < a * inverse a" 

    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)

  thus "0 < inverse a" 

    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)

qed

lemma negative_imp_inverse_negative:

  "a < 0 ==> inverse a < (0::'a::linordered_field)"

by (insert positive_imp_inverse_positive [of "-a"], 

    simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)

lemma inverse_le_imp_le:

assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"

shows "b \<le> (a::'a::linordered_field)"

proof (rule classical)

  assume "~ b \<le> a"

  hence "a < b"  by (simp add: linorder_not_le)

  hence bpos: "0 < b"  by (blast intro: apos order_less_trans)

  hence "a * inverse a \<le> a * inverse b"

    by (simp add: apos invle order_less_imp_le mult_left_mono)

  hence "(a * inverse a) * b \<le> (a * inverse b) * b"

    by (simp add: bpos order_less_imp_le mult_right_mono)

  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)

qed

lemma inverse_positive_imp_positive:

assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"

shows "0 < (a::'a::linordered_field)"

proof -

  have "0 < inverse (inverse a)"

    using inv_gt_0 by (rule positive_imp_inverse_positive)

  thus "0 < a"

    using nz by (simp add: nonzero_inverse_inverse_eq)

qed

lemma inverse_positive_iff_positive [simp]:

  "(0 < inverse a) = (0 < (a::'a::{linordered_field,division_by_zero}))"

apply (cases "a = 0", simp)

apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)

done

lemma inverse_negative_imp_negative:

assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"

shows "a < (0::'a::linordered_field)"

proof -

  have "inverse (inverse a) < 0"

    using inv_less_0 by (rule negative_imp_inverse_negative)

  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)

qed

lemma inverse_negative_iff_negative [simp]:

  "(inverse a < 0) = (a < (0::'a::{linordered_field,division_by_zero}))"

apply (cases "a = 0", simp)

apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)

done

lemma inverse_nonnegative_iff_nonnegative [simp]:

  "(0 \<le> inverse a) = (0 \<le> (a::'a::{linordered_field,division_by_zero}))"

by (simp add: linorder_not_less [symmetric])

lemma inverse_nonpositive_iff_nonpositive [simp]:

  "(inverse a \<le> 0) = (a \<le> (0::'a::{linordered_field,division_by_zero}))"

by (simp add: linorder_not_less [symmetric])

lemma linordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::linordered_field)"

proof

  fix x::'a

  have m1: "- (1::'a) < 0" by simp

  from add_strict_right_mono[OF m1, where c=x] 

  have "(- 1) + x < x" by simp

  thus "\<exists>y. y < x" by blast

qed

lemma linordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::linordered_field)"

proof

  fix x::'a

  have m1: " (1::'a) > 0" by simp

  from add_strict_right_mono[OF m1, where c=x] 

  have "1 + x > x" by simp

  thus "\<exists>y. y > x" by blast

qed

subsection{*Anti-Monotonicity of @{term inverse}*}

lemma less_imp_inverse_less:

assumes less: "a < b" and apos:  "0 < a"

shows "inverse b < inverse (a::'a::linordered_field)"

proof (rule ccontr)

  assume "~ inverse b < inverse a"

  hence "inverse a \<le> inverse b" by (simp add: linorder_not_less)

  hence "~ (a < b)"

    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])

  thus False by (rule notE [OF _ less])

qed

lemma inverse_less_imp_less:

  "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::linordered_field)"

apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])

apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 

done

text{*Both premises are essential. Consider -1 and 1.*}

lemma inverse_less_iff_less [simp,noatp]:

  "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::linordered_field))"

by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 

lemma le_imp_inverse_le:

  "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::linordered_field)"

by (force simp add: order_le_less less_imp_inverse_less)

lemma inverse_le_iff_le [simp,noatp]:

 "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::linordered_field))"

by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 

text{*These results refer to both operands being negative.  The opposite-sign

case is trivial, since inverse preserves signs.*}

lemma inverse_le_imp_le_neg:

  "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::linordered_field)"

apply (rule classical) 

apply (subgoal_tac "a < 0") 

 prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 

apply (insert inverse_le_imp_le [of "-b" "-a"])

apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

done

lemma less_imp_inverse_less_neg:

   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::linordered_field)"

apply (subgoal_tac "a < 0") 

 prefer 2 apply (blast intro: order_less_trans) 

apply (insert less_imp_inverse_less [of "-b" "-a"])

apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

done

lemma inverse_less_imp_less_neg:

   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::linordered_field)"

apply (rule classical) 

apply (subgoal_tac "a < 0") 

 prefer 2

 apply (force simp add: linorder_not_less intro: order_le_less_trans) 

apply (insert inverse_less_imp_less [of "-b" "-a"])

apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

done

lemma inverse_less_iff_less_neg [simp,noatp]:

  "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::linordered_field))"

apply (insert inverse_less_iff_less [of "-b" "-a"])

apply (simp del: inverse_less_iff_less 

            add: order_less_imp_not_eq nonzero_inverse_minus_eq)

done

lemma le_imp_inverse_le_neg:

  "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::linordered_field)"

by (force simp add: order_le_less less_imp_inverse_less_neg)

lemma inverse_le_iff_le_neg [simp,noatp]:

 "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::linordered_field))"

by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 

subsection{*Inverses and the Number One*}

lemma one_less_inverse_iff:

  "(1 < inverse x) = (0 < x & x < (1::'a::{linordered_field,division_by_zero}))"

proof cases

  assume "0 < x"

    with inverse_less_iff_less [OF zero_less_one, of x]

    show ?thesis by simp

next

  assume notless: "~ (0 < x)"

  have "~ (1 < inverse x)"

  proof

    assume "1 < inverse x"

    also with notless have "... \<le> 0" by (simp add: linorder_not_less)

    also have "... < 1" by (rule zero_less_one) 

    finally show False by auto

  qed

  with notless show ?thesis by simp

qed

lemma inverse_eq_1_iff [simp]:

  "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"

by (insert inverse_eq_iff_eq [of x 1], simp) 

lemma one_le_inverse_iff:

  "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{linordered_field,division_by_zero}))"

by (force simp add: order_le_less one_less_inverse_iff zero_less_one 

                    eq_commute [of 1]) 

lemma inverse_less_1_iff:

  "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{linordered_field,division_by_zero}))"

by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 

lemma inverse_le_1_iff:

  "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{linordered_field,division_by_zero}))"

by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 

subsection{*Simplification of Inequalities Involving Literal Divisors*}

lemma pos_le_divide_eq: "0 < (c::'a::linordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"

proof -

  assume less: "0<c"

  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"

    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  also have "... = (a*c \<le> b)"

    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

  finally show ?thesis .

qed

lemma neg_le_divide_eq: "c < (0::'a::linordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"

proof -

  assume less: "c<0"

  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"

    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  also have "... = (b \<le> a*c)"

    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

  finally show ?thesis .

qed

lemma le_divide_eq:

  "(a \<le> b/c) = 

   (if 0 < c then a*c \<le> b

             else if c < 0 then b \<le> a*c

             else  a \<le> (0::'a::{linordered_field,division_by_zero}))"

apply (cases "c=0", simp) 

apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 

done

lemma pos_divide_le_eq: "0 < (c::'a::linordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"

proof -

  assume less: "0<c"

  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"

    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  also have "... = (b \<le> a*c)"

    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

  finally show ?thesis .

qed

lemma neg_divide_le_eq: "c < (0::'a::linordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"

proof -

  assume less: "c<0"

  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"

    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

  also have "... = (a*c \<le> b)"

    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

  finally show ?thesis .

qed

lemma divide_le_eq:

  "(b/c \<le> a) = 

   (if 0 < c then b \<le> a*c

             else if c < 0 then a*c \<le> b

             else 0 \<le> (a::'a::{linordered_field,division_by_zero}))"

apply (cases "c=0", simp) 

apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 

done

lemma pos_less_divide_eq:

     "0 < (c::'a::linordered_field) ==> (a < b/c) = (a*c < b)"

proof -

  assume less: "0<c"

  hence "(a < b/c) = (a*c < (b/c)*c)"

    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  also have "... = (a*c < b)"

    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

  finally show ?thesis .

qed

lemma neg_less_divide_eq:

 "c < (0::'a::linordered_field) ==> (a < b/c) = (b < a*c)"

proof -

  assume less: "c<0"

  hence "(a < b/c) = ((b/c)*c < a*c)"

    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  also have "... = (b < a*c)"

    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

  finally show ?thesis .

qed

lemma less_divide_eq:

  "(a < b/c) = 

   (if 0 < c then a*c < b

             else if c < 0 then b < a*c

             else  a < (0::'a::{linordered_field,division_by_zero}))"

apply (cases "c=0", simp) 

apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 

done

lemma pos_divide_less_eq:

     "0 < (c::'a::linordered_field) ==> (b/c < a) = (b < a*c)"

proof -

  assume less: "0<c"

  hence "(b/c < a) = ((b/c)*c < a*c)"

    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  also have "... = (b < a*c)"

    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

  finally show ?thesis .

qed

lemma neg_divide_less_eq:

 "c < (0::'a::linordered_field) ==> (b/c < a) = (a*c < b)"

proof -

  assume less: "c<0"

  hence "(b/c < a) = (a*c < (b/c)*c)"

    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])

  also have "... = (a*c < b)"

    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

  finally show ?thesis .

qed

lemma divide_less_eq:

  "(b/c < a) = 

   (if 0 < c then b < a*c

             else if c < 0 then a*c < b

             else 0 < (a::'a::{linordered_field,division_by_zero}))"

apply (cases "c=0", simp) 

apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 

done

subsection{*Field simplification*}

text{* Lemmas @{text field_simps} multiply with denominators in in(equations)

if they can be proved to be non-zero (for equations) or positive/negative

(for inequations). Can be too aggressive and is therefore separate from the

more benign @{text algebra_simps}. *}

lemmas field_simps[noatp] = field_eq_simps

  (* multiply ineqn *)

  pos_divide_less_eq neg_divide_less_eq

  pos_less_divide_eq neg_less_divide_eq

  pos_divide_le_eq neg_divide_le_eq

  pos_le_divide_eq neg_le_divide_eq

text{* Lemmas @{text sign_simps} is a first attempt to automate proofs

of positivity/negativity needed for @{text field_simps}. Have not added @{text

sign_simps} to @{text field_simps} because the former can lead to case

explosions. *}

lemmas sign_simps[noatp] = group_simps

  zero_less_mult_iff  mult_less_0_iff

(* Only works once linear arithmetic is installed:

text{*An example:*}

lemma fixes a b c d e f :: "'a::linordered_field"

shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>

 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <

 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"

apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")

 prefer 2 apply(simp add:sign_simps)

apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")

 prefer 2 apply(simp add:sign_simps)

apply(simp add:field_simps)

done

*)

subsection{*Division and Signs*}

lemma zero_less_divide_iff:

     "((0::'a::{linordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"

by (simp add: divide_inverse zero_less_mult_iff)

lemma divide_less_0_iff:

     "(a/b < (0::'a::{linordered_field,division_by_zero})) = 

      (0 < a & b < 0 | a < 0 & 0 < b)"

by (simp add: divide_inverse mult_less_0_iff)

lemma zero_le_divide_iff:

     "((0::'a::{linordered_field,division_by_zero}) \<le> a/b) =

      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"

by (simp add: divide_inverse zero_le_mult_iff)

lemma divide_le_0_iff:

     "(a/b \<le> (0::'a::{linordered_field,division_by_zero})) =

      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"

by (simp add: divide_inverse mult_le_0_iff)

lemma divide_eq_0_iff [simp,noatp]:

     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"

by (simp add: divide_inverse)

lemma divide_pos_pos:

  "0 < (x::'a::linordered_field) ==> 0 < y ==> 0 < x / y"

by(simp add:field_simps)

lemma divide_nonneg_pos:

  "0 <= (x::'a::linordered_field) ==> 0 < y ==> 0 <= x / y"

by(simp add:field_simps)

lemma divide_neg_pos:

  "(x::'a::linordered_field) < 0 ==> 0 < y ==> x / y < 0"

by(simp add:field_simps)

lemma divide_nonpos_pos:

  "(x::'a::linordered_field) <= 0 ==> 0 < y ==> x / y <= 0"

by(simp add:field_simps)

lemma divide_pos_neg:

  "0 < (x::'a::linordered_field) ==> y < 0 ==> x / y < 0"

by(simp add:field_simps)

lemma divide_nonneg_neg:

  "0 <= (x::'a::linordered_field) ==> y < 0 ==> x / y <= 0" 

by(simp add:field_simps)

lemma divide_neg_neg:

  "(x::'a::linordered_field) < 0 ==> y < 0 ==> 0 < x / y"

by(simp add:field_simps)

lemma divide_nonpos_neg:

  "(x::'a::linordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"

by(simp add:field_simps)

subsection{*Cancellation Laws for Division*}

lemma divide_cancel_right [simp,noatp]:

     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"

apply (cases "c=0", simp)

apply (simp add: divide_inverse)

done

lemma divide_cancel_left [simp,noatp]:

     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 

apply (cases "c=0", simp)

apply (simp add: divide_inverse)

done

subsection {* Division and the Number One *}

text{*Simplify expressions equated with 1*}

lemma divide_eq_1_iff [simp,noatp]:

     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"

apply (cases "b=0", simp)

apply (simp add: right_inverse_eq)

done

lemma one_eq_divide_iff [simp,noatp]:

     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"

by (simp add: eq_commute [of 1])

lemma zero_eq_1_divide_iff [simp,noatp]:

     "((0::'a::{linordered_field,division_by_zero}) = 1/a) = (a = 0)"

apply (cases "a=0", simp)

apply (auto simp add: nonzero_eq_divide_eq)

done

lemma one_divide_eq_0_iff [simp,noatp]:

     "(1/a = (0::'a::{linordered_field,division_by_zero})) = (a = 0)"

apply (cases "a=0", simp)

apply (insert zero_neq_one [THEN not_sym])

apply (auto simp add: nonzero_divide_eq_eq)

done

text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}

lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]

lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]

lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]

lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]

declare zero_less_divide_1_iff [simp,noatp]

declare divide_less_0_1_iff [simp,noatp]

declare zero_le_divide_1_iff [simp,noatp]

declare divide_le_0_1_iff [simp,noatp]

subsection {* Ordering Rules for Division *}

lemma divide_strict_right_mono:

     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::linordered_field)"

by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 

              positive_imp_inverse_positive)

lemma divide_right_mono:

     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{linordered_field,division_by_zero})"

by (force simp add: divide_strict_right_mono order_le_less)

lemma divide_right_mono_neg: "(a::'a::{division_by_zero,linordered_field}) <= b 

    ==> c <= 0 ==> b / c <= a / c"

apply (drule divide_right_mono [of _ _ "- c"])

apply auto

done

lemma divide_strict_right_mono_neg:

     "[|b < a; c < 0|] ==> a / c < b / (c::'a::linordered_field)"

apply (drule divide_strict_right_mono [of _ _ "-c"], simp)

apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])

done

text{*The last premise ensures that @{term a} and @{term b} 

      have the same sign*}

lemma divide_strict_left_mono:

  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::linordered_field)"

by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)

lemma divide_left_mono:

  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::linordered_field)"

by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)

lemma divide_left_mono_neg: "(a::'a::{division_by_zero,linordered_field}) <= b 

    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"

  apply (drule divide_left_mono [of _ _ "- c"])

  apply (auto simp add: mult_commute)

done

lemma divide_strict_left_mono_neg:

  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::linordered_field)"

by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)

text{*Simplify quotients that are compared with the value 1.*}

lemma le_divide_eq_1 [noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"

by (auto simp add: le_divide_eq)

lemma divide_le_eq_1 [noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"

by (auto simp add: divide_le_eq)

lemma less_divide_eq_1 [noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"

by (auto simp add: less_divide_eq)

lemma divide_less_eq_1 [noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"

by (auto simp add: divide_less_eq)

subsection{*Conditional Simplification Rules: No Case Splits*}

lemma le_divide_eq_1_pos [simp,noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"

by (auto simp add: le_divide_eq)

lemma le_divide_eq_1_neg [simp,noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"

by (auto simp add: le_divide_eq)

lemma divide_le_eq_1_pos [simp,noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"

by (auto simp add: divide_le_eq)

lemma divide_le_eq_1_neg [simp,noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"

by (auto simp add: divide_le_eq)

lemma less_divide_eq_1_pos [simp,noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"

by (auto simp add: less_divide_eq)

lemma less_divide_eq_1_neg [simp,noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"

by (auto simp add: less_divide_eq)

lemma divide_less_eq_1_pos [simp,noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"

by (auto simp add: divide_less_eq)

lemma divide_less_eq_1_neg [simp,noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"

by (auto simp add: divide_less_eq)

lemma eq_divide_eq_1 [simp,noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"

by (auto simp add: eq_divide_eq)

lemma divide_eq_eq_1 [simp,noatp]:

  fixes a :: "'a :: {linordered_field,division_by_zero}"

  shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"

by (auto simp add: divide_eq_eq)

subsection {* Reasoning about inequalities with division *}

lemma mult_imp_div_pos_le: "0 < (y::'a::linordered_field) ==> x <= z * y ==>

    x / y <= z"

by (subst pos_divide_le_eq, assumption+)

lemma mult_imp_le_div_pos: "0 < (y::'a::linordered_field) ==> z * y <= x ==>

    z <= x / y"

by(simp add:field_simps)

lemma mult_imp_div_pos_less: "0 < (y::'a::linordered_field) ==> x < z * y ==>

    x / y < z"

by(simp add:field_simps)

lemma mult_imp_less_div_pos: "0 < (y::'a::linordered_field) ==> z * y < x ==>

    z < x / y"

by(simp add:field_simps)

lemma frac_le: "(0::'a::linordered_field) <= x ==> 

    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"

  apply (rule mult_imp_div_pos_le)

  apply simp

  apply (subst times_divide_eq_left)

  apply (rule mult_imp_le_div_pos, assumption)

  apply (rule mult_mono)

  apply simp_all

done

lemma frac_less: "(0::'a::linordered_field) <= x ==> 

    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"

  apply (rule mult_imp_div_pos_less)

  apply simp

  apply (subst times_divide_eq_left)

  apply (rule mult_imp_less_div_pos, assumption)

  apply (erule mult_less_le_imp_less)

  apply simp_all

done

lemma frac_less2: "(0::'a::linordered_field) < x ==> 

    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"

  apply (rule mult_imp_div_pos_less)

  apply simp_all

  apply (subst times_divide_eq_left)

  apply (rule mult_imp_less_div_pos, assumption)

  apply (erule mult_le_less_imp_less)

  apply simp_all

done

text{*It's not obvious whether these should be simprules or not. 

  Their effect is to gather terms into one big fraction, like

  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 

  seem to need them.*}

declare times_divide_eq [simp]

subsection {* Ordered Fields are Dense *}

lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::linordered_field)"

by (simp add: field_simps zero_less_two)

lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::linordered_field) < b"

by (simp add: field_simps zero_less_two)

instance linordered_field < dense_linorder

proof

  fix x y :: 'a