(*  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

 subsection {* Division rings *}

 text {*

   A division ring is like a field, but without the commutativity requirement.

 *}

 class inverse =

   fixes inverse :: "'a \<Rightarrow> 'a"

     and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)

 class division_ring = ring_1 + inverse +

   assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"

   assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"

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

 begin

 subclass ring_1_no_zero_divisors

 proof

   fix a b :: 'a

   assume a: "a \<noteq> 0" and b: "b \<noteq> 0"

   show "a * b \<noteq> 0"

   proof

     assume ab: "a * b = 0"

     hence "0 = inverse a * (a * b) * inverse b" by simp

     also have "\<dots> = (inverse a * a) * (b * inverse b)"

       by (simp only: mult_assoc)

     also have "\<dots> = 1" using a b by simp

     finally show False by simp

   qed

 qed

 lemma nonzero_imp_inverse_nonzero:

   "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"

 proof

   assume ianz: "inverse a = 0"

   assume "a \<noteq> 0"

   hence "1 = a * inverse a" by simp

   also have "... = 0" by (simp add: ianz)

   finally have "1 = 0" .

   thus False by (simp add: eq_commute)

 qed

 lemma inverse_zero_imp_zero:

   "inverse a = 0 \<Longrightarrow> a = 0"

 apply (rule classical)

 apply (drule nonzero_imp_inverse_nonzero)

 apply auto

 done

 lemma inverse_unique: 

   assumes ab: "a * b = 1"

   shows "inverse a = b"

 proof -

   have "a \<noteq> 0" using ab by (cases "a = 0") simp_all

   moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)

   ultimately show ?thesis by (simp add: mult_assoc [symmetric])

 qed

 lemma nonzero_inverse_minus_eq:

   "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"

 by (rule inverse_unique) simp

 lemma nonzero_inverse_inverse_eq:

   "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"

 by (rule inverse_unique) simp

 lemma nonzero_inverse_eq_imp_eq:

   assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"

   shows "a = b"

 proof -

   from `inverse a = inverse b`

   have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)

   with `a \<noteq> 0` and `b \<noteq> 0` show "a = b"

     by (simp add: nonzero_inverse_inverse_eq)

 qed

 lemma inverse_1 [simp]: "inverse 1 = 1"

 by (rule inverse_unique) simp

 lemma nonzero_inverse_mult_distrib: 

   assumes "a \<noteq> 0" and "b \<noteq> 0"

   shows "inverse (a * b) = inverse b * inverse a"

 proof -

   have "a * (b * inverse b) * inverse a = 1" using assms by simp

   hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc)

   thus ?thesis by (rule inverse_unique)

 qed

 lemma division_ring_inverse_add:

   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"

 by (simp add: algebra_simps)

 lemma division_ring_inverse_diff:

   "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"

 by (simp add: algebra_simps)

 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_assoc)

     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)

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

   by (simp add: divide_inverse)

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

   by (simp add: divide_inverse mult_assoc)

 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, no_atp]: "(-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 nonzero_eq_divide_eq [field_simps]: "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 [field_simps]: "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 add: divide_inverse mult_assoc)

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

   by (drule sym) (simp add: divide_inverse mult_assoc)

 end

 class division_ring_inverse_zero = division_ring +

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

 begin

 lemma divide_zero [simp]:

   "a / 0 = 0"

   by (simp add: divide_inverse)

 lemma divide_self_if [simp]:

   "a / a = (if a = 0 then 0 else 1)"

   by simp

 lemma inverse_nonzero_iff_nonzero [simp]:

   "inverse a = 0 \<longleftrightarrow> a = 0"

   by rule (fact inverse_zero_imp_zero, simp)

 lemma inverse_minus_eq [simp]:

   "inverse (- a) = - inverse a"

 proof cases

   assume "a=0" thus ?thesis by simp

 next

   assume "a\<noteq>0" 

   thus ?thesis by (simp add: nonzero_inverse_minus_eq)

 qed

 lemma inverse_inverse_eq [simp]:

   "inverse (inverse a) = 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

 lemma inverse_eq_imp_eq:

   "inverse a = inverse b \<Longrightarrow> a = b"

   by (drule arg_cong [where f="inverse"], simp)

 lemma inverse_eq_iff_eq [simp]:

   "inverse a = inverse b \<longleftrightarrow> a = b"

   by (force dest!: inverse_eq_imp_eq)

 end

 subsection {* Fields *}

 class field = comm_ring_1 + inverse +

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

   assumes field_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)

 next

   fix a b :: 'a

   show "a / b = a * inverse b" by (rule field_divide_inverse)

 qed

 subclass idom ..

 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, no_atp]:

 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, no_atp]:

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

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

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

   by (simp add: divide_inverse mult_ac)

 text{*It's not obvious whether @{text times_divide_eq} 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.*}

 lemmas times_divide_eq [no_atp] = 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, no_atp]:

   "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, no_atp]:

   "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, no_atp]:

   "\<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, no_atp]:

   "\<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, no_atp]:

   "\<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, no_atp]:

   "\<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 add_divide_eq_iff [field_simps]:

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

   by (simp add: add_divide_distrib)

 lemma divide_add_eq_iff [field_simps]:

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

   by (simp add: add_divide_distrib)

 lemma diff_divide_eq_iff [field_simps]:

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

   by (simp add: diff_divide_distrib)

 lemma divide_diff_eq_iff [field_simps]:

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

   by (simp add: diff_divide_distrib)

 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_simps)

 lemma frac_eq_eq:

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

   by (simp add: field_simps)

 end

 class field_inverse_zero = field +

   assumes field_inverse_zero: "inverse 0 = 0"

 begin

 subclass division_ring_inverse_zero proof

 qed (fact field_inverse_zero)

 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"

 proof cases

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

   thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac)

 next

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

   thus ?thesis by force

 qed

 lemma inverse_divide [simp]:

   "inverse (a / b) = b / a"

   by (simp add: divide_inverse mult_commute)

 text {* 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 \<Longrightarrow> (c * a) / (c * b) = a / b"

 apply (cases "b = 0")

 apply simp_all

 done

 lemma mult_divide_mult_cancel_right:

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

 apply (cases "b = 0")

 apply simp_all

 done

 lemma divide_divide_eq_right [simp, no_atp]:

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

   by (simp add: divide_inverse mult_ac)

 lemma divide_divide_eq_left [simp, no_atp]:

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

   by (simp add: divide_inverse mult_assoc)

 text {*Special Cancellation Simprules for Division*}

 lemma mult_divide_mult_cancel_left_if [simp,no_atp]:

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

   by (simp add: mult_divide_mult_cancel_left)

 text {* Division and Unary Minus *}

 lemma minus_divide_right:

   "- (a / b) = a / - b"

   by (simp add: divide_inverse)

 lemma divide_minus_right [simp, no_atp]:

   "a / - b = - (a / b)"

   by (simp add: divide_inverse)

 lemma minus_divide_divide:

   "(- a) / (- b) = a / b"

 apply (cases "b=0", simp) 

 apply (simp add: nonzero_minus_divide_divide) 

 done

 lemma eq_divide_eq:

   "a = b / c \<longleftrightarrow> (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 \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"

   by (force simp add: nonzero_divide_eq_eq)

 lemma inverse_eq_1_iff [simp]:

   "inverse x = 1 \<longleftrightarrow> x = 1"

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

 lemma divide_eq_0_iff [simp, no_atp]:

   "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"

   by (simp add: divide_inverse)

 lemma divide_cancel_right [simp, no_atp]:

   "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"

   apply (cases "c=0", simp)

   apply (simp add: divide_inverse)

   done

 lemma divide_cancel_left [simp, no_atp]:

   "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b" 

   apply (cases "c=0", simp)

   apply (simp add: divide_inverse)

   done

 lemma divide_eq_1_iff [simp, no_atp]:

   "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"

   apply (cases "b=0", simp)

   apply (simp add: right_inverse_eq)

   done

 lemma one_eq_divide_iff [simp, no_atp]:

   "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"

   by (simp add: eq_commute [of 1])

 lemma times_divide_times_eq:

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

   by simp

 lemma add_frac_num:

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

   by (simp add: add_divide_distrib)

 lemma add_num_frac:

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

   by (simp add: add_divide_distrib add.commute)

 end

 subsection {* Ordered fields *}

 class linordered_field = field + linordered_idom

 begin

 lemma positive_imp_inverse_positive: 

   assumes a_gt_0: "0 < a" 

   shows "0 < inverse a"

 proof -

   have "0 < a * inverse a" 

     by (simp add: a_gt_0 [THEN less_imp_not_eq2])

   thus "0 < inverse a" 

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

 qed

 lemma negative_imp_inverse_negative:

   "a < 0 \<Longrightarrow> inverse a < 0"

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

     simp add: nonzero_inverse_minus_eq less_imp_not_eq)

 lemma inverse_le_imp_le:

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

   shows "b \<le> a"

 proof (rule classical)

   assume "~ b \<le> a"

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

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

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

     by (simp add: apos invle less_imp_le mult_left_mono)

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

     by (simp add: bpos less_imp_le mult_right_mono)

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

 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_negative_imp_negative:

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

   shows "a < 0"

 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 linordered_field_no_lb:

   "\<forall>x. \<exists>y. y < x"

 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"

 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 less_imp_inverse_less:

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

   shows "inverse b < inverse a"

 proof (rule ccontr)

   assume "~ inverse b < inverse a"

   hence "inverse a \<le> inverse b" by simp

   hence "~ (a < b)"

     by (simp add: 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 \<Longrightarrow> 0 < a \<Longrightarrow> b < a"

 apply (simp add: less_le [of "inverse a"] 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,no_atp]:

   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"

   by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 

 lemma le_imp_inverse_le:

   "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"

   by (force simp add: le_less less_imp_inverse_less)

 lemma inverse_le_iff_le [simp,no_atp]:

   "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"

   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 \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"

 apply (rule classical) 

 apply (subgoal_tac "a < 0") 

  prefer 2 apply force

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

 apply (simp add: nonzero_inverse_minus_eq) 

 done

 lemma less_imp_inverse_less_neg:

    "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"

 apply (subgoal_tac "a < 0") 

  prefer 2 apply (blast intro: less_trans) 

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

 apply (simp add: nonzero_inverse_minus_eq) 

 done

 lemma inverse_less_imp_less_neg:

    "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"

 apply (rule classical) 

 apply (subgoal_tac "a < 0") 

  prefer 2

  apply force

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

 apply (simp add: nonzero_inverse_minus_eq) 

 done

 lemma inverse_less_iff_less_neg [simp,no_atp]:

   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"

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

 apply (simp del: inverse_less_iff_less 

             add: nonzero_inverse_minus_eq)

 done

 lemma le_imp_inverse_le_neg:

   "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"

   by (force simp add: le_less less_imp_inverse_less_neg)

 lemma inverse_le_iff_le_neg [simp,no_atp]:

   "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"

   by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 

 lemma one_less_inverse:

   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"

   using less_imp_inverse_less [of a 1, unfolded inverse_1] .

 lemma one_le_inverse:

   "0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"

   using le_imp_inverse_le [of a 1, unfolded inverse_1] .

 lemma pos_le_divide_eq [field_simps]: "0 < c ==> (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 less_not_sym [OF less] del: times_divide_eq)

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

     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma neg_le_divide_eq [field_simps]: "c < 0 ==> (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 less_not_sym [OF less] del: times_divide_eq)

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

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

   finally show ?thesis .

 qed

 lemma pos_less_divide_eq [field_simps]:

      "0 < c ==> (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 less_not_sym [OF less] del: times_divide_eq)

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

     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma neg_less_divide_eq [field_simps]:

  "c < 0 ==> (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 less_not_sym [OF less] del: times_divide_eq)

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

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

   finally show ?thesis .

 qed

 lemma pos_divide_less_eq [field_simps]:

      "0 < c ==> (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 less_not_sym [OF less] del: times_divide_eq)

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

     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma neg_divide_less_eq [field_simps]:

  "c < 0 ==> (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 less_not_sym [OF less] del: times_divide_eq)

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

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

   finally show ?thesis .

 qed

 lemma pos_divide_le_eq [field_simps]: "0 < c ==> (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 less_not_sym [OF less] del: times_divide_eq)

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

     by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma neg_divide_le_eq [field_simps]: "c < 0 ==> (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 less_not_sym [OF less] del: times_divide_eq)

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

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

   finally show ?thesis .

 qed

 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 [no_atp] = algebra_simps

   zero_less_mult_iff mult_less_0_iff

 lemmas (in -) sign_simps [no_atp] = algebra_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

 *)

 lemma divide_pos_pos:

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

 by(simp add:field_simps)

 lemma divide_nonneg_pos:

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

 by(simp add:field_simps)

 lemma divide_neg_pos:

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

 by(simp add:field_simps)

 lemma divide_nonpos_pos:

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

 by(simp add:field_simps)

 lemma divide_pos_neg:

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

 by(simp add:field_simps)

 lemma divide_nonneg_neg:

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

 by(simp add:field_simps)

 lemma divide_neg_neg:

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

 by(simp add:field_simps)

 lemma divide_nonpos_neg:

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

 by(simp add:field_simps)

 lemma divide_strict_right_mono:

      "[|a < b; 0 < c|] ==> a / c < b / c"

 by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 

               positive_imp_inverse_positive)

 lemma divide_strict_right_mono_neg:

      "[|b < a; c < 0|] ==> a / c < b / c"

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

 apply (simp add: 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"

   by (auto simp: field_simps 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"

   by (auto simp: field_simps zero_less_mult_iff mult_right_mono)

 lemma divide_strict_left_mono_neg:

   "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"

   by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg)

 lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>

     x / y <= z"

 by (subst pos_divide_le_eq, assumption+)

 lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>

     z <= x / y"

 by(simp add:field_simps)

 lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>

     x / y < z"

 by(simp add:field_simps)

 lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>

     z < x / y"

 by(simp add:field_simps)

 lemma frac_le: "0 <= 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 <= 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 < x ==> 

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

   apply (rule mult_imp_div_pos_less)

   apply simp_all

   apply (rule mult_imp_less_div_pos, assumption)

   apply (erule mult_le_less_imp_less)

   apply simp_all

 done

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

 by (simp add: field_simps zero_less_two)

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

 by (simp add: field_simps zero_less_two)

 subclass dense_linorder

 proof

   fix x y :: 'a

   from less_add_one show "\<exists>y. x < y" .. 

   from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)

   then have "x - 1 < x + 1 - 1" by (simp only: diff_minus [symmetric])

   then have "x - 1 < x" by (simp add: algebra_simps)

   then show "\<exists>y. y < x" ..

   show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)

 qed

 lemma nonzero_abs_inverse:

      "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"

 apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 

                       negative_imp_inverse_negative)

 apply (blast intro: positive_imp_inverse_positive elim: less_asym) 

 done

 lemma nonzero_abs_divide:

      "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"

   by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 

 lemma field_le_epsilon:

   assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"

   shows "x \<le> y"

 proof (rule dense_le)

   fix t assume "t < x"

   hence "0 < x - t" by (simp add: less_diff_eq)

   from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)

   then have "0 \<le> y - t" by (simp only: add_le_cancel_left)

   then show "t \<le> y" by (simp add: algebra_simps)

 qed

 end

 class linordered_field_inverse_zero = linordered_field + field_inverse_zero

 begin

 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)"

 apply (cases "c=0", simp) 

 apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 

 done

 lemma inverse_positive_iff_positive [simp]:

   "(0 < inverse a) = (0 < a)"

 apply (cases "a = 0", simp)

 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)

 done

 lemma inverse_negative_iff_negative [simp]:

   "(inverse a < 0) = (a < 0)"

 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 \<longleftrightarrow> 0 \<le> a"

   by (simp add: not_less [symmetric])

 lemma inverse_nonpositive_iff_nonpositive [simp]:

   "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"

   by (simp add: not_less [symmetric])

 lemma one_less_inverse_iff:

   "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"

 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

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

     finally show False by auto

   qed

   with notless show ?thesis by simp

 qed

 lemma one_le_inverse_iff:

   "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"

 proof (cases "x = 1")

   case True then show ?thesis by simp

 next

   case False then have "inverse x \<noteq> 1" by simp

   then have "1 \<noteq> inverse x" by blast

   then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)

   with False show ?thesis by (auto simp add: one_less_inverse_iff)

 qed

 lemma inverse_less_1_iff:

   "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"

   by (simp add: not_le [symmetric] one_le_inverse_iff) 

 lemma inverse_le_1_iff:

   "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"

   by (simp add: not_less [symmetric] one_less_inverse_iff) 

 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)"

 apply (cases "c=0", simp) 

 apply (force simp add: pos_divide_le_eq neg_divide_le_eq) 

 done

 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)"

 apply (cases "c=0", simp) 

 apply (force simp add: pos_less_divide_eq neg_less_divide_eq) 

 done

 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)"

 apply (cases "c=0", simp) 

 apply (force simp add: pos_divide_less_eq neg_divide_less_eq)

 done

 text {*Division and Signs*}

 lemma zero_less_divide_iff:

      "(0 < 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) = 

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

 by (simp add: divide_inverse mult_less_0_iff)

 lemma zero_le_divide_iff:

      "(0 \<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) =

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

 by (simp add: divide_inverse mult_le_0_iff)

 text {* Division and the Number One *}

 text{*Simplify expressions equated with 1*}

 lemma zero_eq_1_divide_iff [simp,no_atp]:

      "(0 = 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,no_atp]:

      "(1/a = 0) = (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"}*}

 lemma zero_le_divide_1_iff [simp, no_atp]:

   "0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"

   by (simp add: zero_le_divide_iff)

 lemma zero_less_divide_1_iff [simp, no_atp]:

   "0 < 1 / a \<longleftrightarrow> 0 < a"

   by (simp add: zero_less_divide_iff)

 lemma divide_le_0_1_iff [simp, no_atp]:

   "1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"

   by (simp add: divide_le_0_iff)

 lemma divide_less_0_1_iff [simp, no_atp]:

   "1 / a < 0 \<longleftrightarrow> a < 0"

   by (simp add: divide_less_0_iff)

 lemma divide_right_mono:

      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"

 by (force simp add: divide_strict_right_mono le_less)

 lemma divide_right_mono_neg: "a <= b 

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

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

 apply auto

 done

 lemma divide_left_mono_neg: "a <= 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 inverse_le_iff:

   "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"

 proof -

   { assume "a < 0"

     then have "inverse a < 0" by simp

     moreover assume "0 < b"

     then have "0 < inverse b" by simp

     ultimately have "inverse a < inverse b" by (rule less_trans)

     then have "inverse a \<le> inverse b" by simp }

   moreover

   { assume "b < 0"

     then have "inverse b < 0" by simp

     moreover assume "0 < a"

     then have "0 < inverse a" by simp

     ultimately have "inverse b < inverse a" by (rule less_trans)

     then have "\<not> inverse a \<le> inverse b" by simp }

   ultimately show ?thesis

     by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])

        (auto simp: not_less zero_less_mult_iff mult_le_0_iff)

 qed

 lemma inverse_less_iff:

   "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"

   by (subst less_le) (auto simp: inverse_le_iff)

 lemma divide_le_cancel:

   "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"

   by (simp add: divide_inverse mult_le_cancel_right)

 lemma divide_less_cancel:

   "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"

   by (auto simp add: divide_inverse mult_less_cancel_right)

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

 lemma le_divide_eq_1 [no_atp]:

   "(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 [no_atp]:

   "(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 [no_atp]:

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

 by (auto simp add: less_divide_eq)

 lemma divide_less_eq_1 [no_atp]:

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

 by (auto simp add: divide_less_eq)

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

 lemma le_divide_eq_1_pos [simp,no_atp]:

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

 by (auto simp add: le_divide_eq)

 lemma le_divide_eq_1_neg [simp,no_atp]:

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

 by (auto simp add: le_divide_eq)

 lemma divide_le_eq_1_pos [simp,no_atp]:

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

 by (auto simp add: divide_le_eq)

 lemma divide_le_eq_1_neg [simp,no_atp]:

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

 by (auto simp add: divide_le_eq)

 lemma less_divide_eq_1_pos [simp,no_atp]:

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

 by (auto simp add: less_divide_eq)

 lemma less_divide_eq_1_neg [simp,no_atp]:

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

 by (auto simp add: less_divide_eq)

 lemma divide_less_eq_1_pos [simp,no_atp]:

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

 by (auto simp add: divide_less_eq)

 lemma divide_less_eq_1_neg [simp,no_atp]:

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

 by (auto simp add: divide_less_eq)

 lemma eq_divide_eq_1 [simp,no_atp]:

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

 by (auto simp add: eq_divide_eq)

 lemma divide_eq_eq_1 [simp,no_atp]:

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

 by (auto simp add: divide_eq_eq)

 lemma abs_inverse [simp]:

      "\<bar>inverse a\<bar> = 

       inverse \<bar>a\<bar>"

 apply (cases "a=0", simp) 

 apply (simp add: nonzero_abs_inverse) 

 done

 lemma abs_divide [simp]:

      "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"

 apply (cases "b=0", simp) 

 apply (simp add: nonzero_abs_divide) 

 done

 lemma abs_div_pos: "0 < y ==> 

     \<bar>x\<bar> / y = \<bar>x / y\<bar>"

   apply (subst abs_divide)

   apply (simp add: order_less_imp_le)

 done

 lemma field_le_mult_one_interval:

   assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"

   shows "x \<le> y"

 proof (cases "0 < x")

   assume "0 < x"

   thus ?thesis

     using dense_le_bounded[of 0 1 "y/x"] *

     unfolding le_divide_eq if_P[OF `0 < x`] by simp

 next

   assume "\<not>0 < x" hence "x \<le> 0" by simp

   obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto

   hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto

   also note *[OF s]

   finally show ?thesis .

 qed

 end

 code_modulename SML

   Fields Arith

 code_modulename OCaml

   Fields Arith

 code_modulename Haskell

   Fields Arith

 end