(*  Title:  HOL/Rational.thy

     Author: Markus Wenzel, TU Muenchen

 *)

 header {* Rational numbers *}

 theory Rational

 imports GCD Archimedean_Field

 begin

 subsection {* Rational numbers as quotient *}

 subsubsection {* Construction of the type of rational numbers *}

 definition

   ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where

   "ratrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"

 lemma ratrel_iff [simp]:

   "(x, y) \<in> ratrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"

   by (simp add: ratrel_def)

 lemma refl_on_ratrel: "refl_on {x. snd x \<noteq> 0} ratrel"

   by (auto simp add: refl_on_def ratrel_def)

 lemma sym_ratrel: "sym ratrel"

   by (simp add: ratrel_def sym_def)

 lemma trans_ratrel: "trans ratrel"

 proof (rule transI, unfold split_paired_all)

   fix a b a' b' a'' b'' :: int

   assume A: "((a, b), (a', b')) \<in> ratrel"

   assume B: "((a', b'), (a'', b'')) \<in> ratrel"

   have "b' * (a * b'') = b'' * (a * b')" by simp

   also from A have "a * b' = a' * b" by auto

   also have "b'' * (a' * b) = b * (a' * b'')" by simp

   also from B have "a' * b'' = a'' * b'" by auto

   also have "b * (a'' * b') = b' * (a'' * b)" by simp

   finally have "b' * (a * b'') = b' * (a'' * b)" .

   moreover from B have "b' \<noteq> 0" by auto

   ultimately have "a * b'' = a'' * b" by simp

   with A B show "((a, b), (a'', b'')) \<in> ratrel" by auto

 qed

 lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel"

   by (rule equiv.intro [OF refl_on_ratrel sym_ratrel trans_ratrel])

 lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]

 lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]

 lemma equiv_ratrel_iff [iff]: 

   assumes "snd x \<noteq> 0" and "snd y \<noteq> 0"

   shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel"

   by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms)

 typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel"

 proof

   have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp

   then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // ratrel" by (rule quotientI)

 qed

 lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel `` {x} \<in> Rat"

   by (simp add: Rat_def quotientI)

 declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp]

 subsubsection {* Representation and basic operations *}

 definition

   Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where

   [code del]: "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"

 code_datatype Fract

 lemma Rat_cases [case_names Fract, cases type: rat]:

   assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C"

   shows C

   using assms by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)

 lemma Rat_induct [case_names Fract, induct type: rat]:

   assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)"

   shows "P q"

   using assms by (cases q) simp

 lemma eq_rat:

   shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b"

   and "\<And>a. Fract a 0 = Fract 0 1"

   and "\<And>a c. Fract 0 a = Fract 0 c"

   by (simp_all add: Fract_def)

 instantiation rat :: comm_ring_1

 begin

 definition

   Zero_rat_def [code, code unfold]: "0 = Fract 0 1"

 definition

   One_rat_def [code, code unfold]: "1 = Fract 1 1"

 definition

   add_rat_def [code del]:

   "q + r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.

     ratrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"

 lemma add_rat [simp]:

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

   shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"

 proof -

   have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})

     respects2 ratrel"

   by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib)

   with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2)

 qed

 definition

   minus_rat_def [code del]:

   "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"

 lemma minus_rat [simp, code]: "- Fract a b = Fract (- a) b"

 proof -

   have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel"

     by (simp add: congruent_def)

   then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel)

 qed

 lemma minus_rat_cancel [simp]: "Fract (- a) (- b) = Fract a b"

   by (cases "b = 0") (simp_all add: eq_rat)

 definition

   diff_rat_def [code del]: "q - r = q + - (r::rat)"

 lemma diff_rat [simp]:

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

   shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"

   using assms by (simp add: diff_rat_def)

 definition

   mult_rat_def [code del]:

   "q * r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.

     ratrel``{(fst x * fst y, snd x * snd y)})"

 lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"

 proof -

   have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel"

     by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all

   then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2)

 qed

 lemma mult_rat_cancel:

   assumes "c \<noteq> 0"

   shows "Fract (c * a) (c * b) = Fract a b"

 proof -

   from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)

   then show ?thesis by (simp add: mult_rat [symmetric])

 qed

 instance proof

   fix q r s :: rat show "(q * r) * s = q * (r * s)" 

     by (cases q, cases r, cases s) (simp add: eq_rat)

 next

   fix q r :: rat show "q * r = r * q"

     by (cases q, cases r) (simp add: eq_rat)

 next

   fix q :: rat show "1 * q = q"

     by (cases q) (simp add: One_rat_def eq_rat)

 next

   fix q r s :: rat show "(q + r) + s = q + (r + s)"

     by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)

 next

   fix q r :: rat show "q + r = r + q"

     by (cases q, cases r) (simp add: eq_rat)

 next

   fix q :: rat show "0 + q = q"

     by (cases q) (simp add: Zero_rat_def eq_rat)

 next

   fix q :: rat show "- q + q = 0"

     by (cases q) (simp add: Zero_rat_def eq_rat)

 next

   fix q r :: rat show "q - r = q + - r"

     by (cases q, cases r) (simp add: eq_rat)

 next

   fix q r s :: rat show "(q + r) * s = q * s + r * s"

     by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)

 next

   show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)

 qed

 end

 lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"

   by (induct k) (simp_all add: Zero_rat_def One_rat_def)

 lemma of_int_rat: "of_int k = Fract k 1"

   by (cases k rule: int_diff_cases) (simp add: of_nat_rat)

 lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"

   by (rule of_nat_rat [symmetric])

 lemma Fract_of_int_eq: "Fract k 1 = of_int k"

   by (rule of_int_rat [symmetric])

 instantiation rat :: number_ring

 begin

 definition

   rat_number_of_def [code del]: "number_of w = Fract w 1"

 instance proof

 qed (simp add: rat_number_of_def of_int_rat)

 end

 lemma rat_number_collapse [code post]:

   "Fract 0 k = 0"

   "Fract 1 1 = 1"

   "Fract (number_of k) 1 = number_of k"

   "Fract k 0 = 0"

   by (cases "k = 0")

     (simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def)

 lemma rat_number_expand [code unfold]:

   "0 = Fract 0 1"

   "1 = Fract 1 1"

   "number_of k = Fract (number_of k) 1"

   by (simp_all add: rat_number_collapse)

 lemma iszero_rat [simp]:

   "iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)"

   by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat)

 lemma Rat_cases_nonzero [case_names Fract 0]:

   assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C"

   assumes 0: "q = 0 \<Longrightarrow> C"

   shows C

 proof (cases "q = 0")

   case True then show C using 0 by auto

 next

   case False

   then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto

   moreover with False have "0 \<noteq> Fract a b" by simp

   with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)

   with Fract `q = Fract a b` `b \<noteq> 0` show C by auto

 qed

 subsubsection {* The field of rational numbers *}

 instantiation rat :: "{field, division_by_zero}"

 begin

 definition

   inverse_rat_def [code del]:

   "inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q.

      ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})"

 lemma inverse_rat [simp]: "inverse (Fract a b) = Fract b a"

 proof -

   have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel"

     by (auto simp add: congruent_def mult_commute)

   then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel)

 qed

 definition

   divide_rat_def [code del]: "q / r = q * inverse (r::rat)"

 lemma divide_rat [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"

   by (simp add: divide_rat_def)

 instance proof

   show "inverse 0 = (0::rat)" by (simp add: rat_number_expand)

     (simp add: rat_number_collapse)

 next

   fix q :: rat

   assume "q \<noteq> 0"

   then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero)

    (simp_all add: mult_rat  inverse_rat rat_number_expand eq_rat)

 next

   fix q r :: rat

   show "q / r = q * inverse r" by (simp add: divide_rat_def)

 qed

 end

 subsubsection {* Various *}

 lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1"

   by (simp add: rat_number_expand)

 lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l"

   by (simp add: Fract_of_int_eq [symmetric])

 lemma Fract_number_of_quotient [code post]:

   "Fract (number_of k) (number_of l) = number_of k / number_of l"

   unfolding Fract_of_int_quotient number_of_is_id number_of_eq ..

 lemma Fract_1_number_of [code post]:

   "Fract 1 (number_of k) = 1 / number_of k"

   unfolding Fract_of_int_quotient number_of_eq by simp

 subsubsection {* The ordered field of rational numbers *}

 instantiation rat :: linorder

 begin

 definition

   le_rat_def [code del]:

    "q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.

       {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})"

 lemma le_rat [simp]:

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

   shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"

 proof -

   have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})

     respects2 ratrel"

   proof (clarsimp simp add: congruent2_def)

     fix a b a' b' c d c' d'::int

     assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"

     assume eq1: "a * b' = a' * b"

     assume eq2: "c * d' = c' * d"

     let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"

     {

       fix a b c d x :: int assume x: "x \<noteq> 0"

       have "?le a b c d = ?le (a * x) (b * x) c d"

       proof -

         from x have "0 < x * x" by (auto simp add: zero_less_mult_iff)

         hence "?le a b c d =

             ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"

           by (simp add: mult_le_cancel_right)

         also have "... = ?le (a * x) (b * x) c d"

           by (simp add: mult_ac)

         finally show ?thesis .

       qed

     } note le_factor = this

     let ?D = "b * d" and ?D' = "b' * d'"

     from neq have D: "?D \<noteq> 0" by simp

     from neq have "?D' \<noteq> 0" by simp

     hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"

       by (rule le_factor)

     also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" 

       by (simp add: mult_ac)

     also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"

       by (simp only: eq1 eq2)

     also have "... = ?le (a' * ?D) (b' * ?D) c' d'"

       by (simp add: mult_ac)

     also from D have "... = ?le a' b' c' d'"

       by (rule le_factor [symmetric])

     finally show "?le a b c d = ?le a' b' c' d'" .

   qed

   with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2)

 qed

 definition

   less_rat_def [code del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"

 lemma less_rat [simp]:

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

   shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"

   using assms by (simp add: less_rat_def eq_rat order_less_le)

 instance proof

   fix q r s :: rat

   {

     assume "q \<le> r" and "r \<le> s"

     show "q \<le> s"

     proof (insert prems, induct q, induct r, induct s)

       fix a b c d e f :: int

       assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"

       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"

       show "Fract a b \<le> Fract e f"

       proof -

         from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"

           by (auto simp add: zero_less_mult_iff linorder_neq_iff)

         have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"

         proof -

           from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"

             by simp

           with ff show ?thesis by (simp add: mult_le_cancel_right)

         qed

         also have "... = (c * f) * (d * f) * (b * b)" by algebra

         also have "... \<le> (e * d) * (d * f) * (b * b)"

         proof -

           from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"

             by simp

           with bb show ?thesis by (simp add: mult_le_cancel_right)

         qed

         finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"

           by (simp only: mult_ac)

         with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"

           by (simp add: mult_le_cancel_right)

         with neq show ?thesis by simp

       qed

     qed

   next

     assume "q \<le> r" and "r \<le> q"

     show "q = r"

     proof (insert prems, induct q, induct r)

       fix a b c d :: int

       assume neq: "b \<noteq> 0"  "d \<noteq> 0"

       assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"

       show "Fract a b = Fract c d"

       proof -

         from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"

           by simp

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

         proof -

           from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"

             by simp

           thus ?thesis by (simp only: mult_ac)

         qed

         finally have "(a * d) * (b * d) = (c * b) * (b * d)" .

         moreover from neq have "b * d \<noteq> 0" by simp

         ultimately have "a * d = c * b" by simp

         with neq show ?thesis by (simp add: eq_rat)

       qed

     qed

   next

     show "q \<le> q"

       by (induct q) simp

     show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"

       by (induct q, induct r) (auto simp add: le_less mult_commute)

     show "q \<le> r \<or> r \<le> q"

       by (induct q, induct r)

          (simp add: mult_commute, rule linorder_linear)

   }

 qed

 end

 instantiation rat :: "{distrib_lattice, abs_if, sgn_if}"

 begin

 definition

   abs_rat_def [code del]: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))"

 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"

   by (auto simp add: abs_rat_def zabs_def Zero_rat_def less_rat not_less le_less minus_rat eq_rat zero_compare_simps)

 definition

   sgn_rat_def [code del]: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"

 lemma sgn_rat [simp, code]: "sgn (Fract a b) = of_int (sgn a * sgn b)"

   unfolding Fract_of_int_eq

   by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat)

     (auto simp: rat_number_collapse not_less le_less zero_less_mult_iff)

 definition

   "(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min"

 definition

   "(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max"

 instance by intro_classes

   (auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)

 end

 instance rat :: ordered_field

 proof

   fix q r s :: rat

   show "q \<le> r ==> s + q \<le> s + r"

   proof (induct q, induct r, induct s)

     fix a b c d e f :: int

     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"

     assume le: "Fract a b \<le> Fract c d"

     show "Fract e f + Fract a b \<le> Fract e f + Fract c d"

     proof -

       let ?F = "f * f" from neq have F: "0 < ?F"

         by (auto simp add: zero_less_mult_iff)

       from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"

         by simp

       with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"

         by (simp add: mult_le_cancel_right)

       with neq show ?thesis by (simp add: mult_ac int_distrib)

     qed

   qed

   show "q < r ==> 0 < s ==> s * q < s * r"

   proof (induct q, induct r, induct s)

     fix a b c d e f :: int

     assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"

     assume le: "Fract a b < Fract c d"

     assume gt: "0 < Fract e f"

     show "Fract e f * Fract a b < Fract e f * Fract c d"

     proof -

       let ?E = "e * f" and ?F = "f * f"

       from neq gt have "0 < ?E"

         by (auto simp add: Zero_rat_def order_less_le eq_rat)

       moreover from neq have "0 < ?F"

         by (auto simp add: zero_less_mult_iff)

       moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"

         by simp

       ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"

         by (simp add: mult_less_cancel_right)

       with neq show ?thesis

         by (simp add: mult_ac)

     qed

   qed

 qed auto

 lemma Rat_induct_pos [case_names Fract, induct type: rat]:

   assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)"

   shows "P q"

 proof (cases q)

   have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)"

   proof -

     fix a::int and b::int

     assume b: "b < 0"

     hence "0 < -b" by simp

     hence "P (Fract (-a) (-b))" by (rule step)

     thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])

   qed

   case (Fract a b)

   thus "P q" by (force simp add: linorder_neq_iff step step')

 qed

 lemma zero_less_Fract_iff:

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

   by (simp add: Zero_rat_def zero_less_mult_iff)

 lemma Fract_less_zero_iff:

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

   by (simp add: Zero_rat_def mult_less_0_iff)

 lemma zero_le_Fract_iff:

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

   by (simp add: Zero_rat_def zero_le_mult_iff)

 lemma Fract_le_zero_iff:

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

   by (simp add: Zero_rat_def mult_le_0_iff)

 lemma one_less_Fract_iff:

   "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a"

   by (simp add: One_rat_def mult_less_cancel_right_disj)

 lemma Fract_less_one_iff:

   "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b"

   by (simp add: One_rat_def mult_less_cancel_right_disj)

 lemma one_le_Fract_iff:

   "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a"

   by (simp add: One_rat_def mult_le_cancel_right)

 lemma Fract_le_one_iff:

   "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b"

   by (simp add: One_rat_def mult_le_cancel_right)

 subsubsection {* Rationals are an Archimedean field *}

 lemma rat_floor_lemma:

   assumes "0 < b"

   shows "of_int (a div b) \<le> Fract a b \<and> Fract a b < of_int (a div b + 1)"

 proof -

   have "Fract a b = of_int (a div b) + Fract (a mod b) b"

     using `0 < b` by (simp add: of_int_rat)

   moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"

     using `0 < b` by (simp add: zero_le_Fract_iff Fract_less_one_iff)

   ultimately show ?thesis by simp

 qed

 instance rat :: archimedean_field

 proof

   fix r :: rat

   show "\<exists>z. r \<le> of_int z"

   proof (induct r)

     case (Fract a b)

     then have "Fract a b \<le> of_int (a div b + 1)"

       using rat_floor_lemma [of b a] by simp

     then show "\<exists>z. Fract a b \<le> of_int z" ..

   qed

 qed

 lemma floor_Fract:

   assumes "0 < b" shows "floor (Fract a b) = a div b"

   using rat_floor_lemma [OF `0 < b`, of a]

   by (simp add: floor_unique)

 subsection {* Linear arithmetic setup *}

 declaration {*

   K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]

     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)

   #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_int_eq_iff} RS iffD2]

     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)

   #> Lin_Arith.add_simps [@{thm neg_less_iff_less},

       @{thm True_implies_equals},

       read_instantiate @{context} [(("a", 0), "(number_of ?v)")] @{thm right_distrib},

       @{thm divide_1}, @{thm divide_zero_left},

       @{thm times_divide_eq_right}, @{thm times_divide_eq_left},

       @{thm minus_divide_left} RS sym, @{thm minus_divide_right} RS sym,

       @{thm of_int_minus}, @{thm of_int_diff},

       @{thm of_int_of_nat_eq}]

   #> Lin_Arith.add_simprocs Numeral_Simprocs.field_cancel_numeral_factors

   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat => rat"})

   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int => rat"}))

 *}

 subsection {* Embedding from Rationals to other Fields *}

 class field_char_0 = field + ring_char_0

 subclass (in ordered_field) field_char_0 ..

 context field_char_0

 begin

 definition of_rat :: "rat \<Rightarrow> 'a" where

   [code del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})"

 end

 lemma of_rat_congruent:

   "(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel"

 apply (rule congruent.intro)

 apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)

 apply (simp only: of_int_mult [symmetric])

 done

 lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b"

   unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent)

 lemma of_rat_0 [simp]: "of_rat 0 = 0"

 by (simp add: Zero_rat_def of_rat_rat)

 lemma of_rat_1 [simp]: "of_rat 1 = 1"

 by (simp add: One_rat_def of_rat_rat)

 lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b"

 by (induct a, induct b, simp add: of_rat_rat add_frac_eq)

 lemma of_rat_minus: "of_rat (- a) = - of_rat a"

 by (induct a, simp add: of_rat_rat)

 lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b"

 by (simp only: diff_minus of_rat_add of_rat_minus)

 lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b"

 apply (induct a, induct b, simp add: of_rat_rat)

 apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac)

 done

 lemma nonzero_of_rat_inverse:

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

 apply (rule inverse_unique [symmetric])

 apply (simp add: of_rat_mult [symmetric])

 done

 lemma of_rat_inverse:

   "(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) =

    inverse (of_rat a)"

 by (cases "a = 0", simp_all add: nonzero_of_rat_inverse)

 lemma nonzero_of_rat_divide:

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

 by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse)

 lemma of_rat_divide:

   "(of_rat (a / b)::'a::{field_char_0,division_by_zero})

    = of_rat a / of_rat b"

 by (cases "b = 0") (simp_all add: nonzero_of_rat_divide)

 lemma of_rat_power:

   "(of_rat (a ^ n)::'a::field_char_0) = of_rat a ^ n"

 by (induct n) (simp_all add: of_rat_mult)

 lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)"

 apply (induct a, induct b)

 apply (simp add: of_rat_rat eq_rat)

 apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)

 apply (simp only: of_int_mult [symmetric] of_int_eq_iff)

 done

 lemma of_rat_less:

   "(of_rat r :: 'a::ordered_field) < of_rat s \<longleftrightarrow> r < s"

 proof (induct r, induct s)

   fix a b c d :: int

   assume not_zero: "b > 0" "d > 0"

   then have "b * d > 0" by (rule mult_pos_pos)

   have of_int_divide_less_eq:

     "(of_int a :: 'a) / of_int b < of_int c / of_int d

       \<longleftrightarrow> (of_int a :: 'a) * of_int d < of_int c * of_int b"

     using not_zero by (simp add: pos_less_divide_eq pos_divide_less_eq)

   show "(of_rat (Fract a b) :: 'a::ordered_field) < of_rat (Fract c d)

     \<longleftrightarrow> Fract a b < Fract c d"

     using not_zero `b * d > 0`

     by (simp add: of_rat_rat of_int_divide_less_eq of_int_mult [symmetric] del: of_int_mult)

 qed

 lemma of_rat_less_eq:

   "(of_rat r :: 'a::ordered_field) \<le> of_rat s \<longleftrightarrow> r \<le> s"

   unfolding le_less by (auto simp add: of_rat_less)

 lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified]

 lemma of_rat_eq_id [simp]: "of_rat = id"

 proof

   fix a

   show "of_rat a = id a"

   by (induct a)

      (simp add: of_rat_rat Fract_of_int_eq [symmetric])

 qed

 text{*Collapse nested embeddings*}

 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"

 by (induct n) (simp_all add: of_rat_add)

 lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z"

 by (cases z rule: int_diff_cases) (simp add: of_rat_diff)

 lemma of_rat_number_of_eq [simp]:

   "of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})"

 by (simp add: number_of_eq)

 lemmas zero_rat = Zero_rat_def

 lemmas one_rat = One_rat_def

 abbreviation

   rat_of_nat :: "nat \<Rightarrow> rat"

 where

   "rat_of_nat \<equiv> of_nat"

 abbreviation

   rat_of_int :: "int \<Rightarrow> rat"

 where

   "rat_of_int \<equiv> of_int"

 subsection {* The Set of Rational Numbers *}

 context field_char_0

 begin

 definition

   Rats  :: "'a set" where

   [code del]: "Rats = range of_rat"

 notation (xsymbols)

   Rats  ("\<rat>")

 end

 lemma Rats_of_rat [simp]: "of_rat r \<in> Rats"

 by (simp add: Rats_def)

 lemma Rats_of_int [simp]: "of_int z \<in> Rats"

 by (subst of_rat_of_int_eq [symmetric], rule Rats_of_rat)

 lemma Rats_of_nat [simp]: "of_nat n \<in> Rats"

 by (subst of_rat_of_nat_eq [symmetric], rule Rats_of_rat)

 lemma Rats_number_of [simp]:

   "(number_of w::'a::{number_ring,field_char_0}) \<in> Rats"

 by (subst of_rat_number_of_eq [symmetric], rule Rats_of_rat)

 lemma Rats_0 [simp]: "0 \<in> Rats"

 apply (unfold Rats_def)

 apply (rule range_eqI)

 apply (rule of_rat_0 [symmetric])

 done

 lemma Rats_1 [simp]: "1 \<in> Rats"

 apply (unfold Rats_def)

 apply (rule range_eqI)

 apply (rule of_rat_1 [symmetric])

 done

 lemma Rats_add [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a + b \<in> Rats"

 apply (auto simp add: Rats_def)

 apply (rule range_eqI)

 apply (rule of_rat_add [symmetric])

 done

 lemma Rats_minus [simp]: "a \<in> Rats \<Longrightarrow> - a \<in> Rats"

 apply (auto simp add: Rats_def)

 apply (rule range_eqI)

 apply (rule of_rat_minus [symmetric])

 done

 lemma Rats_diff [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a - b \<in> Rats"

 apply (auto simp add: Rats_def)

 apply (rule range_eqI)

 apply (rule of_rat_diff [symmetric])

 done

 lemma Rats_mult [simp]: "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a * b \<in> Rats"

 apply (auto simp add: Rats_def)

 apply (rule range_eqI)

 apply (rule of_rat_mult [symmetric])

 done

 lemma nonzero_Rats_inverse:

   fixes a :: "'a::field_char_0"

   shows "\<lbrakk>a \<in> Rats; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Rats"

 apply (auto simp add: Rats_def)

 apply (rule range_eqI)

 apply (erule nonzero_of_rat_inverse [symmetric])

 done

 lemma Rats_inverse [simp]:

   fixes a :: "'a::{field_char_0,division_by_zero}"

   shows "a \<in> Rats \<Longrightarrow> inverse a \<in> Rats"

 apply (auto simp add: Rats_def)

 apply (rule range_eqI)

 apply (rule of_rat_inverse [symmetric])

 done

 lemma nonzero_Rats_divide:

   fixes a b :: "'a::field_char_0"

   shows "\<lbrakk>a \<in> Rats; b \<in> Rats; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Rats"

 apply (auto simp add: Rats_def)

 apply (rule range_eqI)

 apply (erule nonzero_of_rat_divide [symmetric])

 done

 lemma Rats_divide [simp]:

   fixes a b :: "'a::{field_char_0,division_by_zero}"

   shows "\<lbrakk>a \<in> Rats; b \<in> Rats\<rbrakk> \<Longrightarrow> a / b \<in> Rats"

 apply (auto simp add: Rats_def)

 apply (rule range_eqI)

 apply (rule of_rat_divide [symmetric])

 done

 lemma Rats_power [simp]:

   fixes a :: "'a::field_char_0"

   shows "a \<in> Rats \<Longrightarrow> a ^ n \<in> Rats"

 apply (auto simp add: Rats_def)

 apply (rule range_eqI)

 apply (rule of_rat_power [symmetric])

 done

 lemma Rats_cases [cases set: Rats]:

   assumes "q \<in> \<rat>"

   obtains (of_rat) r where "q = of_rat r"

   unfolding Rats_def

 proof -

   from `q \<in> \<rat>` have "q \<in> range of_rat" unfolding Rats_def .

   then obtain r where "q = of_rat r" ..

   then show thesis ..

 qed

 lemma Rats_induct [case_names of_rat, induct set: Rats]:

   "q \<in> \<rat> \<Longrightarrow> (\<And>r. P (of_rat r)) \<Longrightarrow> P q"

   by (rule Rats_cases) auto

 subsection {* Implementation of rational numbers as pairs of integers *}

 lemma Fract_norm: "Fract (a div zgcd a b) (b div zgcd a b) = Fract a b"

 proof (cases "a = 0 \<or> b = 0")

   case True then show ?thesis by (auto simp add: eq_rat)

 next

   let ?c = "zgcd a b"

   case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto

   then have "?c \<noteq> 0" by simp

   then have "Fract ?c ?c = Fract 1 1" by (simp add: eq_rat)

   moreover have "Fract (a div ?c * ?c + a mod ?c) (b div ?c * ?c + b mod ?c) = Fract a b"

     by (simp add: semiring_div_class.mod_div_equality)

   moreover have "a mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])

   moreover have "b mod ?c = 0" by (simp add: dvd_eq_mod_eq_0 [symmetric])

   ultimately show ?thesis

     by (simp add: mult_rat [symmetric])

 qed

 definition Fract_norm :: "int \<Rightarrow> int \<Rightarrow> rat" where

   [simp, code del]: "Fract_norm a b = Fract a b"

 lemma Fract_norm_code [code]: "Fract_norm a b = (if a = 0 \<or> b = 0 then 0 else let c = zgcd a b in

   if b > 0 then Fract (a div c) (b div c) else Fract (- (a div c)) (- (b div c)))"

   by (simp add: eq_rat Zero_rat_def Let_def Fract_norm)

 lemma [code]:

   "of_rat (Fract a b) = (if b \<noteq> 0 then of_int a / of_int b else 0)"

   by (cases "b = 0") (simp_all add: rat_number_collapse of_rat_rat)

 instantiation rat :: eq

 begin

 definition [code del]: "eq_class.eq (a\<Colon>rat) b \<longleftrightarrow> a - b = 0"

 instance by default (simp add: eq_rat_def)

 lemma rat_eq_code [code]:

   "eq_class.eq (Fract a b) (Fract c d) \<longleftrightarrow> (if b = 0

        then c = 0 \<or> d = 0

      else if d = 0

        then a = 0 \<or> b = 0

      else a * d = b * c)"

   by (auto simp add: eq eq_rat)

 lemma rat_eq_refl [code nbe]:

   "eq_class.eq (r::rat) r \<longleftrightarrow> True"

   by (rule HOL.eq_refl)

 end

 lemma le_rat':

   assumes "b \<noteq> 0"

     and "d \<noteq> 0"

   shows "Fract a b \<le> Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"

 proof -

   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp

   have "a * d * (b * d) \<le> c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) \<le> c * b * (sgn b * sgn d)"

   proof (cases "b * d > 0")

     case True

     moreover from True have "sgn b * sgn d = 1"

       by (simp add: sgn_times [symmetric] sgn_1_pos)

     ultimately show ?thesis by (simp add: mult_le_cancel_right)

   next

     case False with assms have "b * d < 0" by (simp add: less_le)

     moreover from this have "sgn b * sgn d = - 1"

       by (simp only: sgn_times [symmetric] sgn_1_neg)

     ultimately show ?thesis by (simp add: mult_le_cancel_right)

   qed

   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d"

     by (simp add: abs_sgn mult_ac)

   finally show ?thesis using assms by simp

 qed

 lemma less_rat': 

   assumes "b \<noteq> 0"

     and "d \<noteq> 0"

   shows "Fract a b < Fract c d \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"

 proof -

   have abs_sgn: "\<And>k::int. \<bar>k\<bar> = k * sgn k" unfolding abs_if sgn_if by simp

   have "a * d * (b * d) < c * b * (b * d) \<longleftrightarrow> a * d * (sgn b * sgn d) < c * b * (sgn b * sgn d)"

   proof (cases "b * d > 0")

     case True

     moreover from True have "sgn b * sgn d = 1"

       by (simp add: sgn_times [symmetric] sgn_1_pos)

     ultimately show ?thesis by (simp add: mult_less_cancel_right)

   next

     case False with assms have "b * d < 0" by (simp add: less_le)

     moreover from this have "sgn b * sgn d = - 1"

       by (simp only: sgn_times [symmetric] sgn_1_neg)

     ultimately show ?thesis by (simp add: mult_less_cancel_right)

   qed

   also have "\<dots> \<longleftrightarrow> a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d"

     by (simp add: abs_sgn mult_ac)

   finally show ?thesis using assms by simp

 qed

 lemma (in ordered_idom) sgn_greater [simp]:

   "0 < sgn a \<longleftrightarrow> 0 < a"

   unfolding sgn_if by auto

 lemma (in ordered_idom) sgn_less [simp]:

   "sgn a < 0 \<longleftrightarrow> a < 0"

   unfolding sgn_if by auto

 lemma rat_le_eq_code [code]:

   "Fract a b < Fract c d \<longleftrightarrow> (if b = 0

        then sgn c * sgn d > 0

      else if d = 0

        then sgn a * sgn b < 0

      else a * \<bar>d\<bar> * sgn b < c * \<bar>b\<bar> * sgn d)"

   by (auto simp add: sgn_times mult_less_0_iff zero_less_mult_iff less_rat' eq_rat simp del: less_rat)

 lemma rat_less_eq_code [code]:

   "Fract a b \<le> Fract c d \<longleftrightarrow> (if b = 0

        then sgn c * sgn d \<ge> 0

      else if d = 0

        then sgn a * sgn b \<le> 0

      else a * \<bar>d\<bar> * sgn b \<le> c * \<bar>b\<bar> * sgn d)"

   by (auto simp add: sgn_times mult_le_0_iff zero_le_mult_iff le_rat' eq_rat simp del: le_rat)

     (auto simp add: le_less not_less sgn_0_0)

 lemma rat_plus_code [code]:

   "Fract a b + Fract c d = (if b = 0

      then Fract c d

    else if d = 0

      then Fract a b

    else Fract_norm (a * d + c * b) (b * d))"

   by (simp add: eq_rat, simp add: Zero_rat_def)

 lemma rat_times_code [code]:

   "Fract a b * Fract c d = Fract_norm (a * c) (b * d)"

   by simp

 lemma rat_minus_code [code]:

   "Fract a b - Fract c d = (if b = 0

      then Fract (- c) d

    else if d = 0

      then Fract a b

    else Fract_norm (a * d - c * b) (b * d))"

   by (simp add: eq_rat, simp add: Zero_rat_def)

 lemma rat_inverse_code [code]:

   "inverse (Fract a b) = (if b = 0 then Fract 1 0

     else if a < 0 then Fract (- b) (- a)

     else Fract b a)"

   by (simp add: eq_rat)

 lemma rat_divide_code [code]:

   "Fract a b / Fract c d = Fract_norm (a * d) (b * c)"

   by simp

 definition (in term_syntax)

   valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Eval.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Eval.term) \<Rightarrow> rat \<times> (unit \<Rightarrow> Code_Eval.term)" where

   [code inline]: "valterm_fract k l = Code_Eval.valtermify Fract {\<cdot>} k {\<cdot>} l"

 notation fcomp (infixl "o>" 60)

 notation scomp (infixl "o\<rightarrow>" 60)

 instantiation rat :: random

 begin

 definition

   "random i = random i o\<rightarrow> (\<lambda>num. Random.range i o\<rightarrow> (\<lambda>denom. Pair (

      let j = Code_Index.int_of (denom + 1)

      in valterm_fract num (j, \<lambda>u. Code_Eval.term_of j))))"

 instance ..

 end

 no_notation fcomp (infixl "o>" 60)

 no_notation scomp (infixl "o\<rightarrow>" 60)

 hide (open) const Fract_norm

 text {* Setup for SML code generator *}

 types_code

   rat ("(int */ int)")

 attach (term_of) {*

 fun term_of_rat (p, q) =

   let

     val rT = Type ("Rational.rat", [])

   in

     if q = 1 orelse p = 0 then HOLogic.mk_number rT p

     else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $

       HOLogic.mk_number rT p $ HOLogic.mk_number rT q

   end;

 *}

 attach (test) {*

 fun gen_rat i =

   let

     val p = random_range 0 i;

     val q = random_range 1 (i + 1);

     val g = Integer.gcd p q;

     val p' = p div g;

     val q' = q div g;

     val r = (if one_of [true, false] then p' else ~ p',

       if p' = 0 then 0 else q')

   in

     (r, fn () => term_of_rat r)

   end;

 *}

 consts_code

   Fract ("(_,/ _)")

 consts_code

   "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")

 attach {*

 fun rat_of_int 0 = (0, 0)

   | rat_of_int i = (i, 1);

 *}

 end