(*  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 {* Function @{text normalize} *}

text{*

Decompose a fraction into normalized, i.e. coprime numerator and denominator:

*}

definition normalize :: "rat \<Rightarrow> int \<times> int" where

"normalize x \<equiv> THE pair. x = Fract (fst pair) (snd pair) &

                   snd pair > 0 & gcd (fst pair) (snd pair) = 1"

declare normalize_def[code del]

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

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

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

next

  let ?c = "gcd 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

text{* Proof by Ren\'e Thiemann: *}

lemma normalize_code[code]:

"normalize (Fract a b) =

 (if b > 0 then (let g = gcd a b in (a div g, b div g))

  else if b = 0 then (0,1)

  else (let g = - gcd a b in (a div g, b div g)))"

proof -

  let ?cond = "% r p. r = Fract (fst p) (snd p) & snd p > 0 &

                 gcd (fst p) (snd p) = 1"

  show ?thesis

  proof (cases "b = 0")

    case True

    thus ?thesis

    proof (simp add: normalize_def)

      show "(THE pair. ?cond (Fract a 0) pair) = (0,1)"

      proof

        show "?cond (Fract a 0) (0,1)"

          by (simp add: rat_number_collapse)

      next

        fix pair

        assume cond: "?cond (Fract a 0) pair"

        show "pair = (0,1)"

        proof (cases pair)

          case (Pair den num)

          with cond have num: "num > 0" by auto

          with Pair cond have den: "den = 0" by (simp add: eq_rat)

          show ?thesis

          proof (cases "num = 1", simp add: Pair den)

            case False

            with num have gr: "num > 1" by auto

            with den have "gcd den num = num" by auto

            with Pair cond False gr show ?thesis by auto

          qed

        qed

      qed

    qed

  next

    { fix a b :: int assume b: "b > 0"

      hence b0: "b \<noteq> 0" and "b >= 0" by auto

      let ?g = "gcd a b"

      from b0 have g0: "?g \<noteq> 0" by auto

      then have gp: "?g > 0" by simp

      then have gs: "?g <= b" by (metis b gcd_le2_int)

      from gcd_dvd1_int[of a b] obtain a' where a': "a = ?g * a'"

        unfolding dvd_def by auto

      from gcd_dvd2_int[of a b] obtain b' where b': "b = ?g * b'"

        unfolding dvd_def by auto

      hence b'2: "b' * ?g = b" by (simp add: ring_simps)

      with b0 have b'0: "b' \<noteq> 0" by auto

      from b b' zero_less_mult_iff[of ?g b'] gp have b'p: "b' > 0" by arith

      have "normalize (Fract a b) = (a div ?g, b div ?g)"

      proof (simp add: normalize_def)

        show "(THE pair. ?cond (Fract a b) pair) = (a div ?g, b div ?g)"

        proof

          have "1 = b div b" using b0 by auto

          also have "\<dots> <= b div ?g" by (rule zdiv_mono2[OF `b >= 0` gp gs])

          finally have div0: "b div ?g > 0" by simp

          show "?cond (Fract a b) (a div ?g, b div ?g)"

            by (simp add: b0 Fract_norm div_gcd_coprime_int div0)

        next

          fix pair assume cond: "?cond (Fract a b) pair"

          show "pair = (a div ?g, b div ?g)"

          proof (cases pair)

            case (Pair den num)

            with cond

            have num: "num > 0" and num0: "num \<noteq> 0" and gcd: "gcd den num = 1"

              by auto

            obtain g where g: "g = ?g" by auto

            with gp have gg0: "g > 0" by auto

            from cond Pair eq_rat(1)[OF b0 num0]

            have eq: "a * num = den * b" by auto

            hence "a' * g * num = den * g * b'"

              using a'[simplified g[symmetric]] b'[simplified g[symmetric]]

              by simp

            hence "a' * num * g = b' * den * g" by (simp add: algebra_simps)

            hence eq2: "a' * num = b' * den" using gg0 by auto

            have "a div ?g = ?g * a' div ?g" using a' by force

            hence adiv: "a div ?g = a'" using g0 by auto

            have "b div ?g = ?g * b' div ?g" using b' by force

            hence bdiv: "b div ?g = b'" using g0 by auto

            from div_gcd_coprime_int[of a b] b0

            have "gcd (a div ?g) (b div ?g) = 1" by auto

            with adiv bdiv have gcd2: "gcd a' b' = 1" by auto

            from gcd have gcd3: "gcd num den = 1"

              by (simp add: gcd_commute_int[of den num])

            from gcd2 have gcd4: "gcd b' a' = 1"

              by (simp add: gcd_commute_int[of a' b'])

            have one: "num dvd b'"

              by (metis coprime_dvd_mult_int[OF gcd3] dvd_triv_right eq2)

            have two: "b' dvd num"

              by (metis coprime_dvd_mult_int[OF gcd4] dvd_triv_left eq2 zmult_commute)

            from zdvd_antisym_abs[OF one two] b'p num

            have numb': "num = b'" by auto

            with eq2 b'0 have "a' = den" by auto

            with numb' adiv bdiv Pair show ?thesis by simp

          qed

        qed

      qed

    }

    note main = this

    assume "b \<noteq> 0"

    hence "b > 0 | b < 0" by arith

    thus ?thesis

    proof

      assume b: "b > 0" thus ?thesis by (simp add: Let_def main[OF b])

    next

      assume b: "b < 0"

      thus ?thesis

        by(simp add:main Let_def minus_rat_cancel[of a b, symmetric]

                    zdiv_zminus2 del:minus_rat_cancel)

    qed

  qed

qed

lemma normalize_id: "normalize (Fract a b) = (p,q) \<Longrightarrow> Fract p q = Fract a b"

by(auto simp add: normalize_code Let_def Fract_norm dvd_div_neg rat_number_collapse

        split:split_if_asm)

lemma normalize_denom_pos: "normalize (Fract a b) = (p,q) \<Longrightarrow> q > 0"

by(auto simp add: normalize_code Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff

        split:split_if_asm)

lemma normalize_coprime: "normalize (Fract a b) = (p,q) \<Longrightarrow> coprime p q"

by(auto simp add: normalize_code Let_def dvd_div_neg div_gcd_coprime_int

        split:split_if_asm)

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