(*  Title: HOL/Library/Rational.thy

    ID:    $Id$

    Author: Markus Wenzel, TU Muenchen

*)

header {* Rational numbers *}

theory Rational

imports Main

uses ("rat_arith.ML")

begin

subsection {* Rational numbers *}

subsubsection {* Equivalence of fractions *}

definition

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

  "fraction = {x. snd x \<noteq> 0}"

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 fraction_iff [simp]: "(x \<in> fraction) = (snd x \<noteq> 0)"

by (simp add: fraction_def)

lemma ratrel_iff [simp]:

  "((x,y) \<in> ratrel) =

   (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_ratrel: "refl fraction ratrel"

by (auto simp add: refl_def fraction_def ratrel_def)

lemma sym_ratrel: "sym ratrel"

by (simp add: ratrel_def sym_def)

lemma trans_ratrel_lemma:

  assumes 1: "a * b' = a' * b"

  assumes 2: "a' * b'' = a'' * b'"

  assumes 3: "b' \<noteq> (0::int)"

  shows "a * b'' = a'' * b"

proof -

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

  also note 1

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

  also note 2

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

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

  with 3 show "a * b'' = a'' * b" by simp

qed

lemma trans_ratrel: "trans ratrel"

by (auto simp add: trans_def elim: trans_ratrel_lemma)

lemma equiv_ratrel: "equiv fraction ratrel"

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

lemmas equiv_ratrel_iff [iff] = eq_equiv_class_iff [OF equiv_ratrel]

lemma equiv_ratrel_iff2:

  "\<lbrakk>snd x \<noteq> 0; snd y \<noteq> 0\<rbrakk>

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

by (rule eq_equiv_class_iff [OF equiv_ratrel], simp_all)

subsubsection {* The type of rational numbers *}

typedef (Rat) rat = "fraction//ratrel"

proof

  have "(0,1) \<in> fraction" by (simp add: fraction_def)

  thus "ratrel``{(0,1)} \<in> fraction//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]

definition

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

  [code func del]: "Fract a b = Abs_Rat (ratrel``{(a,b)})"

lemma Fract_zero:

  "Fract k 0 = Fract l 0"

  by (simp add: Fract_def ratrel_def)

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

    "(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C"

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

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

    "(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q"

  by (cases q) simp

subsubsection {* Congruence lemmas *}

lemma add_congruent2:

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

      respects2 ratrel"

apply (rule equiv_ratrel [THEN congruent2_commuteI])

apply (simp_all add: left_distrib)

done

lemma minus_congruent:

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

by (simp add: congruent_def)

lemma mult_congruent2:

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

by (rule equiv_ratrel [THEN congruent2_commuteI], simp_all)

lemma inverse_congruent:

  "(\<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)

lemma le_congruent2:

  "(\<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

lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]

lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]

subsubsection {* Standard operations on rational numbers *}

instance rat :: zero

  Zero_rat_def: "0 == Fract 0 1" ..

lemmas [code func del] = Zero_rat_def

instance rat :: one

  One_rat_def: "1 == Fract 1 1" ..

lemmas [code func del] = One_rat_def

instance rat :: plus

  add_rat_def:

   "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)})" ..

lemmas [code func del] = add_rat_def

instance rat :: minus

  minus_rat_def:

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

  diff_rat_def:  "q - r == q + - (r::rat)" ..

lemmas [code func del] = minus_rat_def diff_rat_def

instance rat :: times

  mult_rat_def:

   "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)})" ..

lemmas [code func del] = mult_rat_def

instance rat :: inverse

  inverse_rat_def:

    "inverse q ==

        Abs_Rat (\<Union>x \<in> Rep_Rat q.

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

  divide_rat_def:  "q / r == q * inverse (r::rat)" ..

lemmas [code func del] = inverse_rat_def divide_rat_def

instance rat :: ord

  le_rat_def:

   "q \<le> r == 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)})"

  less_rat_def: "(z < (w::rat)) == (z \<le> w & z \<noteq> w)" ..

lemmas [code func del] = le_rat_def less_rat_def

instance rat :: abs

  abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)" ..

instance rat :: power ..

primrec (rat)

  rat_power_0:   "q ^ 0       = 1"

  rat_power_Suc: "q ^ (Suc n) = (q::rat) * (q ^ n)"

theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>

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

by (simp add: Fract_def)

theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>

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

by (simp add: Fract_def add_rat_def add_congruent2 UN_ratrel2)

theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b"

by (simp add: Fract_def minus_rat_def minus_congruent UN_ratrel)

theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>

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

by (simp add: diff_rat_def add_rat minus_rat)

theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>

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

by (simp add: Fract_def mult_rat_def mult_congruent2 UN_ratrel2)

theorem inverse_rat: "a \<noteq> 0 ==> b \<noteq> 0 ==>

  inverse (Fract a b) = Fract b a"

by (simp add: Fract_def inverse_rat_def inverse_congruent UN_ratrel)

theorem divide_rat: "c \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==>

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

by (simp add: divide_rat_def inverse_rat mult_rat)

theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>

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

by (simp add: Fract_def le_rat_def le_congruent2 UN_ratrel2)

theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>

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

by (simp add: less_rat_def le_rat eq_rat order_less_le)

theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"

  by (simp add: abs_rat_def minus_rat Zero_rat_def less_rat eq_rat)

     (auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less

                split: abs_split)

subsubsection {* The ordered field of rational numbers *}

instance rat :: field

proof

  fix q r s :: rat

  show "(q + r) + s = q + (r + s)"

    by (induct q, induct r, induct s)

       (simp add: add_rat add_ac mult_ac int_distrib)

  show "q + r = r + q"

    by (induct q, induct r) (simp add: add_rat add_ac mult_ac)

  show "0 + q = q"

    by (induct q) (simp add: Zero_rat_def add_rat)

  show "(-q) + q = 0"

    by (induct q) (simp add: Zero_rat_def minus_rat add_rat eq_rat)

  show "q - r = q + (-r)"

    by (induct q, induct r) (simp add: add_rat minus_rat diff_rat)

  show "(q * r) * s = q * (r * s)"

    by (induct q, induct r, induct s) (simp add: mult_rat mult_ac)

  show "q * r = r * q"

    by (induct q, induct r) (simp add: mult_rat mult_ac)

  show "1 * q = q"

    by (induct q) (simp add: One_rat_def mult_rat)

  show "(q + r) * s = q * s + r * s"

    by (induct q, induct r, induct s)

       (simp add: add_rat mult_rat eq_rat int_distrib)

  show "q \<noteq> 0 ==> inverse q * q = 1"

    by (induct q) (simp add: inverse_rat mult_rat One_rat_def Zero_rat_def eq_rat)

  show "q / r = q * inverse r"

    by (simp add: divide_rat_def)

  show "0 \<noteq> (1::rat)"

    by (simp add: Zero_rat_def One_rat_def eq_rat)

qed

instance rat :: linorder

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 add: le_rat)

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

        qed

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

          by (simp only: mult_ac)

        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 add: le_rat)

          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 add: le_rat)

      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 add: le_rat)

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

        proof -

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

            by (simp add: le_rat)

          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 add: le_rat)

    show "(q < r) = (q \<le> r \<and> q \<noteq> r)"

      by (simp only: less_rat_def)

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

      by (induct q, induct r)

         (simp add: le_rat mult_commute, rule linorder_linear)

  }

qed

instance rat :: distrib_lattice

  "inf r s \<equiv> min r s"

  "sup r s \<equiv> max r s"

  by default (auto simp add: min_max.sup_inf_distrib1 inf_rat_def sup_rat_def)

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 add: le_rat)

      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: add_rat le_rat 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 less_rat le_rat 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 add: less_rat)

      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: less_rat mult_rat mult_ac)

    qed

  qed

  show "\<bar>q\<bar> = (if q < 0 then -q else q)"

    by (simp only: abs_rat_def)

qed auto

instance rat :: division_by_zero

proof

  show "inverse 0 = (0::rat)"

    by (simp add: Zero_rat_def Fract_def inverse_rat_def

                  inverse_congruent UN_ratrel)

qed

instance rat :: recpower

proof

  fix q :: rat

  fix n :: nat

  show "q ^ 0 = 1" by simp

  show "q ^ (Suc n) = q * (q ^ n)" by simp

qed

subsection {* Various Other Results *}

lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (-a) (-b) = Fract a b"

by (simp add: eq_rat)

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

  assumes step: "!!a b. 0 < b ==> P (Fract a b)"

    shows "P q"

proof (cases q)

  have step': "!!a b. b < 0 ==> 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 ==> (0 < Fract a b) = (0 < a)"

by (simp add: Zero_rat_def less_rat order_less_imp_not_eq2 zero_less_mult_iff)

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

apply (insert add_rat [of concl: m n 1 1])

apply (simp add: One_rat_def [symmetric])

done

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

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

lemma of_int_rat: "of_int k = Fract k 1"

by (cases k rule: int_diff_cases, simp add: of_nat_rat diff_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])

lemma Fract_of_int_quotient: "Fract k l = (if l = 0 then Fract 1 0 else of_int k / of_int l)"

by (auto simp add: Fract_zero Fract_of_int_eq [symmetric] divide_rat)

subsection {* Numerals and Arithmetic *}

instance rat :: number

  rat_number_of_def: "(number_of w :: rat) \<equiv> of_int w" ..

instance rat :: number_ring

  by default (simp add: rat_number_of_def) 

use "rat_arith.ML"

declaration {* K rat_arith_setup *}

subsection {* Embedding from Rationals to other Fields *}

class field_char_0 = field + ring_char_0

instance ordered_field < field_char_0 ..

definition

  of_rat :: "rat \<Rightarrow> 'a::field_char_0"

where

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

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: add_rat of_rat_rat add_frac_eq)

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

by (induct a, simp add: minus_rat 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: mult_rat 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,recpower}) = of_rat a ^ n"

by (induct n) (simp_all add: of_rat_mult power_Suc)

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

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

lemma of_rat_eq_id [simp]: "of_rat = (id :: rat \<Rightarrow> rat)"

proof

  fix a

  show "of_rat a = id a"

  by (induct a)

     (simp add: of_rat_rat divide_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"

end