merged

--- a/NEWS	Thu Feb 25 22:46:52 2010 +0100
+++ b/NEWS	Fri Feb 26 10:57:35 2010 +0100
@@ -54,6 +54,9 @@
 
 *** HOL ***
 
+* Theory "Rational" renamed to "Rat", for consistency with "Nat", "Int" etc.
+INCOMPATIBILITY.
+
 * New set of rules "ac_simps" provides combined assoc / commute rewrites
 for all interpretations of the appropriate generic locales.
 

--- a/src/HOL/Code_Evaluation.thy	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/HOL/Code_Evaluation.thy	Fri Feb 26 10:57:35 2010 +0100
@@ -119,6 +119,44 @@
 end
 *}
 
+setup {*
+let
+  fun mk_term_of_eq thy ty vs tyco abs ty_rep proj =
+    let
+      val arg = Var (("x", 0), ty);
+      val rhs = Abs ("y", @{typ term}, HOLogic.reflect_term (Const (abs, ty_rep --> ty) $ Bound 0)) $
+        (HOLogic.mk_term_of ty_rep (Const (proj, ty --> ty_rep) $ arg))
+        |> Thm.cterm_of thy;
+      val cty = Thm.ctyp_of thy ty;
+    in
+      @{thm term_of_anything}
+      |> Drule.instantiate' [SOME cty] [SOME (Thm.cterm_of thy arg), SOME rhs]
+      |> Thm.varifyT
+    end;
+  fun add_term_of_code tyco raw_vs abs raw_ty_rep proj thy =
+    let
+      val algebra = Sign.classes_of thy;
+      val vs = map (fn (v, sort) =>
+        (v, curry (Sorts.inter_sort algebra) @{sort typerep} sort)) raw_vs;
+      val ty = Type (tyco, map TFree vs);
+      val ty_rep = map_atyps
+        (fn TFree (v, _) => TFree (v, (the o AList.lookup (op =) vs) v)) raw_ty_rep;
+      val const = AxClass.param_of_inst thy (@{const_name term_of}, tyco);
+      val eq = mk_term_of_eq thy ty vs tyco abs ty_rep proj;
+   in
+      thy
+      |> Code.del_eqns const
+      |> Code.add_eqn eq
+    end;
+  fun ensure_term_of_code (tyco, (raw_vs, ((abs, ty), (proj, _)))) thy =
+    let
+      val has_inst = can (Sorts.mg_domain (Sign.classes_of thy) tyco) @{sort term_of};
+    in if has_inst then add_term_of_code tyco raw_vs abs ty proj thy else thy end;
+in
+  Code.abstype_interpretation ensure_term_of_code
+end
+*}
+
 
 subsubsection {* Code generator setup *}
 

--- a/src/HOL/Divides.thy	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/HOL/Divides.thy	Fri Feb 26 10:57:35 2010 +0100
@@ -309,6 +309,15 @@
   apply (simp add: no_zero_divisors)
   done
 
+lemma div_mult_swap:
+  assumes "c dvd b"
+  shows "a * (b div c) = (a * b) div c"
+proof -
+  from assms have "b div c * (a div 1) = b * a div (c * 1)"
+    by (simp only: div_mult_div_if_dvd one_dvd)
+  then show ?thesis by (simp add: mult_commute)
+qed
+   
 lemma div_mult_mult2 [simp]:
   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
   by (drule div_mult_mult1) (simp add: mult_commute)
@@ -347,6 +356,24 @@
 apply(simp add: div_mult_div_if_dvd dvd_power_same)
 done
 
+lemma dvd_div_eq_mult:
+  assumes "a \<noteq> 0" and "a dvd b"  
+  shows "b div a = c \<longleftrightarrow> b = c * a"
+proof
+  assume "b = c * a"
+  then show "b div a = c" by (simp add: assms)
+next
+  assume "b div a = c"
+  then have "b div a * a = c * a" by simp
+  moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self)
+  ultimately show "b = c * a" by simp
+qed
+   
+lemma dvd_div_div_eq_mult:
+  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
+  shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
+  using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)
+
 end
 
 class ring_div = semiring_div + idom

--- a/src/HOL/GCD.thy	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/HOL/GCD.thy	Fri Feb 26 10:57:35 2010 +0100
@@ -412,6 +412,33 @@
 apply (rule gcd_mult_cancel_nat [transferred], auto)
 done
 
+lemma coprime_crossproduct_nat:
+  fixes a b c d :: nat
+  assumes "coprime a d" and "coprime b c"
+  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d" (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume ?rhs then show ?lhs by simp
+next
+  assume ?lhs
+  from `?lhs` have "a dvd b * d" by (auto intro: dvdI dest: sym)
+  with `coprime a d` have "a dvd b" by (simp add: coprime_dvd_mult_iff_nat)
+  from `?lhs` have "b dvd a * c" by (auto intro: dvdI dest: sym)
+  with `coprime b c` have "b dvd a" by (simp add: coprime_dvd_mult_iff_nat)
+  from `?lhs` have "c dvd d * b" by (auto intro: dvdI dest: sym simp add: mult_commute)
+  with `coprime b c` have "c dvd d" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
+  from `?lhs` have "d dvd c * a" by (auto intro: dvdI dest: sym simp add: mult_commute)
+  with `coprime a d` have "d dvd c" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
+  from `a dvd b` `b dvd a` have "a = b" by (rule Nat.dvd.antisym)
+  moreover from `c dvd d` `d dvd c` have "c = d" by (rule Nat.dvd.antisym)
+  ultimately show ?rhs ..
+qed
+
+lemma coprime_crossproduct_int:
+  fixes a b c d :: int
+  assumes "coprime a d" and "coprime b c"
+  shows "\<bar>a\<bar> * \<bar>c\<bar> = \<bar>b\<bar> * \<bar>d\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>b\<bar> \<and> \<bar>c\<bar> = \<bar>d\<bar>"
+  using assms by (intro coprime_crossproduct_nat [transferred]) auto
+
 text {* \medskip Addition laws *}
 
 lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"

--- a/src/HOL/IsaMakefile	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/HOL/IsaMakefile	Fri Feb 26 10:57:35 2010 +0100
@@ -352,7 +352,7 @@
   PReal.thy \
   Parity.thy \
   RComplete.thy \
-  Rational.thy \
+  Rat.thy \
   Real.thy \
   RealDef.thy \
   RealPow.thy \

--- a/src/HOL/Library/Binomial.thy	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/HOL/Library/Binomial.thy	Fri Feb 26 10:57:35 2010 +0100
@@ -1,4 +1,4 @@
-(*  Title:      HOL/Binomial.thy
+(*  Title:      HOL/Library/Binomial.thy
     Author:     Lawrence C Paulson, Amine Chaieb
     Copyright   1997  University of Cambridge
 *)
@@ -6,7 +6,7 @@
 header {* Binomial Coefficients *}
 
 theory Binomial
-imports Fact SetInterval Presburger Main Rational
+imports Complex_Main
 begin
 
 text {* This development is based on the work of Andy Gordon and

--- a/src/HOL/Library/Countable.thy	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/HOL/Library/Countable.thy	Fri Feb 26 10:57:35 2010 +0100
@@ -5,12 +5,7 @@
 header {* Encoding (almost) everything into natural numbers *}
 
 theory Countable
-imports
-  "~~/src/HOL/List"
-  "~~/src/HOL/Hilbert_Choice"
-  "~~/src/HOL/Nat_Int_Bij"
-  "~~/src/HOL/Rational"
-  Main
+imports Main Rat Nat_Int_Bij
 begin
 
 subsection {* The class of countable types *}
@@ -213,8 +208,8 @@
 proof
   fix r::rat
   show "\<exists>n. r = nat_to_rat_surj n"
-  proof(cases r)
-    fix i j assume [simp]: "r = Fract i j" and "j \<noteq> 0"
+  proof (cases r)
+    fix i j assume [simp]: "r = Fract i j" and "j > 0"
     have "r = (let m = inv nat_to_int_bij i; n = inv nat_to_int_bij j
                in nat_to_rat_surj(nat2_to_nat (m,n)))"
       using nat2_to_nat_inj surj_f_inv_f[OF surj_nat_to_int_bij]

--- a/src/HOL/Library/Fraction_Field.thy	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/HOL/Library/Fraction_Field.thy	Fri Feb 26 10:57:35 2010 +0100
@@ -1,12 +1,12 @@
-(*  Title:      Fraction_Field.thy
+(*  Title:      HOL/Library/Fraction_Field.thy
     Author:     Amine Chaieb, University of Cambridge
 *)
 
 header{* A formalization of the fraction field of any integral domain 
-         A generalization of Rational.thy from int to any integral domain *}
+         A generalization of Rat.thy from int to any integral domain *}
 
 theory Fraction_Field
-imports Main (* Equiv_Relations Plain *)
+imports Main
 begin
 
 subsection {* General fractions construction *}

--- a/src/HOL/Nitpick_Examples/Typedef_Nits.thy	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/HOL/Nitpick_Examples/Typedef_Nits.thy	Fri Feb 26 10:57:35 2010 +0100
@@ -8,7 +8,7 @@
 header {* Examples Featuring Nitpick Applied to Typedefs *}
 
 theory Typedef_Nits
-imports Main Rational
+imports Complex_Main
 begin
 
 nitpick_params [card = 1\<midarrow>4, sat_solver = MiniSat_JNI, max_threads = 1,

--- a/src/HOL/PReal.thy	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/HOL/PReal.thy	Fri Feb 26 10:57:35 2010 +0100
@@ -9,7 +9,7 @@
 header {* Positive real numbers *}
 
 theory PReal
-imports Rational 
+imports Rat
 begin
 
 text{*Could be generalized and moved to @{text Groups}*}

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Rat.thy	Fri Feb 26 10:57:35 2010 +0100
@@ -0,0 +1,1210 @@
+(*  Title:  HOL/Rat.thy
+    Author: Markus Wenzel, TU Muenchen
+*)
+
+header {* Rational numbers *}
+
+theory Rat
+imports GCD Archimedean_Field
+uses ("Tools/float_syntax.ML")
+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
+  "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})"
+
+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)
+
+lemma Rat_cases [case_names Fract, cases type: rat]:
+  assumes "\<And>a b. q = Fract a b \<Longrightarrow> b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> C"
+  shows C
+proof -
+  obtain a b :: int where "q = Fract a b" and "b \<noteq> 0"
+    by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def)
+  let ?a = "a div gcd a b"
+  let ?b = "b div gcd a b"
+  from `b \<noteq> 0` have "?b * gcd a b = b"
+    by (simp add: dvd_div_mult_self)
+  with `b \<noteq> 0` have "?b \<noteq> 0" by auto
+  from `q = Fract a b` `b \<noteq> 0` `?b \<noteq> 0` have q: "q = Fract ?a ?b"
+    by (simp add: eq_rat dvd_div_mult mult_commute [of a])
+  from `b \<noteq> 0` have coprime: "coprime ?a ?b"
+    by (auto intro: div_gcd_coprime_int)
+  show C proof (cases "b > 0")
+    case True
+    note assms
+    moreover note q
+    moreover from True have "?b > 0" by (simp add: nonneg1_imp_zdiv_pos_iff)
+    moreover note coprime
+    ultimately show C .
+  next
+    case False
+    note assms
+    moreover from q have "q = Fract (- ?a) (- ?b)" by (simp add: Fract_def)
+    moreover from False `b \<noteq> 0` have "- ?b > 0" by (simp add: pos_imp_zdiv_neg_iff)
+    moreover from coprime have "coprime (- ?a) (- ?b)" by simp
+    ultimately show C .
+  qed
+qed
+
+lemma Rat_induct [case_names Fract, induct type: rat]:
+  assumes "\<And>a b. b > 0 \<Longrightarrow> coprime a b \<Longrightarrow> P (Fract a b)"
+  shows "P q"
+  using assms by (cases q) simp
+
+instantiation rat :: comm_ring_1
+begin
+
+definition
+  Zero_rat_def: "0 = Fract 0 1"
+
+definition
+  One_rat_def: "1 = Fract 1 1"
+
+definition
+  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)})"
+
+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:
+  "- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})"
+
+lemma minus_rat [simp]: "- 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: "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:
+  "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: "number_of w = Fract w 1"
+
+instance proof
+qed (simp add: rat_number_of_def of_int_rat)
+
+end
+
+lemma rat_number_collapse:
+  "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 > 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> coprime a b \<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 > 0" and "coprime a b" by (cases q) auto
+  moreover with False have "0 \<noteq> Fract a b" by simp
+  with `b > 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat)
+  with Fract `q = Fract a b` `b > 0` `coprime a b` show C by blast
+qed
+
+subsubsection {* Function @{text normalize} *}
+
+lemma Fract_coprime: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
+proof (cases "b = 0")
+  case True then show ?thesis by (simp add: eq_rat)
+next
+  case False
+  moreover have "b div gcd a b * gcd a b = b"
+    by (rule dvd_div_mult_self) simp
+  ultimately have "b div gcd a b \<noteq> 0" by auto
+  with False show ?thesis by (simp add: eq_rat dvd_div_mult mult_commute [of a])
+qed
+
+definition normalize :: "int \<times> int \<Rightarrow> int \<times> int" where
+  "normalize p = (if snd p > 0 then (let a = gcd (fst p) (snd p) in (fst p div a, snd p div a))
+    else if snd p = 0 then (0, 1)
+    else (let a = - gcd (fst p) (snd p) in (fst p div a, snd p div a)))"
+
+lemma normalize_crossproduct:
+  assumes "q \<noteq> 0" "s \<noteq> 0"
+  assumes "normalize (p, q) = normalize (r, s)"
+  shows "p * s = r * q"
+proof -
+  have aux: "p * gcd r s = sgn (q * s) * r * gcd p q \<Longrightarrow> q * gcd r s = sgn (q * s) * s * gcd p q \<Longrightarrow> p * s = q * r"
+  proof -
+    assume "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
+    then have "(p * gcd r s) * (sgn (q * s) * s * gcd p q) = (q * gcd r s) * (sgn (q * s) * r * gcd p q)" by simp
+    with assms show "p * s = q * r" by (auto simp add: mult_ac sgn_times sgn_0_0)
+  qed
+  from assms show ?thesis
+    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult_commute sgn_times split: if_splits intro: aux)
+qed
+
+lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
+  by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
+    split:split_if_asm)
+
+lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
+  by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
+    split:split_if_asm)
+
+lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
+  by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime_int
+    split:split_if_asm)
+
+lemma normalize_stable [simp]:
+  "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
+  by (simp add: normalize_def)
+
+lemma normalize_denom_zero [simp]:
+  "normalize (p, 0) = (0, 1)"
+  by (simp add: normalize_def)
+
+lemma normalize_negative [simp]:
+  "q < 0 \<Longrightarrow> normalize (p, q) = normalize (- p, - q)"
+  by (simp add: normalize_def Let_def dvd_div_neg dvd_neg_div)
+
+text{*
+  Decompose a fraction into normalized, i.e. coprime numerator and denominator:
+*}
+
+definition quotient_of :: "rat \<Rightarrow> int \<times> int" where
+  "quotient_of x = (THE pair. x = Fract (fst pair) (snd pair) &
+                   snd pair > 0 & coprime (fst pair) (snd pair))"
+
+lemma quotient_of_unique:
+  "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
+proof (cases r)
+  case (Fract a b)
+  then have "r = Fract (fst (a, b)) (snd (a, b)) \<and> snd (a, b) > 0 \<and> coprime (fst (a, b)) (snd (a, b))" by auto
+  then show ?thesis proof (rule ex1I)
+    fix p
+    obtain c d :: int where p: "p = (c, d)" by (cases p)
+    assume "r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
+    with p have Fract': "r = Fract c d" "d > 0" "coprime c d" by simp_all
+    have "c = a \<and> d = b"
+    proof (cases "a = 0")
+      case True with Fract Fract' show ?thesis by (simp add: eq_rat)
+    next
+      case False
+      with Fract Fract' have *: "c * b = a * d" and "c \<noteq> 0" by (auto simp add: eq_rat)
+      then have "c * b > 0 \<longleftrightarrow> a * d > 0" by auto
+      with `b > 0` `d > 0` have "a > 0 \<longleftrightarrow> c > 0" by (simp add: zero_less_mult_iff)
+      with `a \<noteq> 0` `c \<noteq> 0` have sgn: "sgn a = sgn c" by (auto simp add: not_less)
+      from `coprime a b` `coprime c d` have "\<bar>a\<bar> * \<bar>d\<bar> = \<bar>c\<bar> * \<bar>b\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> \<bar>d\<bar> = \<bar>b\<bar>"
+        by (simp add: coprime_crossproduct_int)
+      with `b > 0` `d > 0` have "\<bar>a\<bar> * d = \<bar>c\<bar> * b \<longleftrightarrow> \<bar>a\<bar> = \<bar>c\<bar> \<and> d = b" by simp
+      then have "a * sgn a * d = c * sgn c * b \<longleftrightarrow> a * sgn a = c * sgn c \<and> d = b" by (simp add: abs_sgn)
+      with sgn * show ?thesis by (auto simp add: sgn_0_0)
+    qed
+    with p show "p = (a, b)" by simp
+  qed
+qed
+
+lemma quotient_of_Fract [code]:
+  "quotient_of (Fract a b) = normalize (a, b)"
+proof -
+  have "Fract a b = Fract (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?Fract)
+    by (rule sym) (auto intro: normalize_eq)
+  moreover have "0 < snd (normalize (a, b))" (is ?denom_pos) 
+    by (cases "normalize (a, b)") (rule normalize_denom_pos, simp)
+  moreover have "coprime (fst (normalize (a, b))) (snd (normalize (a, b)))" (is ?coprime)
+    by (rule normalize_coprime) simp
+  ultimately have "?Fract \<and> ?denom_pos \<and> ?coprime" by blast
+  with quotient_of_unique have
+    "(THE p. Fract a b = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)) = normalize (a, b)"
+    by (rule the1_equality)
+  then show ?thesis by (simp add: quotient_of_def)
+qed
+
+lemma quotient_of_number [simp]:
+  "quotient_of 0 = (0, 1)"
+  "quotient_of 1 = (1, 1)"
+  "quotient_of (number_of k) = (number_of k, 1)"
+  by (simp_all add: rat_number_expand quotient_of_Fract)
+
+lemma quotient_of_eq: "quotient_of (Fract a b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
+  by (simp add: quotient_of_Fract normalize_eq)
+
+lemma quotient_of_denom_pos: "quotient_of r = (p, q) \<Longrightarrow> q > 0"
+  by (cases r) (simp add: quotient_of_Fract normalize_denom_pos)
+
+lemma quotient_of_coprime: "quotient_of r = (p, q) \<Longrightarrow> coprime p q"
+  by (cases r) (simp add: quotient_of_Fract normalize_coprime)
+
+lemma quotient_of_inject:
+  assumes "quotient_of a = quotient_of b"
+  shows "a = b"
+proof -
+  obtain p q r s where a: "a = Fract p q"
+    and b: "b = Fract r s"
+    and "q > 0" and "s > 0" by (cases a, cases b)
+  with assms show ?thesis by (simp add: eq_rat quotient_of_Fract normalize_crossproduct)
+qed
+
+lemma quotient_of_inject_eq:
+  "quotient_of a = quotient_of b \<longleftrightarrow> a = b"
+  by (auto simp add: quotient_of_inject)
+
+
+subsubsection {* The field of rational numbers *}
+
+instantiation rat :: "{field, division_by_zero}"
+begin
+
+definition
+  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)})"
+
+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: "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: 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:
+  "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:
+  "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:
+   "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: "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"
+    then show "q \<le> s" 
+    proof (induct q, induct r, induct s)
+      fix a b c d e f :: int
+      assume neq: "b > 0"  "d > 0"  "f > 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"
+    then show "q = r"
+    proof (induct q, induct r)
+      fix a b c d :: int
+      assume neq: "b > 0"  "d > 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: "\<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 not_less le_less eq_rat zero_less_mult_iff)
+
+definition
+  sgn_rat_def: "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 :: linordered_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 > 0"  "d > 0"  "f > 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 > 0"  "d > 0"  "f > 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:
+  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"
+    by (cases "b = 0", simp, simp add: of_int_rat)
+  moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
+    unfolding Fract_of_int_quotient
+    by (rule linorder_cases [of b 0])
+       (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
+  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)
+    have "Fract a b \<le> of_int (a div b + 1)"
+      using rat_floor_lemma [of a b] by simp
+    then show "\<exists>z. Fract a b \<le> of_int z" ..
+  qed
+qed
+
+lemma floor_Fract: "floor (Fract a b) = a div b"
+  using rat_floor_lemma [of a b]
+  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 linordered_field) field_char_0 ..
+
+context field_char_0
+begin
+
+definition of_rat :: "rat \<Rightarrow> 'a" where
+  "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::linordered_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::linordered_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::linordered_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
+  "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 *}
+
+definition Frct :: "int \<times> int \<Rightarrow> rat" where
+  [simp]: "Frct p = Fract (fst p) (snd p)"
+
+code_abstype Frct quotient_of
+proof (rule eq_reflection)
+  fix r :: rat
+  show "Frct (quotient_of r) = r" by (cases r) (auto intro: quotient_of_eq)
+qed
+
+lemma Frct_code_post [code_post]:
+  "Frct (0, k) = 0"
+  "Frct (k, 0) = 0"
+  "Frct (1, 1) = 1"
+  "Frct (number_of k, 1) = number_of k"
+  "Frct (1, number_of k) = 1 / number_of k"
+  "Frct (number_of k, number_of l) = number_of k / number_of l"
+  by (simp_all add: rat_number_collapse Fract_number_of_quotient Fract_1_number_of)
+
+declare quotient_of_Fract [code abstract]
+
+lemma rat_zero_code [code abstract]:
+  "quotient_of 0 = (0, 1)"
+  by (simp add: Zero_rat_def quotient_of_Fract normalize_def)
+
+lemma rat_one_code [code abstract]:
+  "quotient_of 1 = (1, 1)"
+  by (simp add: One_rat_def quotient_of_Fract normalize_def)
+
+lemma rat_plus_code [code abstract]:
+  "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
+     in normalize (a * d + b * c, c * d))"
+  by (cases p, cases q) (simp add: quotient_of_Fract)
+
+lemma rat_uminus_code [code abstract]:
+  "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
+  by (cases p) (simp add: quotient_of_Fract)
+
+lemma rat_minus_code [code abstract]:
+  "quotient_of (p - q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
+     in normalize (a * d - b * c, c * d))"
+  by (cases p, cases q) (simp add: quotient_of_Fract)
+
+lemma rat_times_code [code abstract]:
+  "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
+     in normalize (a * b, c * d))"
+  by (cases p, cases q) (simp add: quotient_of_Fract)
+
+lemma rat_inverse_code [code abstract]:
+  "quotient_of (inverse p) = (let (a, b) = quotient_of p
+    in if a = 0 then (0, 1) else (sgn a * b, \<bar>a\<bar>))"
+proof (cases p)
+  case (Fract a b) then show ?thesis
+    by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract gcd_int.commute)
+qed
+
+lemma rat_divide_code [code abstract]:
+  "quotient_of (p / q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q
+     in normalize (a * d, c * b))"
+  by (cases p, cases q) (simp add: quotient_of_Fract)
+
+lemma rat_abs_code [code abstract]:
+  "quotient_of \<bar>p\<bar> = (let (a, b) = quotient_of p in (\<bar>a\<bar>, b))"
+  by (cases p) (simp add: quotient_of_Fract)
+
+lemma rat_sgn_code [code abstract]:
+  "quotient_of (sgn p) = (sgn (fst (quotient_of p)), 1)"
+proof (cases p)
+  case (Fract a b) then show ?thesis
+  by (cases "0::int" a rule: linorder_cases) (simp_all add: quotient_of_Fract)
+qed
+
+instantiation rat :: eq
+begin
+
+definition [code]:
+  "eq_class.eq a b \<longleftrightarrow> quotient_of a = quotient_of b"
+
+instance proof
+qed (simp add: eq_rat_def quotient_of_inject_eq)
+
+lemma rat_eq_refl [code nbe]:
+  "eq_class.eq (r::rat) r \<longleftrightarrow> True"
+  by (rule HOL.eq_refl)
+
+end
+
+lemma rat_less_eq_code [code]:
+  "p \<le> q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d \<le> c * b)"
+  by (cases p, cases q) (simp add: quotient_of_Fract times.commute)
+
+lemma rat_less_code [code]:
+  "p < q \<longleftrightarrow> (let (a, c) = quotient_of p; (b, d) = quotient_of q in a * d < c * b)"
+  by (cases p, cases q) (simp add: quotient_of_Fract times.commute)
+
+lemma [code]:
+  "of_rat p = (let (a, b) = quotient_of p in of_int a / of_int b)"
+  by (cases p) (simp add: quotient_of_Fract of_rat_rat)
+
+definition (in term_syntax)
+  valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> rat \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
+  [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
+
+notation fcomp (infixl "o>" 60)
+notation scomp (infixl "o\<rightarrow>" 60)
+
+instantiation rat :: random
+begin
+
+definition
+  "Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>num. Random.range i o\<rightarrow> (\<lambda>denom. Pair (
+     let j = Code_Numeral.int_of (denom + 1)
+     in valterm_fract num (j, \<lambda>u. Code_Evaluation.term_of j))))"
+
+instance ..
+
+end
+
+no_notation fcomp (infixl "o>" 60)
+no_notation scomp (infixl "o\<rightarrow>" 60)
+
+text {* Setup for SML code generator *}
+
+types_code
+  rat ("(int */ int)")
+attach (term_of) {*
+fun term_of_rat (p, q) =
+  let
+    val rT = Type ("Rat.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 1 else q')
+  in
+    (r, fn () => term_of_rat r)
+  end;
+*}
+
+consts_code
+  Fract ("(_,/ _)")
+
+consts_code
+  quotient_of ("{*normalize*}")
+
+consts_code
+  "of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int")
+attach {*
+fun rat_of_int i = (i, 1);
+*}
+
+setup {*
+  Nitpick.register_frac_type @{type_name rat}
+   [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
+    (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
+    (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
+    (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
+    (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
+    (@{const_name number_rat_inst.number_of_rat}, @{const_name Nitpick.number_of_frac}),
+    (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
+    (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
+    (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac}),
+    (@{const_name field_char_0_class.Rats}, @{const_name UNIV})]
+*}
+
+lemmas [nitpick_def] = inverse_rat_inst.inverse_rat
+  number_rat_inst.number_of_rat one_rat_inst.one_rat ord_rat_inst.less_eq_rat
+  plus_rat_inst.plus_rat times_rat_inst.times_rat uminus_rat_inst.uminus_rat
+  zero_rat_inst.zero_rat
+
+subsection{* Float syntax *}
+
+syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
+
+use "Tools/float_syntax.ML"
+setup Float_Syntax.setup
+
+text{* Test: *}
+lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
+by simp
+
+end

--- a/src/HOL/Rational.thy	Thu Feb 25 22:46:52 2010 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1227 +0,0 @@
-(*  Title:  HOL/Rational.thy
-    Author: Markus Wenzel, TU Muenchen
-*)
-
-header {* Rational numbers *}
-
-theory Rational
-imports GCD Archimedean_Field
-uses ("Tools/float_syntax.ML")
-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: 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 not_less le_less eq_rat zero_less_mult_iff)
-
-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 :: linordered_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:
-  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"
-    by (cases "b = 0", simp, simp add: of_int_rat)
-  moreover have "0 \<le> Fract (a mod b) b \<and> Fract (a mod b) b < 1"
-    unfolding Fract_of_int_quotient
-    by (rule linorder_cases [of b 0])
-       (simp add: divide_nonpos_neg, simp, simp add: divide_nonneg_pos)
-  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)
-    have "Fract a b \<le> of_int (a div b + 1)"
-      using rat_floor_lemma [of a b] by simp
-    then show "\<exists>z. Fract a b \<le> of_int z" ..
-  qed
-qed
-
-lemma floor_Fract: "floor (Fract a b) = a div b"
-  using rat_floor_lemma [of a b]
-  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 linordered_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::linordered_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::linordered_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::linordered_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 *}
-
-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 = gcd 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 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_Evaluation.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> rat \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
-  [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
-
-notation fcomp (infixl "o>" 60)
-notation scomp (infixl "o\<rightarrow>" 60)
-
-instantiation rat :: random
-begin
-
-definition
-  "Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>num. Random.range i o\<rightarrow> (\<lambda>denom. Pair (
-     let j = Code_Numeral.int_of (denom + 1)
-     in valterm_fract num (j, \<lambda>u. Code_Evaluation.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 1 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 i = (i, 1);
-*}
-
-setup {*
-  Nitpick.register_frac_type @{type_name rat}
-   [(@{const_name zero_rat_inst.zero_rat}, @{const_name Nitpick.zero_frac}),
-    (@{const_name one_rat_inst.one_rat}, @{const_name Nitpick.one_frac}),
-    (@{const_name plus_rat_inst.plus_rat}, @{const_name Nitpick.plus_frac}),
-    (@{const_name times_rat_inst.times_rat}, @{const_name Nitpick.times_frac}),
-    (@{const_name uminus_rat_inst.uminus_rat}, @{const_name Nitpick.uminus_frac}),
-    (@{const_name number_rat_inst.number_of_rat}, @{const_name Nitpick.number_of_frac}),
-    (@{const_name inverse_rat_inst.inverse_rat}, @{const_name Nitpick.inverse_frac}),
-    (@{const_name ord_rat_inst.less_eq_rat}, @{const_name Nitpick.less_eq_frac}),
-    (@{const_name field_char_0_class.of_rat}, @{const_name Nitpick.of_frac}),
-    (@{const_name field_char_0_class.Rats}, @{const_name UNIV})]
-*}
-
-lemmas [nitpick_def] = inverse_rat_inst.inverse_rat
-  number_rat_inst.number_of_rat one_rat_inst.one_rat ord_rat_inst.less_eq_rat
-  plus_rat_inst.plus_rat times_rat_inst.times_rat uminus_rat_inst.uminus_rat
-  zero_rat_inst.zero_rat
-
-subsection{* Float syntax *}
-
-syntax "_Float" :: "float_const \<Rightarrow> 'a"    ("_")
-
-use "Tools/float_syntax.ML"
-setup Float_Syntax.setup
-
-text{* Test: *}
-lemma "123.456 = -111.111 + 200 + 30 + 4 + 5/10 + 6/100 + (7/1000::rat)"
-by simp
-
-end

--- a/src/HOL/ex/NormalForm.thy	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/HOL/ex/NormalForm.thy	Fri Feb 26 10:57:35 2010 +0100
@@ -3,7 +3,7 @@
 header {* Testing implementation of normalization by evaluation *}
 
 theory NormalForm
-imports Main Rational
+imports Complex_Main
 begin
 
 lemma "True" by normalization

--- a/src/Pure/Isar/code.ML	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/Pure/Isar/code.ML	Fri Feb 26 10:57:35 2010 +0100
@@ -420,10 +420,11 @@
      of Type ("fun", [ty, ty_abs]) => (ty, ty_abs)
       | _ => error ("Not a datatype abstractor:\n" ^ string_of_const thy abs
           ^ " :: " ^ string_of_typ thy ty');
-    val _ = Thm.cterm_of thy (Const (rep, ty_abs --> ty)) handle CTERM _ =>
+    val _ = Thm.cterm_of thy (Const (rep, ty_abs --> ty)) handle TYPE _ =>
       error ("Not a projection:\n" ^ string_of_const thy rep);
-    val cert = Logic.mk_equals (Const (abs, ty --> ty_abs) $ (Const (rep, ty_abs --> ty)
-      $ Free ("x", ty_abs)), Free ("x", ty_abs));
+    val param = Free ("x", ty_abs);
+    val cert = Logic.all param (Logic.mk_equals
+      (Const (abs, ty --> ty_abs) $ (Const (rep, ty_abs --> ty) $ param), param));
   in (tyco, (vs ~~ sorts, ((fst abs_ty, ty), (rep, cert)))) end;    
 
 fun get_type_entry thy tyco = case these (Symtab.lookup ((the_types o the_exec) thy) tyco)
@@ -839,7 +840,9 @@
 fun bare_thms_of_cert thy (cert as Equations _) =
       (map_filter (fn (_, (some_thm, proper)) => if proper then some_thm else NONE)
         o snd o equations_of_cert thy) cert
-  | bare_thms_of_cert thy _ = [];
+  | bare_thms_of_cert thy (Projection _) = []
+  | bare_thms_of_cert thy (Abstract (abs_thm, tyco)) =
+      [Thm.varifyT (snd (concretify_abs thy tyco abs_thm))];
 
 end;
 

--- a/src/Tools/nbe.ML	Thu Feb 25 22:46:52 2010 +0100
+++ b/src/Tools/nbe.ML	Fri Feb 26 10:57:35 2010 +0100
@@ -164,6 +164,7 @@
   | same _ _ = false
 and sames xs ys = length xs = length ys andalso forall (uncurry same) (xs ~~ ys);
 
+
 (* constructor functions *)
 
 fun abss n f = Abs ((n, f), []);
@@ -213,6 +214,7 @@
         |> suffix "\n"
       end;
 
+
 (* nbe specific syntax and sandbox communication *)
 
 val univs_ref = Unsynchronized.ref (NONE : (unit -> Univ list -> Univ list) option);
@@ -255,6 +257,7 @@
 
 open Basic_Code_Thingol;
 
+
 (* code generation *)
 
 fun assemble_eqnss idx_of deps eqnss =
@@ -330,7 +333,7 @@
         val match_cont = if is_eval then NONE else SOME default_rhs;
         val assemble_arg = assemble_iterm
           (fn c => fn _ => fn ts => nbe_apps_constr idx_of c ts) NONE;
-        val assemble_rhs = assemble_iterm assemble_constapp match_cont ;
+        val assemble_rhs = assemble_iterm assemble_constapp match_cont;
         val (samepairs, args') = subst_nonlin_vars args;
         val s_args = map assemble_arg args';
         val s_rhs = if null samepairs then assemble_rhs rhs
@@ -357,6 +360,7 @@
     val deps_vars = ml_list (map (nbe_fun 0) deps);
   in ml_abs deps_vars (ml_Let (ml_fundefs (flat fun_vars)) (ml_list fun_vals)) end;
 
+
 (* code compilation *)
 
 fun compile_eqnss _ gr raw_deps [] = []
@@ -457,6 +461,7 @@
     |> (fn t => apps t (rev dict_frees))
   end;
 
+
 (* reification *)
 
 fun typ_of_itype program vs (ityco `%% itys) =
@@ -480,9 +485,9 @@
       | is_dict (DFree _) = true
       | is_dict _ = false;
     fun const_of_idx idx = (case (Graph.get_node program o the o Inttab.lookup idx_tab) idx
-         of Code_Thingol.Fun (c, _) => c
-          | Code_Thingol.Datatypecons (c, _) => c
-          | Code_Thingol.Classparam (c, _) => c);
+     of Code_Thingol.Fun (c, _) => c
+      | Code_Thingol.Datatypecons (c, _) => c
+      | Code_Thingol.Classparam (c, _) => c);
     fun of_apps bounds (t, ts) =
       fold_map (of_univ bounds) ts
       #>> (fn ts' => list_comb (t, rev ts'))
@@ -503,6 +508,7 @@
           |-> (fn t' => pair (Term.Abs ("u", dummyT, t')))
   in of_univ 0 t 0 |> fst end;
 
+
 (* function store *)
 
 structure Nbe_Functions = Code_Data
@@ -511,6 +517,7 @@
   val empty = (Code_Thingol.empty_naming, (Graph.empty, (0, Inttab.empty)));
 );
 
+
 (* compilation, evaluation and reification *)
 
 fun compile_eval thy naming program vs_t deps =
@@ -524,6 +531,7 @@
     |> term_of_univ thy program idx_tab
   end;
 
+
 (* evaluation with type reconstruction *)
 
 fun normalize thy naming program ((vs0, (vs, ty)), t) deps =
@@ -593,6 +601,7 @@
 
 fun norm thy = lift_triv_classes_rew thy (no_frees_rew (Code_Thingol.eval thy I (normalize thy)));
 
+
 (* evaluation command *)
 
 fun norm_print_term ctxt modes t =