author wenzelm Wed, 06 Dec 2000 20:05:58 +0100 changeset 10614 d5c14e205c24 parent 10613 78b1d6c3ee9c child 10615 163b265d3d83
 src/HOL/IsaMakefile file | annotate | diff | comparison | revisions src/HOL/Library/Rational_Numbers.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/IsaMakefile	Wed Dec 06 19:10:36 2000 +0100
+++ b/src/HOL/IsaMakefile	Wed Dec 06 20:05:58 2000 +0100
@@ -40,7 +40,7 @@
HOL-W0 \
HOL-ex
# ^ this is the sort position
-
+
all: test images

@@ -166,8 +166,8 @@
\$(LOG)/HOL-Library.gz: \$(OUT)/HOL Library/Accessible_Part.thy \
Library/Library.thy Library/List_Prefix.thy Library/Multiset.thy \
-  Library/ROOT.ML Library/While_Combinator.thy \
-  Library/While_Combinator_Example.thy
+  Library/Rational_Numbers.thy Library/ROOT.ML \
+  Library/While_Combinator.thy Library/While_Combinator_Example.thy
@\$(ISATOOL) usedir \$(OUT)/HOL Library

```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Rational_Numbers.thy	Wed Dec 06 20:05:58 2000 +0100
@@ -0,0 +1,657 @@
+
+  \title{Rational numbers}
+  \author{Markus Wenzel}
+*}
+
+theory Rational_Numbers = Quotient + Ring_and_Field:
+
+subsection {* Fractions *}
+
+subsubsection {* The type of fractions *}
+
+typedef fraction = "{(a, b) :: int \<times> int | a b. b \<noteq> 0}"
+proof
+  show "(0, #1) \<in> ?fraction" by simp
+qed
+
+constdefs
+  fract :: "int => int => fraction"
+  "fract a b == Abs_fraction (a, b)"
+  num :: "fraction => int"
+  "num Q == fst (Rep_fraction Q)"
+  den :: "fraction => int"
+  "den Q == snd (Rep_fraction Q)"
+
+lemma fract_num [simp]: "b \<noteq> 0 ==> num (fract a b) = a"
+  by (simp add: fract_def num_def fraction_def Abs_fraction_inverse)
+
+lemma fract_den [simp]: "b \<noteq> 0 ==> den (fract a b) = b"
+  by (simp add: fract_def den_def fraction_def Abs_fraction_inverse)
+
+lemma fraction_cases [case_names fract, cases type: fraction]:
+  "(!!a b. Q = fract a b ==> b \<noteq> 0 ==> C) ==> C"
+proof -
+  assume r: "!!a b. Q = fract a b ==> b \<noteq> 0 ==> C"
+  obtain a b where "Q = fract a b" and "b \<noteq> 0"
+    by (cases Q) (auto simp add: fract_def fraction_def)
+  thus C by (rule r)
+qed
+
+lemma fraction_induct [case_names fract, induct type: fraction]:
+    "(!!a b. b \<noteq> 0 ==> P (fract a b)) ==> P Q"
+  by (cases Q) simp
+
+
+subsubsection {* Equivalence of fractions *}
+
+instance fraction :: eqv ..
+
+  equiv_fraction_def: "Q \<sim> R == num Q * den R = num R * den Q"
+
+lemma equiv_fraction_iff:
+    "b \<noteq> 0 ==> b' \<noteq> 0 ==> (fract a b \<sim> fract a' b') = (a * b' = a' * b)"
+
+lemma equiv_fractionI [intro]:
+    "a * b' = a' * b ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> fract a b \<sim> fract a' b'"
+  by (insert equiv_fraction_iff) blast
+
+lemma equiv_fractionD [dest]:
+    "fract a b \<sim> fract a' b' ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> a * b' = a' * b"
+  by (insert equiv_fraction_iff) blast
+
+instance fraction :: equiv
+proof
+  fix Q R S :: fraction
+  {
+    show "Q \<sim> Q"
+    proof (induct Q)
+      fix a b :: int
+      assume "b \<noteq> 0" and "b \<noteq> 0"
+      with refl show "fract a b \<sim> fract a b" ..
+    qed
+  next
+    assume "Q \<sim> R" and "R \<sim> S"
+    show "Q \<sim> S"
+    proof (insert prems, induct Q, induct R, induct S)
+      fix a b a' b' a'' b'' :: int
+      assume b: "b \<noteq> 0" and b': "b' \<noteq> 0" and b'': "b'' \<noteq> 0"
+      assume "fract a b \<sim> fract a' b'" hence eq1: "a * b' = a' * b" ..
+      assume "fract a' b' \<sim> fract a'' b''" hence eq2: "a' * b'' = a'' * b'" ..
+      have "a * b'' = a'' * b"
+      proof cases
+        assume "a' = 0"
+        with b' eq1 eq2 have "a = 0 \<and> a'' = 0" by auto
+        thus ?thesis by simp
+      next
+        assume a': "a' \<noteq> 0"
+        from eq1 eq2 have "(a * b') * (a' * b'') = (a' * b) * (a'' * b')" by simp
+        hence "(a * b'') * (a' * b') = (a'' * b) * (a' * b')" by (simp only: zmult_ac)
+        with a' b' show ?thesis by simp
+      qed
+      thus "fract a b \<sim> fract a'' b''" ..
+    qed
+  next
+    show "Q \<sim> R ==> R \<sim> Q"
+    proof (induct Q, induct R)
+      fix a b a' b' :: int
+      assume b: "b \<noteq> 0" and b': "b' \<noteq> 0"
+      assume "fract a b \<sim> fract a' b'"
+      hence "a * b' = a' * b" ..
+      hence "a' * b = a * b'" ..
+      thus "fract a' b' \<sim> fract a b" ..
+    qed
+  }
+qed
+
+lemma eq_fraction_iff:
+    "b \<noteq> 0 ==> b' \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>) = (a * b' = a' * b)"
+  by (simp add: equiv_fraction_iff quot_equality)
+
+lemma eq_fractionI [intro]:
+    "a * b' = a' * b ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> \<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>"
+  by (insert eq_fraction_iff) blast
+
+lemma eq_fractionD [dest]:
+    "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> a * b' = a' * b"
+  by (insert eq_fraction_iff) blast
+
+
+subsubsection {* Operations on fractions *}
+
+text {*
+ We define the basic arithmetic operations on fractions and
+ demonstrate their ``well-definedness'', i.e.\ congruence with respect
+ to equivalence of fractions.
+*}
+
+instance fraction :: zero ..
+instance fraction :: plus ..
+instance fraction :: minus ..
+instance fraction :: times ..
+instance fraction :: inverse ..
+instance fraction :: ord ..
+
+  zero_fraction_def: "0 == fract 0 #1"
+  add_fraction_def: "Q + R ==
+    fract (num Q * den R + num R * den Q) (den Q * den R)"
+  minus_fraction_def: "-Q == fract (-(num Q)) (den Q)"
+  mult_fraction_def: "Q * R == fract (num Q * num R) (den Q * den R)"
+  inverse_fraction_def: "inverse Q == fract (den Q) (num Q)"
+  le_fraction_def: "Q \<le> R ==
+    (num Q * den R) * (den Q * den R) \<le> (num R * den Q) * (den Q * den R)"
+
+lemma is_zero_fraction_iff: "b \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>0\<rfloor>) = (a = 0)"
+  by (simp add: zero_fraction_def eq_fraction_iff)
+
+  "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
+    ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
+    ==> \<lfloor>fract a b + fract c d\<rfloor> = \<lfloor>fract a' b' + fract c' d'\<rfloor>"
+proof -
+  assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
+  assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
+  assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
+  have "\<lfloor>fract (a * d + c * b) (b * d)\<rfloor> = \<lfloor>fract (a' * d' + c' * b') (b' * d')\<rfloor>"
+  proof
+    show "(a * d + c * b) * (b' * d') = (a' * d' + c' * b') * (b * d)"
+      (is "?lhs = ?rhs")
+    proof -
+      have "?lhs = (a * b') * (d * d') + (c * d') * (b * b')"
+        by (simp add: int_distrib zmult_ac)
+      also have "... = (a' * b) * (d * d') + (c' * d) * (b * b')"
+        by (simp only: eq1 eq2)
+      also have "... = ?rhs"
+        by (simp add: int_distrib zmult_ac)
+      finally show "?lhs = ?rhs" .
+    qed
+    from neq show "b * d \<noteq> 0" by simp
+    from neq show "b' * d' \<noteq> 0" by simp
+  qed
+qed
+
+theorem minus_fraction_cong:
+  "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0
+    ==> \<lfloor>-(fract a b)\<rfloor> = \<lfloor>-(fract a' b')\<rfloor>"
+proof -
+  assume neq: "b \<noteq> 0"  "b' \<noteq> 0"
+  assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>"
+  hence "a * b' = a' * b" ..
+  hence "-a * b' = -a' * b" by simp
+  hence "\<lfloor>fract (-a) b\<rfloor> = \<lfloor>fract (-a') b'\<rfloor>" ..
+  with neq show ?thesis by (simp add: minus_fraction_def)
+qed
+
+theorem mult_fraction_cong:
+  "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
+    ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
+    ==> \<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>"
+proof -
+  assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
+  assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
+  assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
+  have "\<lfloor>fract (a * c) (b * d)\<rfloor> = \<lfloor>fract (a' * c') (b' * d')\<rfloor>"
+  proof
+    from eq1 eq2 have "(a * b') * (c * d') = (a' * b) * (c' * d)" by simp
+    thus "(a * c) * (b' * d') = (a' * c') * (b * d)" by (simp add: zmult_ac)
+    from neq show "b * d \<noteq> 0" by simp
+    from neq show "b' * d' \<noteq> 0" by simp
+  qed
+  with neq show "\<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>"
+qed
+
+theorem inverse_fraction_cong:
+  "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor> ==> \<lfloor>fract a' b'\<rfloor> \<noteq> \<lfloor>0\<rfloor>
+    ==> b \<noteq> 0 ==> b' \<noteq> 0
+    ==> \<lfloor>inverse (fract a b)\<rfloor> = \<lfloor>inverse (fract a' b')\<rfloor>"
+proof -
+  assume neq: "b \<noteq> 0"  "b' \<noteq> 0"
+  assume "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>" and "\<lfloor>fract a' b'\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
+  with neq obtain "a \<noteq> 0" and "a' \<noteq> 0" by (simp add: is_zero_fraction_iff)
+  assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>"
+  hence "a * b' = a' * b" ..
+  hence "b * a' = b' * a" by (simp only: zmult_ac)
+  hence "\<lfloor>fract b a\<rfloor> = \<lfloor>fract b' a'\<rfloor>" ..
+  with neq show ?thesis by (simp add: inverse_fraction_def)
+qed
+
+theorem le_fraction_cong:
+  "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
+    ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
+    ==> (fract a b \<le> fract c d) = (fract a' b' \<le> fract c' d')"
+proof -
+  assume neq: "b \<noteq> 0"  "b' \<noteq> 0"  "d \<noteq> 0"  "d' \<noteq> 0"
+  assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
+  assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence 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: int_less_le)
+      hence "?le a b c d =
+          ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
+      also have "... = ?le (a * x) (b * x) c d"
+      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')"
+  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'"
+  also from D have "... = ?le a' b' c' d'"
+    by (rule le_factor [symmetric])
+  finally have "?le a b c d = ?le a' b' c' d'" .
+  with neq show ?thesis by (simp add: le_fraction_def)
+qed
+
+
+subsection {* Rational numbers *}
+
+subsubsection {* The type of rational numbers *}
+
+typedef (Rat)
+  rat = "UNIV :: fraction quot set" ..
+
+lemma RatI [intro, simp]: "Q \<in> Rat"
+
+constdefs
+  fraction_of :: "rat => fraction"
+  "fraction_of q == pick (Rep_Rat q)"
+  rat_of :: "fraction => rat"
+  "rat_of Q == Abs_Rat \<lfloor>Q\<rfloor>"
+
+theorem rat_of_equality [iff?]: "(rat_of Q = rat_of Q') = (\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor>)"
+  by (simp add: rat_of_def Abs_Rat_inject)
+
+lemma rat_of: "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> rat_of Q = rat_of Q'" ..
+
+constdefs
+  Fract :: "int => int => rat"
+  "Fract a b == rat_of (fract a b)"
+
+theorem Fract_inverse: "\<lfloor>fraction_of (Fract a b)\<rfloor> = \<lfloor>fract a b\<rfloor>"
+  by (simp add: fraction_of_def rat_of_def Fract_def Abs_Rat_inverse pick_inverse)
+
+theorem Fract_equality [iff?]:
+    "(Fract a b = Fract c d) = (\<lfloor>fract a b\<rfloor> = \<lfloor>fract c d\<rfloor>)"
+  by (simp add: Fract_def rat_of_equality)
+
+theorem eq_rat:
+    "b \<noteq> 0 ==> d \<noteq> 0 ==> (Fract a b = Fract c d) = (a * d = c * b)"
+  by (simp add: Fract_equality eq_fraction_iff)
+
+theorem Rat_cases [case_names Fract, cases type: rat]:
+  "(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C"
+proof -
+  assume r: "!!a b. q = Fract a b ==> b \<noteq> 0 ==> C"
+  obtain x where "q = Abs_Rat x" by (cases q)
+  moreover obtain Q where "x = \<lfloor>Q\<rfloor>" by (cases x)
+  moreover obtain a b where "Q = fract a b" and "b \<noteq> 0" by (cases Q)
+  ultimately have "q = Fract a b" by (simp only: Fract_def rat_of_def)
+  thus ?thesis by (rule r)
+qed
+
+theorem Rat_induct [case_names Fract, induct type: rat]:
+    "(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q"
+  by (cases q) simp
+
+
+subsubsection {* Canonical function definitions *}
+
+text {*
+  Note that the unconditional version below is much easier to read.
+*}
+
+theorem rat_cond_function:
+  "(!!q r. P \<lfloor>fraction_of q\<rfloor> \<lfloor>fraction_of r\<rfloor> ==>
+      f q r == g (fraction_of q) (fraction_of r)) ==>
+    (!!a b a' b' c d c' d'.
+      \<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==>
+      P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==> P \<lfloor>fract a' b'\<rfloor> \<lfloor>fract c' d'\<rfloor> ==>
+      b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==>
+      g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==>
+    P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==>
+      f (Fract a b) (Fract c d) = g (fract a b) (fract c d)"
+  (is "PROP ?eq ==> PROP ?cong ==> ?P ==> _")
+proof -
+  assume eq: "PROP ?eq" and cong: "PROP ?cong" and P: ?P
+  have "f (Abs_Rat \<lfloor>fract a b\<rfloor>) (Abs_Rat \<lfloor>fract c d\<rfloor>) = g (fract a b) (fract c d)"
+  proof (rule quot_cond_function)
+    fix X Y assume "P X Y"
+    with eq show "f (Abs_Rat X) (Abs_Rat Y) == g (pick X) (pick Y)"
+      by (simp add: fraction_of_def pick_inverse Abs_Rat_inverse)
+  next
+    fix Q Q' R R' :: fraction
+    show "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> \<lfloor>R\<rfloor> = \<lfloor>R'\<rfloor> ==>
+        P \<lfloor>Q\<rfloor> \<lfloor>R\<rfloor> ==> P \<lfloor>Q'\<rfloor> \<lfloor>R'\<rfloor> ==> g Q R = g Q' R'"
+      by (induct Q, induct Q', induct R, induct R') (rule cong)
+  qed
+  thus ?thesis by (unfold Fract_def rat_of_def)
+qed
+
+theorem rat_function:
+  "(!!q r. f q r == g (fraction_of q) (fraction_of r)) ==>
+    (!!a b a' b' c d c' d'.
+      \<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==>
+      b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==>
+      g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==>
+    f (Fract a b) (Fract c d) = g (fract a b) (fract c d)"
+proof -
+  case antecedent from this TrueI
+  show ?thesis by (rule rat_cond_function)
+qed
+
+
+subsubsection {* Standard operations on rational numbers *}
+
+instance rat :: zero ..
+instance rat :: plus ..
+instance rat :: minus ..
+instance rat :: times ..
+instance rat :: inverse ..
+instance rat :: ord ..
+instance rat :: number ..
+
+  zero_rat_def: "0 == rat_of 0"
+  add_rat_def: "q + r == rat_of (fraction_of q + fraction_of r)"
+  minus_rat_def: "-q == rat_of (-(fraction_of q))"
+  diff_rat_def: "q - r == q + (-(r::rat))"
+  mult_rat_def: "q * r == rat_of (fraction_of q * fraction_of r)"
+  inverse_rat_def: "q \<noteq> 0 ==> inverse q == rat_of (inverse (fraction_of q))"
+  divide_rat_def: "r \<noteq> 0 ==> q / r == q * inverse (r::rat)"
+  le_rat_def: "q \<le> r == fraction_of q \<le> fraction_of r"
+  less_rat_def: "q < r == q \<le> r \<and> q \<noteq> (r::rat)"
+  abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)"
+  number_of_rat_def: "number_of b == Fract (number_of b) #1"
+
+theorem zero_rat: "0 = Fract 0 #1"
+  by (simp add: zero_rat_def zero_fraction_def rat_of_def Fract_def)
+
+theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
+  Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
+proof -
+  have "Fract a b + Fract c d = rat_of (fract a b + fract c d)"
+  also
+  assume "b \<noteq> 0"  "d \<noteq> 0"
+  hence "fract a b + fract c d = fract (a * d + c * b) (b * d)"
+  finally show ?thesis by (unfold Fract_def)
+qed
+
+theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b"
+proof -
+  have "-(Fract a b) = rat_of (-(fract a b))"
+    by (rule rat_function, rule minus_rat_def, rule rat_of, rule minus_fraction_cong)
+  also assume "b \<noteq> 0" hence "-(fract a b) = fract (-a) b"
+  finally show ?thesis by (unfold Fract_def)
+qed
+
+theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
+    Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
+
+theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
+  Fract a b * Fract c d = Fract (a * c) (b * d)"
+proof -
+  have "Fract a b * Fract c d = rat_of (fract a b * fract c d)"
+    by (rule rat_function, rule mult_rat_def, rule rat_of, rule mult_fraction_cong)
+  also
+  assume "b \<noteq> 0"  "d \<noteq> 0"
+  hence "fract a b * fract c d = fract (a * c) (b * d)"
+  finally show ?thesis by (unfold Fract_def)
+qed
+
+theorem inverse_rat: "Fract a b \<noteq> 0 ==> b \<noteq> 0 ==>
+  inverse (Fract a b) = Fract b a"
+proof -
+  assume neq: "b \<noteq> 0" and nonzero: "Fract a b \<noteq> 0"
+  hence "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
+    by (simp add: zero_rat eq_rat is_zero_fraction_iff)
+  with _ inverse_fraction_cong [THEN rat_of]
+  have "inverse (Fract a b) = rat_of (inverse (fract a b))"
+  proof (rule rat_cond_function)
+    fix q assume cond: "\<lfloor>fraction_of q\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
+    have "q \<noteq> 0"
+    proof (cases q)
+      fix a b assume "b \<noteq> 0" and "q = Fract a b"
+      from this cond show ?thesis
+        by (simp add: Fract_inverse is_zero_fraction_iff zero_rat eq_rat)
+    qed
+    thus "inverse q == rat_of (inverse (fraction_of q))"
+      by (rule inverse_rat_def)
+  qed
+  also from neq nonzero have "inverse (fract a b) = fract b a"
+  finally show ?thesis by (unfold Fract_def)
+qed
+
+theorem divide_rat: "Fract c d \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==>
+  Fract a b / Fract c d = Fract (a * d) (b * c)"
+proof -
+  assume neq: "b \<noteq> 0"  "d \<noteq> 0" and nonzero: "Fract c d \<noteq> 0"
+  hence "c \<noteq> 0" by (simp add: zero_rat eq_rat)
+  with neq nonzero show ?thesis
+    by (simp add: divide_rat_def inverse_rat mult_rat)
+qed
+
+theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
+  (Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
+proof -
+  have "(Fract a b \<le> Fract c d) = (fract a b \<le> fract c d)"
+    by (rule rat_function, rule le_rat_def, rule le_fraction_cong)
+  also
+  assume "b \<noteq> 0"  "d \<noteq> 0"
+  hence "(fract a b \<le> fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
+  finally show ?thesis .
+qed
+
+theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
+    (Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))"
+  by (simp add: less_rat_def le_rat eq_rat int_less_le)
+
+theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
+  by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat)
+    (auto simp add: zmult_less_0_iff int_0_less_mult_iff int_le_less split: zabs_split)
+
+
+subsubsection {* The ordered field of rational numbers *}
+
+instance rat :: field
+proof
+  fix q r s :: rat
+  show "(q + r) + s = q + (r + s)"
+  show "q + r = r + q"
+  show "0 + q = q"
+  show "q + (-q) = 0"
+  show "q - r = q + (-r)"
+  show "(0::rat) = #0"
+    by (simp add: zero_rat number_of_rat_def)
+  show "(q * r) * s = q * (r * s)"
+    by (induct q, induct r, induct s) (simp add: mult_rat zmult_ac)
+  show "q * r = r * q"
+    by (induct q, induct r) (simp add: mult_rat zmult_ac)
+  show "#1 * q = q"
+    by (induct q) (simp add: number_of_rat_def mult_rat)
+  show "(q + r) * s = q * s + r * s"
+    by (induct q, induct r, induct s) (simp add: add_rat mult_rat eq_rat int_distrib)
+  show "q \<noteq> 0 ==> inverse q * q = #1"
+    by (induct q) (simp add: inverse_rat mult_rat number_of_rat_def zero_rat eq_rat)
+  show "r \<noteq> 0 ==> q / r = q * inverse r"
+    by (induct q, induct r) (simp add: mult_rat divide_rat inverse_rat zero_rat eq_rat)
+qed
+
+instance rat :: linorder
+proof
+  fix q r s :: rat
+  {
+    assume "q \<le> r" and "r \<le> s"
+    show "q \<le> s"
+    proof (insert prems, induct q, induct r, induct s)
+      fix a b c d e f :: int
+      assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
+      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
+      show "Fract a b \<le> Fract e f"
+      proof -
+        from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
+          by (auto simp add: int_less_le)
+        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)"
+          with ff show ?thesis by (simp add: zmult_zle_cancel2)
+        qed
+        also have "... = (c * f) * (d * f) * (b * b)"
+          by (simp only: zmult_ac)
+        also have "... \<le> (e * d) * (d * f) * (b * b)"
+        proof -
+          from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
+          with bb show ?thesis by (simp add: zmult_zle_cancel2)
+        qed
+        finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
+          by (simp only: zmult_ac)
+        with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
+        with neq show ?thesis by (simp add: le_rat)
+      qed
+    qed
+  next
+    assume "q \<le> r" and "r \<le> q"
+    show "q = r"
+    proof (insert prems, induct q, induct r)
+      fix a b c d :: int
+      assume neq: "b \<noteq> 0"  "d \<noteq> 0"
+      assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
+      show "Fract a b = Fract c d"
+      proof -
+        from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
+        also have "... \<le> (a * d) * (b * d)"
+        proof -
+          from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
+          thus ?thesis by (simp only: zmult_ac)
+        qed
+        finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
+        moreover from neq have "b * d \<noteq> 0" by simp
+        ultimately have "a * d = c * b" by simp
+        with neq show ?thesis by (simp add: eq_rat)
+      qed
+    qed
+  next
+    show "q \<le> q"
+      by (induct q) (simp add: le_rat)
+    show "(q < r) = (q \<le> r \<and> q \<noteq> r)"
+      by (simp only: less_rat_def)
+    show "q \<le> r \<or> r \<le> q"
+      by (induct q, induct r) (simp add: le_rat zmult_ac, arith)
+  }
+qed
+
+instance rat :: ordered_field
+proof
+  fix q r s :: rat
+  show "q \<le> r ==> s + q \<le> s + r"
+  proof (induct q, induct r, induct s)
+    fix a b c d e f :: int
+    assume neq: "b \<noteq> 0"  "d \<noteq> 0"  "f \<noteq> 0"
+    assume le: "Fract a b \<le> Fract c d"
+    show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
+    proof -
+      let ?F = "f * f" from neq have F: "0 < ?F"
+        by (auto simp add: int_less_le)
+      from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
+      with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
+      with neq show ?thesis by (simp add: add_rat le_rat zmult_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 less_rat le_rat int_less_le eq_rat)
+      moreover from neq have "0 < ?F"
+        by (auto simp add: int_less_le)
+      moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
+      ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
+      with neq show ?thesis
+        by (simp add: less_rat mult_rat zmult_ac)
+    qed
+  qed
+  show "\<bar>q\<bar> = (if q < 0 then -q else q)"
+    by (simp only: abs_rat_def)
+qed
+
+
+subsection {* Embedding integers *}
+
+constdefs    (* FIXME generalize int to any numeric subtype *)
+  rat :: "int => rat"
+  "rat z == Fract z #1"
+  int_set :: "rat set"    ("\<int>")
+  "\<int> == range rat"
+
+lemma rat_inject: "(rat z = rat w) = (z = w)"
+proof
+  assume "rat z = rat w"
+  hence "Fract z #1 = Fract w #1" by (unfold rat_def)
+  hence "\<lfloor>fract z #1\<rfloor> = \<lfloor>fract w #1\<rfloor>" ..
+  thus "z = w" by auto
+next
+  assume "z = w"
+  thus "rat z = rat w" by simp
+qed
+
+lemma int_set_cases [case_names rat, cases set: int_set]:
+  "q \<in> \<int> ==> (!!z. q = rat z ==> C) ==> C"
+proof (unfold int_set_def)
+  assume "!!z. q = rat z ==> C"
+  assume "q \<in> range rat" thus C ..
+qed
+
+lemma int_set_induct [case_names rat, induct set: int_set]:
+  "q \<in> \<int> ==> (!!z. P (rat z)) ==> P q"
+  by (rule int_set_cases) auto
+
+theorem number_of_rat: "number_of b = rat (number_of b)"
+  by (simp only: number_of_rat_def rat_def)
+
+end```