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(* Title: HOL/Library/Rational_Numbers.thy
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ID: $Id$
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Author: Markus Wenzel, TU Muenchen
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License: GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {*
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\title{Rational numbers}
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\author{Markus Wenzel}
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*}
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theory Rational_Numbers = Quotient + Ring_and_Field:
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subsection {* Fractions *}
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subsubsection {* The type of fractions *}
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typedef fraction = "{(a, b) :: int \<times> int | a b. b \<noteq> 0}"
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proof
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show "(0, 1) \<in> ?fraction" by simp
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qed
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constdefs
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fract :: "int => int => fraction"
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"fract a b == Abs_fraction (a, b)"
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num :: "fraction => int"
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"num Q == fst (Rep_fraction Q)"
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den :: "fraction => int"
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"den Q == snd (Rep_fraction Q)"
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lemma fract_num [simp]: "b \<noteq> 0 ==> num (fract a b) = a"
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by (simp add: fract_def num_def fraction_def Abs_fraction_inverse)
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lemma fract_den [simp]: "b \<noteq> 0 ==> den (fract a b) = b"
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by (simp add: fract_def den_def fraction_def Abs_fraction_inverse)
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lemma fraction_cases [case_names fract, cases type: fraction]:
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"(!!a b. Q = fract a b ==> b \<noteq> 0 ==> C) ==> C"
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proof -
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assume r: "!!a b. Q = fract a b ==> b \<noteq> 0 ==> C"
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obtain a b where "Q = fract a b" and "b \<noteq> 0"
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by (cases Q) (auto simp add: fract_def fraction_def)
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thus C by (rule r)
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qed
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lemma fraction_induct [case_names fract, induct type: fraction]:
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"(!!a b. b \<noteq> 0 ==> P (fract a b)) ==> P Q"
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by (cases Q) simp
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subsubsection {* Equivalence of fractions *}
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instance fraction :: eqv ..
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defs (overloaded)
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equiv_fraction_def: "Q \<sim> R == num Q * den R = num R * den Q"
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lemma equiv_fraction_iff [iff]:
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"b \<noteq> 0 ==> b' \<noteq> 0 ==> (fract a b \<sim> fract a' b') = (a * b' = a' * b)"
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by (simp add: equiv_fraction_def)
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instance fraction :: equiv
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proof
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fix Q R S :: fraction
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{
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show "Q \<sim> Q"
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proof (induct Q)
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fix a b :: int
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assume "b \<noteq> 0" and "b \<noteq> 0"
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with refl show "fract a b \<sim> fract a b" ..
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qed
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next
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assume "Q \<sim> R" and "R \<sim> S"
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show "Q \<sim> S"
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proof (insert prems, induct Q, induct R, induct S)
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fix a b a' b' a'' b'' :: int
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assume b: "b \<noteq> 0" and b': "b' \<noteq> 0" and b'': "b'' \<noteq> 0"
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assume "fract a b \<sim> fract a' b'" hence eq1: "a * b' = a' * b" ..
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assume "fract a' b' \<sim> fract a'' b''" hence eq2: "a' * b'' = a'' * b'" ..
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have "a * b'' = a'' * b"
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proof cases
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assume "a' = 0"
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with b' eq1 eq2 have "a = 0 \<and> a'' = 0" by auto
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thus ?thesis by simp
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next
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assume a': "a' \<noteq> 0"
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from eq1 eq2 have "(a * b') * (a' * b'') = (a' * b) * (a'' * b')" by simp
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hence "(a * b'') * (a' * b') = (a'' * b) * (a' * b')" by (simp only: zmult_ac)
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with a' b' show ?thesis by simp
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qed
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thus "fract a b \<sim> fract a'' b''" ..
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qed
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next
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show "Q \<sim> R ==> R \<sim> Q"
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proof (induct Q, induct R)
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fix a b a' b' :: int
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assume b: "b \<noteq> 0" and b': "b' \<noteq> 0"
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assume "fract a b \<sim> fract a' b'"
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hence "a * b' = a' * b" ..
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hence "a' * b = a * b'" ..
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thus "fract a' b' \<sim> fract a b" ..
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qed
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}
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qed
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lemma eq_fraction_iff [iff]:
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"b \<noteq> 0 ==> b' \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>) = (a * b' = a' * b)"
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by (simp add: equiv_fraction_iff quot_equality)
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subsubsection {* Operations on fractions *}
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text {*
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We define the basic arithmetic operations on fractions and
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demonstrate their ``well-definedness'', i.e.\ congruence with respect
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to equivalence of fractions.
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*}
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instance fraction :: zero ..
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instance fraction :: one ..
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instance fraction :: plus ..
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instance fraction :: minus ..
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instance fraction :: times ..
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instance fraction :: inverse ..
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instance fraction :: ord ..
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defs (overloaded)
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zero_fraction_def: "0 == fract 0 1"
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one_fraction_def: "1 == fract 1 1"
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add_fraction_def: "Q + R ==
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fract (num Q * den R + num R * den Q) (den Q * den R)"
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minus_fraction_def: "-Q == fract (-(num Q)) (den Q)"
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mult_fraction_def: "Q * R == fract (num Q * num R) (den Q * den R)"
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inverse_fraction_def: "inverse Q == fract (den Q) (num Q)"
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le_fraction_def: "Q \<le> R ==
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(num Q * den R) * (den Q * den R) \<le> (num R * den Q) * (den Q * den R)"
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lemma is_zero_fraction_iff: "b \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>0\<rfloor>) = (a = 0)"
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by (simp add: zero_fraction_def eq_fraction_iff)
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theorem add_fraction_cong:
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"\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
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==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
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==> \<lfloor>fract a b + fract c d\<rfloor> = \<lfloor>fract a' b' + fract c' d'\<rfloor>"
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proof -
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assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0"
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assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
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assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
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have "\<lfloor>fract (a * d + c * b) (b * d)\<rfloor> = \<lfloor>fract (a' * d' + c' * b') (b' * d')\<rfloor>"
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proof
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show "(a * d + c * b) * (b' * d') = (a' * d' + c' * b') * (b * d)"
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(is "?lhs = ?rhs")
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proof -
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have "?lhs = (a * b') * (d * d') + (c * d') * (b * b')"
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by (simp add: int_distrib zmult_ac)
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also have "... = (a' * b) * (d * d') + (c' * d) * (b * b')"
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by (simp only: eq1 eq2)
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also have "... = ?rhs"
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by (simp add: int_distrib zmult_ac)
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finally show "?lhs = ?rhs" .
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qed
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from neq show "b * d \<noteq> 0" by simp
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from neq show "b' * d' \<noteq> 0" by simp
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qed
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with neq show ?thesis by (simp add: add_fraction_def)
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qed
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theorem minus_fraction_cong:
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"\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0
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==> \<lfloor>-(fract a b)\<rfloor> = \<lfloor>-(fract a' b')\<rfloor>"
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proof -
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assume neq: "b \<noteq> 0" "b' \<noteq> 0"
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assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>"
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hence "a * b' = a' * b" ..
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hence "-a * b' = -a' * b" by simp
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hence "\<lfloor>fract (-a) b\<rfloor> = \<lfloor>fract (-a') b'\<rfloor>" ..
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with neq show ?thesis by (simp add: minus_fraction_def)
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qed
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theorem mult_fraction_cong:
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"\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
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==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
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==> \<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>"
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proof -
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assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0"
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assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
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assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
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have "\<lfloor>fract (a * c) (b * d)\<rfloor> = \<lfloor>fract (a' * c') (b' * d')\<rfloor>"
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proof
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from eq1 eq2 have "(a * b') * (c * d') = (a' * b) * (c' * d)" by simp
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thus "(a * c) * (b' * d') = (a' * c') * (b * d)" by (simp add: zmult_ac)
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from neq show "b * d \<noteq> 0" by simp
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from neq show "b' * d' \<noteq> 0" by simp
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qed
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with neq show "\<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>"
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by (simp add: mult_fraction_def)
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qed
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theorem inverse_fraction_cong:
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"\<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>
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==> b \<noteq> 0 ==> b' \<noteq> 0
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==> \<lfloor>inverse (fract a b)\<rfloor> = \<lfloor>inverse (fract a' b')\<rfloor>"
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proof -
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assume neq: "b \<noteq> 0" "b' \<noteq> 0"
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assume "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>" and "\<lfloor>fract a' b'\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
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with neq obtain "a \<noteq> 0" and "a' \<noteq> 0" by (simp add: is_zero_fraction_iff)
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assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>"
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hence "a * b' = a' * b" ..
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hence "b * a' = b' * a" by (simp only: zmult_ac)
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hence "\<lfloor>fract b a\<rfloor> = \<lfloor>fract b' a'\<rfloor>" ..
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with neq show ?thesis by (simp add: inverse_fraction_def)
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qed
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theorem le_fraction_cong:
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"\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
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==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
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==> (fract a b \<le> fract c d) = (fract a' b' \<le> fract c' d')"
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proof -
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assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0"
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assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
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assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
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let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
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{
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fix a b c d x :: int assume x: "x \<noteq> 0"
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have "?le a b c d = ?le (a * x) (b * x) c d"
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proof -
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from x have "0 < x * x" by (auto simp add: int_less_le)
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hence "?le a b c d =
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((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))"
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by (simp add: zmult_zle_cancel2)
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also have "... = ?le (a * x) (b * x) c d"
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by (simp add: zmult_ac)
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finally show ?thesis .
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qed
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} note le_factor = this
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let ?D = "b * d" and ?D' = "b' * d'"
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from neq have D: "?D \<noteq> 0" by simp
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from neq have "?D' \<noteq> 0" by simp
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hence "?le a b c d = ?le (a * ?D') (b * ?D') c d"
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by (rule le_factor)
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also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')"
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by (simp add: zmult_ac)
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also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')"
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by (simp only: eq1 eq2)
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also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
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by (simp add: zmult_ac)
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also from D have "... = ?le a' b' c' d'"
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by (rule le_factor [symmetric])
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finally have "?le a b c d = ?le a' b' c' d'" .
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with neq show ?thesis by (simp add: le_fraction_def)
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qed
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subsection {* Rational numbers *}
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subsubsection {* The type of rational numbers *}
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typedef (Rat)
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rat = "UNIV :: fraction quot set" ..
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lemma RatI [intro, simp]: "Q \<in> Rat"
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by (simp add: Rat_def)
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constdefs
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fraction_of :: "rat => fraction"
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"fraction_of q == pick (Rep_Rat q)"
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rat_of :: "fraction => rat"
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"rat_of Q == Abs_Rat \<lfloor>Q\<rfloor>"
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theorem rat_of_equality [iff?]: "(rat_of Q = rat_of Q') = (\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor>)"
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by (simp add: rat_of_def Abs_Rat_inject)
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lemma rat_of: "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> rat_of Q = rat_of Q'" ..
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constdefs
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Fract :: "int => int => rat"
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"Fract a b == rat_of (fract a b)"
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theorem Fract_inverse: "\<lfloor>fraction_of (Fract a b)\<rfloor> = \<lfloor>fract a b\<rfloor>"
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by (simp add: fraction_of_def rat_of_def Fract_def Abs_Rat_inverse pick_inverse)
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theorem Fract_equality [iff?]:
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"(Fract a b = Fract c d) = (\<lfloor>fract a b\<rfloor> = \<lfloor>fract c d\<rfloor>)"
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by (simp add: Fract_def rat_of_equality)
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theorem eq_rat:
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"b \<noteq> 0 ==> d \<noteq> 0 ==> (Fract a b = Fract c d) = (a * d = c * b)"
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by (simp add: Fract_equality eq_fraction_iff)
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theorem Rat_cases [case_names Fract, cases type: rat]:
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"(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C"
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proof -
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assume r: "!!a b. q = Fract a b ==> b \<noteq> 0 ==> C"
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obtain x where "q = Abs_Rat x" by (cases q)
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moreover obtain Q where "x = \<lfloor>Q\<rfloor>" by (cases x)
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moreover obtain a b where "Q = fract a b" and "b \<noteq> 0" by (cases Q)
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ultimately have "q = Fract a b" by (simp only: Fract_def rat_of_def)
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thus ?thesis by (rule r)
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qed
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theorem Rat_induct [case_names Fract, induct type: rat]:
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"(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q"
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by (cases q) simp
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subsubsection {* Canonical function definitions *}
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text {*
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Note that the unconditional version below is much easier to read.
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*}
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theorem rat_cond_function:
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315 |
"(!!q r. P \<lfloor>fraction_of q\<rfloor> \<lfloor>fraction_of r\<rfloor> ==>
|
|
316 |
f q r == g (fraction_of q) (fraction_of r)) ==>
|
|
317 |
(!!a b a' b' c d c' d'.
|
|
318 |
\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==>
|
|
319 |
P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==> P \<lfloor>fract a' b'\<rfloor> \<lfloor>fract c' d'\<rfloor> ==>
|
|
320 |
b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==>
|
|
321 |
g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==>
|
|
322 |
P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==>
|
|
323 |
f (Fract a b) (Fract c d) = g (fract a b) (fract c d)"
|
|
324 |
(is "PROP ?eq ==> PROP ?cong ==> ?P ==> _")
|
|
325 |
proof -
|
|
326 |
assume eq: "PROP ?eq" and cong: "PROP ?cong" and P: ?P
|
|
327 |
have "f (Abs_Rat \<lfloor>fract a b\<rfloor>) (Abs_Rat \<lfloor>fract c d\<rfloor>) = g (fract a b) (fract c d)"
|
|
328 |
proof (rule quot_cond_function)
|
|
329 |
fix X Y assume "P X Y"
|
|
330 |
with eq show "f (Abs_Rat X) (Abs_Rat Y) == g (pick X) (pick Y)"
|
|
331 |
by (simp add: fraction_of_def pick_inverse Abs_Rat_inverse)
|
|
332 |
next
|
|
333 |
fix Q Q' R R' :: fraction
|
|
334 |
show "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> \<lfloor>R\<rfloor> = \<lfloor>R'\<rfloor> ==>
|
|
335 |
P \<lfloor>Q\<rfloor> \<lfloor>R\<rfloor> ==> P \<lfloor>Q'\<rfloor> \<lfloor>R'\<rfloor> ==> g Q R = g Q' R'"
|
|
336 |
by (induct Q, induct Q', induct R, induct R') (rule cong)
|
|
337 |
qed
|
|
338 |
thus ?thesis by (unfold Fract_def rat_of_def)
|
|
339 |
qed
|
|
340 |
|
|
341 |
theorem rat_function:
|
|
342 |
"(!!q r. f q r == g (fraction_of q) (fraction_of r)) ==>
|
|
343 |
(!!a b a' b' c d c' d'.
|
|
344 |
\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==>
|
|
345 |
b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==>
|
|
346 |
g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==>
|
|
347 |
f (Fract a b) (Fract c d) = g (fract a b) (fract c d)"
|
|
348 |
proof -
|
11549
|
349 |
case rule_context from this TrueI
|
10614
|
350 |
show ?thesis by (rule rat_cond_function)
|
|
351 |
qed
|
|
352 |
|
|
353 |
|
|
354 |
subsubsection {* Standard operations on rational numbers *}
|
|
355 |
|
|
356 |
instance rat :: zero ..
|
11913
|
357 |
instance rat :: one ..
|
10614
|
358 |
instance rat :: plus ..
|
|
359 |
instance rat :: minus ..
|
|
360 |
instance rat :: times ..
|
|
361 |
instance rat :: inverse ..
|
|
362 |
instance rat :: ord ..
|
|
363 |
instance rat :: number ..
|
|
364 |
|
|
365 |
defs (overloaded)
|
|
366 |
zero_rat_def: "0 == rat_of 0"
|
11913
|
367 |
one_rat_def: "1 == rat_of 1"
|
10614
|
368 |
add_rat_def: "q + r == rat_of (fraction_of q + fraction_of r)"
|
|
369 |
minus_rat_def: "-q == rat_of (-(fraction_of q))"
|
|
370 |
diff_rat_def: "q - r == q + (-(r::rat))"
|
|
371 |
mult_rat_def: "q * r == rat_of (fraction_of q * fraction_of r)"
|
|
372 |
inverse_rat_def: "q \<noteq> 0 ==> inverse q == rat_of (inverse (fraction_of q))"
|
|
373 |
divide_rat_def: "r \<noteq> 0 ==> q / r == q * inverse (r::rat)"
|
|
374 |
le_rat_def: "q \<le> r == fraction_of q \<le> fraction_of r"
|
|
375 |
less_rat_def: "q < r == q \<le> r \<and> q \<noteq> (r::rat)"
|
|
376 |
abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)"
|
11913
|
377 |
number_of_rat_def: "number_of b == Fract (number_of b) 1"
|
10614
|
378 |
|
11913
|
379 |
theorem zero_rat: "0 = Fract 0 1"
|
|
380 |
by (simp add: zero_rat_def zero_fraction_def rat_of_def Fract_def)
|
|
381 |
|
|
382 |
theorem one_rat: "1 = Fract 1 1"
|
|
383 |
by (simp add: one_rat_def one_fraction_def rat_of_def Fract_def)
|
10614
|
384 |
|
|
385 |
theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
|
|
386 |
Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
|
|
387 |
proof -
|
|
388 |
have "Fract a b + Fract c d = rat_of (fract a b + fract c d)"
|
|
389 |
by (rule rat_function, rule add_rat_def, rule rat_of, rule add_fraction_cong)
|
|
390 |
also
|
|
391 |
assume "b \<noteq> 0" "d \<noteq> 0"
|
|
392 |
hence "fract a b + fract c d = fract (a * d + c * b) (b * d)"
|
|
393 |
by (simp add: add_fraction_def)
|
|
394 |
finally show ?thesis by (unfold Fract_def)
|
|
395 |
qed
|
|
396 |
|
|
397 |
theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b"
|
|
398 |
proof -
|
|
399 |
have "-(Fract a b) = rat_of (-(fract a b))"
|
|
400 |
by (rule rat_function, rule minus_rat_def, rule rat_of, rule minus_fraction_cong)
|
|
401 |
also assume "b \<noteq> 0" hence "-(fract a b) = fract (-a) b"
|
|
402 |
by (simp add: minus_fraction_def)
|
|
403 |
finally show ?thesis by (unfold Fract_def)
|
|
404 |
qed
|
|
405 |
|
|
406 |
theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
|
|
407 |
Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
|
|
408 |
by (simp add: diff_rat_def add_rat minus_rat)
|
|
409 |
|
|
410 |
theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
|
|
411 |
Fract a b * Fract c d = Fract (a * c) (b * d)"
|
|
412 |
proof -
|
|
413 |
have "Fract a b * Fract c d = rat_of (fract a b * fract c d)"
|
|
414 |
by (rule rat_function, rule mult_rat_def, rule rat_of, rule mult_fraction_cong)
|
|
415 |
also
|
|
416 |
assume "b \<noteq> 0" "d \<noteq> 0"
|
|
417 |
hence "fract a b * fract c d = fract (a * c) (b * d)"
|
|
418 |
by (simp add: mult_fraction_def)
|
|
419 |
finally show ?thesis by (unfold Fract_def)
|
|
420 |
qed
|
|
421 |
|
|
422 |
theorem inverse_rat: "Fract a b \<noteq> 0 ==> b \<noteq> 0 ==>
|
|
423 |
inverse (Fract a b) = Fract b a"
|
|
424 |
proof -
|
|
425 |
assume neq: "b \<noteq> 0" and nonzero: "Fract a b \<noteq> 0"
|
|
426 |
hence "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
|
|
427 |
by (simp add: zero_rat eq_rat is_zero_fraction_iff)
|
|
428 |
with _ inverse_fraction_cong [THEN rat_of]
|
|
429 |
have "inverse (Fract a b) = rat_of (inverse (fract a b))"
|
|
430 |
proof (rule rat_cond_function)
|
|
431 |
fix q assume cond: "\<lfloor>fraction_of q\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
|
|
432 |
have "q \<noteq> 0"
|
|
433 |
proof (cases q)
|
|
434 |
fix a b assume "b \<noteq> 0" and "q = Fract a b"
|
|
435 |
from this cond show ?thesis
|
|
436 |
by (simp add: Fract_inverse is_zero_fraction_iff zero_rat eq_rat)
|
|
437 |
qed
|
|
438 |
thus "inverse q == rat_of (inverse (fraction_of q))"
|
|
439 |
by (rule inverse_rat_def)
|
|
440 |
qed
|
|
441 |
also from neq nonzero have "inverse (fract a b) = fract b a"
|
|
442 |
by (simp add: inverse_fraction_def)
|
|
443 |
finally show ?thesis by (unfold Fract_def)
|
|
444 |
qed
|
|
445 |
|
|
446 |
theorem divide_rat: "Fract c d \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==>
|
|
447 |
Fract a b / Fract c d = Fract (a * d) (b * c)"
|
|
448 |
proof -
|
|
449 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" and nonzero: "Fract c d \<noteq> 0"
|
|
450 |
hence "c \<noteq> 0" by (simp add: zero_rat eq_rat)
|
|
451 |
with neq nonzero show ?thesis
|
|
452 |
by (simp add: divide_rat_def inverse_rat mult_rat)
|
|
453 |
qed
|
|
454 |
|
|
455 |
theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
|
|
456 |
(Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
|
|
457 |
proof -
|
|
458 |
have "(Fract a b \<le> Fract c d) = (fract a b \<le> fract c d)"
|
|
459 |
by (rule rat_function, rule le_rat_def, rule le_fraction_cong)
|
|
460 |
also
|
|
461 |
assume "b \<noteq> 0" "d \<noteq> 0"
|
|
462 |
hence "(fract a b \<le> fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
|
|
463 |
by (simp add: le_fraction_def)
|
|
464 |
finally show ?thesis .
|
|
465 |
qed
|
|
466 |
|
|
467 |
theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
|
|
468 |
(Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))"
|
|
469 |
by (simp add: less_rat_def le_rat eq_rat int_less_le)
|
|
470 |
|
|
471 |
theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
|
|
472 |
by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat)
|
|
473 |
(auto simp add: zmult_less_0_iff int_0_less_mult_iff int_le_less split: zabs_split)
|
|
474 |
|
|
475 |
|
|
476 |
subsubsection {* The ordered field of rational numbers *}
|
|
477 |
|
|
478 |
instance rat :: field
|
|
479 |
proof
|
|
480 |
fix q r s :: rat
|
|
481 |
show "(q + r) + s = q + (r + s)"
|
|
482 |
by (induct q, induct r, induct s) (simp add: add_rat zadd_ac zmult_ac int_distrib)
|
|
483 |
show "q + r = r + q"
|
|
484 |
by (induct q, induct r) (simp add: add_rat zadd_ac zmult_ac)
|
|
485 |
show "0 + q = q"
|
|
486 |
by (induct q) (simp add: zero_rat add_rat)
|
10621
|
487 |
show "(-q) + q = 0"
|
10614
|
488 |
by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat)
|
|
489 |
show "q - r = q + (-r)"
|
|
490 |
by (induct q, induct r) (simp add: add_rat minus_rat diff_rat)
|
|
491 |
show "(q * r) * s = q * (r * s)"
|
|
492 |
by (induct q, induct r, induct s) (simp add: mult_rat zmult_ac)
|
|
493 |
show "q * r = r * q"
|
|
494 |
by (induct q, induct r) (simp add: mult_rat zmult_ac)
|
11913
|
495 |
show "1 * q = q"
|
|
496 |
by (induct q) (simp add: one_rat mult_rat)
|
10614
|
497 |
show "(q + r) * s = q * s + r * s"
|
|
498 |
by (induct q, induct r, induct s) (simp add: add_rat mult_rat eq_rat int_distrib)
|
11913
|
499 |
show "q \<noteq> 0 ==> inverse q * q = 1"
|
|
500 |
by (induct q) (simp add: inverse_rat mult_rat one_rat zero_rat eq_rat)
|
10614
|
501 |
show "r \<noteq> 0 ==> q / r = q * inverse r"
|
|
502 |
by (induct q, induct r) (simp add: mult_rat divide_rat inverse_rat zero_rat eq_rat)
|
|
503 |
qed
|
|
504 |
|
|
505 |
instance rat :: linorder
|
|
506 |
proof
|
|
507 |
fix q r s :: rat
|
|
508 |
{
|
|
509 |
assume "q \<le> r" and "r \<le> s"
|
|
510 |
show "q \<le> s"
|
|
511 |
proof (insert prems, induct q, induct r, induct s)
|
|
512 |
fix a b c d e f :: int
|
|
513 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
|
|
514 |
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f"
|
|
515 |
show "Fract a b \<le> Fract e f"
|
|
516 |
proof -
|
|
517 |
from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
|
|
518 |
by (auto simp add: int_less_le)
|
|
519 |
have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)"
|
|
520 |
proof -
|
|
521 |
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
|
|
522 |
by (simp add: le_rat)
|
|
523 |
with ff show ?thesis by (simp add: zmult_zle_cancel2)
|
|
524 |
qed
|
|
525 |
also have "... = (c * f) * (d * f) * (b * b)"
|
|
526 |
by (simp only: zmult_ac)
|
|
527 |
also have "... \<le> (e * d) * (d * f) * (b * b)"
|
|
528 |
proof -
|
|
529 |
from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)"
|
|
530 |
by (simp add: le_rat)
|
|
531 |
with bb show ?thesis by (simp add: zmult_zle_cancel2)
|
|
532 |
qed
|
|
533 |
finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)"
|
|
534 |
by (simp only: zmult_ac)
|
|
535 |
with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)"
|
|
536 |
by (simp add: zmult_zle_cancel2)
|
|
537 |
with neq show ?thesis by (simp add: le_rat)
|
|
538 |
qed
|
|
539 |
qed
|
|
540 |
next
|
|
541 |
assume "q \<le> r" and "r \<le> q"
|
|
542 |
show "q = r"
|
|
543 |
proof (insert prems, induct q, induct r)
|
|
544 |
fix a b c d :: int
|
|
545 |
assume neq: "b \<noteq> 0" "d \<noteq> 0"
|
|
546 |
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b"
|
|
547 |
show "Fract a b = Fract c d"
|
|
548 |
proof -
|
|
549 |
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
|
|
550 |
by (simp add: le_rat)
|
|
551 |
also have "... \<le> (a * d) * (b * d)"
|
|
552 |
proof -
|
|
553 |
from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)"
|
|
554 |
by (simp add: le_rat)
|
|
555 |
thus ?thesis by (simp only: zmult_ac)
|
|
556 |
qed
|
|
557 |
finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
|
|
558 |
moreover from neq have "b * d \<noteq> 0" by simp
|
|
559 |
ultimately have "a * d = c * b" by simp
|
|
560 |
with neq show ?thesis by (simp add: eq_rat)
|
|
561 |
qed
|
|
562 |
qed
|
|
563 |
next
|
|
564 |
show "q \<le> q"
|
|
565 |
by (induct q) (simp add: le_rat)
|
|
566 |
show "(q < r) = (q \<le> r \<and> q \<noteq> r)"
|
|
567 |
by (simp only: less_rat_def)
|
|
568 |
show "q \<le> r \<or> r \<le> q"
|
|
569 |
by (induct q, induct r) (simp add: le_rat zmult_ac, arith)
|
|
570 |
}
|
|
571 |
qed
|
|
572 |
|
|
573 |
instance rat :: ordered_field
|
|
574 |
proof
|
|
575 |
fix q r s :: rat
|
|
576 |
show "q \<le> r ==> s + q \<le> s + r"
|
|
577 |
proof (induct q, induct r, induct s)
|
|
578 |
fix a b c d e f :: int
|
|
579 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
|
|
580 |
assume le: "Fract a b \<le> Fract c d"
|
|
581 |
show "Fract e f + Fract a b \<le> Fract e f + Fract c d"
|
|
582 |
proof -
|
|
583 |
let ?F = "f * f" from neq have F: "0 < ?F"
|
|
584 |
by (auto simp add: int_less_le)
|
|
585 |
from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)"
|
|
586 |
by (simp add: le_rat)
|
|
587 |
with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F"
|
|
588 |
by (simp add: zmult_zle_cancel2)
|
|
589 |
with neq show ?thesis by (simp add: add_rat le_rat zmult_ac int_distrib)
|
|
590 |
qed
|
|
591 |
qed
|
|
592 |
show "q < r ==> 0 < s ==> s * q < s * r"
|
|
593 |
proof (induct q, induct r, induct s)
|
|
594 |
fix a b c d e f :: int
|
|
595 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
|
|
596 |
assume le: "Fract a b < Fract c d"
|
|
597 |
assume gt: "0 < Fract e f"
|
|
598 |
show "Fract e f * Fract a b < Fract e f * Fract c d"
|
|
599 |
proof -
|
|
600 |
let ?E = "e * f" and ?F = "f * f"
|
|
601 |
from neq gt have "0 < ?E"
|
|
602 |
by (auto simp add: zero_rat less_rat le_rat int_less_le eq_rat)
|
|
603 |
moreover from neq have "0 < ?F"
|
|
604 |
by (auto simp add: int_less_le)
|
|
605 |
moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
|
|
606 |
by (simp add: less_rat)
|
|
607 |
ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
|
|
608 |
by (simp add: zmult_zless_cancel2)
|
|
609 |
with neq show ?thesis
|
|
610 |
by (simp add: less_rat mult_rat zmult_ac)
|
|
611 |
qed
|
|
612 |
qed
|
|
613 |
show "\<bar>q\<bar> = (if q < 0 then -q else q)"
|
|
614 |
by (simp only: abs_rat_def)
|
|
615 |
qed
|
|
616 |
|
|
617 |
|
|
618 |
subsection {* Embedding integers *}
|
|
619 |
|
10665
|
620 |
constdefs
|
11913
|
621 |
rat :: "int => rat" (* FIXME generalize int to any numeric subtype (?) *)
|
|
622 |
"rat z == Fract z 1"
|
|
623 |
int_set :: "rat set" ("\<int>") (* FIXME generalize rat to any numeric supertype (?) *)
|
10614
|
624 |
"\<int> == range rat"
|
|
625 |
|
|
626 |
lemma rat_inject: "(rat z = rat w) = (z = w)"
|
|
627 |
proof
|
|
628 |
assume "rat z = rat w"
|
11913
|
629 |
hence "Fract z 1 = Fract w 1" by (unfold rat_def)
|
|
630 |
hence "\<lfloor>fract z 1\<rfloor> = \<lfloor>fract w 1\<rfloor>" ..
|
10614
|
631 |
thus "z = w" by auto
|
|
632 |
next
|
|
633 |
assume "z = w"
|
|
634 |
thus "rat z = rat w" by simp
|
|
635 |
qed
|
|
636 |
|
|
637 |
lemma int_set_cases [case_names rat, cases set: int_set]:
|
|
638 |
"q \<in> \<int> ==> (!!z. q = rat z ==> C) ==> C"
|
|
639 |
proof (unfold int_set_def)
|
|
640 |
assume "!!z. q = rat z ==> C"
|
|
641 |
assume "q \<in> range rat" thus C ..
|
|
642 |
qed
|
|
643 |
|
|
644 |
lemma int_set_induct [case_names rat, induct set: int_set]:
|
|
645 |
"q \<in> \<int> ==> (!!z. P (rat z)) ==> P q"
|
|
646 |
by (rule int_set_cases) auto
|
|
647 |
|
|
648 |
theorem number_of_rat: "number_of b = rat (number_of b)"
|
11913
|
649 |
by (simp add: number_of_rat_def rat_def)
|
10614
|
650 |
|
|
651 |
end
|