1.1 --- a/src/HOL/Real/Rational.thy Thu Feb 02 18:04:10 2006 +0100
1.2 +++ b/src/HOL/Real/Rational.thy Thu Feb 02 19:57:13 2006 +0100
1.3 @@ -6,212 +6,120 @@
1.4 header {* Rational numbers *}
1.5
1.6 theory Rational
1.7 -imports Quotient
1.8 +imports Main
1.9 uses ("rat_arith.ML")
1.10 begin
1.11
1.12 -subsection {* Fractions *}
1.13 -
1.14 -subsubsection {* The type of fractions *}
1.15 -
1.16 -typedef fraction = "{(a, b) :: int \<times> int | a b. b \<noteq> 0}"
1.17 -proof
1.18 - show "(0, 1) \<in> ?fraction" by simp
1.19 -qed
1.20 -
1.21 -constdefs
1.22 - fract :: "int => int => fraction"
1.23 - "fract a b == Abs_fraction (a, b)"
1.24 - num :: "fraction => int"
1.25 - "num Q == fst (Rep_fraction Q)"
1.26 - den :: "fraction => int"
1.27 - "den Q == snd (Rep_fraction Q)"
1.28 -
1.29 -lemma fract_num [simp]: "b \<noteq> 0 ==> num (fract a b) = a"
1.30 - by (simp add: fract_def num_def fraction_def Abs_fraction_inverse)
1.31 -
1.32 -lemma fract_den [simp]: "b \<noteq> 0 ==> den (fract a b) = b"
1.33 - by (simp add: fract_def den_def fraction_def Abs_fraction_inverse)
1.34 -
1.35 -lemma fraction_cases [case_names fract, cases type: fraction]:
1.36 - "(!!a b. Q = fract a b ==> b \<noteq> 0 ==> C) ==> C"
1.37 -proof -
1.38 - assume r: "!!a b. Q = fract a b ==> b \<noteq> 0 ==> C"
1.39 - obtain a b where "Q = fract a b" and "b \<noteq> 0"
1.40 - by (cases Q) (auto simp add: fract_def fraction_def)
1.41 - thus C by (rule r)
1.42 -qed
1.43 -
1.44 -lemma fraction_induct [case_names fract, induct type: fraction]:
1.45 - "(!!a b. b \<noteq> 0 ==> P (fract a b)) ==> P Q"
1.46 - by (cases Q) simp
1.47 -
1.48 +subsection {* Rational numbers *}
1.49
1.50 subsubsection {* Equivalence of fractions *}
1.51
1.52 -instance fraction :: eqv ..
1.53 +constdefs
1.54 + fraction :: "(int \<times> int) set"
1.55 + "fraction \<equiv> {x. snd x \<noteq> 0}"
1.56
1.57 -defs (overloaded)
1.58 - equiv_fraction_def: "Q \<sim> R == num Q * den R = num R * den Q"
1.59 + ratrel :: "((int \<times> int) \<times> (int \<times> int)) set"
1.60 + "ratrel \<equiv> {(x,y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}"
1.61
1.62 -lemma equiv_fraction_iff [iff]:
1.63 - "b \<noteq> 0 ==> b' \<noteq> 0 ==> (fract a b \<sim> fract a' b') = (a * b' = a' * b)"
1.64 - by (simp add: equiv_fraction_def)
1.65 +lemma fraction_iff [simp]: "(x \<in> fraction) = (snd x \<noteq> 0)"
1.66 +by (simp add: fraction_def)
1.67
1.68 -instance fraction :: equiv
1.69 -proof
1.70 - fix Q R S :: fraction
1.71 - {
1.72 - show "Q \<sim> Q"
1.73 - proof (induct Q)
1.74 - fix a b :: int
1.75 - assume "b \<noteq> 0" and "b \<noteq> 0"
1.76 - with refl show "fract a b \<sim> fract a b" ..
1.77 - qed
1.78 - next
1.79 - assume "Q \<sim> R" and "R \<sim> S"
1.80 - show "Q \<sim> S"
1.81 - proof (insert prems, induct Q, induct R, induct S)
1.82 - fix a b a' b' a'' b'' :: int
1.83 - assume b: "b \<noteq> 0" and b': "b' \<noteq> 0" and b'': "b'' \<noteq> 0"
1.84 - assume "fract a b \<sim> fract a' b'" hence eq1: "a * b' = a' * b" ..
1.85 - assume "fract a' b' \<sim> fract a'' b''" hence eq2: "a' * b'' = a'' * b'" ..
1.86 - have "a * b'' = a'' * b"
1.87 - proof cases
1.88 - assume "a' = 0"
1.89 - with b' eq1 eq2 have "a = 0 \<and> a'' = 0" by auto
1.90 - thus ?thesis by simp
1.91 - next
1.92 - assume a': "a' \<noteq> 0"
1.93 - from eq1 eq2 have "(a * b') * (a' * b'') = (a' * b) * (a'' * b')" by simp
1.94 - hence "(a * b'') * (a' * b') = (a'' * b) * (a' * b')" by (simp only: mult_ac)
1.95 - with a' b' show ?thesis by simp
1.96 - qed
1.97 - thus "fract a b \<sim> fract a'' b''" ..
1.98 - qed
1.99 - next
1.100 - show "Q \<sim> R ==> R \<sim> Q"
1.101 - proof (induct Q, induct R)
1.102 - fix a b a' b' :: int
1.103 - assume b: "b \<noteq> 0" and b': "b' \<noteq> 0"
1.104 - assume "fract a b \<sim> fract a' b'"
1.105 - hence "a * b' = a' * b" ..
1.106 - hence "a' * b = a * b'" ..
1.107 - thus "fract a' b' \<sim> fract a b" ..
1.108 - qed
1.109 - }
1.110 +lemma ratrel_iff [simp]:
1.111 + "((x,y) \<in> ratrel) =
1.112 + (snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)"
1.113 +by (simp add: ratrel_def)
1.114 +
1.115 +lemma refl_ratrel: "refl fraction ratrel"
1.116 +by (auto simp add: refl_def fraction_def ratrel_def)
1.117 +
1.118 +lemma sym_ratrel: "sym ratrel"
1.119 +by (simp add: ratrel_def sym_def)
1.120 +
1.121 +lemma trans_ratrel_lemma:
1.122 + assumes 1: "a * b' = a' * b"
1.123 + assumes 2: "a' * b'' = a'' * b'"
1.124 + assumes 3: "b' \<noteq> (0::int)"
1.125 + shows "a * b'' = a'' * b"
1.126 +proof -
1.127 + have "b' * (a * b'') = b'' * (a * b')" by simp
1.128 + also note 1
1.129 + also have "b'' * (a' * b) = b * (a' * b'')" by simp
1.130 + also note 2
1.131 + also have "b * (a'' * b') = b' * (a'' * b)" by simp
1.132 + finally have "b' * (a * b'') = b' * (a'' * b)" .
1.133 + with 3 show "a * b'' = a'' * b" by simp
1.134 qed
1.135
1.136 -lemma eq_fraction_iff [iff]:
1.137 - "b \<noteq> 0 ==> b' \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>) = (a * b' = a' * b)"
1.138 - by (simp add: equiv_fraction_iff quot_equality)
1.139 +lemma trans_ratrel: "trans ratrel"
1.140 +by (auto simp add: trans_def elim: trans_ratrel_lemma)
1.141
1.142 +lemma equiv_ratrel: "equiv fraction ratrel"
1.143 +by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel])
1.144
1.145 -subsubsection {* Operations on fractions *}
1.146 +lemmas equiv_ratrel_iff [iff] = eq_equiv_class_iff [OF equiv_ratrel]
1.147
1.148 -text {*
1.149 - We define the basic arithmetic operations on fractions and
1.150 - demonstrate their ``well-definedness'', i.e.\ congruence with respect
1.151 - to equivalence of fractions.
1.152 -*}
1.153 +lemma equiv_ratrel_iff2:
1.154 + "\<lbrakk>snd x \<noteq> 0; snd y \<noteq> 0\<rbrakk>
1.155 + \<Longrightarrow> (ratrel `` {x} = ratrel `` {y}) = ((x,y) \<in> ratrel)"
1.156 +by (rule eq_equiv_class_iff [OF equiv_ratrel], simp_all)
1.157
1.158 -instance fraction :: "{zero, one, plus, minus, times, inverse, ord}" ..
1.159
1.160 -defs (overloaded)
1.161 - zero_fraction_def: "0 == fract 0 1"
1.162 - one_fraction_def: "1 == fract 1 1"
1.163 - add_fraction_def: "Q + R ==
1.164 - fract (num Q * den R + num R * den Q) (den Q * den R)"
1.165 - minus_fraction_def: "-Q == fract (-(num Q)) (den Q)"
1.166 - mult_fraction_def: "Q * R == fract (num Q * num R) (den Q * den R)"
1.167 - inverse_fraction_def: "inverse Q == fract (den Q) (num Q)"
1.168 - le_fraction_def: "Q \<le> R ==
1.169 - (num Q * den R) * (den Q * den R) \<le> (num R * den Q) * (den Q * den R)"
1.170 +subsubsection {* The type of rational numbers *}
1.171
1.172 -lemma is_zero_fraction_iff: "b \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>0\<rfloor>) = (a = 0)"
1.173 - by (simp add: zero_fraction_def eq_fraction_iff)
1.174 -
1.175 -theorem add_fraction_cong:
1.176 - "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
1.177 - ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
1.178 - ==> \<lfloor>fract a b + fract c d\<rfloor> = \<lfloor>fract a' b' + fract c' d'\<rfloor>"
1.179 -proof -
1.180 - assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0"
1.181 - assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
1.182 - assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
1.183 - have "\<lfloor>fract (a * d + c * b) (b * d)\<rfloor> = \<lfloor>fract (a' * d' + c' * b') (b' * d')\<rfloor>"
1.184 - proof
1.185 - show "(a * d + c * b) * (b' * d') = (a' * d' + c' * b') * (b * d)"
1.186 - (is "?lhs = ?rhs")
1.187 - proof -
1.188 - have "?lhs = (a * b') * (d * d') + (c * d') * (b * b')"
1.189 - by (simp add: int_distrib mult_ac)
1.190 - also have "... = (a' * b) * (d * d') + (c' * d) * (b * b')"
1.191 - by (simp only: eq1 eq2)
1.192 - also have "... = ?rhs"
1.193 - by (simp add: int_distrib mult_ac)
1.194 - finally show "?lhs = ?rhs" .
1.195 - qed
1.196 - from neq show "b * d \<noteq> 0" by simp
1.197 - from neq show "b' * d' \<noteq> 0" by simp
1.198 - qed
1.199 - with neq show ?thesis by (simp add: add_fraction_def)
1.200 +typedef (Rat) rat = "fraction//ratrel"
1.201 +proof
1.202 + have "(0,1) \<in> fraction" by (simp add: fraction_def)
1.203 + thus "ratrel``{(0,1)} \<in> fraction//ratrel" by (rule quotientI)
1.204 qed
1.205
1.206 -theorem minus_fraction_cong:
1.207 - "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0
1.208 - ==> \<lfloor>-(fract a b)\<rfloor> = \<lfloor>-(fract a' b')\<rfloor>"
1.209 -proof -
1.210 - assume neq: "b \<noteq> 0" "b' \<noteq> 0"
1.211 - assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>"
1.212 - hence "a * b' = a' * b" ..
1.213 - hence "-a * b' = -a' * b" by simp
1.214 - hence "\<lfloor>fract (-a) b\<rfloor> = \<lfloor>fract (-a') b'\<rfloor>" ..
1.215 - with neq show ?thesis by (simp add: minus_fraction_def)
1.216 -qed
1.217 +lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel``{x} \<in> Rat"
1.218 +by (simp add: Rat_def quotientI)
1.219
1.220 -theorem mult_fraction_cong:
1.221 - "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
1.222 - ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
1.223 - ==> \<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>"
1.224 -proof -
1.225 +declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp]
1.226 +
1.227 +
1.228 +constdefs
1.229 + Fract :: "int \<Rightarrow> int \<Rightarrow> rat"
1.230 + "Fract a b \<equiv> Abs_Rat (ratrel``{(a,b)})"
1.231 +
1.232 +theorem Rat_cases [case_names Fract, cases type: rat]:
1.233 + "(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C"
1.234 +by (cases q, clarsimp simp add: Fract_def Rat_def fraction_def quotient_def)
1.235 +
1.236 +theorem Rat_induct [case_names Fract, induct type: rat]:
1.237 + "(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q"
1.238 + by (cases q) simp
1.239 +
1.240 +
1.241 +subsubsection {* Congruence lemmas *}
1.242 +
1.243 +lemma add_congruent2:
1.244 + "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})
1.245 + respects2 ratrel"
1.246 +apply (rule equiv_ratrel [THEN congruent2_commuteI])
1.247 +apply (simp_all add: left_distrib)
1.248 +done
1.249 +
1.250 +lemma minus_congruent:
1.251 + "(\<lambda>x. ratrel``{(- fst x, snd x)}) respects ratrel"
1.252 +by (simp add: congruent_def)
1.253 +
1.254 +lemma mult_congruent2:
1.255 + "(\<lambda>x y. ratrel``{(fst x * fst y, snd x * snd y)}) respects2 ratrel"
1.256 +by (rule equiv_ratrel [THEN congruent2_commuteI], simp_all)
1.257 +
1.258 +lemma inverse_congruent:
1.259 + "(\<lambda>x. ratrel``{if fst x=0 then (0,1) else (snd x, fst x)}) respects ratrel"
1.260 +by (auto simp add: congruent_def mult_commute)
1.261 +
1.262 +lemma le_congruent2:
1.263 + "(\<lambda>x y. (fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y))
1.264 + respects2 ratrel"
1.265 +proof (clarsimp simp add: congruent2_def)
1.266 + fix a b a' b' c d c' d'::int
1.267 assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0"
1.268 - assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
1.269 - assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
1.270 - have "\<lfloor>fract (a * c) (b * d)\<rfloor> = \<lfloor>fract (a' * c') (b' * d')\<rfloor>"
1.271 - proof
1.272 - from eq1 eq2 have "(a * b') * (c * d') = (a' * b) * (c' * d)" by simp
1.273 - thus "(a * c) * (b' * d') = (a' * c') * (b * d)" by (simp add: mult_ac)
1.274 - from neq show "b * d \<noteq> 0" by simp
1.275 - from neq show "b' * d' \<noteq> 0" by simp
1.276 - qed
1.277 - with neq show "\<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>"
1.278 - by (simp add: mult_fraction_def)
1.279 -qed
1.280 -
1.281 -theorem inverse_fraction_cong:
1.282 - "\<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>
1.283 - ==> b \<noteq> 0 ==> b' \<noteq> 0
1.284 - ==> \<lfloor>inverse (fract a b)\<rfloor> = \<lfloor>inverse (fract a' b')\<rfloor>"
1.285 -proof -
1.286 - assume neq: "b \<noteq> 0" "b' \<noteq> 0"
1.287 - assume "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>" and "\<lfloor>fract a' b'\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
1.288 - with neq obtain "a \<noteq> 0" and "a' \<noteq> 0" by (simp add: is_zero_fraction_iff)
1.289 - assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>"
1.290 - hence "a * b' = a' * b" ..
1.291 - hence "b * a' = b' * a" by (simp only: mult_ac)
1.292 - hence "\<lfloor>fract b a\<rfloor> = \<lfloor>fract b' a'\<rfloor>" ..
1.293 - with neq show ?thesis by (simp add: inverse_fraction_def)
1.294 -qed
1.295 -
1.296 -theorem le_fraction_cong:
1.297 - "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>
1.298 - ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0
1.299 - ==> (fract a b \<le> fract c d) = (fract a' b' \<le> fract c' d')"
1.300 -proof -
1.301 - assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0"
1.302 - assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" ..
1.303 - assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" ..
1.304 + assume eq1: "a * b' = a' * b"
1.305 + assume eq2: "c * d' = c' * d"
1.306
1.307 let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))"
1.308 {
1.309 @@ -241,215 +149,120 @@
1.310 by (simp add: mult_ac)
1.311 also from D have "... = ?le a' b' c' d'"
1.312 by (rule le_factor [symmetric])
1.313 - finally have "?le a b c d = ?le a' b' c' d'" .
1.314 - with neq show ?thesis by (simp add: le_fraction_def)
1.315 + finally show "?le a b c d = ?le a' b' c' d'" .
1.316 qed
1.317
1.318 +lemma All_equiv_class:
1.319 + "equiv A r ==> f respects r ==> a \<in> A
1.320 + ==> (\<forall>x \<in> r``{a}. f x) = f a"
1.321 +apply safe
1.322 +apply (drule (1) equiv_class_self)
1.323 +apply simp
1.324 +apply (drule (1) congruent.congruent)
1.325 +apply simp
1.326 +done
1.327
1.328 -subsection {* Rational numbers *}
1.329 +lemma congruent2_implies_congruent_All:
1.330 + "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
1.331 + congruent r1 (\<lambda>x1. \<forall>x2 \<in> r2``{a}. f x1 x2)"
1.332 + apply (unfold congruent_def)
1.333 + apply clarify
1.334 + apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
1.335 + apply (simp add: UN_equiv_class congruent2_implies_congruent)
1.336 + apply (unfold congruent2_def equiv_def refl_def)
1.337 + apply (blast del: equalityI)
1.338 + done
1.339
1.340 -subsubsection {* The type of rational numbers *}
1.341 +lemma All_equiv_class2:
1.342 + "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
1.343 + ==> (\<forall>x1 \<in> r1``{a1}. \<forall>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
1.344 + by (simp add: All_equiv_class congruent2_implies_congruent
1.345 + congruent2_implies_congruent_All)
1.346
1.347 -typedef (Rat)
1.348 - rat = "UNIV :: fraction quot set" ..
1.349 -
1.350 -lemma RatI [intro, simp]: "Q \<in> Rat"
1.351 - by (simp add: Rat_def)
1.352 -
1.353 -constdefs
1.354 - fraction_of :: "rat => fraction"
1.355 - "fraction_of q == pick (Rep_Rat q)"
1.356 - rat_of :: "fraction => rat"
1.357 - "rat_of Q == Abs_Rat \<lfloor>Q\<rfloor>"
1.358 -
1.359 -theorem rat_of_equality [iff?]: "(rat_of Q = rat_of Q') = (\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor>)"
1.360 - by (simp add: rat_of_def Abs_Rat_inject)
1.361 -
1.362 -lemma rat_of: "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> rat_of Q = rat_of Q'" ..
1.363 -
1.364 -constdefs
1.365 - Fract :: "int => int => rat"
1.366 - "Fract a b == rat_of (fract a b)"
1.367 -
1.368 -theorem Fract_inverse: "\<lfloor>fraction_of (Fract a b)\<rfloor> = \<lfloor>fract a b\<rfloor>"
1.369 - by (simp add: fraction_of_def rat_of_def Fract_def Abs_Rat_inverse pick_inverse)
1.370 -
1.371 -theorem Fract_equality [iff?]:
1.372 - "(Fract a b = Fract c d) = (\<lfloor>fract a b\<rfloor> = \<lfloor>fract c d\<rfloor>)"
1.373 - by (simp add: Fract_def rat_of_equality)
1.374 -
1.375 -theorem eq_rat:
1.376 - "b \<noteq> 0 ==> d \<noteq> 0 ==> (Fract a b = Fract c d) = (a * d = c * b)"
1.377 - by (simp add: Fract_equality eq_fraction_iff)
1.378 -
1.379 -theorem Rat_cases [case_names Fract, cases type: rat]:
1.380 - "(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C"
1.381 -proof -
1.382 - assume r: "!!a b. q = Fract a b ==> b \<noteq> 0 ==> C"
1.383 - obtain x where "q = Abs_Rat x" by (cases q)
1.384 - moreover obtain Q where "x = \<lfloor>Q\<rfloor>" by (cases x)
1.385 - moreover obtain a b where "Q = fract a b" and "b \<noteq> 0" by (cases Q)
1.386 - ultimately have "q = Fract a b" by (simp only: Fract_def rat_of_def)
1.387 - thus ?thesis by (rule r)
1.388 -qed
1.389 -
1.390 -theorem Rat_induct [case_names Fract, induct type: rat]:
1.391 - "(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q"
1.392 - by (cases q) simp
1.393 -
1.394 -
1.395 -subsubsection {* Canonical function definitions *}
1.396 -
1.397 -text {*
1.398 - Note that the unconditional version below is much easier to read.
1.399 -*}
1.400 -
1.401 -theorem rat_cond_function:
1.402 - "(!!q r. P \<lfloor>fraction_of q\<rfloor> \<lfloor>fraction_of r\<rfloor> ==>
1.403 - f q r == g (fraction_of q) (fraction_of r)) ==>
1.404 - (!!a b a' b' c d c' d'.
1.405 - \<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==>
1.406 - P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==> P \<lfloor>fract a' b'\<rfloor> \<lfloor>fract c' d'\<rfloor> ==>
1.407 - b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==>
1.408 - g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==>
1.409 - P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==>
1.410 - f (Fract a b) (Fract c d) = g (fract a b) (fract c d)"
1.411 - (is "PROP ?eq ==> PROP ?cong ==> ?P ==> _")
1.412 -proof -
1.413 - assume eq: "PROP ?eq" and cong: "PROP ?cong" and P: ?P
1.414 - have "f (Abs_Rat \<lfloor>fract a b\<rfloor>) (Abs_Rat \<lfloor>fract c d\<rfloor>) = g (fract a b) (fract c d)"
1.415 - proof (rule quot_cond_function)
1.416 - fix X Y assume "P X Y"
1.417 - with eq show "f (Abs_Rat X) (Abs_Rat Y) == g (pick X) (pick Y)"
1.418 - by (simp add: fraction_of_def pick_inverse Abs_Rat_inverse)
1.419 - next
1.420 - fix Q Q' R R' :: fraction
1.421 - show "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> \<lfloor>R\<rfloor> = \<lfloor>R'\<rfloor> ==>
1.422 - P \<lfloor>Q\<rfloor> \<lfloor>R\<rfloor> ==> P \<lfloor>Q'\<rfloor> \<lfloor>R'\<rfloor> ==> g Q R = g Q' R'"
1.423 - by (induct Q, induct Q', induct R, induct R') (rule cong)
1.424 - qed
1.425 - thus ?thesis by (unfold Fract_def rat_of_def)
1.426 -qed
1.427 -
1.428 -theorem rat_function:
1.429 - assumes "!!q r. f q r == g (fraction_of q) (fraction_of r)"
1.430 - and "!!a b a' b' c d c' d'.
1.431 - \<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==>
1.432 - b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==>
1.433 - g (fract a b) (fract c d) = g (fract a' b') (fract c' d')"
1.434 - shows "f (Fract a b) (Fract c d) = g (fract a b) (fract c d)"
1.435 - using prems and TrueI by (rule rat_cond_function)
1.436 +lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel]
1.437 +lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel]
1.438 +lemmas All_ratrel2 = All_equiv_class2 [OF equiv_ratrel equiv_ratrel]
1.439
1.440
1.441 subsubsection {* Standard operations on rational numbers *}
1.442
1.443 -instance rat :: "{zero, one, plus, minus, times, inverse, ord}" ..
1.444 +instance rat :: "{ord, zero, one, plus, times, minus, inverse}" ..
1.445
1.446 defs (overloaded)
1.447 - zero_rat_def: "0 == rat_of 0"
1.448 - one_rat_def: "1 == rat_of 1"
1.449 - add_rat_def: "q + r == rat_of (fraction_of q + fraction_of r)"
1.450 - minus_rat_def: "-q == rat_of (-(fraction_of q))"
1.451 - diff_rat_def: "q - r == q + (-(r::rat))"
1.452 - mult_rat_def: "q * r == rat_of (fraction_of q * fraction_of r)"
1.453 - inverse_rat_def: "inverse q ==
1.454 - if q=0 then 0 else rat_of (inverse (fraction_of q))"
1.455 - divide_rat_def: "q / r == q * inverse (r::rat)"
1.456 - le_rat_def: "q \<le> r == fraction_of q \<le> fraction_of r"
1.457 - less_rat_def: "q < r == q \<le> r \<and> q \<noteq> (r::rat)"
1.458 + Zero_rat_def: "0 == Fract 0 1"
1.459 + One_rat_def: "1 == Fract 1 1"
1.460 +
1.461 + add_rat_def:
1.462 + "q + r ==
1.463 + Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
1.464 + ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})"
1.465 +
1.466 + minus_rat_def:
1.467 + "- q == Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel``{(- fst x, snd x)})"
1.468 +
1.469 + diff_rat_def: "q - r == q + - (r::rat)"
1.470 +
1.471 + mult_rat_def:
1.472 + "q * r ==
1.473 + Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r.
1.474 + ratrel``{(fst x * fst y, snd x * snd y)})"
1.475 +
1.476 + inverse_rat_def:
1.477 + "inverse q ==
1.478 + Abs_Rat (\<Union>x \<in> Rep_Rat q.
1.479 + ratrel``{if fst x=0 then (0,1) else (snd x, fst x)})"
1.480 +
1.481 + divide_rat_def: "q / r == q * inverse (r::rat)"
1.482 +
1.483 + le_rat_def:
1.484 + "q \<le> (r::rat) ==
1.485 + \<forall>x \<in> Rep_Rat q. \<forall>y \<in> Rep_Rat r.
1.486 + (fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)"
1.487 +
1.488 + less_rat_def: "(z < (w::rat)) == (z \<le> w & z \<noteq> w)"
1.489 +
1.490 abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)"
1.491
1.492 -theorem zero_rat: "0 = Fract 0 1"
1.493 - by (simp add: zero_rat_def zero_fraction_def rat_of_def Fract_def)
1.494 +lemma zero_rat: "0 = Fract 0 1"
1.495 +by (simp add: Zero_rat_def)
1.496
1.497 -theorem one_rat: "1 = Fract 1 1"
1.498 - by (simp add: one_rat_def one_fraction_def rat_of_def Fract_def)
1.499 +lemma one_rat: "1 = Fract 1 1"
1.500 +by (simp add: One_rat_def)
1.501 +
1.502 +theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
1.503 + (Fract a b = Fract c d) = (a * d = c * b)"
1.504 +by (simp add: Fract_def)
1.505
1.506 theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
1.507 Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
1.508 -proof -
1.509 - have "Fract a b + Fract c d = rat_of (fract a b + fract c d)"
1.510 - by (rule rat_function, rule add_rat_def, rule rat_of, rule add_fraction_cong)
1.511 - also
1.512 - assume "b \<noteq> 0" "d \<noteq> 0"
1.513 - hence "fract a b + fract c d = fract (a * d + c * b) (b * d)"
1.514 - by (simp add: add_fraction_def)
1.515 - finally show ?thesis by (unfold Fract_def)
1.516 -qed
1.517 +by (simp add: Fract_def add_rat_def add_congruent2 UN_ratrel2)
1.518
1.519 theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b"
1.520 -proof -
1.521 - have "-(Fract a b) = rat_of (-(fract a b))"
1.522 - by (rule rat_function, rule minus_rat_def, rule rat_of, rule minus_fraction_cong)
1.523 - also assume "b \<noteq> 0" hence "-(fract a b) = fract (-a) b"
1.524 - by (simp add: minus_fraction_def)
1.525 - finally show ?thesis by (unfold Fract_def)
1.526 -qed
1.527 +by (simp add: Fract_def minus_rat_def minus_congruent UN_ratrel)
1.528
1.529 theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
1.530 Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
1.531 - by (simp add: diff_rat_def add_rat minus_rat)
1.532 +by (simp add: diff_rat_def add_rat minus_rat)
1.533
1.534 theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
1.535 Fract a b * Fract c d = Fract (a * c) (b * d)"
1.536 -proof -
1.537 - have "Fract a b * Fract c d = rat_of (fract a b * fract c d)"
1.538 - by (rule rat_function, rule mult_rat_def, rule rat_of, rule mult_fraction_cong)
1.539 - also
1.540 - assume "b \<noteq> 0" "d \<noteq> 0"
1.541 - hence "fract a b * fract c d = fract (a * c) (b * d)"
1.542 - by (simp add: mult_fraction_def)
1.543 - finally show ?thesis by (unfold Fract_def)
1.544 -qed
1.545 +by (simp add: Fract_def mult_rat_def mult_congruent2 UN_ratrel2)
1.546
1.547 -theorem inverse_rat: "Fract a b \<noteq> 0 ==> b \<noteq> 0 ==>
1.548 +theorem inverse_rat: "a \<noteq> 0 ==> b \<noteq> 0 ==>
1.549 inverse (Fract a b) = Fract b a"
1.550 -proof -
1.551 - assume neq: "b \<noteq> 0" and nonzero: "Fract a b \<noteq> 0"
1.552 - hence "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
1.553 - by (simp add: zero_rat eq_rat is_zero_fraction_iff)
1.554 - with _ inverse_fraction_cong [THEN rat_of]
1.555 - have "inverse (Fract a b) = rat_of (inverse (fract a b))"
1.556 - proof (rule rat_cond_function)
1.557 - fix q assume cond: "\<lfloor>fraction_of q\<rfloor> \<noteq> \<lfloor>0\<rfloor>"
1.558 - have "q \<noteq> 0"
1.559 - proof (cases q)
1.560 - fix a b assume "b \<noteq> 0" and "q = Fract a b"
1.561 - from this cond show ?thesis
1.562 - by (simp add: Fract_inverse is_zero_fraction_iff zero_rat eq_rat)
1.563 - qed
1.564 - thus "inverse q == rat_of (inverse (fraction_of q))"
1.565 - by (simp add: inverse_rat_def)
1.566 - qed
1.567 - also from neq nonzero have "inverse (fract a b) = fract b a"
1.568 - by (simp add: inverse_fraction_def)
1.569 - finally show ?thesis by (unfold Fract_def)
1.570 -qed
1.571 +by (simp add: Fract_def inverse_rat_def inverse_congruent UN_ratrel)
1.572
1.573 -theorem divide_rat: "Fract c d \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==>
1.574 +theorem divide_rat: "c \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==>
1.575 Fract a b / Fract c d = Fract (a * d) (b * c)"
1.576 -proof -
1.577 - assume neq: "b \<noteq> 0" "d \<noteq> 0" and nonzero: "Fract c d \<noteq> 0"
1.578 - hence "c \<noteq> 0" by (simp add: zero_rat eq_rat)
1.579 - with neq nonzero show ?thesis
1.580 - by (simp add: divide_rat_def inverse_rat mult_rat)
1.581 -qed
1.582 +by (simp add: divide_rat_def inverse_rat mult_rat)
1.583
1.584 theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
1.585 (Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
1.586 -proof -
1.587 - have "(Fract a b \<le> Fract c d) = (fract a b \<le> fract c d)"
1.588 - by (rule rat_function, rule le_rat_def, rule le_fraction_cong)
1.589 - also
1.590 - assume "b \<noteq> 0" "d \<noteq> 0"
1.591 - hence "(fract a b \<le> fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))"
1.592 - by (simp add: le_fraction_def)
1.593 - finally show ?thesis .
1.594 -qed
1.595 +by (simp add: Fract_def le_rat_def le_congruent2 All_ratrel2)
1.596
1.597 theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==>
1.598 (Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))"
1.599 - by (simp add: less_rat_def le_rat eq_rat order_less_le)
1.600 +by (simp add: less_rat_def le_rat eq_rat order_less_le)
1.601
1.602 theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
1.603 by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat)
1.604 @@ -474,13 +287,14 @@
1.605 proof
1.606 fix q r s :: rat
1.607 show "(q + r) + s = q + (r + s)"
1.608 - by (rule rat_add_assoc)
1.609 + by (induct q, induct r, induct s)
1.610 + (simp add: add_rat add_ac mult_ac int_distrib)
1.611 show "q + r = r + q"
1.612 by (induct q, induct r) (simp add: add_rat add_ac mult_ac)
1.613 show "0 + q = q"
1.614 by (induct q) (simp add: zero_rat add_rat)
1.615 show "(-q) + q = 0"
1.616 - by (rule rat_left_minus)
1.617 + by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat)
1.618 show "q - r = q + (-r)"
1.619 by (induct q, induct r) (simp add: add_rat minus_rat diff_rat)
1.620 show "(q * r) * s = q * (r * s)"
1.621 @@ -564,7 +378,8 @@
1.622 show "(q < r) = (q \<le> r \<and> q \<noteq> r)"
1.623 by (simp only: less_rat_def)
1.624 show "q \<le> r \<or> r \<le> q"
1.625 - by (induct q, induct r) (simp add: le_rat mult_ac, arith)
1.626 + by (induct q, induct r)
1.627 + (simp add: le_rat mult_commute, rule linorder_linear)
1.628 }
1.629 qed
1.630
1.631 @@ -614,14 +429,16 @@
1.632
1.633 instance rat :: division_by_zero
1.634 proof
1.635 - show "inverse 0 = (0::rat)" by (simp add: inverse_rat_def)
1.636 + show "inverse 0 = (0::rat)"
1.637 + by (simp add: zero_rat Fract_def inverse_rat_def
1.638 + inverse_congruent UN_ratrel)
1.639 qed
1.640
1.641
1.642 subsection {* Various Other Results *}
1.643
1.644 lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (-a) (-b) = Fract a b"
1.645 -by (simp add: Fract_equality eq_fraction_iff)
1.646 +by (simp add: eq_rat)
1.647
1.648 theorem Rat_induct_pos [case_names Fract, induct type: rat]:
1.649 assumes step: "!!a b. 0 < b ==> P (Fract a b)"