author  haftmann 
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child 26732  6ea9de67e576 
permissions  rwrr 
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(* Title: HOL/Library/Rational.thy 
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ID: $Id$ 
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Author: Markus Wenzel, TU Muenchen 
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*) 
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14691  6 
header {* Rational numbers *} 
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15131  8 
theory Rational 
25762  9 
imports "~~/src/HOL/Library/Abstract_Rat" 
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uses ("rat_arith.ML") 
15131  11 
begin 
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18913  13 
subsection {* Rational numbers *} 
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subsubsection {* Equivalence of fractions *} 
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19765  17 
definition 
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fraction :: "(int \<times> int) set" where 
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"fraction = {x. snd x \<noteq> 0}" 
18913  20 

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definition 
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ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where 
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"ratrel = {(x,y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}" 
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18913  25 
lemma fraction_iff [simp]: "(x \<in> fraction) = (snd x \<noteq> 0)" 
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by (simp add: fraction_def) 

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18913  28 
lemma ratrel_iff [simp]: 
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"((x,y) \<in> ratrel) = 

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(snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)" 

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by (simp add: ratrel_def) 

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18913  33 
lemma refl_ratrel: "refl fraction ratrel" 
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by (auto simp add: refl_def fraction_def ratrel_def) 

35 

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lemma sym_ratrel: "sym ratrel" 

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by (simp add: ratrel_def sym_def) 

38 

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lemma trans_ratrel_lemma: 

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

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

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

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

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proof  

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

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also note 1 

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

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also note 2 

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

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

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

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qed 
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18913  54 
lemma trans_ratrel: "trans ratrel" 
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by (auto simp add: trans_def elim: trans_ratrel_lemma) 

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lemma equiv_ratrel: "equiv fraction ratrel" 

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

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lemmas equiv_ratrel_iff [iff] = eq_equiv_class_iff [OF equiv_ratrel] 

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lemma equiv_ratrel_iff2: 

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

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

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by (rule eq_equiv_class_iff [OF equiv_ratrel], simp_all) 

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18913  68 
subsubsection {* The type of rational numbers *} 
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18913  70 
typedef (Rat) rat = "fraction//ratrel" 
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proof 

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

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

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qed 
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18913  76 
lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel``{x} \<in> Rat" 
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by (simp add: Rat_def quotientI) 

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declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp] 

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definition 
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Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where 
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[code func del]: "Fract a b = Abs_Rat (ratrel``{(a,b)})" 
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lemma Fract_zero: 

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"Fract k 0 = Fract l 0" 

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by (simp add: Fract_def ratrel_def) 

18913  89 

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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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by (cases q) (clarsimp simp add: Fract_def Rat_def fraction_def quotient_def) 
18913  93 

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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 {* Congruence lemmas *} 

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18913  101 
lemma add_congruent2: 
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"(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)}) 

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respects2 ratrel" 

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apply (rule equiv_ratrel [THEN congruent2_commuteI]) 

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apply (simp_all add: left_distrib) 

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done 

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lemma minus_congruent: 

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

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by (simp add: congruent_def) 

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lemma mult_congruent2: 

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

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by (rule equiv_ratrel [THEN congruent2_commuteI], simp_all) 

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lemma inverse_congruent: 

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"(\<lambda>x. ratrel``{if fst x=0 then (0,1) else (snd x, fst x)}) respects ratrel" 

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by (auto simp add: congruent_def mult_commute) 

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lemma le_congruent2: 

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"(\<lambda>x y. {(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)}) 
18913  122 
respects2 ratrel" 
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proof (clarsimp simp add: congruent2_def) 

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fix a b a' b' c d c' d'::int 

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assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" 
18913  126 
assume eq1: "a * b' = a' * b" 
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assume 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: zero_less_mult_iff) 
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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: mult_le_cancel_right) 
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also have "... = ?le (a * x) (b * x) c d" 
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by (simp add: mult_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: mult_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: mult_ac) 
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also from D have "... = ?le a' b' c' d'" 
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by (rule le_factor [symmetric]) 
18913  157 
finally show "?le a b c d = ?le a' b' c' d'" . 
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qed 
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18913  160 
lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel] 
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lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel] 

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subsubsection {* Standard operations on rational numbers *} 
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25762  166 
instantiation rat :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}" 
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begin 
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definition 
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Zero_rat_def [code func del]: "0 = Fract 0 1" 
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definition 
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One_rat_def [code func del]: "1 = Fract 1 1" 
18913  174 

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definition 
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add_rat_def [code func del]: 
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"q + r = 
18913  178 
Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. 
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ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})" 
18913  180 

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definition 
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minus_rat_def [code func del]: 
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" q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel``{( fst x, snd x)})" 
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diff_rat_def [code func del]: "q  r = q +  (r::rat)" 
18913  187 

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definition 
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mult_rat_def [code func del]: 
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"q * r = 
18913  191 
Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. 
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ratrel``{(fst x * fst y, snd x * snd y)})" 
18913  193 

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definition 
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inverse_rat_def [code func del]: 
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"inverse q = 
18913  197 
Abs_Rat (\<Union>x \<in> Rep_Rat q. 
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ratrel``{if fst x=0 then (0,1) else (snd x, fst x)})" 

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definition 
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divide_rat_def [code func del]: "q / r = q * inverse (r::rat)" 
18913  202 

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definition 
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le_rat_def [code func del]: 
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"q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. 
18982  206 
{(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)})" 
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definition 
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less_rat_def [code func del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w" 
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definition 
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abs_rat_def: "\<bar>q\<bar> = (if q < 0 then q else (q::rat))" 
18913  213 

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definition 
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sgn_rat_def: "sgn (q::rat) = (if q=0 then 0 else if 0<q then 1 else  1)" 
18913  216 

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instance .. 
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219 
end 
24506  220 

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221 
instantiation rat :: power 
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222 
begin 
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223 

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primrec power_rat 
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225 
where 
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rat_power_0: "q ^ 0 = (1\<Colon>rat)" 
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 rat_power_Suc: "q ^ (Suc n) = (q\<Colon>rat) * (q ^ n)" 
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instance .. 
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230 

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231 
end 
20522  232 

18913  233 
theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 
234 
(Fract a b = Fract c d) = (a * d = c * b)" 

235 
by (simp add: Fract_def) 

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236 

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237 
theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 
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Fract a b + Fract c d = Fract (a * d + c * b) (b * d)" 
18913  239 
by (simp add: Fract_def add_rat_def add_congruent2 UN_ratrel2) 
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240 

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241 
theorem minus_rat: "b \<noteq> 0 ==> (Fract a b) = Fract (a) b" 
18913  242 
by (simp add: Fract_def minus_rat_def minus_congruent UN_ratrel) 
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243 

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244 
theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 
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Fract a b  Fract c d = Fract (a * d  c * b) (b * d)" 
18913  246 
by (simp add: diff_rat_def add_rat minus_rat) 
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247 

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248 
theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 
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Fract a b * Fract c d = Fract (a * c) (b * d)" 
18913  250 
by (simp add: Fract_def mult_rat_def mult_congruent2 UN_ratrel2) 
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251 

18913  252 
theorem inverse_rat: "a \<noteq> 0 ==> b \<noteq> 0 ==> 
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253 
inverse (Fract a b) = Fract b a" 
18913  254 
by (simp add: Fract_def inverse_rat_def inverse_congruent UN_ratrel) 
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255 

18913  256 
theorem divide_rat: "c \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==> 
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257 
Fract a b / Fract c d = Fract (a * d) (b * c)" 
18913  258 
by (simp add: divide_rat_def inverse_rat mult_rat) 
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259 

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260 
theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 
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261 
(Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))" 
18982  262 
by (simp add: Fract_def le_rat_def le_congruent2 UN_ratrel2) 
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263 

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264 
theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 
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265 
(Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))" 
18913  266 
by (simp add: less_rat_def le_rat eq_rat order_less_le) 
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267 

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268 
theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>" 
23879  269 
by (simp add: abs_rat_def minus_rat Zero_rat_def less_rat eq_rat) 
14691  270 
(auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less 
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271 
split: abs_split) 
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272 

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273 

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274 
subsubsection {* The ordered field of rational numbers *} 
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275 

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276 
instance rat :: field 
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277 
proof 
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278 
fix q r s :: rat 
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279 
show "(q + r) + s = q + (r + s)" 
18913  280 
by (induct q, induct r, induct s) 
281 
(simp add: add_rat add_ac mult_ac int_distrib) 

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282 
show "q + r = r + q" 
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283 
by (induct q, induct r) (simp add: add_rat add_ac mult_ac) 
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284 
show "0 + q = q" 
23879  285 
by (induct q) (simp add: Zero_rat_def add_rat) 
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286 
show "(q) + q = 0" 
23879  287 
by (induct q) (simp add: Zero_rat_def minus_rat add_rat eq_rat) 
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288 
show "q  r = q + (r)" 
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289 
by (induct q, induct r) (simp add: add_rat minus_rat diff_rat) 
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290 
show "(q * r) * s = q * (r * s)" 
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changeset

291 
by (induct q, induct r, induct s) (simp add: mult_rat mult_ac) 
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292 
show "q * r = r * q" 
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changeset

293 
by (induct q, induct r) (simp add: mult_rat mult_ac) 
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294 
show "1 * q = q" 
23879  295 
by (induct q) (simp add: One_rat_def mult_rat) 
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296 
show "(q + r) * s = q * s + r * s" 
14691  297 
by (induct q, induct r, induct s) 
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298 
(simp add: add_rat mult_rat eq_rat int_distrib) 
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299 
show "q \<noteq> 0 ==> inverse q * q = 1" 
23879  300 
by (induct q) (simp add: inverse_rat mult_rat One_rat_def Zero_rat_def eq_rat) 
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changeset

301 
show "q / r = q * inverse r" 
14691  302 
by (simp add: divide_rat_def) 
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303 
show "0 \<noteq> (1::rat)" 
23879  304 
by (simp add: Zero_rat_def One_rat_def eq_rat) 
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305 
qed 
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306 

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307 
instance rat :: linorder 
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308 
proof 
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309 
fix q r s :: rat 
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310 
{ 
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311 
assume "q \<le> r" and "r \<le> s" 
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312 
show "q \<le> s" 
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313 
proof (insert prems, induct q, induct r, induct s) 
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parents:
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changeset

314 
fix a b c d e f :: int 
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changeset

315 
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" 
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changeset

316 
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f" 
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parents:
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changeset

317 
show "Fract a b \<le> Fract e f" 
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parents:
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changeset

318 
proof  
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changeset

319 
from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f" 
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320 
by (auto simp add: zero_less_mult_iff linorder_neq_iff) 
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changeset

321 
have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)" 
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parents:
diff
changeset

322 
proof  
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parents:
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changeset

323 
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)" 
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parents:
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changeset

324 
by (simp add: le_rat) 
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325 
with ff show ?thesis by (simp add: mult_le_cancel_right) 
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326 
qed 
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changeset

327 
also have "... = (c * f) * (d * f) * (b * b)" 
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changeset

328 
by (simp only: mult_ac) 
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329 
also have "... \<le> (e * d) * (d * f) * (b * b)" 
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parents:
diff
changeset

330 
proof  
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parents:
diff
changeset

331 
from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)" 
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parents:
diff
changeset

332 
by (simp add: le_rat) 
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parents:
diff
changeset

333 
with bb show ?thesis by (simp add: mult_le_cancel_right) 
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parents:
diff
changeset

334 
qed 
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paulson
parents:
diff
changeset

335 
finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)" 
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parents:
diff
changeset

336 
by (simp only: mult_ac) 
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changeset

337 
with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)" 
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paulson
parents:
diff
changeset

338 
by (simp add: mult_le_cancel_right) 
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diff
changeset

339 
with neq show ?thesis by (simp add: le_rat) 
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parents:
diff
changeset

340 
qed 
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paulson
parents:
diff
changeset

341 
qed 
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paulson
parents:
diff
changeset

342 
next 
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343 
assume "q \<le> r" and "r \<le> q" 
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diff
changeset

344 
show "q = r" 
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diff
changeset

345 
proof (insert prems, induct q, induct r) 
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changeset

346 
fix a b c d :: int 
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347 
assume neq: "b \<noteq> 0" "d \<noteq> 0" 
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changeset

348 
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b" 
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parents:
diff
changeset

349 
show "Fract a b = Fract c d" 
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paulson
parents:
diff
changeset

350 
proof  
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parents:
diff
changeset

351 
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)" 
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parents:
diff
changeset

352 
by (simp add: le_rat) 
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parents:
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changeset

353 
also have "... \<le> (a * d) * (b * d)" 
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parents:
diff
changeset

354 
proof  
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parents:
diff
changeset

355 
from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)" 
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paulson
parents:
diff
changeset

356 
by (simp add: le_rat) 
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paulson
parents:
diff
changeset

357 
thus ?thesis by (simp only: mult_ac) 
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paulson
parents:
diff
changeset

358 
qed 
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paulson
parents:
diff
changeset

359 
finally have "(a * d) * (b * d) = (c * b) * (b * d)" . 
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paulson
parents:
diff
changeset

360 
moreover from neq have "b * d \<noteq> 0" by simp 
3d4df8c166ae
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paulson
parents:
diff
changeset

361 
ultimately have "a * d = c * b" by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

362 
with neq show ?thesis by (simp add: eq_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

363 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

364 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

365 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

366 
show "q \<le> q" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

367 
by (induct q) (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

368 
show "(q < r) = (q \<le> r \<and> q \<noteq> r)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

369 
by (simp only: less_rat_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

370 
show "q \<le> r \<or> r \<le> q" 
18913  371 
by (induct q, induct r) 
372 
(simp add: le_rat mult_commute, rule linorder_linear) 

14365
3d4df8c166ae
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paulson
parents:
diff
changeset

373 
} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

374 
qed 
3d4df8c166ae
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paulson
parents:
diff
changeset

375 

25571
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376 
instantiation rat :: distrib_lattice 
c9e39eafc7a0
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haftmann
parents:
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377 
begin 
c9e39eafc7a0
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parents:
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378 

c9e39eafc7a0
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379 
definition 
c9e39eafc7a0
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380 
"(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min" 
c9e39eafc7a0
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parents:
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381 

c9e39eafc7a0
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382 
definition 
c9e39eafc7a0
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parents:
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383 
"(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max" 
c9e39eafc7a0
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384 

c9e39eafc7a0
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385 
instance 
22456  386 
by default (auto simp add: min_max.sup_inf_distrib1 inf_rat_def sup_rat_def) 
387 

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388 
end 
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parents:
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389 

14365
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390 
instance rat :: ordered_field 
3d4df8c166ae
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paulson
parents:
diff
changeset

391 
proof 
3d4df8c166ae
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parents:
diff
changeset

392 
fix q r s :: rat 
3d4df8c166ae
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paulson
parents:
diff
changeset

393 
show "q \<le> r ==> s + q \<le> s + r" 
3d4df8c166ae
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paulson
parents:
diff
changeset

394 
proof (induct q, induct r, induct s) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

395 
fix a b c d e f :: int 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

396 
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" 
3d4df8c166ae
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paulson
parents:
diff
changeset

397 
assume le: "Fract a b \<le> Fract c d" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

398 
show "Fract e f + Fract a b \<le> Fract e f + Fract c d" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

399 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

400 
let ?F = "f * f" from neq have F: "0 < ?F" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

401 
by (auto simp add: zero_less_mult_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

402 
from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

403 
by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

404 
with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

405 
by (simp add: mult_le_cancel_right) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

406 
with neq show ?thesis by (simp add: add_rat le_rat mult_ac int_distrib) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

407 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

408 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

409 
show "q < r ==> 0 < s ==> s * q < s * r" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

410 
proof (induct q, induct r, induct s) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

411 
fix a b c d e f :: int 
3d4df8c166ae
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paulson
parents:
diff
changeset

412 
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

413 
assume le: "Fract a b < Fract c d" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

414 
assume gt: "0 < Fract e f" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

415 
show "Fract e f * Fract a b < Fract e f * Fract c d" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

416 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

417 
let ?E = "e * f" and ?F = "f * f" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

418 
from neq gt have "0 < ?E" 
23879  419 
by (auto simp add: Zero_rat_def less_rat le_rat order_less_le eq_rat) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

420 
moreover from neq have "0 < ?F" 
3d4df8c166ae
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paulson
parents:
diff
changeset

421 
by (auto simp add: zero_less_mult_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

422 
moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

423 
by (simp add: less_rat) 
3d4df8c166ae
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paulson
parents:
diff
changeset

424 
ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

425 
by (simp add: mult_less_cancel_right) 
3d4df8c166ae
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paulson
parents:
diff
changeset

426 
with neq show ?thesis 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

427 
by (simp add: less_rat mult_rat mult_ac) 
3d4df8c166ae
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paulson
parents:
diff
changeset

428 
qed 
3d4df8c166ae
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paulson
parents:
diff
changeset

429 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

430 
show "\<bar>q\<bar> = (if q < 0 then q else q)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

431 
by (simp only: abs_rat_def) 
24506  432 
qed (auto simp: sgn_rat_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

433 

3d4df8c166ae
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paulson
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diff
changeset

434 
instance rat :: division_by_zero 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

435 
proof 
18913  436 
show "inverse 0 = (0::rat)" 
23879  437 
by (simp add: Zero_rat_def Fract_def inverse_rat_def 
18913  438 
inverse_congruent UN_ratrel) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

439 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

440 

20522  441 
instance rat :: recpower 
442 
proof 

443 
fix q :: rat 

444 
fix n :: nat 

445 
show "q ^ 0 = 1" by simp 

446 
show "q ^ (Suc n) = q * (q ^ n)" by simp 

447 
qed 

448 

14365
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paulson
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diff
changeset

449 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
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diff
changeset

450 
subsection {* Various Other Results *} 
3d4df8c166ae
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paulson
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diff
changeset

451 

3d4df8c166ae
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paulson
parents:
diff
changeset

452 
lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (a) (b) = Fract a b" 
18913  453 
by (simp add: eq_rat) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

454 

3d4df8c166ae
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paulson
parents:
diff
changeset

455 
theorem Rat_induct_pos [case_names Fract, induct type: rat]: 
3d4df8c166ae
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paulson
parents:
diff
changeset

456 
assumes step: "!!a b. 0 < b ==> P (Fract a b)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

457 
shows "P q" 
3d4df8c166ae
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paulson
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diff
changeset

458 
proof (cases q) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

459 
have step': "!!a b. b < 0 ==> P (Fract a b)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

460 
proof  
3d4df8c166ae
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paulson
parents:
diff
changeset

461 
fix a::int and b::int 
3d4df8c166ae
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paulson
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diff
changeset

462 
assume b: "b < 0" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

463 
hence "0 < b" by simp 
3d4df8c166ae
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paulson
parents:
diff
changeset

464 
hence "P (Fract (a) (b))" by (rule step) 
3d4df8c166ae
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paulson
parents:
diff
changeset

465 
thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

466 
qed 
3d4df8c166ae
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paulson
parents:
diff
changeset

467 
case (Fract a b) 
3d4df8c166ae
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paulson
parents:
diff
changeset

468 
thus "P q" by (force simp add: linorder_neq_iff step step') 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

469 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

470 

3d4df8c166ae
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paulson
parents:
diff
changeset

471 
lemma zero_less_Fract_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

472 
"0 < b ==> (0 < Fract a b) = (0 < a)" 
23879  473 
by (simp add: Zero_rat_def less_rat order_less_imp_not_eq2 zero_less_mult_iff) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

474 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

475 
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

476 
apply (insert add_rat [of concl: m n 1 1]) 
23879  477 
apply (simp add: One_rat_def [symmetric]) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

478 
done 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

479 

23429  480 
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1" 
23879  481 
by (induct k) (simp_all add: Zero_rat_def One_rat_def add_rat) 
23429  482 

483 
lemma of_int_rat: "of_int k = Fract k 1" 

484 
by (cases k rule: int_diff_cases, simp add: of_nat_rat diff_rat) 

485 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

486 
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k" 
23429  487 
by (rule of_nat_rat [symmetric]) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

488 

69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

489 
lemma Fract_of_int_eq: "Fract k 1 = of_int k" 
23429  490 
by (rule of_int_rat [symmetric]) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

491 

24198  492 
lemma Fract_of_int_quotient: "Fract k l = (if l = 0 then Fract 1 0 else of_int k / of_int l)" 
493 
by (auto simp add: Fract_zero Fract_of_int_eq [symmetric] divide_rat) 

494 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

495 

14691  496 
subsection {* Numerals and Arithmetic *} 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
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14378
diff
changeset

497 

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498 
instantiation rat :: number_ring 
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haftmann
parents:
25502
diff
changeset

499 
begin 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

500 

25571
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501 
definition 
25965  502 
rat_number_of_def [code func del]: "number_of w = (of_int w \<Colon> rat)" 
25571
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changeset

503 

c9e39eafc7a0
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changeset

504 
instance 
c9e39eafc7a0
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parents:
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changeset

505 
by default (simp add: rat_number_of_def) 
c9e39eafc7a0
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haftmann
parents:
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diff
changeset

506 

c9e39eafc7a0
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parents:
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changeset

507 
end 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

508 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

509 
use "rat_arith.ML" 
24075  510 
declaration {* K rat_arith_setup *} 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

511 

23342  512 

513 
subsection {* Embedding from Rationals to other Fields *} 

514 

24198  515 
class field_char_0 = field + ring_char_0 
23342  516 

25571
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changeset

517 
instance ordered_field < field_char_0 .. 
23342  518 

519 
definition 

520 
of_rat :: "rat \<Rightarrow> 'a::field_char_0" 

521 
where 

24198  522 
[code func del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})" 
23342  523 

524 
lemma of_rat_congruent: 

525 
"(\<lambda>(a, b). {of_int a / of_int b::'a::field_char_0}) respects ratrel" 

526 
apply (rule congruent.intro) 

527 
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq) 

528 
apply (simp only: of_int_mult [symmetric]) 

529 
done 

530 

531 
lemma of_rat_rat: 

532 
"b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b" 

533 
unfolding Fract_def of_rat_def 

534 
by (simp add: UN_ratrel of_rat_congruent) 

535 

536 
lemma of_rat_0 [simp]: "of_rat 0 = 0" 

537 
by (simp add: Zero_rat_def of_rat_rat) 

538 

539 
lemma of_rat_1 [simp]: "of_rat 1 = 1" 

540 
by (simp add: One_rat_def of_rat_rat) 

541 

542 
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b" 

543 
by (induct a, induct b, simp add: add_rat of_rat_rat add_frac_eq) 

544 

23343  545 
lemma of_rat_minus: "of_rat ( a) =  of_rat a" 
546 
by (induct a, simp add: minus_rat of_rat_rat) 

547 

548 
lemma of_rat_diff: "of_rat (a  b) = of_rat a  of_rat b" 

549 
by (simp only: diff_minus of_rat_add of_rat_minus) 

550 

23342  551 
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b" 
552 
apply (induct a, induct b, simp add: mult_rat of_rat_rat) 

553 
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac) 

554 
done 

555 

556 
lemma nonzero_of_rat_inverse: 

557 
"a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)" 

23343  558 
apply (rule inverse_unique [symmetric]) 
559 
apply (simp add: of_rat_mult [symmetric]) 

23342  560 
done 
561 

562 
lemma of_rat_inverse: 

563 
"(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) = 

564 
inverse (of_rat a)" 

565 
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse) 

566 

567 
lemma nonzero_of_rat_divide: 

568 
"b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b" 

569 
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse) 

570 

571 
lemma of_rat_divide: 

572 
"(of_rat (a / b)::'a::{field_char_0,division_by_zero}) 

573 
= of_rat a / of_rat b" 

574 
by (cases "b = 0", simp_all add: nonzero_of_rat_divide) 

575 

23343  576 
lemma of_rat_power: 
577 
"(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n" 

578 
by (induct n) (simp_all add: of_rat_mult power_Suc) 

579 

580 
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)" 

581 
apply (induct a, induct b) 

582 
apply (simp add: of_rat_rat eq_rat) 

583 
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq) 

584 
apply (simp only: of_int_mult [symmetric] of_int_eq_iff) 

585 
done 

586 

587 
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified] 

588 

589 
lemma of_rat_eq_id [simp]: "of_rat = (id :: rat \<Rightarrow> rat)" 

590 
proof 

591 
fix a 

592 
show "of_rat a = id a" 

593 
by (induct a) 

594 
(simp add: of_rat_rat divide_rat Fract_of_int_eq [symmetric]) 

595 
qed 

596 

597 
text{*Collapse nested embeddings*} 

598 
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n" 

599 
by (induct n) (simp_all add: of_rat_add) 

600 

601 
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z" 

23365  602 
by (cases z rule: int_diff_cases, simp add: of_rat_diff) 
23343  603 

604 
lemma of_rat_number_of_eq [simp]: 

605 
"of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})" 

606 
by (simp add: number_of_eq) 

607 

23879  608 
lemmas zero_rat = Zero_rat_def 
609 
lemmas one_rat = One_rat_def 

610 

24198  611 
abbreviation 
612 
rat_of_nat :: "nat \<Rightarrow> rat" 

613 
where 

614 
"rat_of_nat \<equiv> of_nat" 

615 

616 
abbreviation 

617 
rat_of_int :: "int \<Rightarrow> rat" 

618 
where 

619 
"rat_of_int \<equiv> of_int" 

620 

24533
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621 

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622 
subsection {* Implementation of rational numbers as pairs of integers *} 
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623 

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624 
definition 
24622  625 
Rational :: "int \<times> int \<Rightarrow> rat" 
24533
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626 
where 
24622  627 
"Rational = INum" 
24533
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628 

24622  629 
code_datatype Rational 
24533
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630 

24622  631 
lemma Rational_simp: 
632 
"Rational (k, l) = rat_of_int k / rat_of_int l" 

633 
unfolding Rational_def INum_def by simp 

24533
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634 

24622  635 
lemma Rational_zero [simp]: "Rational 0\<^sub>N = 0" 
636 
by (simp add: Rational_simp) 

24533
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637 

24622  638 
lemma Rational_lit [simp]: "Rational i\<^sub>N = rat_of_int i" 
639 
by (simp add: Rational_simp) 

24533
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640 

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641 
lemma zero_rat_code [code, code unfold]: 
24622  642 
"0 = Rational 0\<^sub>N" by simp 
25965  643 
declare zero_rat_code [symmetric, code post] 
24533
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644 

25965  645 
lemma one_rat_code [code, code unfold]: 
24622  646 
"1 = Rational 1\<^sub>N" by simp 
25965  647 
declare one_rat_code [symmetric, code post] 
24533
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648 

25965  649 
lemma [code unfold, symmetric, code post]: 
24533
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650 
"number_of k = rat_of_int (number_of k)" 
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651 
by (simp add: number_of_is_id rat_number_of_def) 
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Added code generator setup (taken from Library/Executable_Rat.thy,
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652 

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653 
definition 
fe1f93f6a15a
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654 
[code func del]: "Fract' (b\<Colon>bool) k l = Fract k l" 
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Added code generator setup (taken from Library/Executable_Rat.thy,
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changeset

655 

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656 
lemma [code]: 
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657 
"Fract k l = Fract' (l \<noteq> 0) k l" 
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658 
unfolding Fract'_def .. 
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659 

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660 
lemma [code]: 
24622  661 
"Fract' True k l = (if l \<noteq> 0 then Rational (k, l) else Fract 1 0)" 
662 
by (simp add: Fract'_def Rational_simp Fract_of_int_quotient [of k l]) 

24533
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663 

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664 
lemma [code]: 
24622  665 
"of_rat (Rational (k, l)) = (if l \<noteq> 0 then of_int k / of_int l else 0)" 
24533
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666 
by (cases "l = 0") 
24622  667 
(auto simp add: Rational_simp of_rat_rat [simplified Fract_of_int_quotient [of k l], symmetric]) 
24533
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668 

26513  669 
instantiation rat :: eq 
670 
begin 

671 

672 
definition [code func del]: "eq (r\<Colon>rat) s \<longleftrightarrow> r = s" 

24533
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673 

26513  674 
instance by default (simp add: eq_rat_def) 
675 

676 
lemma rat_eq_code [code]: "eq (Rational x) (Rational y) \<longleftrightarrow> eq (normNum x) (normNum y)" 

677 
unfolding Rational_def INum_normNum_iff eq .. 

678 

679 
end 

24533
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680 

24622  681 
lemma rat_less_eq_code [code]: "Rational x \<le> Rational y \<longleftrightarrow> normNum x \<le>\<^sub>N normNum y" 
24533
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682 
proof  
24622  683 
have "normNum x \<le>\<^sub>N normNum y \<longleftrightarrow> Rational (normNum x) \<le> Rational (normNum y)" 
684 
by (simp add: Rational_def del: normNum) 

685 
also have "\<dots> = (Rational x \<le> Rational y)" by (simp add: Rational_def) 

24533
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686 
finally show ?thesis by simp 
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687 
qed 
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688 

24622  689 
lemma rat_less_code [code]: "Rational x < Rational y \<longleftrightarrow> normNum x <\<^sub>N normNum y" 
24533
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690 
proof  
24622  691 
have "normNum x <\<^sub>N normNum y \<longleftrightarrow> Rational (normNum x) < Rational (normNum y)" 
692 
by (simp add: Rational_def del: normNum) 

693 
also have "\<dots> = (Rational x < Rational y)" by (simp add: Rational_def) 

24533
fe1f93f6a15a
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694 
finally show ?thesis by simp 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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diff
changeset

695 
qed 
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Added code generator setup (taken from Library/Executable_Rat.thy,
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24506
diff
changeset

696 

24622  697 
lemma rat_add_code [code]: "Rational x + Rational y = Rational (x +\<^sub>N y)" 
698 
unfolding Rational_def by simp 

24533
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699 

24622  700 
lemma rat_mul_code [code]: "Rational x * Rational y = Rational (x *\<^sub>N y)" 
701 
unfolding Rational_def by simp 

24533
fe1f93f6a15a
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changeset

702 

24622  703 
lemma rat_neg_code [code]: " Rational x = Rational (~\<^sub>N x)" 
704 
unfolding Rational_def by simp 

24533
fe1f93f6a15a
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parents:
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705 

24622  706 
lemma rat_sub_code [code]: "Rational x  Rational y = Rational (x \<^sub>N y)" 
707 
unfolding Rational_def by simp 

24533
fe1f93f6a15a
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708 

24622  709 
lemma rat_inv_code [code]: "inverse (Rational x) = Rational (Ninv x)" 
710 
unfolding Rational_def Ninv divide_rat_def by simp 

24533
fe1f93f6a15a
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changeset

711 

24622  712 
lemma rat_div_code [code]: "Rational x / Rational y = Rational (x \<div>\<^sub>N y)" 
713 
unfolding Rational_def by simp 

24533
fe1f93f6a15a
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714 

24622  715 
text {* Setup for SML code generator *} 
24533
fe1f93f6a15a
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24506
diff
changeset

716 

fe1f93f6a15a
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diff
changeset

717 
types_code 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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24506
diff
changeset

718 
rat ("(int */ int)") 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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parents:
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diff
changeset

719 
attach (term_of) {* 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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diff
changeset

720 
fun term_of_rat (p, q) = 
24622  721 
let 
24661  722 
val rT = Type ("Rational.rat", []) 
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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diff
changeset

723 
in 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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parents:
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diff
changeset

724 
if q = 1 orelse p = 0 then HOLogic.mk_number rT p 
25885  725 
else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $ 
24533
fe1f93f6a15a
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parents:
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changeset

726 
HOLogic.mk_number rT p $ HOLogic.mk_number rT q 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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changeset

727 
end; 
fe1f93f6a15a
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diff
changeset

728 
*} 
fe1f93f6a15a
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diff
changeset

729 
attach (test) {* 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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parents:
24506
diff
changeset

730 
fun gen_rat i = 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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diff
changeset

731 
let 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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parents:
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diff
changeset

732 
val p = random_range 0 i; 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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parents:
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diff
changeset

733 
val q = random_range 1 (i + 1); 
fe1f93f6a15a
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changeset

734 
val g = Integer.gcd p q; 
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24622
diff
changeset

735 
val p' = p div g; 
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24622
diff
changeset

736 
val q' = q div g; 
25885  737 
val r = (if one_of [true, false] then p' else ~ p', 
738 
if p' = 0 then 0 else q') 

24533
fe1f93f6a15a
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changeset

739 
in 
25885  740 
(r, fn () => term_of_rat r) 
24533
fe1f93f6a15a
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changeset

741 
end; 
fe1f93f6a15a
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742 
*} 
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
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diff
changeset

743 

fe1f93f6a15a
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changeset

744 
consts_code 
24622  745 
Rational ("(_)") 
24533
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changeset

746 

fe1f93f6a15a
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changeset

747 
consts_code 
fe1f93f6a15a
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748 
"of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int") 
fe1f93f6a15a
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changeset

749 
attach {* 
fe1f93f6a15a
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changeset

750 
fun rat_of_int 0 = (0, 0) 
fe1f93f6a15a
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changeset

751 
 rat_of_int i = (i, 1); 
fe1f93f6a15a
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752 
*} 
fe1f93f6a15a
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changeset

753 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

754 
end 