author  huffman 
Wed, 20 Jun 2007 05:06:16 +0200  
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parent 23365  f31794033ae1 
child 23879  4776af8be741 
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 
18913  9 
imports Main 
16417  10 
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 
19765  19 
"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 
19765  23 
"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 

36 
lemma sym_ratrel: "sym ratrel" 

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by (simp add: ratrel_def sym_def) 

38 

39 
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)" 

43 
shows "a * b'' = a'' * b" 

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proof  

45 
have "b' * (a * b'') = b'' * (a * b')" by simp 

46 
also note 1 

47 
also have "b'' * (a' * b) = b * (a' * b'')" by simp 

48 
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)" . 

51 
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) 

56 

57 
lemma equiv_ratrel: "equiv fraction ratrel" 

58 
by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel]) 

59 

60 
lemmas equiv_ratrel_iff [iff] = eq_equiv_class_iff [OF equiv_ratrel] 

61 

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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) 

78 

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declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp] 

80 

81 

19765  82 
definition 
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Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where 
19765  84 
"Fract a b = Abs_Rat (ratrel``{(a,b)})" 
18913  85 

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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  89 

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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  97 
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 

103 

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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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116 
lemma le_congruent2: 

18982  117 
"(\<lambda>x y. {(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)}) 
18913  118 
respects2 ratrel" 
119 
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  122 
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  153 
finally show "?le a b c d = ?le a' b' c' d'" . 
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qed 
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18913  156 
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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20522  162 
instance rat :: "{ord, zero, one, plus, times, minus, inverse, power}" .. 
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defs (overloaded) 
18913  165 
Zero_rat_def: "0 == Fract 0 1" 
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One_rat_def: "1 == Fract 1 1" 

167 

168 
add_rat_def: 

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"q + r == 

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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)})" 

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minus_rat_def: 

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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: "q  r == q +  (r::rat)" 

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mult_rat_def: 

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"q * r == 

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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)})" 

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183 
inverse_rat_def: 

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"inverse q == 

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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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188 
divide_rat_def: "q / r == q * inverse (r::rat)" 

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190 
le_rat_def: 

18982  191 
"q \<le> r == contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. 
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{(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)})" 

18913  193 

194 
less_rat_def: "(z < (w::rat)) == (z \<le> w & z \<noteq> w)" 

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abs_rat_def: "\<bar>q\<bar> == if q < 0 then q else (q::rat)" 
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20522  198 
primrec (rat) 
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rat_power_0: "q ^ 0 = 1" 

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rat_power_Suc: "q ^ (Suc n) = (q::rat) * (q ^ n)" 

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18913  202 
lemma zero_rat: "0 = Fract 0 1" 
203 
by (simp add: Zero_rat_def) 

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18913  205 
lemma one_rat: "1 = Fract 1 1" 
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by (simp add: One_rat_def) 

207 

208 
theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 

209 
(Fract a b = Fract c d) = (a * d = c * b)" 

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by (simp add: Fract_def) 

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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  214 
by (simp add: Fract_def add_rat_def add_congruent2 UN_ratrel2) 
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theorem minus_rat: "b \<noteq> 0 ==> (Fract a b) = Fract (a) b" 
18913  217 
by (simp add: Fract_def minus_rat_def minus_congruent UN_ratrel) 
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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  221 
by (simp add: diff_rat_def add_rat minus_rat) 
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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  225 
by (simp add: Fract_def mult_rat_def mult_congruent2 UN_ratrel2) 
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18913  227 
theorem inverse_rat: "a \<noteq> 0 ==> b \<noteq> 0 ==> 
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inverse (Fract a b) = Fract b a" 
18913  229 
by (simp add: Fract_def inverse_rat_def inverse_congruent UN_ratrel) 
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18913  231 
theorem divide_rat: "c \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==> 
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Fract a b / Fract c d = Fract (a * d) (b * c)" 
18913  233 
by (simp add: divide_rat_def inverse_rat mult_rat) 
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theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 
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236 
(Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))" 
18982  237 
by (simp add: Fract_def le_rat_def le_congruent2 UN_ratrel2) 
14365
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paulson
parents:
diff
changeset

238 

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paulson
parents:
diff
changeset

239 
theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

240 
(Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))" 
18913  241 
by (simp add: less_rat_def le_rat eq_rat order_less_le) 
14365
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paulson
parents:
diff
changeset

242 

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

243 
theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>" 
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paulson
parents:
diff
changeset

244 
by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat) 
14691  245 
(auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less 
14365
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paulson
parents:
diff
changeset

246 
split: abs_split) 
3d4df8c166ae
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paulson
parents:
diff
changeset

247 

3d4df8c166ae
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paulson
parents:
diff
changeset

248 

3d4df8c166ae
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paulson
parents:
diff
changeset

249 
subsubsection {* The ordered field of rational numbers *} 
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parents:
diff
changeset

250 

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parents:
diff
changeset

251 
instance rat :: field 
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paulson
parents:
diff
changeset

252 
proof 
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paulson
parents:
diff
changeset

253 
fix q r s :: rat 
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paulson
parents:
diff
changeset

254 
show "(q + r) + s = q + (r + s)" 
18913  255 
by (induct q, induct r, induct s) 
256 
(simp add: add_rat add_ac mult_ac int_distrib) 

14365
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paulson
parents:
diff
changeset

257 
show "q + r = r + q" 
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paulson
parents:
diff
changeset

258 
by (induct q, induct r) (simp add: add_rat add_ac mult_ac) 
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paulson
parents:
diff
changeset

259 
show "0 + q = q" 
3d4df8c166ae
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paulson
parents:
diff
changeset

260 
by (induct q) (simp add: zero_rat add_rat) 
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paulson
parents:
diff
changeset

261 
show "(q) + q = 0" 
18913  262 
by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

263 
show "q  r = q + (r)" 
3d4df8c166ae
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paulson
parents:
diff
changeset

264 
by (induct q, induct r) (simp add: add_rat minus_rat diff_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

265 
show "(q * r) * s = q * (r * s)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

266 
by (induct q, induct r, induct s) (simp add: mult_rat mult_ac) 
3d4df8c166ae
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paulson
parents:
diff
changeset

267 
show "q * r = r * q" 
3d4df8c166ae
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paulson
parents:
diff
changeset

268 
by (induct q, induct r) (simp add: mult_rat mult_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

269 
show "1 * q = q" 
3d4df8c166ae
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paulson
parents:
diff
changeset

270 
by (induct q) (simp add: one_rat mult_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

271 
show "(q + r) * s = q * s + r * s" 
14691  272 
by (induct q, induct r, induct s) 
14365
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paulson
parents:
diff
changeset

273 
(simp add: add_rat mult_rat eq_rat int_distrib) 
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paulson
parents:
diff
changeset

274 
show "q \<noteq> 0 ==> inverse q * q = 1" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

275 
by (induct q) (simp add: inverse_rat mult_rat one_rat zero_rat eq_rat) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

276 
show "q / r = q * inverse r" 
14691  277 
by (simp add: divide_rat_def) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

278 
show "0 \<noteq> (1::rat)" 
14691  279 
by (simp add: zero_rat one_rat eq_rat) 
14365
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paulson
parents:
diff
changeset

280 
qed 
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paulson
parents:
diff
changeset

281 

3d4df8c166ae
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paulson
parents:
diff
changeset

282 
instance rat :: linorder 
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paulson
parents:
diff
changeset

283 
proof 
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

284 
fix q r s :: rat 
3d4df8c166ae
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paulson
parents:
diff
changeset

285 
{ 
3d4df8c166ae
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paulson
parents:
diff
changeset

286 
assume "q \<le> r" and "r \<le> s" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

287 
show "q \<le> s" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

288 
proof (insert prems, induct q, induct r, induct s) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

289 
fix a b c d e f :: int 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

290 
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

291 
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

292 
show "Fract a b \<le> Fract e f" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

293 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

294 
from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

295 
by (auto simp add: zero_less_mult_iff linorder_neq_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

296 
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

297 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

298 
from neq 1 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

299 
by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

300 
with ff show ?thesis by (simp add: mult_le_cancel_right) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

301 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

302 
also have "... = (c * f) * (d * f) * (b * b)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

303 
by (simp only: mult_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

304 
also have "... \<le> (e * d) * (d * f) * (b * b)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

305 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

306 
from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

307 
by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

308 
with bb show ?thesis by (simp add: mult_le_cancel_right) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

309 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

310 
finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

311 
by (simp only: mult_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

312 
with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

313 
by (simp add: mult_le_cancel_right) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

314 
with neq show ?thesis by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

315 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

316 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

317 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

318 
assume "q \<le> r" and "r \<le> q" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

319 
show "q = r" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

320 
proof (insert prems, induct q, induct r) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

321 
fix a b c d :: int 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

322 
assume neq: "b \<noteq> 0" "d \<noteq> 0" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

323 
assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

324 
show "Fract a b = Fract c d" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

325 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

326 
from neq 1 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

327 
by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

328 
also have "... \<le> (a * d) * (b * d)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

329 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

330 
from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

331 
by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

332 
thus ?thesis by (simp only: mult_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

333 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

334 
finally have "(a * d) * (b * d) = (c * b) * (b * d)" . 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

335 
moreover from neq have "b * d \<noteq> 0" by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

336 
ultimately have "a * d = c * b" by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

337 
with neq show ?thesis by (simp add: eq_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

338 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

339 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

340 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

341 
show "q \<le> q" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

342 
by (induct q) (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

343 
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

344 
by (simp only: less_rat_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

345 
show "q \<le> r \<or> r \<le> q" 
18913  346 
by (induct q, induct r) 
347 
(simp add: le_rat mult_commute, rule linorder_linear) 

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

348 
} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

349 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

350 

22456  351 
instance rat :: distrib_lattice 
352 
"inf r s \<equiv> min r s" 

353 
"sup r s \<equiv> max r s" 

354 
by default (auto simp add: min_max.sup_inf_distrib1 inf_rat_def sup_rat_def) 

355 

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

356 
instance rat :: ordered_field 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

357 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

358 
fix q r s :: rat 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

359 
show "q \<le> r ==> s + q \<le> s + r" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

360 
proof (induct q, induct r, induct s) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

361 
fix a b c d e f :: int 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

362 
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

363 
assume le: "Fract a b \<le> Fract c d" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

364 
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

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

366 
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

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

368 
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

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

370 
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

371 
by (simp add: mult_le_cancel_right) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

372 
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

373 
qed 
3d4df8c166ae
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parents:
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374 
qed 
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375 
show "q < r ==> 0 < s ==> s * q < s * r" 
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376 
proof (induct q, induct r, induct s) 
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377 
fix a b c d e f :: int 
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378 
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" 
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379 
assume le: "Fract a b < Fract c d" 
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parents:
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changeset

380 
assume gt: "0 < Fract e f" 
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parents:
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381 
show "Fract e f * Fract a b < Fract e f * Fract c d" 
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382 
proof  
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383 
let ?E = "e * f" and ?F = "f * f" 
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384 
from neq gt have "0 < ?E" 
14378
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diff
changeset

385 
by (auto simp add: zero_rat less_rat le_rat order_less_le eq_rat) 
14365
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386 
moreover from neq have "0 < ?F" 
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387 
by (auto simp add: zero_less_mult_iff) 
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parents:
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changeset

388 
moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)" 
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389 
by (simp add: less_rat) 
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390 
ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F" 
3d4df8c166ae
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parents:
diff
changeset

391 
by (simp add: mult_less_cancel_right) 
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changeset

392 
with neq show ?thesis 
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diff
changeset

393 
by (simp add: less_rat mult_rat mult_ac) 
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parents:
diff
changeset

394 
qed 
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changeset

395 
qed 
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parents:
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changeset

396 
show "\<bar>q\<bar> = (if q < 0 then q else q)" 
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parents:
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changeset

397 
by (simp only: abs_rat_def) 
22456  398 
qed auto 
14365
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parents:
diff
changeset

399 

3d4df8c166ae
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400 
instance rat :: division_by_zero 
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401 
proof 
18913  402 
show "inverse 0 = (0::rat)" 
403 
by (simp add: zero_rat Fract_def inverse_rat_def 

404 
inverse_congruent UN_ratrel) 

14365
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405 
qed 
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406 

20522  407 
instance rat :: recpower 
408 
proof 

409 
fix q :: rat 

410 
fix n :: nat 

411 
show "q ^ 0 = 1" by simp 

412 
show "q ^ (Suc n) = q * (q ^ n)" by simp 

413 
qed 

414 

14365
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parents:
diff
changeset

415 

3d4df8c166ae
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416 
subsection {* Various Other Results *} 
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417 

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418 
lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (a) (b) = Fract a b" 
18913  419 
by (simp add: eq_rat) 
14365
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paulson
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420 

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421 
theorem Rat_induct_pos [case_names Fract, induct type: rat]: 
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422 
assumes step: "!!a b. 0 < b ==> P (Fract a b)" 
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423 
shows "P q" 
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424 
proof (cases q) 
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425 
have step': "!!a b. b < 0 ==> P (Fract a b)" 
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426 
proof  
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diff
changeset

427 
fix a::int and b::int 
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428 
assume b: "b < 0" 
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429 
hence "0 < b" by simp 
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430 
hence "P (Fract (a) (b))" by (rule step) 
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431 
thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b]) 
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changeset

432 
qed 
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433 
case (Fract a b) 
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434 
thus "P q" by (force simp add: linorder_neq_iff step step') 
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parents:
diff
changeset

435 
qed 
3d4df8c166ae
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paulson
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diff
changeset

436 

3d4df8c166ae
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437 
lemma zero_less_Fract_iff: 
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438 
"0 < b ==> (0 < Fract a b) = (0 < a)" 
14691  439 
by (simp add: zero_rat less_rat order_less_imp_not_eq2 zero_less_mult_iff) 
14365
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paulson
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diff
changeset

440 

14378
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changeset

441 
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1" 
69c4d5997669
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442 
apply (insert add_rat [of concl: m n 1 1]) 
14691  443 
apply (simp add: one_rat [symmetric]) 
14378
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444 
done 
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445 

23429  446 
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1" 
447 
by (induct k) (simp_all add: zero_rat one_rat add_rat) 

448 

449 
lemma of_int_rat: "of_int k = Fract k 1" 

450 
by (cases k rule: int_diff_cases, simp add: of_nat_rat diff_rat) 

451 

14378
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452 
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k" 
23429  453 
by (rule of_nat_rat [symmetric]) 
14378
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changeset

454 

69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
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parents:
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455 
lemma Fract_of_int_eq: "Fract k 1 = of_int k" 
23429  456 
by (rule of_int_rat [symmetric]) 
14378
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diff
changeset

457 

69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
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458 

14691  459 
subsection {* Numerals and Arithmetic *} 
14387
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Polymorphic treatment of binary arithmetic using axclasses
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460 

22456  461 
instance rat :: number 
462 
rat_number_of_def: "(number_of w :: rat) \<equiv> of_int w" .. 

14387
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463 

e96d5c42c4b0
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464 
instance rat :: number_ring 
19765  465 
by default (simp add: rat_number_of_def) 
14387
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Polymorphic treatment of binary arithmetic using axclasses
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466 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
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467 
use "rat_arith.ML" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
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468 
setup rat_arith_setup 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
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469 

23342  470 

471 
subsection {* Embedding from Rationals to other Fields *} 

472 

473 
axclass field_char_0 < field, ring_char_0 

474 

475 
instance ordered_field < field_char_0 .. 

476 

477 
definition 

478 
of_rat :: "rat \<Rightarrow> 'a::field_char_0" 

479 
where 

480 
"of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})" 

481 

482 
lemma of_rat_congruent: 

483 
"(\<lambda>(a, b). {of_int a / of_int b::'a::field_char_0}) respects ratrel" 

484 
apply (rule congruent.intro) 

485 
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq) 

486 
apply (simp only: of_int_mult [symmetric]) 

487 
done 

488 

489 
lemma of_rat_rat: 

490 
"b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b" 

491 
unfolding Fract_def of_rat_def 

492 
by (simp add: UN_ratrel of_rat_congruent) 

493 

494 
lemma of_rat_0 [simp]: "of_rat 0 = 0" 

495 
by (simp add: Zero_rat_def of_rat_rat) 

496 

497 
lemma of_rat_1 [simp]: "of_rat 1 = 1" 

498 
by (simp add: One_rat_def of_rat_rat) 

499 

500 
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b" 

501 
by (induct a, induct b, simp add: add_rat of_rat_rat add_frac_eq) 

502 

23343  503 
lemma of_rat_minus: "of_rat ( a) =  of_rat a" 
504 
by (induct a, simp add: minus_rat of_rat_rat) 

505 

506 
lemma of_rat_diff: "of_rat (a  b) = of_rat a  of_rat b" 

507 
by (simp only: diff_minus of_rat_add of_rat_minus) 

508 

23342  509 
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b" 
510 
apply (induct a, induct b, simp add: mult_rat of_rat_rat) 

511 
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac) 

512 
done 

513 

514 
lemma nonzero_of_rat_inverse: 

515 
"a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)" 

23343  516 
apply (rule inverse_unique [symmetric]) 
517 
apply (simp add: of_rat_mult [symmetric]) 

23342  518 
done 
519 

520 
lemma of_rat_inverse: 

521 
"(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) = 

522 
inverse (of_rat a)" 

523 
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse) 

524 

525 
lemma nonzero_of_rat_divide: 

526 
"b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b" 

527 
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse) 

528 

529 
lemma of_rat_divide: 

530 
"(of_rat (a / b)::'a::{field_char_0,division_by_zero}) 

531 
= of_rat a / of_rat b" 

532 
by (cases "b = 0", simp_all add: nonzero_of_rat_divide) 

533 

23343  534 
lemma of_rat_power: 
535 
"(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n" 

536 
by (induct n) (simp_all add: of_rat_mult power_Suc) 

537 

538 
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)" 

539 
apply (induct a, induct b) 

540 
apply (simp add: of_rat_rat eq_rat) 

541 
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq) 

542 
apply (simp only: of_int_mult [symmetric] of_int_eq_iff) 

543 
done 

544 

545 
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified] 

546 

547 
lemma of_rat_eq_id [simp]: "of_rat = (id :: rat \<Rightarrow> rat)" 

548 
proof 

549 
fix a 

550 
show "of_rat a = id a" 

551 
by (induct a) 

552 
(simp add: of_rat_rat divide_rat Fract_of_int_eq [symmetric]) 

553 
qed 

554 

555 
text{*Collapse nested embeddings*} 

556 
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n" 

557 
by (induct n) (simp_all add: of_rat_add) 

558 

559 
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z" 

23365  560 
by (cases z rule: int_diff_cases, simp add: of_rat_diff) 
23343  561 

562 
lemma of_rat_number_of_eq [simp]: 

563 
"of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})" 

564 
by (simp add: number_of_eq) 

565 

14365
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paulson
parents:
diff
changeset

566 
end 