author  huffman 
Thu, 02 Feb 2006 19:57:13 +0100  
changeset 18913  57f19fad8c2a 
parent 18372  2bffdf62fe7f 
child 18982  a2950f748445 
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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18913  17 
constdefs 
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fraction :: "(int \<times> int) set" 

19 
"fraction \<equiv> {x. snd x \<noteq> 0}" 

20 

21 
ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" 

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"ratrel \<equiv> {(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  24 
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  27 
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  32 
lemma refl_ratrel: "refl fraction ratrel" 
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by (auto simp add: refl_def fraction_def ratrel_def) 

34 

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lemma sym_ratrel: "sym ratrel" 

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by (simp add: ratrel_def sym_def) 

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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  53 
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]) 

58 

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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  67 
subsubsection {* The type of rational numbers *} 
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18913  69 
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  75 
lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel``{x} \<in> Rat" 
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by (simp add: Rat_def quotientI) 

77 

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declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp] 

79 

80 

81 
constdefs 

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Fract :: "int \<Rightarrow> int \<Rightarrow> rat" 

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"Fract a b \<equiv> Abs_Rat (ratrel``{(a,b)})" 

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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) 

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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  96 
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)) 

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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  121 
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  152 
finally show "?le a b c d = ?le a' b' c' d'" . 
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qed 
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18913  155 
lemma All_equiv_class: 
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"equiv A r ==> f respects r ==> a \<in> A 

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==> (\<forall>x \<in> r``{a}. f x) = f a" 

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apply safe 

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apply (drule (1) equiv_class_self) 

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apply simp 

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apply (drule (1) congruent.congruent) 

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apply simp 

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done 

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18913  165 
lemma congruent2_implies_congruent_All: 
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"equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==> 

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congruent r1 (\<lambda>x1. \<forall>x2 \<in> r2``{a}. f x1 x2)" 

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apply (unfold congruent_def) 

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apply clarify 

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apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+) 

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apply (simp add: UN_equiv_class congruent2_implies_congruent) 

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apply (unfold congruent2_def equiv_def refl_def) 

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apply (blast del: equalityI) 

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done 

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18913  176 
lemma All_equiv_class2: 
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"equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2 

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==> (\<forall>x1 \<in> r1``{a1}. \<forall>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2" 

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by (simp add: All_equiv_class congruent2_implies_congruent 

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congruent2_implies_congruent_All) 

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18913  182 
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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lemmas All_ratrel2 = All_equiv_class2 [OF equiv_ratrel equiv_ratrel] 

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subsubsection {* Standard operations on rational numbers *} 
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18913  189 
instance rat :: "{ord, zero, one, plus, times, minus, inverse}" .. 
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defs (overloaded) 
18913  192 
Zero_rat_def: "0 == Fract 0 1" 
193 
One_rat_def: "1 == Fract 1 1" 

194 

195 
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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200 
minus_rat_def: 

201 
" q == Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel``{( fst x, snd x)})" 

202 

203 
diff_rat_def: "q  r == q +  (r::rat)" 

204 

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mult_rat_def: 

206 
"q * r == 

207 
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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inverse_rat_def: 

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"inverse q == 

212 
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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215 
divide_rat_def: "q / r == q * inverse (r::rat)" 

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217 
le_rat_def: 

218 
"q \<le> (r::rat) == 

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\<forall>x \<in> Rep_Rat q. \<forall>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)" 

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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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18913  226 
lemma zero_rat: "0 = Fract 0 1" 
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by (simp add: Zero_rat_def) 

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18913  229 
lemma one_rat: "1 = Fract 1 1" 
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by (simp add: One_rat_def) 

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232 
theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 

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(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  238 
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  241 
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  245 
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  249 
by (simp add: Fract_def mult_rat_def mult_congruent2 UN_ratrel2) 
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18913  251 
theorem inverse_rat: "a \<noteq> 0 ==> b \<noteq> 0 ==> 
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inverse (Fract a b) = Fract b a" 
18913  253 
by (simp add: Fract_def inverse_rat_def inverse_congruent UN_ratrel) 
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254 

18913  255 
theorem divide_rat: "c \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==> 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

256 
Fract a b / Fract c d = Fract (a * d) (b * c)" 
18913  257 
by (simp add: divide_rat_def inverse_rat mult_rat) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

258 

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

259 
theorem le_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

260 
(Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))" 
18913  261 
by (simp add: Fract_def le_rat_def le_congruent2 All_ratrel2) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

262 

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

263 
theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

264 
(Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))" 
18913  265 
by (simp add: less_rat_def le_rat eq_rat order_less_le) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

266 

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

267 
theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>" 
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

268 
by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat) 
14691  269 
(auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

270 
split: abs_split) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

271 

3d4df8c166ae
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paulson
parents:
diff
changeset

272 

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

273 
subsubsection {* The ordered field of rational numbers *} 
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

274 

3d4df8c166ae
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paulson
parents:
diff
changeset

275 
lemma rat_add_assoc: "(q + r) + s = q + (r + (s::rat))" 
14691  276 
by (induct q, induct r, induct s) 
14365
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paulson
parents:
diff
changeset

277 
(simp add: add_rat add_ac mult_ac int_distrib) 
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paulson
parents:
diff
changeset

278 

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

279 
lemma rat_add_0: "0 + q = (q::rat)" 
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

280 
by (induct q) (simp add: zero_rat add_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

281 

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

282 
lemma rat_left_minus: "(q) + q = (0::rat)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

283 
by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

284 

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

285 

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

286 
instance rat :: field 
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

287 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

288 
fix q r s :: rat 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

289 
show "(q + r) + s = q + (r + s)" 
18913  290 
by (induct q, induct r, induct s) 
291 
(simp add: add_rat add_ac mult_ac int_distrib) 

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

292 
show "q + r = r + q" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

293 
by (induct q, induct r) (simp add: add_rat add_ac mult_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

294 
show "0 + q = q" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

295 
by (induct q) (simp add: zero_rat add_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

296 
show "(q) + q = 0" 
18913  297 
by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

298 
show "q  r = q + (r)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

299 
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

300 
show "(q * r) * s = q * (r * s)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

301 
by (induct q, induct r, induct s) (simp add: mult_rat mult_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

302 
show "q * r = r * q" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

303 
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

304 
show "1 * q = q" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

305 
by (induct q) (simp add: one_rat mult_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

306 
show "(q + r) * s = q * s + r * s" 
14691  307 
by (induct q, induct r, induct s) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

308 
(simp add: add_rat mult_rat eq_rat int_distrib) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

309 
show "q \<noteq> 0 ==> inverse q * q = 1" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

310 
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

311 
show "q / r = q * inverse r" 
14691  312 
by (simp add: divide_rat_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

313 
show "0 \<noteq> (1::rat)" 
14691  314 
by (simp add: zero_rat one_rat eq_rat) 
14365
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 

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

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

318 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

319 
fix q r s :: rat 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

320 
{ 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

321 
assume "q \<le> r" and "r \<le> s" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

322 
show "q \<le> s" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

323 
proof (insert prems, induct q, induct r, induct s) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

324 
fix a b c d e f :: int 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

325 
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

326 
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

327 
show "Fract a b \<le> Fract e f" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

328 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

329 
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

330 
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

331 
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

332 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

333 
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

334 
by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

335 
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

336 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

337 
also have "... = (c * f) * (d * f) * (b * b)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

338 
by (simp only: mult_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

339 
also have "... \<le> (e * d) * (d * f) * (b * b)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

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

341 
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

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

343 
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

344 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

345 
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

346 
by (simp only: mult_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

347 
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

348 
by (simp add: mult_le_cancel_right) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

349 
with neq show ?thesis by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

350 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

351 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

352 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

353 
assume "q \<le> r" and "r \<le> q" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

354 
show "q = r" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

355 
proof (insert prems, induct q, induct r) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

356 
fix a b c d :: int 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

357 
assume neq: "b \<noteq> 0" "d \<noteq> 0" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

358 
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

359 
show "Fract a b = Fract c d" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

360 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

361 
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

362 
by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

363 
also have "... \<le> (a * d) * (b * d)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

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

365 
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

366 
by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

367 
thus ?thesis by (simp only: mult_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

368 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

369 
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

370 
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

371 
ultimately have "a * d = c * b" by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

372 
with neq show ?thesis by (simp add: eq_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

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

374 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

375 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

376 
show "q \<le> q" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

377 
by (induct q) (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

378 
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

379 
by (simp only: less_rat_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

380 
show "q \<le> r \<or> r \<le> q" 
18913  381 
by (induct q, induct r) 
382 
(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

383 
} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

384 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

385 

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

386 
instance rat :: ordered_field 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

387 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

388 
fix q r s :: rat 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

389 
show "q \<le> r ==> s + q \<le> s + r" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

390 
proof (induct q, induct r, induct s) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

391 
fix a b c d e f :: int 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

392 
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

393 
assume le: "Fract a b \<le> Fract c d" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

394 
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

395 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

396 
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

397 
by (auto simp add: zero_less_mult_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

398 
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

399 
by (simp add: le_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

400 
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

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

402 
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

403 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

404 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

405 
show "q < r ==> 0 < s ==> s * q < s * r" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

406 
proof (induct q, induct r, induct s) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

407 
fix a b c d e f :: int 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

408 
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

409 
assume le: "Fract a b < Fract c d" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

410 
assume gt: "0 < Fract e f" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

411 
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

412 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

413 
let ?E = "e * f" and ?F = "f * f" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

414 
from neq gt have "0 < ?E" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

415 
by (auto simp add: zero_rat 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

416 
moreover from neq have "0 < ?F" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

417 
by (auto simp add: zero_less_mult_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

418 
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

419 
by (simp add: less_rat) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

420 
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

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

422 
with neq show ?thesis 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

423 
by (simp add: less_rat mult_rat mult_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

424 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

425 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

426 
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

427 
by (simp only: abs_rat_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

428 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

429 

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

430 
instance rat :: division_by_zero 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

431 
proof 
18913  432 
show "inverse 0 = (0::rat)" 
433 
by (simp add: zero_rat Fract_def inverse_rat_def 

434 
inverse_congruent UN_ratrel) 

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

435 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

436 

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

437 

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

438 
subsection {* Various Other Results *} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

439 

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

440 
lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (a) (b) = Fract a b" 
18913  441 
by (simp add: eq_rat) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

442 

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

443 
theorem Rat_induct_pos [case_names Fract, induct type: rat]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

444 
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

445 
shows "P q" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

446 
proof (cases q) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

447 
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

448 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

449 
fix a::int and b::int 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

450 
assume b: "b < 0" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

451 
hence "0 < b" by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

452 
hence "P (Fract (a) (b))" by (rule step) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

453 
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

454 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

455 
case (Fract a b) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

456 
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

457 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

458 

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

459 
lemma zero_less_Fract_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset

460 
"0 < b ==> (0 < Fract a b) = (0 < a)" 
14691  461 
by (simp add: zero_rat 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

462 

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

463 
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

464 
apply (insert add_rat [of concl: m n 1 1]) 
14691  465 
apply (simp add: one_rat [symmetric]) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

466 
done 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

467 

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

468 
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k" 
14691  469 
apply (induct k) 
470 
apply (simp add: zero_rat) 

471 
apply (simp add: Fract_add_one) 

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

472 
done 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

473 

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

474 
lemma Fract_of_int_eq: "Fract k 1 = of_int k" 
14691  475 
proof (cases k rule: int_cases) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

476 
case (nonneg n) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

477 
thus ?thesis by (simp add: int_eq_of_nat Fract_of_nat_eq) 
14691  478 
next 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

479 
case (neg n) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

480 
hence "Fract k 1 =  (Fract (of_nat (Suc n)) 1)" 
14691  481 
by (simp only: minus_rat int_eq_of_nat) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

482 
also have "... =  (of_nat (Suc n))" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset

483 
by (simp only: Fract_of_nat_eq) 
14691  484 
finally show ?thesis 
485 
by (simp add: only: prems int_eq_of_nat of_int_minus of_int_of_nat_eq) 

486 
qed 

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

487 

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

488 

14691  489 
subsection {* Numerals and Arithmetic *} 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

490 

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

491 
instance rat :: number .. 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

492 

15013  493 
defs (overloaded) 
494 
rat_number_of_def: "(number_of w :: rat) == of_int (Rep_Bin w)" 

495 
{*the type constraint is essential!*} 

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

496 

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

497 
instance rat :: number_ring 
15013  498 
by (intro_classes, simp add: rat_number_of_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

499 

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

500 
declare diff_rat_def [symmetric] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

501 

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

502 
use "rat_arith.ML" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

503 

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

504 
setup rat_arith_setup 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

505 

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

506 
end 