author | haftmann |
Thu, 09 Aug 2007 15:52:53 +0200 | |
changeset 24198 | 4031da6d8ba3 |
parent 24075 | 366d4d234814 |
child 24506 | 020db6ec334a |
permissions | -rw-r--r-- |
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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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|
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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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|
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subsubsection {* Equivalence of fractions *} |
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|
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definition |
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fraction :: "(int \<times> int) set" where |
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"fraction = {x. snd x \<noteq> 0}" |
18913 | 20 |
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definition |
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ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where |
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"ratrel = {(x,y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}" |
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|
18913 | 25 |
lemma fraction_iff [simp]: "(x \<in> fraction) = (snd x \<noteq> 0)" |
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by (simp add: fraction_def) |
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18913 | 28 |
lemma ratrel_iff [simp]: |
29 |
"((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) |
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35 |
||
36 |
lemma sym_ratrel: "sym ratrel" |
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by (simp add: ratrel_def sym_def) |
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||
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lemma trans_ratrel_lemma: |
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assumes 1: "a * b' = a' * b" |
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assumes 2: "a' * b'' = a'' * b'" |
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assumes 3: "b' \<noteq> (0::int)" |
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shows "a * b'' = a'' * b" |
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proof - |
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have "b' * (a * b'') = b'' * (a * b')" by simp |
|
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also note 1 |
|
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also have "b'' * (a' * b) = b * (a' * b'')" by simp |
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also note 2 |
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also have "b * (a'' * b') = b' * (a'' * b)" by simp |
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finally have "b' * (a * b'') = b' * (a'' * b)" . |
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with 3 show "a * b'' = a'' * b" by simp |
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qed |
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18913 | 54 |
lemma trans_ratrel: "trans ratrel" |
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by (auto simp add: trans_def elim: trans_ratrel_lemma) |
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||
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lemma equiv_ratrel: "equiv fraction ratrel" |
|
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by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel]) |
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||
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lemmas equiv_ratrel_iff [iff] = eq_equiv_class_iff [OF equiv_ratrel] |
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||
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lemma equiv_ratrel_iff2: |
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"\<lbrakk>snd x \<noteq> 0; snd y \<noteq> 0\<rbrakk> |
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\<Longrightarrow> (ratrel `` {x} = ratrel `` {y}) = ((x,y) \<in> ratrel)" |
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by (rule eq_equiv_class_iff [OF equiv_ratrel], simp_all) |
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18913 | 68 |
subsubsection {* The type of rational numbers *} |
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18913 | 70 |
typedef (Rat) rat = "fraction//ratrel" |
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proof |
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have "(0,1) \<in> fraction" by (simp add: fraction_def) |
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thus "ratrel``{(0,1)} \<in> fraction//ratrel" by (rule quotientI) |
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qed |
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18913 | 76 |
lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel``{x} \<in> Rat" |
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by (simp add: Rat_def quotientI) |
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||
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declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp] |
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definition |
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Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where |
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[code func del]: "Fract a b = Abs_Rat (ratrel``{(a,b)})" |
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lemma Fract_zero: |
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"Fract k 0 = Fract l 0" |
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by (simp add: Fract_def ratrel_def) |
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18913 | 89 |
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theorem Rat_cases [case_names Fract, cases type: rat]: |
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"(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C" |
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by (cases q) (clarsimp simp add: Fract_def Rat_def fraction_def quotient_def) |
18913 | 93 |
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theorem Rat_induct [case_names Fract, induct type: rat]: |
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"(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q" |
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by (cases q) simp |
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subsubsection {* Congruence lemmas *} |
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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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||
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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" |
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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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|
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let ?D = "b * d" and ?D' = "b' * d'" |
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from neq have D: "?D \<noteq> 0" by simp |
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from neq have "?D' \<noteq> 0" by simp |
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hence "?le a b c d = ?le (a * ?D') (b * ?D') c d" |
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by (rule le_factor) |
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also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" |
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by (simp add: mult_ac) |
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also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')" |
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by (simp only: eq1 eq2) |
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also have "... = ?le (a' * ?D) (b' * ?D) c' d'" |
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by (simp add: mult_ac) |
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also from D have "... = ?le a' b' c' d'" |
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by (rule le_factor [symmetric]) |
18913 | 157 |
finally show "?le a b c d = ?le a' b' c' d'" . |
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qed |
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|
18913 | 160 |
lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel] |
161 |
lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel] |
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|
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subsubsection {* Standard operations on rational numbers *} |
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|
23879 | 166 |
instance rat :: zero |
167 |
Zero_rat_def: "0 == Fract 0 1" .. |
|
24198 | 168 |
lemmas [code func del] = Zero_rat_def |
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169 |
|
23879 | 170 |
instance rat :: one |
171 |
One_rat_def: "1 == Fract 1 1" .. |
|
24198 | 172 |
lemmas [code func del] = One_rat_def |
18913 | 173 |
|
23879 | 174 |
instance rat :: plus |
18913 | 175 |
add_rat_def: |
176 |
"q + r == |
|
177 |
Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. |
|
23879 | 178 |
ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)})" .. |
179 |
lemmas [code func del] = add_rat_def |
|
18913 | 180 |
|
23879 | 181 |
instance rat :: minus |
18913 | 182 |
minus_rat_def: |
183 |
"- q == Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel``{(- fst x, snd x)})" |
|
23879 | 184 |
diff_rat_def: "q - r == q + - (r::rat)" .. |
24198 | 185 |
lemmas [code func del] = minus_rat_def diff_rat_def |
18913 | 186 |
|
23879 | 187 |
instance rat :: times |
18913 | 188 |
mult_rat_def: |
189 |
"q * r == |
|
190 |
Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. |
|
23879 | 191 |
ratrel``{(fst x * fst y, snd x * snd y)})" .. |
192 |
lemmas [code func del] = mult_rat_def |
|
18913 | 193 |
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23879 | 194 |
instance rat :: inverse |
18913 | 195 |
inverse_rat_def: |
196 |
"inverse q == |
|
197 |
Abs_Rat (\<Union>x \<in> Rep_Rat q. |
|
198 |
ratrel``{if fst x=0 then (0,1) else (snd x, fst x)})" |
|
23879 | 199 |
divide_rat_def: "q / r == q * inverse (r::rat)" .. |
24198 | 200 |
lemmas [code func del] = inverse_rat_def divide_rat_def |
18913 | 201 |
|
23879 | 202 |
instance rat :: ord |
18913 | 203 |
le_rat_def: |
18982 | 204 |
"q \<le> r == contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. |
205 |
{(fst x * snd y)*(snd x * snd y) \<le> (fst y * snd x)*(snd x * snd y)})" |
|
23879 | 206 |
less_rat_def: "(z < (w::rat)) == (z \<le> w & z \<noteq> w)" .. |
207 |
lemmas [code func del] = le_rat_def less_rat_def |
|
18913 | 208 |
|
23879 | 209 |
instance rat :: abs |
210 |
abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)" .. |
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18913 | 211 |
|
23879 | 212 |
instance rat :: power .. |
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213 |
|
20522 | 214 |
primrec (rat) |
215 |
rat_power_0: "q ^ 0 = 1" |
|
216 |
rat_power_Suc: "q ^ (Suc n) = (q::rat) * (q ^ n)" |
|
217 |
||
18913 | 218 |
theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> |
219 |
(Fract a b = Fract c d) = (a * d = c * b)" |
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220 |
by (simp add: Fract_def) |
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221 |
|
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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 | 224 |
by (simp add: Fract_def add_rat_def add_congruent2 UN_ratrel2) |
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225 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
226 |
theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b" |
18913 | 227 |
by (simp add: Fract_def minus_rat_def minus_congruent UN_ratrel) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
228 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
229 |
theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
230 |
Fract a b - Fract c d = Fract (a * d - c * b) (b * d)" |
18913 | 231 |
by (simp add: diff_rat_def add_rat minus_rat) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
232 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
233 |
theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
234 |
Fract a b * Fract c d = Fract (a * c) (b * d)" |
18913 | 235 |
by (simp add: Fract_def mult_rat_def mult_congruent2 UN_ratrel2) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
236 |
|
18913 | 237 |
theorem inverse_rat: "a \<noteq> 0 ==> b \<noteq> 0 ==> |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
238 |
inverse (Fract a b) = Fract b a" |
18913 | 239 |
by (simp add: Fract_def inverse_rat_def inverse_congruent UN_ratrel) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
240 |
|
18913 | 241 |
theorem divide_rat: "c \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==> |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
242 |
Fract a b / Fract c d = Fract (a * d) (b * c)" |
18913 | 243 |
by (simp add: divide_rat_def inverse_rat mult_rat) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
244 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
245 |
theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
246 |
(Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))" |
18982 | 247 |
by (simp add: Fract_def le_rat_def le_congruent2 UN_ratrel2) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
248 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
249 |
theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
250 |
(Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))" |
18913 | 251 |
by (simp add: less_rat_def le_rat eq_rat order_less_le) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
252 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
253 |
theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>" |
23879 | 254 |
by (simp add: abs_rat_def minus_rat Zero_rat_def less_rat eq_rat) |
14691 | 255 |
(auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
256 |
split: abs_split) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
257 |
|
3d4df8c166ae
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 |
subsubsection {* The ordered field of rational numbers *} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
260 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
261 |
instance rat :: field |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
262 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
263 |
fix q r s :: rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
264 |
show "(q + r) + s = q + (r + s)" |
18913 | 265 |
by (induct q, induct r, induct s) |
266 |
(simp add: add_rat add_ac mult_ac int_distrib) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
267 |
show "q + r = r + q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
268 |
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
|
269 |
show "0 + q = q" |
23879 | 270 |
by (induct q) (simp add: Zero_rat_def add_rat) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
271 |
show "(-q) + q = 0" |
23879 | 272 |
by (induct q) (simp add: Zero_rat_def minus_rat add_rat eq_rat) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
273 |
show "q - r = q + (-r)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
274 |
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
|
275 |
show "(q * r) * s = q * (r * s)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
276 |
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
|
277 |
show "q * r = r * q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
278 |
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
|
279 |
show "1 * q = q" |
23879 | 280 |
by (induct q) (simp add: One_rat_def mult_rat) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
281 |
show "(q + r) * s = q * s + r * s" |
14691 | 282 |
by (induct q, induct r, induct s) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
283 |
(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
|
284 |
show "q \<noteq> 0 ==> inverse q * q = 1" |
23879 | 285 |
by (induct q) (simp add: inverse_rat mult_rat One_rat_def Zero_rat_def eq_rat) |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
286 |
show "q / r = q * inverse r" |
14691 | 287 |
by (simp add: divide_rat_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
288 |
show "0 \<noteq> (1::rat)" |
23879 | 289 |
by (simp add: Zero_rat_def One_rat_def eq_rat) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
290 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
291 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
292 |
instance rat :: linorder |
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 |
fix q r s :: rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
295 |
{ |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
296 |
assume "q \<le> r" and "r \<le> s" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
297 |
show "q \<le> s" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
298 |
proof (insert prems, induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
299 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
300 |
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
|
301 |
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
|
302 |
show "Fract a b \<le> Fract e f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
303 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
304 |
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
|
305 |
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
|
306 |
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
|
307 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
308 |
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
|
309 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
310 |
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
|
311 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
312 |
also have "... = (c * f) * (d * f) * (b * b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
313 |
by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
314 |
also have "... \<le> (e * d) * (d * f) * (b * b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
315 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
316 |
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
|
317 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
318 |
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
|
319 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
320 |
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
|
321 |
by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
322 |
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
|
323 |
by (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
324 |
with neq show ?thesis by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
325 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
326 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
327 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
328 |
assume "q \<le> r" and "r \<le> q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
329 |
show "q = r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
330 |
proof (insert prems, induct q, induct r) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
331 |
fix a b c d :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
332 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
333 |
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
|
334 |
show "Fract a b = Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
335 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
336 |
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
|
337 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
338 |
also have "... \<le> (a * d) * (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
339 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
340 |
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
|
341 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
342 |
thus ?thesis by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
343 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
344 |
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
|
345 |
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
|
346 |
ultimately have "a * d = c * b" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
347 |
with neq show ?thesis by (simp add: eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
348 |
qed |
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 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
351 |
show "q \<le> q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
352 |
by (induct q) (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
353 |
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
|
354 |
by (simp only: less_rat_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
355 |
show "q \<le> r \<or> r \<le> q" |
18913 | 356 |
by (induct q, induct r) |
357 |
(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
|
358 |
} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
359 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
360 |
|
22456 | 361 |
instance rat :: distrib_lattice |
362 |
"inf r s \<equiv> min r s" |
|
363 |
"sup r s \<equiv> max r s" |
|
364 |
by default (auto simp add: min_max.sup_inf_distrib1 inf_rat_def sup_rat_def) |
|
365 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
366 |
instance rat :: ordered_field |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
367 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
368 |
fix q r s :: rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
369 |
show "q \<le> r ==> s + q \<le> s + r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
370 |
proof (induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
371 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
372 |
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
|
373 |
assume le: "Fract a b \<le> Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
374 |
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
|
375 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
376 |
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
|
377 |
by (auto simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
378 |
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
|
379 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
380 |
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
|
381 |
by (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
382 |
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
|
383 |
qed |
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 |
show "q < r ==> 0 < s ==> s * q < s * r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
386 |
proof (induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
387 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
388 |
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
|
389 |
assume le: "Fract a b < Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
390 |
assume gt: "0 < Fract e f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
391 |
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
|
392 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
393 |
let ?E = "e * f" and ?F = "f * f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
394 |
from neq gt have "0 < ?E" |
23879 | 395 |
by (auto simp add: Zero_rat_def less_rat le_rat order_less_le eq_rat) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
396 |
moreover from neq have "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 |
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
|
399 |
by (simp add: less_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
400 |
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
|
401 |
by (simp add: mult_less_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
402 |
with neq show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
403 |
by (simp add: less_rat mult_rat mult_ac) |
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 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
406 |
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
|
407 |
by (simp only: abs_rat_def) |
22456 | 408 |
qed auto |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
409 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
410 |
instance rat :: division_by_zero |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
411 |
proof |
18913 | 412 |
show "inverse 0 = (0::rat)" |
23879 | 413 |
by (simp add: Zero_rat_def Fract_def inverse_rat_def |
18913 | 414 |
inverse_congruent UN_ratrel) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
415 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
416 |
|
20522 | 417 |
instance rat :: recpower |
418 |
proof |
|
419 |
fix q :: rat |
|
420 |
fix n :: nat |
|
421 |
show "q ^ 0 = 1" by simp |
|
422 |
show "q ^ (Suc n) = q * (q ^ n)" by simp |
|
423 |
qed |
|
424 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
425 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
426 |
subsection {* Various Other Results *} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
427 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
428 |
lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (-a) (-b) = Fract a b" |
18913 | 429 |
by (simp add: eq_rat) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
430 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
431 |
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
|
432 |
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
|
433 |
shows "P q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
434 |
proof (cases q) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
435 |
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
|
436 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
437 |
fix a::int and b::int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
438 |
assume b: "b < 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
439 |
hence "0 < -b" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
440 |
hence "P (Fract (-a) (-b))" by (rule step) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
441 |
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
|
442 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
443 |
case (Fract a b) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
444 |
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
|
445 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
446 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
447 |
lemma zero_less_Fract_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
448 |
"0 < b ==> (0 < Fract a b) = (0 < a)" |
23879 | 449 |
by (simp add: Zero_rat_def less_rat order_less_imp_not_eq2 zero_less_mult_iff) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
450 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
451 |
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
|
452 |
apply (insert add_rat [of concl: m n 1 1]) |
23879 | 453 |
apply (simp add: One_rat_def [symmetric]) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
454 |
done |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
455 |
|
23429 | 456 |
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1" |
23879 | 457 |
by (induct k) (simp_all add: Zero_rat_def One_rat_def add_rat) |
23429 | 458 |
|
459 |
lemma of_int_rat: "of_int k = Fract k 1" |
|
460 |
by (cases k rule: int_diff_cases, simp add: of_nat_rat diff_rat) |
|
461 |
||
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
462 |
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k" |
23429 | 463 |
by (rule of_nat_rat [symmetric]) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
464 |
|
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
465 |
lemma Fract_of_int_eq: "Fract k 1 = of_int k" |
23429 | 466 |
by (rule of_int_rat [symmetric]) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
467 |
|
24198 | 468 |
lemma Fract_of_int_quotient: "Fract k l = (if l = 0 then Fract 1 0 else of_int k / of_int l)" |
469 |
by (auto simp add: Fract_zero Fract_of_int_eq [symmetric] divide_rat) |
|
470 |
||
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
471 |
|
14691 | 472 |
subsection {* Numerals and Arithmetic *} |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
473 |
|
22456 | 474 |
instance rat :: number |
475 |
rat_number_of_def: "(number_of w :: rat) \<equiv> of_int w" .. |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
476 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
477 |
instance rat :: number_ring |
19765 | 478 |
by default (simp add: rat_number_of_def) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
479 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
480 |
use "rat_arith.ML" |
24075 | 481 |
declaration {* K rat_arith_setup *} |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
482 |
|
23342 | 483 |
|
484 |
subsection {* Embedding from Rationals to other Fields *} |
|
485 |
||
24198 | 486 |
class field_char_0 = field + ring_char_0 |
23342 | 487 |
|
488 |
instance ordered_field < field_char_0 .. |
|
489 |
||
490 |
definition |
|
491 |
of_rat :: "rat \<Rightarrow> 'a::field_char_0" |
|
492 |
where |
|
24198 | 493 |
[code func del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})" |
23342 | 494 |
|
495 |
lemma of_rat_congruent: |
|
496 |
"(\<lambda>(a, b). {of_int a / of_int b::'a::field_char_0}) respects ratrel" |
|
497 |
apply (rule congruent.intro) |
|
498 |
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq) |
|
499 |
apply (simp only: of_int_mult [symmetric]) |
|
500 |
done |
|
501 |
||
502 |
lemma of_rat_rat: |
|
503 |
"b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b" |
|
504 |
unfolding Fract_def of_rat_def |
|
505 |
by (simp add: UN_ratrel of_rat_congruent) |
|
506 |
||
507 |
lemma of_rat_0 [simp]: "of_rat 0 = 0" |
|
508 |
by (simp add: Zero_rat_def of_rat_rat) |
|
509 |
||
510 |
lemma of_rat_1 [simp]: "of_rat 1 = 1" |
|
511 |
by (simp add: One_rat_def of_rat_rat) |
|
512 |
||
513 |
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b" |
|
514 |
by (induct a, induct b, simp add: add_rat of_rat_rat add_frac_eq) |
|
515 |
||
23343 | 516 |
lemma of_rat_minus: "of_rat (- a) = - of_rat a" |
517 |
by (induct a, simp add: minus_rat of_rat_rat) |
|
518 |
||
519 |
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b" |
|
520 |
by (simp only: diff_minus of_rat_add of_rat_minus) |
|
521 |
||
23342 | 522 |
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b" |
523 |
apply (induct a, induct b, simp add: mult_rat of_rat_rat) |
|
524 |
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac) |
|
525 |
done |
|
526 |
||
527 |
lemma nonzero_of_rat_inverse: |
|
528 |
"a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)" |
|
23343 | 529 |
apply (rule inverse_unique [symmetric]) |
530 |
apply (simp add: of_rat_mult [symmetric]) |
|
23342 | 531 |
done |
532 |
||
533 |
lemma of_rat_inverse: |
|
534 |
"(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) = |
|
535 |
inverse (of_rat a)" |
|
536 |
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse) |
|
537 |
||
538 |
lemma nonzero_of_rat_divide: |
|
539 |
"b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b" |
|
540 |
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse) |
|
541 |
||
542 |
lemma of_rat_divide: |
|
543 |
"(of_rat (a / b)::'a::{field_char_0,division_by_zero}) |
|
544 |
= of_rat a / of_rat b" |
|
545 |
by (cases "b = 0", simp_all add: nonzero_of_rat_divide) |
|
546 |
||
23343 | 547 |
lemma of_rat_power: |
548 |
"(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n" |
|
549 |
by (induct n) (simp_all add: of_rat_mult power_Suc) |
|
550 |
||
551 |
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)" |
|
552 |
apply (induct a, induct b) |
|
553 |
apply (simp add: of_rat_rat eq_rat) |
|
554 |
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq) |
|
555 |
apply (simp only: of_int_mult [symmetric] of_int_eq_iff) |
|
556 |
done |
|
557 |
||
558 |
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified] |
|
559 |
||
560 |
lemma of_rat_eq_id [simp]: "of_rat = (id :: rat \<Rightarrow> rat)" |
|
561 |
proof |
|
562 |
fix a |
|
563 |
show "of_rat a = id a" |
|
564 |
by (induct a) |
|
565 |
(simp add: of_rat_rat divide_rat Fract_of_int_eq [symmetric]) |
|
566 |
qed |
|
567 |
||
568 |
text{*Collapse nested embeddings*} |
|
569 |
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n" |
|
570 |
by (induct n) (simp_all add: of_rat_add) |
|
571 |
||
572 |
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z" |
|
23365 | 573 |
by (cases z rule: int_diff_cases, simp add: of_rat_diff) |
23343 | 574 |
|
575 |
lemma of_rat_number_of_eq [simp]: |
|
576 |
"of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})" |
|
577 |
by (simp add: number_of_eq) |
|
578 |
||
23879 | 579 |
lemmas zero_rat = Zero_rat_def |
580 |
lemmas one_rat = One_rat_def |
|
581 |
||
24198 | 582 |
abbreviation |
583 |
rat_of_nat :: "nat \<Rightarrow> rat" |
|
584 |
where |
|
585 |
"rat_of_nat \<equiv> of_nat" |
|
586 |
||
587 |
abbreviation |
|
588 |
rat_of_int :: "int \<Rightarrow> rat" |
|
589 |
where |
|
590 |
"rat_of_int \<equiv> of_int" |
|
591 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
592 |
end |