author | haftmann |
Fri, 11 Jul 2008 09:03:11 +0200 | |
changeset 27551 | 9a5543d4cc24 |
parent 27509 | 63161d5f8f29 |
child 27652 | 818666de6c24 |
permissions | -rw-r--r-- |
27551 | 1 |
(* Title: HOL/Library/Rational.thy |
2 |
ID: $Id$ |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
3 |
Author: Markus Wenzel, TU Muenchen |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
4 |
*) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
5 |
|
14691 | 6 |
header {* Rational numbers *} |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
7 |
|
15131 | 8 |
theory Rational |
27551 | 9 |
imports "../Presburger" GCD Abstract_Rat |
16417 | 10 |
uses ("rat_arith.ML") |
15131 | 11 |
begin |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
12 |
|
27551 | 13 |
subsection {* Rational numbers as quotient *} |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
14 |
|
27551 | 15 |
subsubsection {* Construction of the type of rational numbers *} |
18913 | 16 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20522
diff
changeset
|
17 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20522
diff
changeset
|
18 |
ratrel :: "((int \<times> int) \<times> (int \<times> int)) set" where |
27551 | 19 |
"ratrel = {(x, y). snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x}" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
20 |
|
18913 | 21 |
lemma ratrel_iff [simp]: |
27551 | 22 |
"(x, y) \<in> ratrel \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x" |
23 |
by (simp add: ratrel_def) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
24 |
|
27551 | 25 |
lemma refl_ratrel: "refl {x. snd x \<noteq> 0} ratrel" |
26 |
by (auto simp add: refl_def ratrel_def) |
|
18913 | 27 |
|
28 |
lemma sym_ratrel: "sym ratrel" |
|
27551 | 29 |
by (simp add: ratrel_def sym_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
30 |
|
18913 | 31 |
lemma trans_ratrel: "trans ratrel" |
27551 | 32 |
proof (rule transI, unfold split_paired_all) |
33 |
fix a b a' b' a'' b'' :: int |
|
34 |
assume A: "((a, b), (a', b')) \<in> ratrel" |
|
35 |
assume B: "((a', b'), (a'', b'')) \<in> ratrel" |
|
36 |
have "b' * (a * b'') = b'' * (a * b')" by simp |
|
37 |
also from A have "a * b' = a' * b" by auto |
|
38 |
also have "b'' * (a' * b) = b * (a' * b'')" by simp |
|
39 |
also from B have "a' * b'' = a'' * b'" by auto |
|
40 |
also have "b * (a'' * b') = b' * (a'' * b)" by simp |
|
41 |
finally have "b' * (a * b'') = b' * (a'' * b)" . |
|
42 |
moreover from B have "b' \<noteq> 0" by auto |
|
43 |
ultimately have "a * b'' = a'' * b" by simp |
|
44 |
with A B show "((a, b), (a'', b'')) \<in> ratrel" by auto |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
45 |
qed |
27551 | 46 |
|
47 |
lemma equiv_ratrel: "equiv {x. snd x \<noteq> 0} ratrel" |
|
48 |
by (rule equiv.intro [OF refl_ratrel sym_ratrel trans_ratrel]) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
49 |
|
18913 | 50 |
lemmas UN_ratrel = UN_equiv_class [OF equiv_ratrel] |
51 |
lemmas UN_ratrel2 = UN_equiv_class2 [OF equiv_ratrel equiv_ratrel] |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
52 |
|
27551 | 53 |
lemma equiv_ratrel_iff [iff]: |
54 |
assumes "snd x \<noteq> 0" and "snd y \<noteq> 0" |
|
55 |
shows "ratrel `` {x} = ratrel `` {y} \<longleftrightarrow> (x, y) \<in> ratrel" |
|
56 |
by (rule eq_equiv_class_iff, rule equiv_ratrel) (auto simp add: assms) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
57 |
|
27551 | 58 |
typedef (Rat) rat = "{x. snd x \<noteq> 0} // ratrel" |
59 |
proof |
|
60 |
have "(0::int, 1::int) \<in> {x. snd x \<noteq> 0}" by simp |
|
61 |
then show "ratrel `` {(0, 1)} \<in> {x. snd x \<noteq> 0} // ratrel" by (rule quotientI) |
|
62 |
qed |
|
63 |
||
64 |
lemma ratrel_in_Rat [simp]: "snd x \<noteq> 0 \<Longrightarrow> ratrel `` {x} \<in> Rat" |
|
65 |
by (simp add: Rat_def quotientI) |
|
66 |
||
67 |
declare Abs_Rat_inject [simp] Abs_Rat_inverse [simp] |
|
68 |
||
69 |
||
70 |
subsubsection {* Representation and basic operations *} |
|
71 |
||
72 |
definition |
|
73 |
Fract :: "int \<Rightarrow> int \<Rightarrow> rat" where |
|
74 |
[code func del]: "Fract a b = Abs_Rat (ratrel `` {if b = 0 then (0, 1) else (a, b)})" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
75 |
|
27551 | 76 |
code_datatype Fract |
77 |
||
78 |
lemma Rat_cases [case_names Fract, cases type: rat]: |
|
79 |
assumes "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> C" |
|
80 |
shows C |
|
81 |
using assms by (cases q) (clarsimp simp add: Fract_def Rat_def quotient_def) |
|
82 |
||
83 |
lemma Rat_induct [case_names Fract, induct type: rat]: |
|
84 |
assumes "\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)" |
|
85 |
shows "P q" |
|
86 |
using assms by (cases q) simp |
|
87 |
||
88 |
lemma eq_rat: |
|
89 |
shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b" |
|
90 |
and "\<And>a c. Fract a 0 = Fract c 0" |
|
91 |
by (simp_all add: Fract_def) |
|
92 |
||
93 |
instantiation rat :: "{comm_ring_1, recpower}" |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
94 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
95 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
96 |
definition |
27551 | 97 |
Zero_rat_def [code, code unfold]: "0 = Fract 0 1" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
98 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
99 |
definition |
27551 | 100 |
One_rat_def [code, code unfold]: "1 = Fract 1 1" |
18913 | 101 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
102 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
103 |
add_rat_def [code func del]: |
27551 | 104 |
"q + r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. |
105 |
ratrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})" |
|
106 |
||
107 |
lemma add_rat: |
|
108 |
assumes "b \<noteq> 0" and "d \<noteq> 0" |
|
109 |
shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)" |
|
110 |
proof - |
|
111 |
have "(\<lambda>x y. ratrel``{(fst x * snd y + fst y * snd x, snd x * snd y)}) |
|
112 |
respects2 ratrel" |
|
113 |
by (rule equiv_ratrel [THEN congruent2_commuteI]) (simp_all add: left_distrib) |
|
114 |
with assms show ?thesis by (simp add: Fract_def add_rat_def UN_ratrel2) |
|
115 |
qed |
|
18913 | 116 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
117 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
118 |
minus_rat_def [code func del]: |
27551 | 119 |
"- q = Abs_Rat (\<Union>x \<in> Rep_Rat q. ratrel `` {(- fst x, snd x)})" |
120 |
||
121 |
lemma minus_rat: "- Fract a b = Fract (- a) b" |
|
122 |
proof - |
|
123 |
have "(\<lambda>x. ratrel `` {(- fst x, snd x)}) respects ratrel" |
|
124 |
by (simp add: congruent_def) |
|
125 |
then show ?thesis by (simp add: Fract_def minus_rat_def UN_ratrel) |
|
126 |
qed |
|
127 |
||
128 |
lemma minus_rat_cancel [simp]: |
|
129 |
"Fract (- a) (- b) = Fract a b" |
|
130 |
by (cases "b = 0") (simp_all add: eq_rat) |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
131 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
132 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
133 |
diff_rat_def [code func del]: "q - r = q + - (r::rat)" |
18913 | 134 |
|
27551 | 135 |
lemma diff_rat: |
136 |
assumes "b \<noteq> 0" and "d \<noteq> 0" |
|
137 |
shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)" |
|
138 |
using assms by (simp add: diff_rat_def add_rat minus_rat) |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
139 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
140 |
definition |
27551 | 141 |
mult_rat_def [code func del]: |
142 |
"q * r = Abs_Rat (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. |
|
143 |
ratrel``{(fst x * fst y, snd x * snd y)})" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
144 |
|
27551 | 145 |
lemma mult_rat: "Fract a b * Fract c d = Fract (a * c) (b * d)" |
146 |
proof - |
|
147 |
have "(\<lambda>x y. ratrel `` {(fst x * fst y, snd x * snd y)}) respects2 ratrel" |
|
148 |
by (rule equiv_ratrel [THEN congruent2_commuteI]) simp_all |
|
149 |
then show ?thesis by (simp add: Fract_def mult_rat_def UN_ratrel2) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
150 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
151 |
|
27551 | 152 |
lemma mult_rat_cancel [simp]: |
153 |
assumes "c \<noteq> 0" |
|
154 |
shows "Fract (c * a) (c * b) = Fract a b" |
|
155 |
proof - |
|
156 |
from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def) |
|
157 |
then show ?thesis by (simp add: mult_rat [symmetric] mult_rat) |
|
158 |
qed |
|
27509 | 159 |
|
160 |
primrec power_rat |
|
161 |
where |
|
27551 | 162 |
rat_power_0: "q ^ 0 = (1\<Colon>rat)" |
163 |
| rat_power_Suc: "q ^ Suc n = (q\<Colon>rat) * (q ^ n)" |
|
27509 | 164 |
|
165 |
instance proof |
|
27551 | 166 |
fix q r s :: rat show "(q * r) * s = q * (r * s)" |
167 |
by (cases q, cases r, cases s) (simp add: mult_rat eq_rat) |
|
168 |
next |
|
169 |
fix q r :: rat show "q * r = r * q" |
|
170 |
by (cases q, cases r) (simp add: mult_rat eq_rat) |
|
171 |
next |
|
172 |
fix q :: rat show "1 * q = q" |
|
173 |
by (cases q) (simp add: One_rat_def mult_rat eq_rat) |
|
174 |
next |
|
175 |
fix q r s :: rat show "(q + r) + s = q + (r + s)" |
|
176 |
by (cases q, cases r, cases s) (simp add: add_rat eq_rat ring_simps) |
|
177 |
next |
|
178 |
fix q r :: rat show "q + r = r + q" |
|
179 |
by (cases q, cases r) (simp add: add_rat eq_rat) |
|
180 |
next |
|
181 |
fix q :: rat show "0 + q = q" |
|
182 |
by (cases q) (simp add: Zero_rat_def add_rat eq_rat) |
|
183 |
next |
|
184 |
fix q :: rat show "- q + q = 0" |
|
185 |
by (cases q) (simp add: Zero_rat_def add_rat minus_rat eq_rat) |
|
186 |
next |
|
187 |
fix q r :: rat show "q - r = q + - r" |
|
188 |
by (cases q, cases r) (simp add: diff_rat add_rat minus_rat eq_rat) |
|
189 |
next |
|
190 |
fix q r s :: rat show "(q + r) * s = q * s + r * s" |
|
191 |
by (cases q, cases r, cases s) (simp add: add_rat mult_rat eq_rat ring_simps) |
|
192 |
next |
|
193 |
show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat) |
|
194 |
next |
|
195 |
fix q :: rat show "q * 1 = q" |
|
196 |
by (cases q) (simp add: One_rat_def mult_rat eq_rat) |
|
197 |
next |
|
27509 | 198 |
fix q :: rat |
199 |
fix n :: nat |
|
200 |
show "q ^ 0 = 1" by simp |
|
201 |
show "q ^ (Suc n) = q * (q ^ n)" by simp |
|
202 |
qed |
|
203 |
||
204 |
end |
|
205 |
||
27551 | 206 |
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1" |
207 |
by (induct k) (simp_all add: Zero_rat_def One_rat_def add_rat) |
|
208 |
||
209 |
lemma of_int_rat: "of_int k = Fract k 1" |
|
210 |
by (cases k rule: int_diff_cases, simp add: of_nat_rat diff_rat) |
|
211 |
||
212 |
lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k" |
|
213 |
by (rule of_nat_rat [symmetric]) |
|
214 |
||
215 |
lemma Fract_of_int_eq: "Fract k 1 = of_int k" |
|
216 |
by (rule of_int_rat [symmetric]) |
|
217 |
||
218 |
instantiation rat :: number_ring |
|
219 |
begin |
|
220 |
||
221 |
definition |
|
222 |
rat_number_of_def [code func del]: "number_of w = Fract w 1" |
|
223 |
||
224 |
instance by intro_classes (simp add: rat_number_of_def of_int_rat) |
|
225 |
||
226 |
end |
|
227 |
||
228 |
lemma rat_number_collapse [code post]: |
|
229 |
"Fract 0 k = 0" |
|
230 |
"Fract 1 1 = 1" |
|
231 |
"Fract (number_of k) 1 = number_of k" |
|
232 |
"Fract k 0 = 0" |
|
233 |
by (cases "k = 0") |
|
234 |
(simp_all add: Zero_rat_def One_rat_def number_of_is_id number_of_eq of_int_rat eq_rat Fract_def) |
|
235 |
||
236 |
lemma rat_number_expand [code unfold]: |
|
237 |
"0 = Fract 0 1" |
|
238 |
"1 = Fract 1 1" |
|
239 |
"number_of k = Fract (number_of k) 1" |
|
240 |
by (simp_all add: rat_number_collapse) |
|
241 |
||
242 |
lemma iszero_rat [simp]: |
|
243 |
"iszero (number_of k :: rat) \<longleftrightarrow> iszero (number_of k :: int)" |
|
244 |
by (simp add: iszero_def rat_number_expand number_of_is_id eq_rat) |
|
245 |
||
246 |
lemma Rat_cases_nonzero [case_names Fract 0]: |
|
247 |
assumes Fract: "\<And>a b. q = Fract a b \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> C" |
|
248 |
assumes 0: "q = 0 \<Longrightarrow> C" |
|
249 |
shows C |
|
250 |
proof (cases "q = 0") |
|
251 |
case True then show C using 0 by auto |
|
252 |
next |
|
253 |
case False |
|
254 |
then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto |
|
255 |
moreover with False have "0 \<noteq> Fract a b" by simp |
|
256 |
with `b \<noteq> 0` have "a \<noteq> 0" by (simp add: Zero_rat_def eq_rat) |
|
257 |
with Fract `q = Fract a b` `b \<noteq> 0` show C by auto |
|
258 |
qed |
|
259 |
||
260 |
||
261 |
||
262 |
subsubsection {* The field of rational numbers *} |
|
263 |
||
264 |
instantiation rat :: "{field, division_by_zero}" |
|
265 |
begin |
|
266 |
||
267 |
definition |
|
268 |
inverse_rat_def [code func del]: |
|
269 |
"inverse q = Abs_Rat (\<Union>x \<in> Rep_Rat q. |
|
270 |
ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)})" |
|
271 |
||
272 |
lemma inverse_rat: "inverse (Fract a b) = Fract b a" |
|
273 |
proof - |
|
274 |
have "(\<lambda>x. ratrel `` {if fst x = 0 then (0, 1) else (snd x, fst x)}) respects ratrel" |
|
275 |
by (auto simp add: congruent_def mult_commute) |
|
276 |
then show ?thesis by (simp add: Fract_def inverse_rat_def UN_ratrel) |
|
27509 | 277 |
qed |
278 |
||
27551 | 279 |
definition |
280 |
divide_rat_def [code func del]: "q / r = q * inverse (r::rat)" |
|
281 |
||
282 |
lemma divide_rat: "Fract a b / Fract c d = Fract (a * d) (b * c)" |
|
283 |
by (simp add: divide_rat_def inverse_rat mult_rat) |
|
284 |
||
285 |
instance proof |
|
286 |
show "inverse 0 = (0::rat)" by (simp add: rat_number_expand inverse_rat) |
|
287 |
(simp add: rat_number_collapse) |
|
288 |
next |
|
289 |
fix q :: rat |
|
290 |
assume "q \<noteq> 0" |
|
291 |
then show "inverse q * q = 1" by (cases q rule: Rat_cases_nonzero) |
|
292 |
(simp_all add: mult_rat inverse_rat rat_number_expand eq_rat) |
|
293 |
next |
|
294 |
fix q r :: rat |
|
295 |
show "q / r = q * inverse r" by (simp add: divide_rat_def) |
|
296 |
qed |
|
297 |
||
298 |
end |
|
299 |
||
300 |
||
301 |
subsubsection {* Various *} |
|
302 |
||
303 |
lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1" |
|
304 |
by (simp add: rat_number_expand add_rat) |
|
305 |
||
306 |
lemma Fract_of_int_quotient: "Fract k l = of_int k / of_int l" |
|
307 |
by (simp add: Fract_of_int_eq [symmetric] divide_rat) |
|
308 |
||
309 |
lemma Fract_number_of_quotient [code post]: |
|
310 |
"Fract (number_of k) (number_of l) = number_of k / number_of l" |
|
311 |
unfolding Fract_of_int_quotient number_of_is_id number_of_eq .. |
|
312 |
||
313 |
||
314 |
subsubsection {* The ordered field of rational numbers *} |
|
27509 | 315 |
|
316 |
instantiation rat :: linorder |
|
317 |
begin |
|
318 |
||
319 |
definition |
|
320 |
le_rat_def [code func del]: |
|
321 |
"q \<le> r \<longleftrightarrow> contents (\<Union>x \<in> Rep_Rat q. \<Union>y \<in> Rep_Rat r. |
|
27551 | 322 |
{(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)})" |
323 |
||
324 |
lemma le_rat: |
|
325 |
assumes "b \<noteq> 0" and "d \<noteq> 0" |
|
326 |
shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)" |
|
327 |
proof - |
|
328 |
have "(\<lambda>x y. {(fst x * snd y) * (snd x * snd y) \<le> (fst y * snd x) * (snd x * snd y)}) |
|
329 |
respects2 ratrel" |
|
330 |
proof (clarsimp simp add: congruent2_def) |
|
331 |
fix a b a' b' c d c' d'::int |
|
332 |
assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" |
|
333 |
assume eq1: "a * b' = a' * b" |
|
334 |
assume eq2: "c * d' = c' * d" |
|
335 |
||
336 |
let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))" |
|
337 |
{ |
|
338 |
fix a b c d x :: int assume x: "x \<noteq> 0" |
|
339 |
have "?le a b c d = ?le (a * x) (b * x) c d" |
|
340 |
proof - |
|
341 |
from x have "0 < x * x" by (auto simp add: zero_less_mult_iff) |
|
342 |
hence "?le a b c d = |
|
343 |
((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))" |
|
344 |
by (simp add: mult_le_cancel_right) |
|
345 |
also have "... = ?le (a * x) (b * x) c d" |
|
346 |
by (simp add: mult_ac) |
|
347 |
finally show ?thesis . |
|
348 |
qed |
|
349 |
} note le_factor = this |
|
350 |
||
351 |
let ?D = "b * d" and ?D' = "b' * d'" |
|
352 |
from neq have D: "?D \<noteq> 0" by simp |
|
353 |
from neq have "?D' \<noteq> 0" by simp |
|
354 |
hence "?le a b c d = ?le (a * ?D') (b * ?D') c d" |
|
355 |
by (rule le_factor) |
|
356 |
also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" |
|
357 |
by (simp add: mult_ac) |
|
358 |
also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')" |
|
359 |
by (simp only: eq1 eq2) |
|
360 |
also have "... = ?le (a' * ?D) (b' * ?D) c' d'" |
|
361 |
by (simp add: mult_ac) |
|
362 |
also from D have "... = ?le a' b' c' d'" |
|
363 |
by (rule le_factor [symmetric]) |
|
364 |
finally show "?le a b c d = ?le a' b' c' d'" . |
|
365 |
qed |
|
366 |
with assms show ?thesis by (simp add: Fract_def le_rat_def UN_ratrel2) |
|
367 |
qed |
|
27509 | 368 |
|
369 |
definition |
|
370 |
less_rat_def [code func del]: "z < (w::rat) \<longleftrightarrow> z \<le> w \<and> z \<noteq> w" |
|
371 |
||
27551 | 372 |
lemma less_rat: |
373 |
assumes "b \<noteq> 0" and "d \<noteq> 0" |
|
374 |
shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)" |
|
375 |
using assms by (simp add: less_rat_def le_rat eq_rat order_less_le) |
|
27509 | 376 |
|
377 |
instance proof |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
378 |
fix q r s :: rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
379 |
{ |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
380 |
assume "q \<le> r" and "r \<le> s" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
381 |
show "q \<le> s" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
382 |
proof (insert prems, induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
383 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
384 |
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
|
385 |
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
|
386 |
show "Fract a b \<le> Fract e f" |
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 |
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
|
389 |
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
|
390 |
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
|
391 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
392 |
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
|
393 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
394 |
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
|
395 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
396 |
also have "... = (c * f) * (d * f) * (b * b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
397 |
by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
398 |
also have "... \<le> (e * d) * (d * f) * (b * b)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
399 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
400 |
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
|
401 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
402 |
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
|
403 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
404 |
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
|
405 |
by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
406 |
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
|
407 |
by (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
408 |
with neq show ?thesis by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
409 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
410 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
411 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
412 |
assume "q \<le> r" and "r \<le> q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
413 |
show "q = r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
414 |
proof (insert prems, induct q, induct r) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
415 |
fix a b c d :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
416 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
417 |
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
|
418 |
show "Fract a b = Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
419 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
420 |
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
|
421 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
422 |
also have "... \<le> (a * d) * (b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
423 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
424 |
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
|
425 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
426 |
thus ?thesis by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
427 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
428 |
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
|
429 |
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
|
430 |
ultimately have "a * d = c * b" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
431 |
with neq show ?thesis by (simp add: eq_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
432 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
433 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
434 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
435 |
show "q \<le> q" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
436 |
by (induct q) (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
437 |
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
|
438 |
by (simp only: less_rat_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
439 |
show "q \<le> r \<or> r \<le> q" |
18913 | 440 |
by (induct q, induct r) |
441 |
(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
|
442 |
} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
443 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
444 |
|
27509 | 445 |
end |
446 |
||
27551 | 447 |
instantiation rat :: "{distrib_lattice, abs_if, sgn_if}" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
448 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
449 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
450 |
definition |
27551 | 451 |
abs_rat_def: "\<bar>q\<bar> = (if q < 0 then -q else (q::rat))" |
452 |
||
453 |
lemma abs_rat: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>" |
|
454 |
by (auto simp add: abs_rat_def zabs_def Zero_rat_def less_rat not_less le_less minus_rat eq_rat zero_compare_simps) |
|
455 |
||
456 |
definition |
|
457 |
sgn_rat_def: "sgn (q::rat) = (if q = 0 then 0 else if 0 < q then 1 else - 1)" |
|
458 |
||
459 |
lemma sgn_rat: "sgn (Fract a b) = Fract (sgn a * sgn b) 1" |
|
460 |
unfolding Fract_of_int_eq |
|
461 |
by (auto simp: zsgn_def sgn_rat_def Zero_rat_def eq_rat less_rat) |
|
462 |
(auto simp: rat_number_collapse not_less le_less zero_less_mult_iff) |
|
463 |
||
464 |
definition |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
465 |
"(inf \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = min" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
466 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
467 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
468 |
"(sup \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat) = max" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
469 |
|
27551 | 470 |
instance by intro_classes |
471 |
(auto simp add: abs_rat_def sgn_rat_def min_max.sup_inf_distrib1 inf_rat_def sup_rat_def) |
|
22456 | 472 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
473 |
end |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
474 |
|
27551 | 475 |
instance rat :: ordered_field |
476 |
proof |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
477 |
fix q r s :: rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
478 |
show "q \<le> r ==> s + q \<le> s + r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
479 |
proof (induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
480 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
481 |
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
|
482 |
assume le: "Fract a b \<le> Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
483 |
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
|
484 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
485 |
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
|
486 |
by (auto simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
487 |
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
|
488 |
by (simp add: le_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
489 |
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
|
490 |
by (simp add: mult_le_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
491 |
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
|
492 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
493 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
494 |
show "q < r ==> 0 < s ==> s * q < s * r" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
495 |
proof (induct q, induct r, induct s) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
496 |
fix a b c d e f :: int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
497 |
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
|
498 |
assume le: "Fract a b < Fract c d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
499 |
assume gt: "0 < Fract e f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
500 |
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
|
501 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
502 |
let ?E = "e * f" and ?F = "f * f" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
503 |
from neq gt have "0 < ?E" |
23879 | 504 |
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
|
505 |
moreover from neq have "0 < ?F" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
506 |
by (auto simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
507 |
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
|
508 |
by (simp add: less_rat) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
509 |
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
|
510 |
by (simp add: mult_less_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
511 |
with neq show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
512 |
by (simp add: less_rat mult_rat mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
513 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
514 |
qed |
27551 | 515 |
qed auto |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
516 |
|
27551 | 517 |
lemma Rat_induct_pos [case_names Fract, induct type: rat]: |
518 |
assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)" |
|
519 |
shows "P q" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
520 |
proof (cases q) |
27551 | 521 |
have step': "\<And>a b. b < 0 \<Longrightarrow> P (Fract a b)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
522 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
523 |
fix a::int and b::int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
524 |
assume b: "b < 0" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
525 |
hence "0 < -b" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
526 |
hence "P (Fract (-a) (-b))" by (rule step) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
527 |
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
|
528 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
529 |
case (Fract a b) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
530 |
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
|
531 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
532 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
533 |
lemma zero_less_Fract_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
534 |
"0 < b ==> (0 < Fract a b) = (0 < a)" |
23879 | 535 |
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
|
536 |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14365
diff
changeset
|
537 |
|
27551 | 538 |
subsection {* Arithmetic setup *} |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
539 |
|
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
540 |
use "rat_arith.ML" |
24075 | 541 |
declaration {* K rat_arith_setup *} |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
542 |
|
23342 | 543 |
|
544 |
subsection {* Embedding from Rationals to other Fields *} |
|
545 |
||
24198 | 546 |
class field_char_0 = field + ring_char_0 |
23342 | 547 |
|
27551 | 548 |
subclass (in ordered_field) field_char_0 .. |
23342 | 549 |
|
27551 | 550 |
context field_char_0 |
551 |
begin |
|
552 |
||
553 |
definition of_rat :: "rat \<Rightarrow> 'a" where |
|
24198 | 554 |
[code func del]: "of_rat q = contents (\<Union>(a,b) \<in> Rep_Rat q. {of_int a / of_int b})" |
23342 | 555 |
|
27551 | 556 |
end |
557 |
||
23342 | 558 |
lemma of_rat_congruent: |
27551 | 559 |
"(\<lambda>(a, b). {of_int a / of_int b :: 'a::field_char_0}) respects ratrel" |
23342 | 560 |
apply (rule congruent.intro) |
561 |
apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq) |
|
562 |
apply (simp only: of_int_mult [symmetric]) |
|
563 |
done |
|
564 |
||
27551 | 565 |
lemma of_rat_rat: "b \<noteq> 0 \<Longrightarrow> of_rat (Fract a b) = of_int a / of_int b" |
566 |
unfolding Fract_def of_rat_def by (simp add: UN_ratrel of_rat_congruent) |
|
23342 | 567 |
|
568 |
lemma of_rat_0 [simp]: "of_rat 0 = 0" |
|
569 |
by (simp add: Zero_rat_def of_rat_rat) |
|
570 |
||
571 |
lemma of_rat_1 [simp]: "of_rat 1 = 1" |
|
572 |
by (simp add: One_rat_def of_rat_rat) |
|
573 |
||
574 |
lemma of_rat_add: "of_rat (a + b) = of_rat a + of_rat b" |
|
575 |
by (induct a, induct b, simp add: add_rat of_rat_rat add_frac_eq) |
|
576 |
||
23343 | 577 |
lemma of_rat_minus: "of_rat (- a) = - of_rat a" |
578 |
by (induct a, simp add: minus_rat of_rat_rat) |
|
579 |
||
580 |
lemma of_rat_diff: "of_rat (a - b) = of_rat a - of_rat b" |
|
581 |
by (simp only: diff_minus of_rat_add of_rat_minus) |
|
582 |
||
23342 | 583 |
lemma of_rat_mult: "of_rat (a * b) = of_rat a * of_rat b" |
584 |
apply (induct a, induct b, simp add: mult_rat of_rat_rat) |
|
585 |
apply (simp add: divide_inverse nonzero_inverse_mult_distrib mult_ac) |
|
586 |
done |
|
587 |
||
588 |
lemma nonzero_of_rat_inverse: |
|
589 |
"a \<noteq> 0 \<Longrightarrow> of_rat (inverse a) = inverse (of_rat a)" |
|
23343 | 590 |
apply (rule inverse_unique [symmetric]) |
591 |
apply (simp add: of_rat_mult [symmetric]) |
|
23342 | 592 |
done |
593 |
||
594 |
lemma of_rat_inverse: |
|
595 |
"(of_rat (inverse a)::'a::{field_char_0,division_by_zero}) = |
|
596 |
inverse (of_rat a)" |
|
597 |
by (cases "a = 0", simp_all add: nonzero_of_rat_inverse) |
|
598 |
||
599 |
lemma nonzero_of_rat_divide: |
|
600 |
"b \<noteq> 0 \<Longrightarrow> of_rat (a / b) = of_rat a / of_rat b" |
|
601 |
by (simp add: divide_inverse of_rat_mult nonzero_of_rat_inverse) |
|
602 |
||
603 |
lemma of_rat_divide: |
|
604 |
"(of_rat (a / b)::'a::{field_char_0,division_by_zero}) |
|
605 |
= of_rat a / of_rat b" |
|
606 |
by (cases "b = 0", simp_all add: nonzero_of_rat_divide) |
|
607 |
||
23343 | 608 |
lemma of_rat_power: |
609 |
"(of_rat (a ^ n)::'a::{field_char_0,recpower}) = of_rat a ^ n" |
|
610 |
by (induct n) (simp_all add: of_rat_mult power_Suc) |
|
611 |
||
612 |
lemma of_rat_eq_iff [simp]: "(of_rat a = of_rat b) = (a = b)" |
|
613 |
apply (induct a, induct b) |
|
614 |
apply (simp add: of_rat_rat eq_rat) |
|
615 |
apply (simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq) |
|
616 |
apply (simp only: of_int_mult [symmetric] of_int_eq_iff) |
|
617 |
done |
|
618 |
||
619 |
lemmas of_rat_eq_0_iff [simp] = of_rat_eq_iff [of _ 0, simplified] |
|
620 |
||
621 |
lemma of_rat_eq_id [simp]: "of_rat = (id :: rat \<Rightarrow> rat)" |
|
622 |
proof |
|
623 |
fix a |
|
624 |
show "of_rat a = id a" |
|
625 |
by (induct a) |
|
626 |
(simp add: of_rat_rat divide_rat Fract_of_int_eq [symmetric]) |
|
627 |
qed |
|
628 |
||
629 |
text{*Collapse nested embeddings*} |
|
630 |
lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n" |
|
631 |
by (induct n) (simp_all add: of_rat_add) |
|
632 |
||
633 |
lemma of_rat_of_int_eq [simp]: "of_rat (of_int z) = of_int z" |
|
23365 | 634 |
by (cases z rule: int_diff_cases, simp add: of_rat_diff) |
23343 | 635 |
|
636 |
lemma of_rat_number_of_eq [simp]: |
|
637 |
"of_rat (number_of w) = (number_of w :: 'a::{number_ring,field_char_0})" |
|
638 |
by (simp add: number_of_eq) |
|
639 |
||
23879 | 640 |
lemmas zero_rat = Zero_rat_def |
641 |
lemmas one_rat = One_rat_def |
|
642 |
||
24198 | 643 |
abbreviation |
644 |
rat_of_nat :: "nat \<Rightarrow> rat" |
|
645 |
where |
|
646 |
"rat_of_nat \<equiv> of_nat" |
|
647 |
||
648 |
abbreviation |
|
649 |
rat_of_int :: "int \<Rightarrow> rat" |
|
650 |
where |
|
651 |
"rat_of_int \<equiv> of_int" |
|
652 |
||
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
653 |
|
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
654 |
subsection {* Implementation of rational numbers as pairs of integers *} |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
655 |
|
27551 | 656 |
lemma INum_Fract [simp]: "INum = split Fract" |
657 |
by (auto simp add: expand_fun_eq INum_def Fract_of_int_quotient) |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
658 |
|
27551 | 659 |
lemma split_Fract_normNum [simp]: "split Fract (normNum (k, l)) = Fract k l" |
660 |
unfolding INum_Fract [symmetric] normNum by simp |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
661 |
|
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
662 |
lemma [code]: |
27551 | 663 |
"of_rat (Fract k l) = (if l \<noteq> 0 then of_int k / of_int l else 0)" |
664 |
by (cases "l = 0") (simp_all add: rat_number_collapse of_rat_rat) |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
665 |
|
26513 | 666 |
instantiation rat :: eq |
667 |
begin |
|
668 |
||
27551 | 669 |
definition [code func del]: "eq_class.eq (r\<Colon>rat) s \<longleftrightarrow> r - s = 0" |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
670 |
|
26513 | 671 |
instance by default (simp add: eq_rat_def) |
672 |
||
27551 | 673 |
lemma rat_eq_code [code]: "eq_class.eq (Fract k l) (Fract r s) \<longleftrightarrow> eq_class.eq (normNum (k, l)) (normNum (r, s))" |
674 |
by (simp add: eq INum_normNum_iff [where ?'a = rat, symmetric]) |
|
26513 | 675 |
|
676 |
end |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
677 |
|
27551 | 678 |
lemma rat_less_eq_code [code]: "Fract k l \<le> Fract r s \<longleftrightarrow> normNum (k, l) \<le>\<^sub>N normNum (r, s)" |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
679 |
proof - |
27551 | 680 |
have "normNum (k, l) \<le>\<^sub>N normNum (r, s) \<longleftrightarrow> split Fract (normNum (k, l)) \<le> split Fract (normNum (r, s))" |
681 |
by (simp add: INum_Fract [symmetric] del: INum_Fract normNum) |
|
682 |
also have "\<dots> = (Fract k l \<le> Fract r s)" by simp |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
683 |
finally show ?thesis by simp |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
684 |
qed |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
685 |
|
27551 | 686 |
lemma rat_less_code [code]: "Fract k l < Fract r s \<longleftrightarrow> normNum (k, l) <\<^sub>N normNum (r, s)" |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
687 |
proof - |
27551 | 688 |
have "normNum (k, l) <\<^sub>N normNum (r, s) \<longleftrightarrow> split Fract (normNum (k, l)) < split Fract (normNum (r, s))" |
689 |
by (simp add: INum_Fract [symmetric] del: INum_Fract normNum) |
|
690 |
also have "\<dots> = (Fract k l < Fract r s)" by simp |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
691 |
finally show ?thesis by simp |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
692 |
qed |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
693 |
|
27551 | 694 |
lemma rat_add_code [code]: "Fract k l + Fract r s = split Fract ((k, l) +\<^sub>N (r, s))" |
695 |
by (simp add: INum_Fract [symmetric] del: INum_Fract, simp) |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
696 |
|
27551 | 697 |
lemma rat_mul_code [code]: "Fract k l * Fract r s = split Fract ((k, l) *\<^sub>N (r, s))" |
698 |
by (simp add: INum_Fract [symmetric] del: INum_Fract, simp) |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
699 |
|
27551 | 700 |
lemma rat_neg_code [code]: "- Fract k l = split Fract (~\<^sub>N (k, l))" |
701 |
by (simp add: INum_Fract [symmetric] del: INum_Fract, simp) |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
702 |
|
27551 | 703 |
lemma rat_sub_code [code]: "Fract k l - Fract r s = split Fract ((k, l) -\<^sub>N (r, s))" |
704 |
by (simp add: INum_Fract [symmetric] del: INum_Fract, simp) |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
705 |
|
27551 | 706 |
lemma rat_inv_code [code]: "inverse (Fract k l) = split Fract (Ninv (k, l))" |
707 |
by (simp add: INum_Fract [symmetric] del: INum_Fract, simp add: divide_rat_def) |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
708 |
|
27551 | 709 |
lemma rat_div_code [code]: "Fract k l / Fract r s = split Fract ((k, l) \<div>\<^sub>N (r, s))" |
710 |
by (simp add: INum_Fract [symmetric] del: INum_Fract, simp) |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
711 |
|
24622 | 712 |
text {* Setup for SML code generator *} |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
713 |
|
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
714 |
types_code |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
715 |
rat ("(int */ int)") |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
716 |
attach (term_of) {* |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
717 |
fun term_of_rat (p, q) = |
24622 | 718 |
let |
24661 | 719 |
val rT = Type ("Rational.rat", []) |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
720 |
in |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
721 |
if q = 1 orelse p = 0 then HOLogic.mk_number rT p |
25885 | 722 |
else @{term "op / \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat"} $ |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
723 |
HOLogic.mk_number rT p $ HOLogic.mk_number rT q |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
724 |
end; |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
725 |
*} |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
726 |
attach (test) {* |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
727 |
fun gen_rat i = |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
728 |
let |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
729 |
val p = random_range 0 i; |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
730 |
val q = random_range 1 (i + 1); |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
731 |
val g = Integer.gcd p q; |
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24622
diff
changeset
|
732 |
val p' = p div g; |
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24622
diff
changeset
|
733 |
val q' = q div g; |
25885 | 734 |
val r = (if one_of [true, false] then p' else ~ p', |
735 |
if p' = 0 then 0 else q') |
|
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
736 |
in |
25885 | 737 |
(r, fn () => term_of_rat r) |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
738 |
end; |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
739 |
*} |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
740 |
|
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
741 |
consts_code |
27551 | 742 |
Fract ("(_,/ _)") |
24533
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
743 |
|
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
744 |
consts_code |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
745 |
"of_int :: int \<Rightarrow> rat" ("\<module>rat'_of'_int") |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
746 |
attach {* |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
747 |
fun rat_of_int 0 = (0, 0) |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
748 |
| rat_of_int i = (i, 1); |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
749 |
*} |
fe1f93f6a15a
Added code generator setup (taken from Library/Executable_Rat.thy,
berghofe
parents:
24506
diff
changeset
|
750 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
diff
changeset
|
751 |
end |