author | wenzelm |
Mon, 15 Feb 2016 14:55:44 +0100 | |
changeset 62337 | d3996d5873dd |
parent 62046 | 2c9f68fbf047 |
child 63625 | 1e7c5bbea36d |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Library/Nat_Bijection.thy |
35700 | 2 |
Author: Brian Huffman |
3 |
Author: Florian Haftmann |
|
4 |
Author: Stefan Richter |
|
5 |
Author: Tobias Nipkow |
|
6 |
Author: Alexander Krauss |
|
7 |
*) |
|
8 |
||
60500 | 9 |
section \<open>Bijections between natural numbers and other types\<close> |
35700 | 10 |
|
11 |
theory Nat_Bijection |
|
58770 | 12 |
imports Main |
35700 | 13 |
begin |
14 |
||
60500 | 15 |
subsection \<open>Type @{typ "nat \<times> nat"}\<close> |
35700 | 16 |
|
17 |
text "Triangle numbers: 0, 1, 3, 6, 10, 15, ..." |
|
18 |
||
62046 | 19 |
definition triangle :: "nat \<Rightarrow> nat" |
20 |
where "triangle n = (n * Suc n) div 2" |
|
35700 | 21 |
|
22 |
lemma triangle_0 [simp]: "triangle 0 = 0" |
|
23 |
unfolding triangle_def by simp |
|
24 |
||
25 |
lemma triangle_Suc [simp]: "triangle (Suc n) = triangle n + Suc n" |
|
26 |
unfolding triangle_def by simp |
|
27 |
||
62046 | 28 |
definition prod_encode :: "nat \<times> nat \<Rightarrow> nat" |
29 |
where "prod_encode = (\<lambda>(m, n). triangle (m + n) + m)" |
|
35700 | 30 |
|
60500 | 31 |
text \<open>In this auxiliary function, @{term "triangle k + m"} is an invariant.\<close> |
35700 | 32 |
|
62046 | 33 |
fun prod_decode_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" |
35700 | 34 |
where |
35 |
"prod_decode_aux k m = |
|
36 |
(if m \<le> k then (m, k - m) else prod_decode_aux (Suc k) (m - Suc k))" |
|
37 |
||
38 |
declare prod_decode_aux.simps [simp del] |
|
39 |
||
62046 | 40 |
definition prod_decode :: "nat \<Rightarrow> nat \<times> nat" |
41 |
where "prod_decode = prod_decode_aux 0" |
|
35700 | 42 |
|
43 |
lemma prod_encode_prod_decode_aux: |
|
44 |
"prod_encode (prod_decode_aux k m) = triangle k + m" |
|
45 |
apply (induct k m rule: prod_decode_aux.induct) |
|
46 |
apply (subst prod_decode_aux.simps) |
|
47 |
apply (simp add: prod_encode_def) |
|
48 |
done |
|
49 |
||
50 |
lemma prod_decode_inverse [simp]: "prod_encode (prod_decode n) = n" |
|
51 |
unfolding prod_decode_def by (simp add: prod_encode_prod_decode_aux) |
|
52 |
||
62046 | 53 |
lemma prod_decode_triangle_add: "prod_decode (triangle k + m) = prod_decode_aux k m" |
35700 | 54 |
apply (induct k arbitrary: m) |
55 |
apply (simp add: prod_decode_def) |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
51489
diff
changeset
|
56 |
apply (simp only: triangle_Suc add.assoc) |
35700 | 57 |
apply (subst prod_decode_aux.simps, simp) |
58 |
done |
|
59 |
||
60 |
lemma prod_encode_inverse [simp]: "prod_decode (prod_encode x) = x" |
|
61 |
unfolding prod_encode_def |
|
62 |
apply (induct x) |
|
63 |
apply (simp add: prod_decode_triangle_add) |
|
64 |
apply (subst prod_decode_aux.simps, simp) |
|
65 |
done |
|
66 |
||
67 |
lemma inj_prod_encode: "inj_on prod_encode A" |
|
68 |
by (rule inj_on_inverseI, rule prod_encode_inverse) |
|
69 |
||
70 |
lemma inj_prod_decode: "inj_on prod_decode A" |
|
71 |
by (rule inj_on_inverseI, rule prod_decode_inverse) |
|
72 |
||
73 |
lemma surj_prod_encode: "surj prod_encode" |
|
74 |
by (rule surjI, rule prod_decode_inverse) |
|
75 |
||
76 |
lemma surj_prod_decode: "surj prod_decode" |
|
77 |
by (rule surjI, rule prod_encode_inverse) |
|
78 |
||
79 |
lemma bij_prod_encode: "bij prod_encode" |
|
80 |
by (rule bijI [OF inj_prod_encode surj_prod_encode]) |
|
81 |
||
82 |
lemma bij_prod_decode: "bij prod_decode" |
|
83 |
by (rule bijI [OF inj_prod_decode surj_prod_decode]) |
|
84 |
||
85 |
lemma prod_encode_eq: "prod_encode x = prod_encode y \<longleftrightarrow> x = y" |
|
86 |
by (rule inj_prod_encode [THEN inj_eq]) |
|
87 |
||
88 |
lemma prod_decode_eq: "prod_decode x = prod_decode y \<longleftrightarrow> x = y" |
|
89 |
by (rule inj_prod_decode [THEN inj_eq]) |
|
90 |
||
62046 | 91 |
|
60500 | 92 |
text \<open>Ordering properties\<close> |
35700 | 93 |
|
94 |
lemma le_prod_encode_1: "a \<le> prod_encode (a, b)" |
|
95 |
unfolding prod_encode_def by simp |
|
96 |
||
97 |
lemma le_prod_encode_2: "b \<le> prod_encode (a, b)" |
|
98 |
unfolding prod_encode_def by (induct b, simp_all) |
|
99 |
||
100 |
||
60500 | 101 |
subsection \<open>Type @{typ "nat + nat"}\<close> |
35700 | 102 |
|
62046 | 103 |
definition sum_encode :: "nat + nat \<Rightarrow> nat" |
35700 | 104 |
where |
105 |
"sum_encode x = (case x of Inl a \<Rightarrow> 2 * a | Inr b \<Rightarrow> Suc (2 * b))" |
|
106 |
||
62046 | 107 |
definition sum_decode :: "nat \<Rightarrow> nat + nat" |
35700 | 108 |
where |
109 |
"sum_decode n = (if even n then Inl (n div 2) else Inr (n div 2))" |
|
110 |
||
111 |
lemma sum_encode_inverse [simp]: "sum_decode (sum_encode x) = x" |
|
112 |
unfolding sum_decode_def sum_encode_def |
|
113 |
by (induct x) simp_all |
|
114 |
||
115 |
lemma sum_decode_inverse [simp]: "sum_encode (sum_decode n) = n" |
|
58834 | 116 |
by (simp add: even_two_times_div_two sum_decode_def sum_encode_def) |
35700 | 117 |
|
118 |
lemma inj_sum_encode: "inj_on sum_encode A" |
|
119 |
by (rule inj_on_inverseI, rule sum_encode_inverse) |
|
120 |
||
121 |
lemma inj_sum_decode: "inj_on sum_decode A" |
|
122 |
by (rule inj_on_inverseI, rule sum_decode_inverse) |
|
123 |
||
124 |
lemma surj_sum_encode: "surj sum_encode" |
|
125 |
by (rule surjI, rule sum_decode_inverse) |
|
126 |
||
127 |
lemma surj_sum_decode: "surj sum_decode" |
|
128 |
by (rule surjI, rule sum_encode_inverse) |
|
129 |
||
130 |
lemma bij_sum_encode: "bij sum_encode" |
|
131 |
by (rule bijI [OF inj_sum_encode surj_sum_encode]) |
|
132 |
||
133 |
lemma bij_sum_decode: "bij sum_decode" |
|
134 |
by (rule bijI [OF inj_sum_decode surj_sum_decode]) |
|
135 |
||
136 |
lemma sum_encode_eq: "sum_encode x = sum_encode y \<longleftrightarrow> x = y" |
|
137 |
by (rule inj_sum_encode [THEN inj_eq]) |
|
138 |
||
139 |
lemma sum_decode_eq: "sum_decode x = sum_decode y \<longleftrightarrow> x = y" |
|
140 |
by (rule inj_sum_decode [THEN inj_eq]) |
|
141 |
||
142 |
||
60500 | 143 |
subsection \<open>Type @{typ "int"}\<close> |
35700 | 144 |
|
62046 | 145 |
definition int_encode :: "int \<Rightarrow> nat" |
35700 | 146 |
where |
147 |
"int_encode i = sum_encode (if 0 \<le> i then Inl (nat i) else Inr (nat (- i - 1)))" |
|
148 |
||
62046 | 149 |
definition int_decode :: "nat \<Rightarrow> int" |
35700 | 150 |
where |
151 |
"int_decode n = (case sum_decode n of Inl a \<Rightarrow> int a | Inr b \<Rightarrow> - int b - 1)" |
|
152 |
||
153 |
lemma int_encode_inverse [simp]: "int_decode (int_encode x) = x" |
|
154 |
unfolding int_decode_def int_encode_def by simp |
|
155 |
||
156 |
lemma int_decode_inverse [simp]: "int_encode (int_decode n) = n" |
|
157 |
unfolding int_decode_def int_encode_def using sum_decode_inverse [of n] |
|
158 |
by (cases "sum_decode n", simp_all) |
|
159 |
||
160 |
lemma inj_int_encode: "inj_on int_encode A" |
|
161 |
by (rule inj_on_inverseI, rule int_encode_inverse) |
|
162 |
||
163 |
lemma inj_int_decode: "inj_on int_decode A" |
|
164 |
by (rule inj_on_inverseI, rule int_decode_inverse) |
|
165 |
||
166 |
lemma surj_int_encode: "surj int_encode" |
|
167 |
by (rule surjI, rule int_decode_inverse) |
|
168 |
||
169 |
lemma surj_int_decode: "surj int_decode" |
|
170 |
by (rule surjI, rule int_encode_inverse) |
|
171 |
||
172 |
lemma bij_int_encode: "bij int_encode" |
|
173 |
by (rule bijI [OF inj_int_encode surj_int_encode]) |
|
174 |
||
175 |
lemma bij_int_decode: "bij int_decode" |
|
176 |
by (rule bijI [OF inj_int_decode surj_int_decode]) |
|
177 |
||
178 |
lemma int_encode_eq: "int_encode x = int_encode y \<longleftrightarrow> x = y" |
|
179 |
by (rule inj_int_encode [THEN inj_eq]) |
|
180 |
||
181 |
lemma int_decode_eq: "int_decode x = int_decode y \<longleftrightarrow> x = y" |
|
182 |
by (rule inj_int_decode [THEN inj_eq]) |
|
183 |
||
184 |
||
60500 | 185 |
subsection \<open>Type @{typ "nat list"}\<close> |
35700 | 186 |
|
62046 | 187 |
fun list_encode :: "nat list \<Rightarrow> nat" |
35700 | 188 |
where |
189 |
"list_encode [] = 0" |
|
190 |
| "list_encode (x # xs) = Suc (prod_encode (x, list_encode xs))" |
|
191 |
||
62046 | 192 |
function list_decode :: "nat \<Rightarrow> nat list" |
35700 | 193 |
where |
194 |
"list_decode 0 = []" |
|
195 |
| "list_decode (Suc n) = (case prod_decode n of (x, y) \<Rightarrow> x # list_decode y)" |
|
196 |
by pat_completeness auto |
|
197 |
||
198 |
termination list_decode |
|
199 |
apply (relation "measure id", simp_all) |
|
200 |
apply (drule arg_cong [where f="prod_encode"]) |
|
37591 | 201 |
apply (drule sym) |
35700 | 202 |
apply (simp add: le_imp_less_Suc le_prod_encode_2) |
203 |
done |
|
204 |
||
205 |
lemma list_encode_inverse [simp]: "list_decode (list_encode x) = x" |
|
206 |
by (induct x rule: list_encode.induct) simp_all |
|
207 |
||
208 |
lemma list_decode_inverse [simp]: "list_encode (list_decode n) = n" |
|
209 |
apply (induct n rule: list_decode.induct, simp) |
|
210 |
apply (simp split: prod.split) |
|
211 |
apply (simp add: prod_decode_eq [symmetric]) |
|
212 |
done |
|
213 |
||
214 |
lemma inj_list_encode: "inj_on list_encode A" |
|
215 |
by (rule inj_on_inverseI, rule list_encode_inverse) |
|
216 |
||
217 |
lemma inj_list_decode: "inj_on list_decode A" |
|
218 |
by (rule inj_on_inverseI, rule list_decode_inverse) |
|
219 |
||
220 |
lemma surj_list_encode: "surj list_encode" |
|
221 |
by (rule surjI, rule list_decode_inverse) |
|
222 |
||
223 |
lemma surj_list_decode: "surj list_decode" |
|
224 |
by (rule surjI, rule list_encode_inverse) |
|
225 |
||
226 |
lemma bij_list_encode: "bij list_encode" |
|
227 |
by (rule bijI [OF inj_list_encode surj_list_encode]) |
|
228 |
||
229 |
lemma bij_list_decode: "bij list_decode" |
|
230 |
by (rule bijI [OF inj_list_decode surj_list_decode]) |
|
231 |
||
232 |
lemma list_encode_eq: "list_encode x = list_encode y \<longleftrightarrow> x = y" |
|
233 |
by (rule inj_list_encode [THEN inj_eq]) |
|
234 |
||
235 |
lemma list_decode_eq: "list_decode x = list_decode y \<longleftrightarrow> x = y" |
|
236 |
by (rule inj_list_decode [THEN inj_eq]) |
|
237 |
||
238 |
||
60500 | 239 |
subsection \<open>Finite sets of naturals\<close> |
35700 | 240 |
|
60500 | 241 |
subsubsection \<open>Preliminaries\<close> |
35700 | 242 |
|
243 |
lemma finite_vimage_Suc_iff: "finite (Suc -` F) \<longleftrightarrow> finite F" |
|
244 |
apply (safe intro!: finite_vimageI inj_Suc) |
|
245 |
apply (rule finite_subset [where B="insert 0 (Suc ` Suc -` F)"]) |
|
246 |
apply (rule subsetI, case_tac x, simp, simp) |
|
247 |
apply (rule finite_insert [THEN iffD2]) |
|
248 |
apply (erule finite_imageI) |
|
249 |
done |
|
250 |
||
251 |
lemma vimage_Suc_insert_0: "Suc -` insert 0 A = Suc -` A" |
|
252 |
by auto |
|
253 |
||
254 |
lemma vimage_Suc_insert_Suc: |
|
255 |
"Suc -` insert (Suc n) A = insert n (Suc -` A)" |
|
256 |
by auto |
|
257 |
||
258 |
lemma div2_even_ext_nat: |
|
58834 | 259 |
fixes x y :: nat |
260 |
assumes "x div 2 = y div 2" |
|
261 |
and "even x \<longleftrightarrow> even y" |
|
262 |
shows "x = y" |
|
263 |
proof - |
|
60500 | 264 |
from \<open>even x \<longleftrightarrow> even y\<close> have "x mod 2 = y mod 2" |
58834 | 265 |
by (simp only: even_iff_mod_2_eq_zero) auto |
266 |
with assms have "x div 2 * 2 + x mod 2 = y div 2 * 2 + y mod 2" |
|
267 |
by simp |
|
268 |
then show ?thesis |
|
269 |
by simp |
|
270 |
qed |
|
35700 | 271 |
|
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
57512
diff
changeset
|
272 |
|
60500 | 273 |
subsubsection \<open>From sets to naturals\<close> |
35700 | 274 |
|
62046 | 275 |
definition set_encode :: "nat set \<Rightarrow> nat" |
276 |
where "set_encode = setsum (op ^ 2)" |
|
35700 | 277 |
|
278 |
lemma set_encode_empty [simp]: "set_encode {} = 0" |
|
279 |
by (simp add: set_encode_def) |
|
280 |
||
59506
4af607652318
Not a simprule, as it complicates proofs
paulson <lp15@cam.ac.uk>
parents:
58881
diff
changeset
|
281 |
lemma set_encode_inf: "~ finite A \<Longrightarrow> set_encode A = 0" |
4af607652318
Not a simprule, as it complicates proofs
paulson <lp15@cam.ac.uk>
parents:
58881
diff
changeset
|
282 |
by (simp add: set_encode_def) |
4af607652318
Not a simprule, as it complicates proofs
paulson <lp15@cam.ac.uk>
parents:
58881
diff
changeset
|
283 |
|
35700 | 284 |
lemma set_encode_insert [simp]: |
285 |
"\<lbrakk>finite A; n \<notin> A\<rbrakk> \<Longrightarrow> set_encode (insert n A) = 2^n + set_encode A" |
|
286 |
by (simp add: set_encode_def) |
|
287 |
||
288 |
lemma even_set_encode_iff: "finite A \<Longrightarrow> even (set_encode A) \<longleftrightarrow> 0 \<notin> A" |
|
289 |
unfolding set_encode_def by (induct set: finite, auto) |
|
290 |
||
291 |
lemma set_encode_vimage_Suc: "set_encode (Suc -` A) = set_encode A div 2" |
|
292 |
apply (cases "finite A") |
|
293 |
apply (erule finite_induct, simp) |
|
294 |
apply (case_tac x) |
|
58710
7216a10d69ba
augmented and tuned facts on even/odd and division
haftmann
parents:
57512
diff
changeset
|
295 |
apply (simp add: even_set_encode_iff vimage_Suc_insert_0) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
51489
diff
changeset
|
296 |
apply (simp add: finite_vimageI add.commute vimage_Suc_insert_Suc) |
35700 | 297 |
apply (simp add: set_encode_def finite_vimage_Suc_iff) |
298 |
done |
|
299 |
||
300 |
lemmas set_encode_div_2 = set_encode_vimage_Suc [symmetric] |
|
301 |
||
62046 | 302 |
|
60500 | 303 |
subsubsection \<open>From naturals to sets\<close> |
35700 | 304 |
|
62046 | 305 |
definition set_decode :: "nat \<Rightarrow> nat set" |
306 |
where "set_decode x = {n. odd (x div 2 ^ n)}" |
|
35700 | 307 |
|
308 |
lemma set_decode_0 [simp]: "0 \<in> set_decode x \<longleftrightarrow> odd x" |
|
309 |
by (simp add: set_decode_def) |
|
310 |
||
311 |
lemma set_decode_Suc [simp]: |
|
312 |
"Suc n \<in> set_decode x \<longleftrightarrow> n \<in> set_decode (x div 2)" |
|
313 |
by (simp add: set_decode_def div_mult2_eq) |
|
314 |
||
315 |
lemma set_decode_zero [simp]: "set_decode 0 = {}" |
|
316 |
by (simp add: set_decode_def) |
|
317 |
||
318 |
lemma set_decode_div_2: "set_decode (x div 2) = Suc -` set_decode x" |
|
319 |
by auto |
|
320 |
||
321 |
lemma set_decode_plus_power_2: |
|
322 |
"n \<notin> set_decode z \<Longrightarrow> set_decode (2 ^ n + z) = insert n (set_decode z)" |
|
60352
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
323 |
proof (induct n arbitrary: z) |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
324 |
case 0 show ?case |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
325 |
proof (rule set_eqI) |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
326 |
fix q show "q \<in> set_decode (2 ^ 0 + z) \<longleftrightarrow> q \<in> insert 0 (set_decode z)" |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
327 |
by (induct q) (insert 0, simp_all) |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
328 |
qed |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
329 |
next |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
330 |
case (Suc n) show ?case |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
331 |
proof (rule set_eqI) |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
332 |
fix q show "q \<in> set_decode (2 ^ Suc n + z) \<longleftrightarrow> q \<in> insert (Suc n) (set_decode z)" |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
333 |
by (induct q) (insert Suc, simp_all) |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
334 |
qed |
d46de31a50c4
separate class for division operator, with particular syntax added in more specific classes
haftmann
parents:
59506
diff
changeset
|
335 |
qed |
35700 | 336 |
|
337 |
lemma finite_set_decode [simp]: "finite (set_decode n)" |
|
338 |
apply (induct n rule: nat_less_induct) |
|
339 |
apply (case_tac "n = 0", simp) |
|
340 |
apply (drule_tac x="n div 2" in spec, simp) |
|
341 |
apply (simp add: set_decode_div_2) |
|
342 |
apply (simp add: finite_vimage_Suc_iff) |
|
343 |
done |
|
344 |
||
62046 | 345 |
|
60500 | 346 |
subsubsection \<open>Proof of isomorphism\<close> |
35700 | 347 |
|
348 |
lemma set_decode_inverse [simp]: "set_encode (set_decode n) = n" |
|
349 |
apply (induct n rule: nat_less_induct) |
|
350 |
apply (case_tac "n = 0", simp) |
|
351 |
apply (drule_tac x="n div 2" in spec, simp) |
|
352 |
apply (simp add: set_decode_div_2 set_encode_vimage_Suc) |
|
353 |
apply (erule div2_even_ext_nat) |
|
354 |
apply (simp add: even_set_encode_iff) |
|
355 |
done |
|
356 |
||
357 |
lemma set_encode_inverse [simp]: "finite A \<Longrightarrow> set_decode (set_encode A) = A" |
|
358 |
apply (erule finite_induct, simp_all) |
|
359 |
apply (simp add: set_decode_plus_power_2) |
|
360 |
done |
|
361 |
||
362 |
lemma inj_on_set_encode: "inj_on set_encode (Collect finite)" |
|
363 |
by (rule inj_on_inverseI [where g="set_decode"], simp) |
|
364 |
||
365 |
lemma set_encode_eq: |
|
366 |
"\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow> set_encode A = set_encode B \<longleftrightarrow> A = B" |
|
367 |
by (rule iffI, simp add: inj_onD [OF inj_on_set_encode], simp) |
|
368 |
||
62046 | 369 |
lemma subset_decode_imp_le: |
370 |
assumes "set_decode m \<subseteq> set_decode n" |
|
371 |
shows "m \<le> n" |
|
51414 | 372 |
proof - |
373 |
have "n = m + set_encode (set_decode n - set_decode m)" |
|
374 |
proof - |
|
375 |
obtain A B where "m = set_encode A" "finite A" |
|
376 |
"n = set_encode B" "finite B" |
|
377 |
by (metis finite_set_decode set_decode_inverse) |
|
378 |
thus ?thesis using assms |
|
379 |
apply auto |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
51489
diff
changeset
|
380 |
apply (simp add: set_encode_def add.commute setsum.subset_diff) |
51414 | 381 |
done |
382 |
qed |
|
383 |
thus ?thesis |
|
384 |
by (metis le_add1) |
|
385 |
qed |
|
386 |
||
35700 | 387 |
end |