src/HOL/Library/Nat_Bijection.thy
 author wenzelm Thu Feb 16 22:53:24 2012 +0100 (2012-02-16) changeset 46507 1b24c24017dd parent 41959 b460124855b8 child 51414 587f493447d9 permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Library/Nat_Bijection.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Florian Haftmann
```
```     4     Author:     Stefan Richter
```
```     5     Author:     Tobias Nipkow
```
```     6     Author:     Alexander Krauss
```
```     7 *)
```
```     8
```
```     9 header {* Bijections between natural numbers and other types *}
```
```    10
```
```    11 theory Nat_Bijection
```
```    12 imports Main Parity
```
```    13 begin
```
```    14
```
```    15 subsection {* Type @{typ "nat \<times> nat"} *}
```
```    16
```
```    17 text "Triangle numbers: 0, 1, 3, 6, 10, 15, ..."
```
```    18
```
```    19 definition
```
```    20   triangle :: "nat \<Rightarrow> nat"
```
```    21 where
```
```    22   "triangle n = n * Suc n div 2"
```
```    23
```
```    24 lemma triangle_0 [simp]: "triangle 0 = 0"
```
```    25 unfolding triangle_def by simp
```
```    26
```
```    27 lemma triangle_Suc [simp]: "triangle (Suc n) = triangle n + Suc n"
```
```    28 unfolding triangle_def by simp
```
```    29
```
```    30 definition
```
```    31   prod_encode :: "nat \<times> nat \<Rightarrow> nat"
```
```    32 where
```
```    33   "prod_encode = (\<lambda>(m, n). triangle (m + n) + m)"
```
```    34
```
```    35 text {* In this auxiliary function, @{term "triangle k + m"} is an invariant. *}
```
```    36
```
```    37 fun
```
```    38   prod_decode_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
```
```    39 where
```
```    40   "prod_decode_aux k m =
```
```    41     (if m \<le> k then (m, k - m) else prod_decode_aux (Suc k) (m - Suc k))"
```
```    42
```
```    43 declare prod_decode_aux.simps [simp del]
```
```    44
```
```    45 definition
```
```    46   prod_decode :: "nat \<Rightarrow> nat \<times> nat"
```
```    47 where
```
```    48   "prod_decode = prod_decode_aux 0"
```
```    49
```
```    50 lemma prod_encode_prod_decode_aux:
```
```    51   "prod_encode (prod_decode_aux k m) = triangle k + m"
```
```    52 apply (induct k m rule: prod_decode_aux.induct)
```
```    53 apply (subst prod_decode_aux.simps)
```
```    54 apply (simp add: prod_encode_def)
```
```    55 done
```
```    56
```
```    57 lemma prod_decode_inverse [simp]: "prod_encode (prod_decode n) = n"
```
```    58 unfolding prod_decode_def by (simp add: prod_encode_prod_decode_aux)
```
```    59
```
```    60 lemma prod_decode_triangle_add:
```
```    61   "prod_decode (triangle k + m) = prod_decode_aux k m"
```
```    62 apply (induct k arbitrary: m)
```
```    63 apply (simp add: prod_decode_def)
```
```    64 apply (simp only: triangle_Suc add_assoc)
```
```    65 apply (subst prod_decode_aux.simps, simp)
```
```    66 done
```
```    67
```
```    68 lemma prod_encode_inverse [simp]: "prod_decode (prod_encode x) = x"
```
```    69 unfolding prod_encode_def
```
```    70 apply (induct x)
```
```    71 apply (simp add: prod_decode_triangle_add)
```
```    72 apply (subst prod_decode_aux.simps, simp)
```
```    73 done
```
```    74
```
```    75 lemma inj_prod_encode: "inj_on prod_encode A"
```
```    76 by (rule inj_on_inverseI, rule prod_encode_inverse)
```
```    77
```
```    78 lemma inj_prod_decode: "inj_on prod_decode A"
```
```    79 by (rule inj_on_inverseI, rule prod_decode_inverse)
```
```    80
```
```    81 lemma surj_prod_encode: "surj prod_encode"
```
```    82 by (rule surjI, rule prod_decode_inverse)
```
```    83
```
```    84 lemma surj_prod_decode: "surj prod_decode"
```
```    85 by (rule surjI, rule prod_encode_inverse)
```
```    86
```
```    87 lemma bij_prod_encode: "bij prod_encode"
```
```    88 by (rule bijI [OF inj_prod_encode surj_prod_encode])
```
```    89
```
```    90 lemma bij_prod_decode: "bij prod_decode"
```
```    91 by (rule bijI [OF inj_prod_decode surj_prod_decode])
```
```    92
```
```    93 lemma prod_encode_eq: "prod_encode x = prod_encode y \<longleftrightarrow> x = y"
```
```    94 by (rule inj_prod_encode [THEN inj_eq])
```
```    95
```
```    96 lemma prod_decode_eq: "prod_decode x = prod_decode y \<longleftrightarrow> x = y"
```
```    97 by (rule inj_prod_decode [THEN inj_eq])
```
```    98
```
```    99 text {* Ordering properties *}
```
```   100
```
```   101 lemma le_prod_encode_1: "a \<le> prod_encode (a, b)"
```
```   102 unfolding prod_encode_def by simp
```
```   103
```
```   104 lemma le_prod_encode_2: "b \<le> prod_encode (a, b)"
```
```   105 unfolding prod_encode_def by (induct b, simp_all)
```
```   106
```
```   107
```
```   108 subsection {* Type @{typ "nat + nat"} *}
```
```   109
```
```   110 definition
```
```   111   sum_encode  :: "nat + nat \<Rightarrow> nat"
```
```   112 where
```
```   113   "sum_encode x = (case x of Inl a \<Rightarrow> 2 * a | Inr b \<Rightarrow> Suc (2 * b))"
```
```   114
```
```   115 definition
```
```   116   sum_decode  :: "nat \<Rightarrow> nat + nat"
```
```   117 where
```
```   118   "sum_decode n = (if even n then Inl (n div 2) else Inr (n div 2))"
```
```   119
```
```   120 lemma sum_encode_inverse [simp]: "sum_decode (sum_encode x) = x"
```
```   121 unfolding sum_decode_def sum_encode_def
```
```   122 by (induct x) simp_all
```
```   123
```
```   124 lemma sum_decode_inverse [simp]: "sum_encode (sum_decode n) = n"
```
```   125 unfolding sum_decode_def sum_encode_def numeral_2_eq_2
```
```   126 by (simp add: even_nat_div_two_times_two odd_nat_div_two_times_two_plus_one
```
```   127          del: mult_Suc)
```
```   128
```
```   129 lemma inj_sum_encode: "inj_on sum_encode A"
```
```   130 by (rule inj_on_inverseI, rule sum_encode_inverse)
```
```   131
```
```   132 lemma inj_sum_decode: "inj_on sum_decode A"
```
```   133 by (rule inj_on_inverseI, rule sum_decode_inverse)
```
```   134
```
```   135 lemma surj_sum_encode: "surj sum_encode"
```
```   136 by (rule surjI, rule sum_decode_inverse)
```
```   137
```
```   138 lemma surj_sum_decode: "surj sum_decode"
```
```   139 by (rule surjI, rule sum_encode_inverse)
```
```   140
```
```   141 lemma bij_sum_encode: "bij sum_encode"
```
```   142 by (rule bijI [OF inj_sum_encode surj_sum_encode])
```
```   143
```
```   144 lemma bij_sum_decode: "bij sum_decode"
```
```   145 by (rule bijI [OF inj_sum_decode surj_sum_decode])
```
```   146
```
```   147 lemma sum_encode_eq: "sum_encode x = sum_encode y \<longleftrightarrow> x = y"
```
```   148 by (rule inj_sum_encode [THEN inj_eq])
```
```   149
```
```   150 lemma sum_decode_eq: "sum_decode x = sum_decode y \<longleftrightarrow> x = y"
```
```   151 by (rule inj_sum_decode [THEN inj_eq])
```
```   152
```
```   153
```
```   154 subsection {* Type @{typ "int"} *}
```
```   155
```
```   156 definition
```
```   157   int_encode :: "int \<Rightarrow> nat"
```
```   158 where
```
```   159   "int_encode i = sum_encode (if 0 \<le> i then Inl (nat i) else Inr (nat (- i - 1)))"
```
```   160
```
```   161 definition
```
```   162   int_decode :: "nat \<Rightarrow> int"
```
```   163 where
```
```   164   "int_decode n = (case sum_decode n of Inl a \<Rightarrow> int a | Inr b \<Rightarrow> - int b - 1)"
```
```   165
```
```   166 lemma int_encode_inverse [simp]: "int_decode (int_encode x) = x"
```
```   167 unfolding int_decode_def int_encode_def by simp
```
```   168
```
```   169 lemma int_decode_inverse [simp]: "int_encode (int_decode n) = n"
```
```   170 unfolding int_decode_def int_encode_def using sum_decode_inverse [of n]
```
```   171 by (cases "sum_decode n", simp_all)
```
```   172
```
```   173 lemma inj_int_encode: "inj_on int_encode A"
```
```   174 by (rule inj_on_inverseI, rule int_encode_inverse)
```
```   175
```
```   176 lemma inj_int_decode: "inj_on int_decode A"
```
```   177 by (rule inj_on_inverseI, rule int_decode_inverse)
```
```   178
```
```   179 lemma surj_int_encode: "surj int_encode"
```
```   180 by (rule surjI, rule int_decode_inverse)
```
```   181
```
```   182 lemma surj_int_decode: "surj int_decode"
```
```   183 by (rule surjI, rule int_encode_inverse)
```
```   184
```
```   185 lemma bij_int_encode: "bij int_encode"
```
```   186 by (rule bijI [OF inj_int_encode surj_int_encode])
```
```   187
```
```   188 lemma bij_int_decode: "bij int_decode"
```
```   189 by (rule bijI [OF inj_int_decode surj_int_decode])
```
```   190
```
```   191 lemma int_encode_eq: "int_encode x = int_encode y \<longleftrightarrow> x = y"
```
```   192 by (rule inj_int_encode [THEN inj_eq])
```
```   193
```
```   194 lemma int_decode_eq: "int_decode x = int_decode y \<longleftrightarrow> x = y"
```
```   195 by (rule inj_int_decode [THEN inj_eq])
```
```   196
```
```   197
```
```   198 subsection {* Type @{typ "nat list"} *}
```
```   199
```
```   200 fun
```
```   201   list_encode :: "nat list \<Rightarrow> nat"
```
```   202 where
```
```   203   "list_encode [] = 0"
```
```   204 | "list_encode (x # xs) = Suc (prod_encode (x, list_encode xs))"
```
```   205
```
```   206 function
```
```   207   list_decode :: "nat \<Rightarrow> nat list"
```
```   208 where
```
```   209   "list_decode 0 = []"
```
```   210 | "list_decode (Suc n) = (case prod_decode n of (x, y) \<Rightarrow> x # list_decode y)"
```
```   211 by pat_completeness auto
```
```   212
```
```   213 termination list_decode
```
```   214 apply (relation "measure id", simp_all)
```
```   215 apply (drule arg_cong [where f="prod_encode"])
```
```   216 apply (drule sym)
```
```   217 apply (simp add: le_imp_less_Suc le_prod_encode_2)
```
```   218 done
```
```   219
```
```   220 lemma list_encode_inverse [simp]: "list_decode (list_encode x) = x"
```
```   221 by (induct x rule: list_encode.induct) simp_all
```
```   222
```
```   223 lemma list_decode_inverse [simp]: "list_encode (list_decode n) = n"
```
```   224 apply (induct n rule: list_decode.induct, simp)
```
```   225 apply (simp split: prod.split)
```
```   226 apply (simp add: prod_decode_eq [symmetric])
```
```   227 done
```
```   228
```
```   229 lemma inj_list_encode: "inj_on list_encode A"
```
```   230 by (rule inj_on_inverseI, rule list_encode_inverse)
```
```   231
```
```   232 lemma inj_list_decode: "inj_on list_decode A"
```
```   233 by (rule inj_on_inverseI, rule list_decode_inverse)
```
```   234
```
```   235 lemma surj_list_encode: "surj list_encode"
```
```   236 by (rule surjI, rule list_decode_inverse)
```
```   237
```
```   238 lemma surj_list_decode: "surj list_decode"
```
```   239 by (rule surjI, rule list_encode_inverse)
```
```   240
```
```   241 lemma bij_list_encode: "bij list_encode"
```
```   242 by (rule bijI [OF inj_list_encode surj_list_encode])
```
```   243
```
```   244 lemma bij_list_decode: "bij list_decode"
```
```   245 by (rule bijI [OF inj_list_decode surj_list_decode])
```
```   246
```
```   247 lemma list_encode_eq: "list_encode x = list_encode y \<longleftrightarrow> x = y"
```
```   248 by (rule inj_list_encode [THEN inj_eq])
```
```   249
```
```   250 lemma list_decode_eq: "list_decode x = list_decode y \<longleftrightarrow> x = y"
```
```   251 by (rule inj_list_decode [THEN inj_eq])
```
```   252
```
```   253
```
```   254 subsection {* Finite sets of naturals *}
```
```   255
```
```   256 subsubsection {* Preliminaries *}
```
```   257
```
```   258 lemma finite_vimage_Suc_iff: "finite (Suc -` F) \<longleftrightarrow> finite F"
```
```   259 apply (safe intro!: finite_vimageI inj_Suc)
```
```   260 apply (rule finite_subset [where B="insert 0 (Suc ` Suc -` F)"])
```
```   261 apply (rule subsetI, case_tac x, simp, simp)
```
```   262 apply (rule finite_insert [THEN iffD2])
```
```   263 apply (erule finite_imageI)
```
```   264 done
```
```   265
```
```   266 lemma vimage_Suc_insert_0: "Suc -` insert 0 A = Suc -` A"
```
```   267 by auto
```
```   268
```
```   269 lemma vimage_Suc_insert_Suc:
```
```   270   "Suc -` insert (Suc n) A = insert n (Suc -` A)"
```
```   271 by auto
```
```   272
```
```   273 lemma even_nat_Suc_div_2: "even x \<Longrightarrow> Suc x div 2 = x div 2"
```
```   274 by (simp only: numeral_2_eq_2 even_nat_plus_one_div_two)
```
```   275
```
```   276 lemma div2_even_ext_nat:
```
```   277   "\<lbrakk>x div 2 = y div 2; even x = even y\<rbrakk> \<Longrightarrow> (x::nat) = y"
```
```   278 apply (rule mod_div_equality [of x 2, THEN subst])
```
```   279 apply (rule mod_div_equality [of y 2, THEN subst])
```
```   280 apply (case_tac "even x")
```
```   281 apply (simp add: numeral_2_eq_2 even_nat_equiv_def)
```
```   282 apply (simp add: numeral_2_eq_2 odd_nat_equiv_def)
```
```   283 done
```
```   284
```
```   285 subsubsection {* From sets to naturals *}
```
```   286
```
```   287 definition
```
```   288   set_encode :: "nat set \<Rightarrow> nat"
```
```   289 where
```
```   290   "set_encode = setsum (op ^ 2)"
```
```   291
```
```   292 lemma set_encode_empty [simp]: "set_encode {} = 0"
```
```   293 by (simp add: set_encode_def)
```
```   294
```
```   295 lemma set_encode_insert [simp]:
```
```   296   "\<lbrakk>finite A; n \<notin> A\<rbrakk> \<Longrightarrow> set_encode (insert n A) = 2^n + set_encode A"
```
```   297 by (simp add: set_encode_def)
```
```   298
```
```   299 lemma even_set_encode_iff: "finite A \<Longrightarrow> even (set_encode A) \<longleftrightarrow> 0 \<notin> A"
```
```   300 unfolding set_encode_def by (induct set: finite, auto)
```
```   301
```
```   302 lemma set_encode_vimage_Suc: "set_encode (Suc -` A) = set_encode A div 2"
```
```   303 apply (cases "finite A")
```
```   304 apply (erule finite_induct, simp)
```
```   305 apply (case_tac x)
```
```   306 apply (simp add: even_nat_Suc_div_2 even_set_encode_iff vimage_Suc_insert_0)
```
```   307 apply (simp add: finite_vimageI add_commute vimage_Suc_insert_Suc)
```
```   308 apply (simp add: set_encode_def finite_vimage_Suc_iff)
```
```   309 done
```
```   310
```
```   311 lemmas set_encode_div_2 = set_encode_vimage_Suc [symmetric]
```
```   312
```
```   313 subsubsection {* From naturals to sets *}
```
```   314
```
```   315 definition
```
```   316   set_decode :: "nat \<Rightarrow> nat set"
```
```   317 where
```
```   318   "set_decode x = {n. odd (x div 2 ^ n)}"
```
```   319
```
```   320 lemma set_decode_0 [simp]: "0 \<in> set_decode x \<longleftrightarrow> odd x"
```
```   321 by (simp add: set_decode_def)
```
```   322
```
```   323 lemma set_decode_Suc [simp]:
```
```   324   "Suc n \<in> set_decode x \<longleftrightarrow> n \<in> set_decode (x div 2)"
```
```   325 by (simp add: set_decode_def div_mult2_eq)
```
```   326
```
```   327 lemma set_decode_zero [simp]: "set_decode 0 = {}"
```
```   328 by (simp add: set_decode_def)
```
```   329
```
```   330 lemma set_decode_div_2: "set_decode (x div 2) = Suc -` set_decode x"
```
```   331 by auto
```
```   332
```
```   333 lemma set_decode_plus_power_2:
```
```   334   "n \<notin> set_decode z \<Longrightarrow> set_decode (2 ^ n + z) = insert n (set_decode z)"
```
```   335  apply (induct n arbitrary: z, simp_all)
```
```   336   apply (rule set_eqI, induct_tac x, simp, simp add: even_nat_Suc_div_2)
```
```   337  apply (rule set_eqI, induct_tac x, simp, simp add: add_commute)
```
```   338 done
```
```   339
```
```   340 lemma finite_set_decode [simp]: "finite (set_decode n)"
```
```   341 apply (induct n rule: nat_less_induct)
```
```   342 apply (case_tac "n = 0", simp)
```
```   343 apply (drule_tac x="n div 2" in spec, simp)
```
```   344 apply (simp add: set_decode_div_2)
```
```   345 apply (simp add: finite_vimage_Suc_iff)
```
```   346 done
```
```   347
```
```   348 subsubsection {* Proof of isomorphism *}
```
```   349
```
```   350 lemma set_decode_inverse [simp]: "set_encode (set_decode n) = n"
```
```   351 apply (induct n rule: nat_less_induct)
```
```   352 apply (case_tac "n = 0", simp)
```
```   353 apply (drule_tac x="n div 2" in spec, simp)
```
```   354 apply (simp add: set_decode_div_2 set_encode_vimage_Suc)
```
```   355 apply (erule div2_even_ext_nat)
```
```   356 apply (simp add: even_set_encode_iff)
```
```   357 done
```
```   358
```
```   359 lemma set_encode_inverse [simp]: "finite A \<Longrightarrow> set_decode (set_encode A) = A"
```
```   360 apply (erule finite_induct, simp_all)
```
```   361 apply (simp add: set_decode_plus_power_2)
```
```   362 done
```
```   363
```
```   364 lemma inj_on_set_encode: "inj_on set_encode (Collect finite)"
```
```   365 by (rule inj_on_inverseI [where g="set_decode"], simp)
```
```   366
```
```   367 lemma set_encode_eq:
```
```   368   "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow> set_encode A = set_encode B \<longleftrightarrow> A = B"
```
```   369 by (rule iffI, simp add: inj_onD [OF inj_on_set_encode], simp)
```
```   370
```
```   371 end
```