src/HOL/Library/Nat_Infinity.thy
 author haftmann Wed Apr 22 19:09:21 2009 +0200 (2009-04-22) changeset 30960 fec1a04b7220 parent 30663 0b6aff7451b2 child 31077 28dd6fd3d184 permissions -rw-r--r--
power operation defined generic
```     1 (*  Title:      HOL/Library/Nat_Infinity.thy
```
```     2     Author:     David von Oheimb, TU Muenchen;  Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Natural numbers with infinity *}
```
```     6
```
```     7 theory Nat_Infinity
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 subsection {* Type definition *}
```
```    12
```
```    13 text {*
```
```    14   We extend the standard natural numbers by a special value indicating
```
```    15   infinity.
```
```    16 *}
```
```    17
```
```    18 datatype inat = Fin nat | Infty
```
```    19
```
```    20 notation (xsymbols)
```
```    21   Infty  ("\<infinity>")
```
```    22
```
```    23 notation (HTML output)
```
```    24   Infty  ("\<infinity>")
```
```    25
```
```    26
```
```    27 subsection {* Constructors and numbers *}
```
```    28
```
```    29 instantiation inat :: "{zero, one, number}"
```
```    30 begin
```
```    31
```
```    32 definition
```
```    33   "0 = Fin 0"
```
```    34
```
```    35 definition
```
```    36   [code inline]: "1 = Fin 1"
```
```    37
```
```    38 definition
```
```    39   [code inline, code del]: "number_of k = Fin (number_of k)"
```
```    40
```
```    41 instance ..
```
```    42
```
```    43 end
```
```    44
```
```    45 definition iSuc :: "inat \<Rightarrow> inat" where
```
```    46   "iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
```
```    47
```
```    48 lemma Fin_0: "Fin 0 = 0"
```
```    49   by (simp add: zero_inat_def)
```
```    50
```
```    51 lemma Fin_1: "Fin 1 = 1"
```
```    52   by (simp add: one_inat_def)
```
```    53
```
```    54 lemma Fin_number: "Fin (number_of k) = number_of k"
```
```    55   by (simp add: number_of_inat_def)
```
```    56
```
```    57 lemma one_iSuc: "1 = iSuc 0"
```
```    58   by (simp add: zero_inat_def one_inat_def iSuc_def)
```
```    59
```
```    60 lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
```
```    61   by (simp add: zero_inat_def)
```
```    62
```
```    63 lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
```
```    64   by (simp add: zero_inat_def)
```
```    65
```
```    66 lemma zero_inat_eq [simp]:
```
```    67   "number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)"
```
```    68   "(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)"
```
```    69   unfolding zero_inat_def number_of_inat_def by simp_all
```
```    70
```
```    71 lemma one_inat_eq [simp]:
```
```    72   "number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)"
```
```    73   "(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)"
```
```    74   unfolding one_inat_def number_of_inat_def by simp_all
```
```    75
```
```    76 lemma zero_one_inat_neq [simp]:
```
```    77   "\<not> 0 = (1\<Colon>inat)"
```
```    78   "\<not> 1 = (0\<Colon>inat)"
```
```    79   unfolding zero_inat_def one_inat_def by simp_all
```
```    80
```
```    81 lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1"
```
```    82   by (simp add: one_inat_def)
```
```    83
```
```    84 lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>"
```
```    85   by (simp add: one_inat_def)
```
```    86
```
```    87 lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k"
```
```    88   by (simp add: number_of_inat_def)
```
```    89
```
```    90 lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>"
```
```    91   by (simp add: number_of_inat_def)
```
```    92
```
```    93 lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)"
```
```    94   by (simp add: iSuc_def)
```
```    95
```
```    96 lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))"
```
```    97   by (simp add: iSuc_Fin number_of_inat_def)
```
```    98
```
```    99 lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
```
```   100   by (simp add: iSuc_def)
```
```   101
```
```   102 lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
```
```   103   by (simp add: iSuc_def zero_inat_def split: inat.splits)
```
```   104
```
```   105 lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n"
```
```   106   by (rule iSuc_ne_0 [symmetric])
```
```   107
```
```   108 lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n"
```
```   109   by (simp add: iSuc_def split: inat.splits)
```
```   110
```
```   111 lemma number_of_inat_inject [simp]:
```
```   112   "(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l"
```
```   113   by (simp add: number_of_inat_def)
```
```   114
```
```   115
```
```   116 subsection {* Addition *}
```
```   117
```
```   118 instantiation inat :: comm_monoid_add
```
```   119 begin
```
```   120
```
```   121 definition
```
```   122   [code del]: "m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | Fin m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | Fin n \<Rightarrow> Fin (m + n)))"
```
```   123
```
```   124 lemma plus_inat_simps [simp, code]:
```
```   125   "Fin m + Fin n = Fin (m + n)"
```
```   126   "\<infinity> + q = \<infinity>"
```
```   127   "q + \<infinity> = \<infinity>"
```
```   128   by (simp_all add: plus_inat_def split: inat.splits)
```
```   129
```
```   130 instance proof
```
```   131   fix n m q :: inat
```
```   132   show "n + m + q = n + (m + q)"
```
```   133     by (cases n, auto, cases m, auto, cases q, auto)
```
```   134   show "n + m = m + n"
```
```   135     by (cases n, auto, cases m, auto)
```
```   136   show "0 + n = n"
```
```   137     by (cases n) (simp_all add: zero_inat_def)
```
```   138 qed
```
```   139
```
```   140 end
```
```   141
```
```   142 lemma plus_inat_0 [simp]:
```
```   143   "0 + (q\<Colon>inat) = q"
```
```   144   "(q\<Colon>inat) + 0 = q"
```
```   145   by (simp_all add: plus_inat_def zero_inat_def split: inat.splits)
```
```   146
```
```   147 lemma plus_inat_number [simp]:
```
```   148   "(number_of k \<Colon> inat) + number_of l = (if k < Int.Pls then number_of l
```
```   149     else if l < Int.Pls then number_of k else number_of (k + l))"
```
```   150   unfolding number_of_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] ..
```
```   151
```
```   152 lemma iSuc_number [simp]:
```
```   153   "iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))"
```
```   154   unfolding iSuc_number_of
```
```   155   unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] ..
```
```   156
```
```   157 lemma iSuc_plus_1:
```
```   158   "iSuc n = n + 1"
```
```   159   by (cases n) (simp_all add: iSuc_Fin one_inat_def)
```
```   160
```
```   161 lemma plus_1_iSuc:
```
```   162   "1 + q = iSuc q"
```
```   163   "q + 1 = iSuc q"
```
```   164   unfolding iSuc_plus_1 by (simp_all add: add_ac)
```
```   165
```
```   166
```
```   167 subsection {* Multiplication *}
```
```   168
```
```   169 instantiation inat :: comm_semiring_1
```
```   170 begin
```
```   171
```
```   172 definition
```
```   173   times_inat_def [code del]:
```
```   174   "m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | Fin m \<Rightarrow>
```
```   175     (case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | Fin n \<Rightarrow> Fin (m * n)))"
```
```   176
```
```   177 lemma times_inat_simps [simp, code]:
```
```   178   "Fin m * Fin n = Fin (m * n)"
```
```   179   "\<infinity> * \<infinity> = \<infinity>"
```
```   180   "\<infinity> * Fin n = (if n = 0 then 0 else \<infinity>)"
```
```   181   "Fin m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
```
```   182   unfolding times_inat_def zero_inat_def
```
```   183   by (simp_all split: inat.split)
```
```   184
```
```   185 instance proof
```
```   186   fix a b c :: inat
```
```   187   show "(a * b) * c = a * (b * c)"
```
```   188     unfolding times_inat_def zero_inat_def
```
```   189     by (simp split: inat.split)
```
```   190   show "a * b = b * a"
```
```   191     unfolding times_inat_def zero_inat_def
```
```   192     by (simp split: inat.split)
```
```   193   show "1 * a = a"
```
```   194     unfolding times_inat_def zero_inat_def one_inat_def
```
```   195     by (simp split: inat.split)
```
```   196   show "(a + b) * c = a * c + b * c"
```
```   197     unfolding times_inat_def zero_inat_def
```
```   198     by (simp split: inat.split add: left_distrib)
```
```   199   show "0 * a = 0"
```
```   200     unfolding times_inat_def zero_inat_def
```
```   201     by (simp split: inat.split)
```
```   202   show "a * 0 = 0"
```
```   203     unfolding times_inat_def zero_inat_def
```
```   204     by (simp split: inat.split)
```
```   205   show "(0::inat) \<noteq> 1"
```
```   206     unfolding zero_inat_def one_inat_def
```
```   207     by simp
```
```   208 qed
```
```   209
```
```   210 end
```
```   211
```
```   212 lemma mult_iSuc: "iSuc m * n = n + m * n"
```
```   213   unfolding iSuc_plus_1 by (simp add: algebra_simps)
```
```   214
```
```   215 lemma mult_iSuc_right: "m * iSuc n = m + m * n"
```
```   216   unfolding iSuc_plus_1 by (simp add: algebra_simps)
```
```   217
```
```   218 lemma of_nat_eq_Fin: "of_nat n = Fin n"
```
```   219   apply (induct n)
```
```   220   apply (simp add: Fin_0)
```
```   221   apply (simp add: plus_1_iSuc iSuc_Fin)
```
```   222   done
```
```   223
```
```   224 instance inat :: semiring_char_0
```
```   225   by default (simp add: of_nat_eq_Fin)
```
```   226
```
```   227
```
```   228 subsection {* Ordering *}
```
```   229
```
```   230 instantiation inat :: ordered_ab_semigroup_add
```
```   231 begin
```
```   232
```
```   233 definition
```
```   234   [code del]: "m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
```
```   235     | \<infinity> \<Rightarrow> True)"
```
```   236
```
```   237 definition
```
```   238   [code del]: "m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
```
```   239     | \<infinity> \<Rightarrow> False)"
```
```   240
```
```   241 lemma inat_ord_simps [simp]:
```
```   242   "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
```
```   243   "Fin m < Fin n \<longleftrightarrow> m < n"
```
```   244   "q \<le> \<infinity>"
```
```   245   "q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>"
```
```   246   "\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>"
```
```   247   "\<infinity> < q \<longleftrightarrow> False"
```
```   248   by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits)
```
```   249
```
```   250 lemma inat_ord_code [code]:
```
```   251   "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
```
```   252   "Fin m < Fin n \<longleftrightarrow> m < n"
```
```   253   "q \<le> \<infinity> \<longleftrightarrow> True"
```
```   254   "Fin m < \<infinity> \<longleftrightarrow> True"
```
```   255   "\<infinity> \<le> Fin n \<longleftrightarrow> False"
```
```   256   "\<infinity> < q \<longleftrightarrow> False"
```
```   257   by simp_all
```
```   258
```
```   259 instance by default
```
```   260   (auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits)
```
```   261
```
```   262 end
```
```   263
```
```   264 instance inat :: pordered_comm_semiring
```
```   265 proof
```
```   266   fix a b c :: inat
```
```   267   assume "a \<le> b" and "0 \<le> c"
```
```   268   thus "c * a \<le> c * b"
```
```   269     unfolding times_inat_def less_eq_inat_def zero_inat_def
```
```   270     by (simp split: inat.splits)
```
```   271 qed
```
```   272
```
```   273 lemma inat_ord_number [simp]:
```
```   274   "(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n"
```
```   275   "(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
```
```   276   by (simp_all add: number_of_inat_def)
```
```   277
```
```   278 lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n"
```
```   279   by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
```
```   280
```
```   281 lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0"
```
```   282   by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
```
```   283
```
```   284 lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R"
```
```   285   by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
```
```   286
```
```   287 lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R"
```
```   288   by simp
```
```   289
```
```   290 lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)"
```
```   291   by (simp add: zero_inat_def less_inat_def split: inat.splits)
```
```   292
```
```   293 lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0"
```
```   294   by (simp add: zero_inat_def less_inat_def split: inat.splits)
```
```   295
```
```   296 lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m"
```
```   297   by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
```
```   298
```
```   299 lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m"
```
```   300   by (simp add: iSuc_def less_inat_def split: inat.splits)
```
```   301
```
```   302 lemma ile_iSuc [simp]: "n \<le> iSuc n"
```
```   303   by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
```
```   304
```
```   305 lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
```
```   306   by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits)
```
```   307
```
```   308 lemma i0_iless_iSuc [simp]: "0 < iSuc n"
```
```   309   by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits)
```
```   310
```
```   311 lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n"
```
```   312   by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits)
```
```   313
```
```   314 lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n"
```
```   315   by (cases n) auto
```
```   316
```
```   317 lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n"
```
```   318   by (auto simp add: iSuc_def less_inat_def split: inat.splits)
```
```   319
```
```   320 lemma min_inat_simps [simp]:
```
```   321   "min (Fin m) (Fin n) = Fin (min m n)"
```
```   322   "min q 0 = 0"
```
```   323   "min 0 q = 0"
```
```   324   "min q \<infinity> = q"
```
```   325   "min \<infinity> q = q"
```
```   326   by (auto simp add: min_def)
```
```   327
```
```   328 lemma max_inat_simps [simp]:
```
```   329   "max (Fin m) (Fin n) = Fin (max m n)"
```
```   330   "max q 0 = q"
```
```   331   "max 0 q = q"
```
```   332   "max q \<infinity> = \<infinity>"
```
```   333   "max \<infinity> q = \<infinity>"
```
```   334   by (simp_all add: max_def)
```
```   335
```
```   336 lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k"
```
```   337   by (cases n) simp_all
```
```   338
```
```   339 lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k"
```
```   340   by (cases n) simp_all
```
```   341
```
```   342 lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
```
```   343 apply (induct_tac k)
```
```   344  apply (simp (no_asm) only: Fin_0)
```
```   345  apply (fast intro: le_less_trans [OF i0_lb])
```
```   346 apply (erule exE)
```
```   347 apply (drule spec)
```
```   348 apply (erule exE)
```
```   349 apply (drule ileI1)
```
```   350 apply (rule iSuc_Fin [THEN subst])
```
```   351 apply (rule exI)
```
```   352 apply (erule (1) le_less_trans)
```
```   353 done
```
```   354
```
```   355 instantiation inat :: "{bot, top}"
```
```   356 begin
```
```   357
```
```   358 definition bot_inat :: inat where
```
```   359   "bot_inat = 0"
```
```   360
```
```   361 definition top_inat :: inat where
```
```   362   "top_inat = \<infinity>"
```
```   363
```
```   364 instance proof
```
```   365 qed (simp_all add: bot_inat_def top_inat_def)
```
```   366
```
```   367 end
```
```   368
```
```   369
```
```   370 subsection {* Well-ordering *}
```
```   371
```
```   372 lemma less_FinE:
```
```   373   "[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P"
```
```   374 by (induct n) auto
```
```   375
```
```   376 lemma less_InftyE:
```
```   377   "[| n < Infty; !!k. n = Fin k ==> P |] ==> P"
```
```   378 by (induct n) auto
```
```   379
```
```   380 lemma inat_less_induct:
```
```   381   assumes prem: "!!n. \<forall>m::inat. m < n --> P m ==> P n" shows "P n"
```
```   382 proof -
```
```   383   have P_Fin: "!!k. P (Fin k)"
```
```   384     apply (rule nat_less_induct)
```
```   385     apply (rule prem, clarify)
```
```   386     apply (erule less_FinE, simp)
```
```   387     done
```
```   388   show ?thesis
```
```   389   proof (induct n)
```
```   390     fix nat
```
```   391     show "P (Fin nat)" by (rule P_Fin)
```
```   392   next
```
```   393     show "P Infty"
```
```   394       apply (rule prem, clarify)
```
```   395       apply (erule less_InftyE)
```
```   396       apply (simp add: P_Fin)
```
```   397       done
```
```   398   qed
```
```   399 qed
```
```   400
```
```   401 instance inat :: wellorder
```
```   402 proof
```
```   403   fix P and n
```
```   404   assume hyp: "(\<And>n\<Colon>inat. (\<And>m\<Colon>inat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
```
```   405   show "P n" by (blast intro: inat_less_induct hyp)
```
```   406 qed
```
```   407
```
```   408
```
```   409 subsection {* Traditional theorem names *}
```
```   410
```
```   411 lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def
```
```   412   plus_inat_def less_eq_inat_def less_inat_def
```
```   413
```
```   414 lemmas inat_splits = inat.splits
```
```   415
```
```   416 end
```