src/HOL/Library/Nat_Infinity.thy
 author huffman Thu Jun 11 09:03:24 2009 -0700 (2009-06-11) changeset 31563 ded2364d14d4 parent 31094 7d6edb28bdbc child 31998 2c7a24f74db9 permissions -rw-r--r--
cleaned up some proofs
```     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 lemma not_Infty_eq[iff]: "(x ~= Infty) = (EX i. x = Fin i)"
```
```    28 by (cases x) auto
```
```    29
```
```    30 lemma not_Fin_eq [iff]: "(ALL y. x ~= Fin y) = (x = Infty)"
```
```    31 by (cases x) auto
```
```    32
```
```    33
```
```    34 subsection {* Constructors and numbers *}
```
```    35
```
```    36 instantiation inat :: "{zero, one, number}"
```
```    37 begin
```
```    38
```
```    39 definition
```
```    40   "0 = Fin 0"
```
```    41
```
```    42 definition
```
```    43   [code inline]: "1 = Fin 1"
```
```    44
```
```    45 definition
```
```    46   [code inline, code del]: "number_of k = Fin (number_of k)"
```
```    47
```
```    48 instance ..
```
```    49
```
```    50 end
```
```    51
```
```    52 definition iSuc :: "inat \<Rightarrow> inat" where
```
```    53   "iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
```
```    54
```
```    55 lemma Fin_0: "Fin 0 = 0"
```
```    56   by (simp add: zero_inat_def)
```
```    57
```
```    58 lemma Fin_1: "Fin 1 = 1"
```
```    59   by (simp add: one_inat_def)
```
```    60
```
```    61 lemma Fin_number: "Fin (number_of k) = number_of k"
```
```    62   by (simp add: number_of_inat_def)
```
```    63
```
```    64 lemma one_iSuc: "1 = iSuc 0"
```
```    65   by (simp add: zero_inat_def one_inat_def iSuc_def)
```
```    66
```
```    67 lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
```
```    68   by (simp add: zero_inat_def)
```
```    69
```
```    70 lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
```
```    71   by (simp add: zero_inat_def)
```
```    72
```
```    73 lemma zero_inat_eq [simp]:
```
```    74   "number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)"
```
```    75   "(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)"
```
```    76   unfolding zero_inat_def number_of_inat_def by simp_all
```
```    77
```
```    78 lemma one_inat_eq [simp]:
```
```    79   "number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)"
```
```    80   "(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)"
```
```    81   unfolding one_inat_def number_of_inat_def by simp_all
```
```    82
```
```    83 lemma zero_one_inat_neq [simp]:
```
```    84   "\<not> 0 = (1\<Colon>inat)"
```
```    85   "\<not> 1 = (0\<Colon>inat)"
```
```    86   unfolding zero_inat_def one_inat_def by simp_all
```
```    87
```
```    88 lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1"
```
```    89   by (simp add: one_inat_def)
```
```    90
```
```    91 lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>"
```
```    92   by (simp add: one_inat_def)
```
```    93
```
```    94 lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k"
```
```    95   by (simp add: number_of_inat_def)
```
```    96
```
```    97 lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>"
```
```    98   by (simp add: number_of_inat_def)
```
```    99
```
```   100 lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)"
```
```   101   by (simp add: iSuc_def)
```
```   102
```
```   103 lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))"
```
```   104   by (simp add: iSuc_Fin number_of_inat_def)
```
```   105
```
```   106 lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
```
```   107   by (simp add: iSuc_def)
```
```   108
```
```   109 lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
```
```   110   by (simp add: iSuc_def zero_inat_def split: inat.splits)
```
```   111
```
```   112 lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n"
```
```   113   by (rule iSuc_ne_0 [symmetric])
```
```   114
```
```   115 lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n"
```
```   116   by (simp add: iSuc_def split: inat.splits)
```
```   117
```
```   118 lemma number_of_inat_inject [simp]:
```
```   119   "(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l"
```
```   120   by (simp add: number_of_inat_def)
```
```   121
```
```   122
```
```   123 subsection {* Addition *}
```
```   124
```
```   125 instantiation inat :: comm_monoid_add
```
```   126 begin
```
```   127
```
```   128 definition
```
```   129   [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)))"
```
```   130
```
```   131 lemma plus_inat_simps [simp, code]:
```
```   132   "Fin m + Fin n = Fin (m + n)"
```
```   133   "\<infinity> + q = \<infinity>"
```
```   134   "q + \<infinity> = \<infinity>"
```
```   135   by (simp_all add: plus_inat_def split: inat.splits)
```
```   136
```
```   137 instance proof
```
```   138   fix n m q :: inat
```
```   139   show "n + m + q = n + (m + q)"
```
```   140     by (cases n, auto, cases m, auto, cases q, auto)
```
```   141   show "n + m = m + n"
```
```   142     by (cases n, auto, cases m, auto)
```
```   143   show "0 + n = n"
```
```   144     by (cases n) (simp_all add: zero_inat_def)
```
```   145 qed
```
```   146
```
```   147 end
```
```   148
```
```   149 lemma plus_inat_0 [simp]:
```
```   150   "0 + (q\<Colon>inat) = q"
```
```   151   "(q\<Colon>inat) + 0 = q"
```
```   152   by (simp_all add: plus_inat_def zero_inat_def split: inat.splits)
```
```   153
```
```   154 lemma plus_inat_number [simp]:
```
```   155   "(number_of k \<Colon> inat) + number_of l = (if k < Int.Pls then number_of l
```
```   156     else if l < Int.Pls then number_of k else number_of (k + l))"
```
```   157   unfolding number_of_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] ..
```
```   158
```
```   159 lemma iSuc_number [simp]:
```
```   160   "iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))"
```
```   161   unfolding iSuc_number_of
```
```   162   unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] ..
```
```   163
```
```   164 lemma iSuc_plus_1:
```
```   165   "iSuc n = n + 1"
```
```   166   by (cases n) (simp_all add: iSuc_Fin one_inat_def)
```
```   167
```
```   168 lemma plus_1_iSuc:
```
```   169   "1 + q = iSuc q"
```
```   170   "q + 1 = iSuc q"
```
```   171   unfolding iSuc_plus_1 by (simp_all add: add_ac)
```
```   172
```
```   173
```
```   174 subsection {* Multiplication *}
```
```   175
```
```   176 instantiation inat :: comm_semiring_1
```
```   177 begin
```
```   178
```
```   179 definition
```
```   180   times_inat_def [code del]:
```
```   181   "m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | Fin m \<Rightarrow>
```
```   182     (case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | Fin n \<Rightarrow> Fin (m * n)))"
```
```   183
```
```   184 lemma times_inat_simps [simp, code]:
```
```   185   "Fin m * Fin n = Fin (m * n)"
```
```   186   "\<infinity> * \<infinity> = \<infinity>"
```
```   187   "\<infinity> * Fin n = (if n = 0 then 0 else \<infinity>)"
```
```   188   "Fin m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
```
```   189   unfolding times_inat_def zero_inat_def
```
```   190   by (simp_all split: inat.split)
```
```   191
```
```   192 instance proof
```
```   193   fix a b c :: inat
```
```   194   show "(a * b) * c = a * (b * c)"
```
```   195     unfolding times_inat_def zero_inat_def
```
```   196     by (simp split: inat.split)
```
```   197   show "a * b = b * a"
```
```   198     unfolding times_inat_def zero_inat_def
```
```   199     by (simp split: inat.split)
```
```   200   show "1 * a = a"
```
```   201     unfolding times_inat_def zero_inat_def one_inat_def
```
```   202     by (simp split: inat.split)
```
```   203   show "(a + b) * c = a * c + b * c"
```
```   204     unfolding times_inat_def zero_inat_def
```
```   205     by (simp split: inat.split add: left_distrib)
```
```   206   show "0 * a = 0"
```
```   207     unfolding times_inat_def zero_inat_def
```
```   208     by (simp split: inat.split)
```
```   209   show "a * 0 = 0"
```
```   210     unfolding times_inat_def zero_inat_def
```
```   211     by (simp split: inat.split)
```
```   212   show "(0::inat) \<noteq> 1"
```
```   213     unfolding zero_inat_def one_inat_def
```
```   214     by simp
```
```   215 qed
```
```   216
```
```   217 end
```
```   218
```
```   219 lemma mult_iSuc: "iSuc m * n = n + m * n"
```
```   220   unfolding iSuc_plus_1 by (simp add: algebra_simps)
```
```   221
```
```   222 lemma mult_iSuc_right: "m * iSuc n = m + m * n"
```
```   223   unfolding iSuc_plus_1 by (simp add: algebra_simps)
```
```   224
```
```   225 lemma of_nat_eq_Fin: "of_nat n = Fin n"
```
```   226   apply (induct n)
```
```   227   apply (simp add: Fin_0)
```
```   228   apply (simp add: plus_1_iSuc iSuc_Fin)
```
```   229   done
```
```   230
```
```   231 instance inat :: semiring_char_0
```
```   232   by default (simp add: of_nat_eq_Fin)
```
```   233
```
```   234
```
```   235 subsection {* Ordering *}
```
```   236
```
```   237 instantiation inat :: ordered_ab_semigroup_add
```
```   238 begin
```
```   239
```
```   240 definition
```
```   241   [code del]: "m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
```
```   242     | \<infinity> \<Rightarrow> True)"
```
```   243
```
```   244 definition
```
```   245   [code del]: "m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
```
```   246     | \<infinity> \<Rightarrow> False)"
```
```   247
```
```   248 lemma inat_ord_simps [simp]:
```
```   249   "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
```
```   250   "Fin m < Fin n \<longleftrightarrow> m < n"
```
```   251   "q \<le> \<infinity>"
```
```   252   "q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>"
```
```   253   "\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>"
```
```   254   "\<infinity> < q \<longleftrightarrow> False"
```
```   255   by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits)
```
```   256
```
```   257 lemma inat_ord_code [code]:
```
```   258   "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
```
```   259   "Fin m < Fin n \<longleftrightarrow> m < n"
```
```   260   "q \<le> \<infinity> \<longleftrightarrow> True"
```
```   261   "Fin m < \<infinity> \<longleftrightarrow> True"
```
```   262   "\<infinity> \<le> Fin n \<longleftrightarrow> False"
```
```   263   "\<infinity> < q \<longleftrightarrow> False"
```
```   264   by simp_all
```
```   265
```
```   266 instance by default
```
```   267   (auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits)
```
```   268
```
```   269 end
```
```   270
```
```   271 instance inat :: pordered_comm_semiring
```
```   272 proof
```
```   273   fix a b c :: inat
```
```   274   assume "a \<le> b" and "0 \<le> c"
```
```   275   thus "c * a \<le> c * b"
```
```   276     unfolding times_inat_def less_eq_inat_def zero_inat_def
```
```   277     by (simp split: inat.splits)
```
```   278 qed
```
```   279
```
```   280 lemma inat_ord_number [simp]:
```
```   281   "(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n"
```
```   282   "(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
```
```   283   by (simp_all add: number_of_inat_def)
```
```   284
```
```   285 lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n"
```
```   286   by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
```
```   287
```
```   288 lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0"
```
```   289   by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
```
```   290
```
```   291 lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R"
```
```   292   by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
```
```   293
```
```   294 lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R"
```
```   295   by simp
```
```   296
```
```   297 lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)"
```
```   298   by (simp add: zero_inat_def less_inat_def split: inat.splits)
```
```   299
```
```   300 lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0"
```
```   301   by (simp add: zero_inat_def less_inat_def split: inat.splits)
```
```   302
```
```   303 lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m"
```
```   304   by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
```
```   305
```
```   306 lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m"
```
```   307   by (simp add: iSuc_def less_inat_def split: inat.splits)
```
```   308
```
```   309 lemma ile_iSuc [simp]: "n \<le> iSuc n"
```
```   310   by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
```
```   311
```
```   312 lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
```
```   313   by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits)
```
```   314
```
```   315 lemma i0_iless_iSuc [simp]: "0 < iSuc n"
```
```   316   by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits)
```
```   317
```
```   318 lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n"
```
```   319   by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits)
```
```   320
```
```   321 lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n"
```
```   322   by (cases n) auto
```
```   323
```
```   324 lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n"
```
```   325   by (auto simp add: iSuc_def less_inat_def split: inat.splits)
```
```   326
```
```   327 lemma min_inat_simps [simp]:
```
```   328   "min (Fin m) (Fin n) = Fin (min m n)"
```
```   329   "min q 0 = 0"
```
```   330   "min 0 q = 0"
```
```   331   "min q \<infinity> = q"
```
```   332   "min \<infinity> q = q"
```
```   333   by (auto simp add: min_def)
```
```   334
```
```   335 lemma max_inat_simps [simp]:
```
```   336   "max (Fin m) (Fin n) = Fin (max m n)"
```
```   337   "max q 0 = q"
```
```   338   "max 0 q = q"
```
```   339   "max q \<infinity> = \<infinity>"
```
```   340   "max \<infinity> q = \<infinity>"
```
```   341   by (simp_all add: max_def)
```
```   342
```
```   343 lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k"
```
```   344   by (cases n) simp_all
```
```   345
```
```   346 lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k"
```
```   347   by (cases n) simp_all
```
```   348
```
```   349 lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
```
```   350 apply (induct_tac k)
```
```   351  apply (simp (no_asm) only: Fin_0)
```
```   352  apply (fast intro: le_less_trans [OF i0_lb])
```
```   353 apply (erule exE)
```
```   354 apply (drule spec)
```
```   355 apply (erule exE)
```
```   356 apply (drule ileI1)
```
```   357 apply (rule iSuc_Fin [THEN subst])
```
```   358 apply (rule exI)
```
```   359 apply (erule (1) le_less_trans)
```
```   360 done
```
```   361
```
```   362 instantiation inat :: "{bot, top}"
```
```   363 begin
```
```   364
```
```   365 definition bot_inat :: inat where
```
```   366   "bot_inat = 0"
```
```   367
```
```   368 definition top_inat :: inat where
```
```   369   "top_inat = \<infinity>"
```
```   370
```
```   371 instance proof
```
```   372 qed (simp_all add: bot_inat_def top_inat_def)
```
```   373
```
```   374 end
```
```   375
```
```   376
```
```   377 subsection {* Well-ordering *}
```
```   378
```
```   379 lemma less_FinE:
```
```   380   "[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P"
```
```   381 by (induct n) auto
```
```   382
```
```   383 lemma less_InftyE:
```
```   384   "[| n < Infty; !!k. n = Fin k ==> P |] ==> P"
```
```   385 by (induct n) auto
```
```   386
```
```   387 lemma inat_less_induct:
```
```   388   assumes prem: "!!n. \<forall>m::inat. m < n --> P m ==> P n" shows "P n"
```
```   389 proof -
```
```   390   have P_Fin: "!!k. P (Fin k)"
```
```   391     apply (rule nat_less_induct)
```
```   392     apply (rule prem, clarify)
```
```   393     apply (erule less_FinE, simp)
```
```   394     done
```
```   395   show ?thesis
```
```   396   proof (induct n)
```
```   397     fix nat
```
```   398     show "P (Fin nat)" by (rule P_Fin)
```
```   399   next
```
```   400     show "P Infty"
```
```   401       apply (rule prem, clarify)
```
```   402       apply (erule less_InftyE)
```
```   403       apply (simp add: P_Fin)
```
```   404       done
```
```   405   qed
```
```   406 qed
```
```   407
```
```   408 instance inat :: wellorder
```
```   409 proof
```
```   410   fix P and n
```
```   411   assume hyp: "(\<And>n\<Colon>inat. (\<And>m\<Colon>inat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
```
```   412   show "P n" by (blast intro: inat_less_induct hyp)
```
```   413 qed
```
```   414
```
```   415
```
```   416 subsection {* Traditional theorem names *}
```
```   417
```
```   418 lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def
```
```   419   plus_inat_def less_eq_inat_def less_inat_def
```
```   420
```
```   421 lemmas inat_splits = inat.splits
```
```   422
```
```   423 end
```