src/HOL/Library/Extended_Nat.thy
 author huffman Thu Nov 17 07:15:30 2011 +0100 (2011-11-17) changeset 45539 787a1a097465 parent 44890 22f665a2e91c child 45775 6c340de26a0d permissions -rw-r--r--
remove redundant simp rules plus_enat_0
```     1 (*  Title:      HOL/Library/Extended_Nat.thy
```
```     2     Author:     David von Oheimb, TU Muenchen;  Florian Haftmann, TU Muenchen
```
```     3     Contributions: David Trachtenherz, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 header {* Extended natural numbers (i.e. with infinity) *}
```
```     7
```
```     8 theory Extended_Nat
```
```     9 imports Main
```
```    10 begin
```
```    11
```
```    12 class infinity =
```
```    13   fixes infinity :: "'a"
```
```    14
```
```    15 notation (xsymbols)
```
```    16   infinity  ("\<infinity>")
```
```    17
```
```    18 notation (HTML output)
```
```    19   infinity  ("\<infinity>")
```
```    20
```
```    21 subsection {* Type definition *}
```
```    22
```
```    23 text {*
```
```    24   We extend the standard natural numbers by a special value indicating
```
```    25   infinity.
```
```    26 *}
```
```    27
```
```    28 typedef (open) enat = "UNIV :: nat option set" ..
```
```    29
```
```    30 definition enat :: "nat \<Rightarrow> enat" where
```
```    31   "enat n = Abs_enat (Some n)"
```
```    32
```
```    33 instantiation enat :: infinity
```
```    34 begin
```
```    35   definition "\<infinity> = Abs_enat None"
```
```    36   instance proof qed
```
```    37 end
```
```    38
```
```    39 rep_datatype enat "\<infinity> :: enat"
```
```    40 proof -
```
```    41   fix P i assume "\<And>j. P (enat j)" "P \<infinity>"
```
```    42   then show "P i"
```
```    43   proof induct
```
```    44     case (Abs_enat y) then show ?case
```
```    45       by (cases y rule: option.exhaust)
```
```    46          (auto simp: enat_def infinity_enat_def)
```
```    47   qed
```
```    48 qed (auto simp add: enat_def infinity_enat_def Abs_enat_inject)
```
```    49
```
```    50 declare [[coercion "enat::nat\<Rightarrow>enat"]]
```
```    51
```
```    52 lemma not_infinity_eq [iff]: "(x \<noteq> \<infinity>) = (EX i. x = enat i)"
```
```    53   by (cases x) auto
```
```    54
```
```    55 lemma not_enat_eq [iff]: "(ALL y. x ~= enat y) = (x = \<infinity>)"
```
```    56   by (cases x) auto
```
```    57
```
```    58 primrec the_enat :: "enat \<Rightarrow> nat"
```
```    59   where "the_enat (enat n) = n"
```
```    60
```
```    61 subsection {* Constructors and numbers *}
```
```    62
```
```    63 instantiation enat :: "{zero, one, number}"
```
```    64 begin
```
```    65
```
```    66 definition
```
```    67   "0 = enat 0"
```
```    68
```
```    69 definition
```
```    70   [code_unfold]: "1 = enat 1"
```
```    71
```
```    72 definition
```
```    73   [code_unfold, code del]: "number_of k = enat (number_of k)"
```
```    74
```
```    75 instance ..
```
```    76
```
```    77 end
```
```    78
```
```    79 definition eSuc :: "enat \<Rightarrow> enat" where
```
```    80   "eSuc i = (case i of enat n \<Rightarrow> enat (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
```
```    81
```
```    82 lemma enat_0: "enat 0 = 0"
```
```    83   by (simp add: zero_enat_def)
```
```    84
```
```    85 lemma enat_1: "enat 1 = 1"
```
```    86   by (simp add: one_enat_def)
```
```    87
```
```    88 lemma enat_number: "enat (number_of k) = number_of k"
```
```    89   by (simp add: number_of_enat_def)
```
```    90
```
```    91 lemma one_eSuc: "1 = eSuc 0"
```
```    92   by (simp add: zero_enat_def one_enat_def eSuc_def)
```
```    93
```
```    94 lemma infinity_ne_i0 [simp]: "(\<infinity>::enat) \<noteq> 0"
```
```    95   by (simp add: zero_enat_def)
```
```    96
```
```    97 lemma i0_ne_infinity [simp]: "0 \<noteq> (\<infinity>::enat)"
```
```    98   by (simp add: zero_enat_def)
```
```    99
```
```   100 lemma zero_enat_eq [simp]:
```
```   101   "number_of k = (0\<Colon>enat) \<longleftrightarrow> number_of k = (0\<Colon>nat)"
```
```   102   "(0\<Colon>enat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)"
```
```   103   unfolding zero_enat_def number_of_enat_def by simp_all
```
```   104
```
```   105 lemma one_enat_eq [simp]:
```
```   106   "number_of k = (1\<Colon>enat) \<longleftrightarrow> number_of k = (1\<Colon>nat)"
```
```   107   "(1\<Colon>enat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)"
```
```   108   unfolding one_enat_def number_of_enat_def by simp_all
```
```   109
```
```   110 lemma zero_one_enat_neq [simp]:
```
```   111   "\<not> 0 = (1\<Colon>enat)"
```
```   112   "\<not> 1 = (0\<Colon>enat)"
```
```   113   unfolding zero_enat_def one_enat_def by simp_all
```
```   114
```
```   115 lemma infinity_ne_i1 [simp]: "(\<infinity>::enat) \<noteq> 1"
```
```   116   by (simp add: one_enat_def)
```
```   117
```
```   118 lemma i1_ne_infinity [simp]: "1 \<noteq> (\<infinity>::enat)"
```
```   119   by (simp add: one_enat_def)
```
```   120
```
```   121 lemma infinity_ne_number [simp]: "(\<infinity>::enat) \<noteq> number_of k"
```
```   122   by (simp add: number_of_enat_def)
```
```   123
```
```   124 lemma number_ne_infinity [simp]: "number_of k \<noteq> (\<infinity>::enat)"
```
```   125   by (simp add: number_of_enat_def)
```
```   126
```
```   127 lemma eSuc_enat: "eSuc (enat n) = enat (Suc n)"
```
```   128   by (simp add: eSuc_def)
```
```   129
```
```   130 lemma eSuc_number_of: "eSuc (number_of k) = enat (Suc (number_of k))"
```
```   131   by (simp add: eSuc_enat number_of_enat_def)
```
```   132
```
```   133 lemma eSuc_infinity [simp]: "eSuc \<infinity> = \<infinity>"
```
```   134   by (simp add: eSuc_def)
```
```   135
```
```   136 lemma eSuc_ne_0 [simp]: "eSuc n \<noteq> 0"
```
```   137   by (simp add: eSuc_def zero_enat_def split: enat.splits)
```
```   138
```
```   139 lemma zero_ne_eSuc [simp]: "0 \<noteq> eSuc n"
```
```   140   by (rule eSuc_ne_0 [symmetric])
```
```   141
```
```   142 lemma eSuc_inject [simp]: "eSuc m = eSuc n \<longleftrightarrow> m = n"
```
```   143   by (simp add: eSuc_def split: enat.splits)
```
```   144
```
```   145 lemma number_of_enat_inject [simp]:
```
```   146   "(number_of k \<Colon> enat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l"
```
```   147   by (simp add: number_of_enat_def)
```
```   148
```
```   149
```
```   150 subsection {* Addition *}
```
```   151
```
```   152 instantiation enat :: comm_monoid_add
```
```   153 begin
```
```   154
```
```   155 definition [nitpick_simp]:
```
```   156   "m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | enat m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | enat n \<Rightarrow> enat (m + n)))"
```
```   157
```
```   158 lemma plus_enat_simps [simp, code]:
```
```   159   fixes q :: enat
```
```   160   shows "enat m + enat n = enat (m + n)"
```
```   161     and "\<infinity> + q = \<infinity>"
```
```   162     and "q + \<infinity> = \<infinity>"
```
```   163   by (simp_all add: plus_enat_def split: enat.splits)
```
```   164
```
```   165 instance proof
```
```   166   fix n m q :: enat
```
```   167   show "n + m + q = n + (m + q)"
```
```   168     by (cases n, auto, cases m, auto, cases q, auto)
```
```   169   show "n + m = m + n"
```
```   170     by (cases n, auto, cases m, auto)
```
```   171   show "0 + n = n"
```
```   172     by (cases n) (simp_all add: zero_enat_def)
```
```   173 qed
```
```   174
```
```   175 end
```
```   176
```
```   177 lemma plus_enat_number [simp]:
```
```   178   "(number_of k \<Colon> enat) + number_of l = (if k < Int.Pls then number_of l
```
```   179     else if l < Int.Pls then number_of k else number_of (k + l))"
```
```   180   unfolding number_of_enat_def plus_enat_simps nat_arith(1) if_distrib [symmetric, of _ enat] ..
```
```   181
```
```   182 lemma eSuc_number [simp]:
```
```   183   "eSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))"
```
```   184   unfolding eSuc_number_of
```
```   185   unfolding one_enat_def number_of_enat_def Suc_nat_number_of if_distrib [symmetric] ..
```
```   186
```
```   187 lemma eSuc_plus_1:
```
```   188   "eSuc n = n + 1"
```
```   189   by (cases n) (simp_all add: eSuc_enat one_enat_def)
```
```   190
```
```   191 lemma plus_1_eSuc:
```
```   192   "1 + q = eSuc q"
```
```   193   "q + 1 = eSuc q"
```
```   194   by (simp_all add: eSuc_plus_1 add_ac)
```
```   195
```
```   196 lemma iadd_Suc: "eSuc m + n = eSuc (m + n)"
```
```   197   by (simp_all add: eSuc_plus_1 add_ac)
```
```   198
```
```   199 lemma iadd_Suc_right: "m + eSuc n = eSuc (m + n)"
```
```   200   by (simp only: add_commute[of m] iadd_Suc)
```
```   201
```
```   202 lemma iadd_is_0: "(m + n = (0::enat)) = (m = 0 \<and> n = 0)"
```
```   203   by (cases m, cases n, simp_all add: zero_enat_def)
```
```   204
```
```   205 subsection {* Multiplication *}
```
```   206
```
```   207 instantiation enat :: comm_semiring_1
```
```   208 begin
```
```   209
```
```   210 definition times_enat_def [nitpick_simp]:
```
```   211   "m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | enat m \<Rightarrow>
```
```   212     (case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | enat n \<Rightarrow> enat (m * n)))"
```
```   213
```
```   214 lemma times_enat_simps [simp, code]:
```
```   215   "enat m * enat n = enat (m * n)"
```
```   216   "\<infinity> * \<infinity> = (\<infinity>::enat)"
```
```   217   "\<infinity> * enat n = (if n = 0 then 0 else \<infinity>)"
```
```   218   "enat m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
```
```   219   unfolding times_enat_def zero_enat_def
```
```   220   by (simp_all split: enat.split)
```
```   221
```
```   222 instance proof
```
```   223   fix a b c :: enat
```
```   224   show "(a * b) * c = a * (b * c)"
```
```   225     unfolding times_enat_def zero_enat_def
```
```   226     by (simp split: enat.split)
```
```   227   show "a * b = b * a"
```
```   228     unfolding times_enat_def zero_enat_def
```
```   229     by (simp split: enat.split)
```
```   230   show "1 * a = a"
```
```   231     unfolding times_enat_def zero_enat_def one_enat_def
```
```   232     by (simp split: enat.split)
```
```   233   show "(a + b) * c = a * c + b * c"
```
```   234     unfolding times_enat_def zero_enat_def
```
```   235     by (simp split: enat.split add: left_distrib)
```
```   236   show "0 * a = 0"
```
```   237     unfolding times_enat_def zero_enat_def
```
```   238     by (simp split: enat.split)
```
```   239   show "a * 0 = 0"
```
```   240     unfolding times_enat_def zero_enat_def
```
```   241     by (simp split: enat.split)
```
```   242   show "(0::enat) \<noteq> 1"
```
```   243     unfolding zero_enat_def one_enat_def
```
```   244     by simp
```
```   245 qed
```
```   246
```
```   247 end
```
```   248
```
```   249 lemma mult_eSuc: "eSuc m * n = n + m * n"
```
```   250   unfolding eSuc_plus_1 by (simp add: algebra_simps)
```
```   251
```
```   252 lemma mult_eSuc_right: "m * eSuc n = m + m * n"
```
```   253   unfolding eSuc_plus_1 by (simp add: algebra_simps)
```
```   254
```
```   255 lemma of_nat_eq_enat: "of_nat n = enat n"
```
```   256   apply (induct n)
```
```   257   apply (simp add: enat_0)
```
```   258   apply (simp add: plus_1_eSuc eSuc_enat)
```
```   259   done
```
```   260
```
```   261 instance enat :: number_semiring
```
```   262 proof
```
```   263   fix n show "number_of (int n) = (of_nat n :: enat)"
```
```   264     unfolding number_of_enat_def number_of_int of_nat_id of_nat_eq_enat ..
```
```   265 qed
```
```   266
```
```   267 instance enat :: semiring_char_0 proof
```
```   268   have "inj enat" by (rule injI) simp
```
```   269   then show "inj (\<lambda>n. of_nat n :: enat)" by (simp add: of_nat_eq_enat)
```
```   270 qed
```
```   271
```
```   272 lemma imult_is_0 [simp]: "((m::enat) * n = 0) = (m = 0 \<or> n = 0)"
```
```   273   by (auto simp add: times_enat_def zero_enat_def split: enat.split)
```
```   274
```
```   275 lemma imult_is_infinity: "((a::enat) * b = \<infinity>) = (a = \<infinity> \<and> b \<noteq> 0 \<or> b = \<infinity> \<and> a \<noteq> 0)"
```
```   276   by (auto simp add: times_enat_def zero_enat_def split: enat.split)
```
```   277
```
```   278
```
```   279 subsection {* Subtraction *}
```
```   280
```
```   281 instantiation enat :: minus
```
```   282 begin
```
```   283
```
```   284 definition diff_enat_def:
```
```   285 "a - b = (case a of (enat x) \<Rightarrow> (case b of (enat y) \<Rightarrow> enat (x - y) | \<infinity> \<Rightarrow> 0)
```
```   286           | \<infinity> \<Rightarrow> \<infinity>)"
```
```   287
```
```   288 instance ..
```
```   289
```
```   290 end
```
```   291
```
```   292 lemma idiff_enat_enat [simp,code]: "enat a - enat b = enat (a - b)"
```
```   293   by (simp add: diff_enat_def)
```
```   294
```
```   295 lemma idiff_infinity [simp,code]: "\<infinity> - n = (\<infinity>::enat)"
```
```   296   by (simp add: diff_enat_def)
```
```   297
```
```   298 lemma idiff_infinity_right [simp,code]: "enat a - \<infinity> = 0"
```
```   299   by (simp add: diff_enat_def)
```
```   300
```
```   301 lemma idiff_0 [simp]: "(0::enat) - n = 0"
```
```   302   by (cases n, simp_all add: zero_enat_def)
```
```   303
```
```   304 lemmas idiff_enat_0 [simp] = idiff_0 [unfolded zero_enat_def]
```
```   305
```
```   306 lemma idiff_0_right [simp]: "(n::enat) - 0 = n"
```
```   307   by (cases n) (simp_all add: zero_enat_def)
```
```   308
```
```   309 lemmas idiff_enat_0_right [simp] = idiff_0_right [unfolded zero_enat_def]
```
```   310
```
```   311 lemma idiff_self [simp]: "n \<noteq> \<infinity> \<Longrightarrow> (n::enat) - n = 0"
```
```   312   by (auto simp: zero_enat_def)
```
```   313
```
```   314 lemma eSuc_minus_eSuc [simp]: "eSuc n - eSuc m = n - m"
```
```   315   by (simp add: eSuc_def split: enat.split)
```
```   316
```
```   317 lemma eSuc_minus_1 [simp]: "eSuc n - 1 = n"
```
```   318   by (simp add: one_enat_def eSuc_enat[symmetric] zero_enat_def[symmetric])
```
```   319
```
```   320 (*lemmas idiff_self_eq_0_enat = idiff_self_eq_0[unfolded zero_enat_def]*)
```
```   321
```
```   322 subsection {* Ordering *}
```
```   323
```
```   324 instantiation enat :: linordered_ab_semigroup_add
```
```   325 begin
```
```   326
```
```   327 definition [nitpick_simp]:
```
```   328   "m \<le> n = (case n of enat n1 \<Rightarrow> (case m of enat m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
```
```   329     | \<infinity> \<Rightarrow> True)"
```
```   330
```
```   331 definition [nitpick_simp]:
```
```   332   "m < n = (case m of enat m1 \<Rightarrow> (case n of enat n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
```
```   333     | \<infinity> \<Rightarrow> False)"
```
```   334
```
```   335 lemma enat_ord_simps [simp]:
```
```   336   "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
```
```   337   "enat m < enat n \<longleftrightarrow> m < n"
```
```   338   "q \<le> (\<infinity>::enat)"
```
```   339   "q < (\<infinity>::enat) \<longleftrightarrow> q \<noteq> \<infinity>"
```
```   340   "(\<infinity>::enat) \<le> q \<longleftrightarrow> q = \<infinity>"
```
```   341   "(\<infinity>::enat) < q \<longleftrightarrow> False"
```
```   342   by (simp_all add: less_eq_enat_def less_enat_def split: enat.splits)
```
```   343
```
```   344 lemma enat_ord_code [code]:
```
```   345   "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
```
```   346   "enat m < enat n \<longleftrightarrow> m < n"
```
```   347   "q \<le> (\<infinity>::enat) \<longleftrightarrow> True"
```
```   348   "enat m < \<infinity> \<longleftrightarrow> True"
```
```   349   "\<infinity> \<le> enat n \<longleftrightarrow> False"
```
```   350   "(\<infinity>::enat) < q \<longleftrightarrow> False"
```
```   351   by simp_all
```
```   352
```
```   353 instance by default
```
```   354   (auto simp add: less_eq_enat_def less_enat_def plus_enat_def split: enat.splits)
```
```   355
```
```   356 end
```
```   357
```
```   358 instance enat :: ordered_comm_semiring
```
```   359 proof
```
```   360   fix a b c :: enat
```
```   361   assume "a \<le> b" and "0 \<le> c"
```
```   362   thus "c * a \<le> c * b"
```
```   363     unfolding times_enat_def less_eq_enat_def zero_enat_def
```
```   364     by (simp split: enat.splits)
```
```   365 qed
```
```   366
```
```   367 lemma enat_ord_number [simp]:
```
```   368   "(number_of m \<Colon> enat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n"
```
```   369   "(number_of m \<Colon> enat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
```
```   370   by (simp_all add: number_of_enat_def)
```
```   371
```
```   372 lemma i0_lb [simp]: "(0\<Colon>enat) \<le> n"
```
```   373   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
```
```   374
```
```   375 lemma ile0_eq [simp]: "n \<le> (0\<Colon>enat) \<longleftrightarrow> n = 0"
```
```   376   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
```
```   377
```
```   378 lemma infinity_ileE [elim!]: "\<infinity> \<le> enat m \<Longrightarrow> R"
```
```   379   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
```
```   380
```
```   381 lemma infinity_ilessE [elim!]: "\<infinity> < enat m \<Longrightarrow> R"
```
```   382   by simp
```
```   383
```
```   384 lemma not_iless0 [simp]: "\<not> n < (0\<Colon>enat)"
```
```   385   by (simp add: zero_enat_def less_enat_def split: enat.splits)
```
```   386
```
```   387 lemma i0_less [simp]: "(0\<Colon>enat) < n \<longleftrightarrow> n \<noteq> 0"
```
```   388   by (simp add: zero_enat_def less_enat_def split: enat.splits)
```
```   389
```
```   390 lemma eSuc_ile_mono [simp]: "eSuc n \<le> eSuc m \<longleftrightarrow> n \<le> m"
```
```   391   by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
```
```   392
```
```   393 lemma eSuc_mono [simp]: "eSuc n < eSuc m \<longleftrightarrow> n < m"
```
```   394   by (simp add: eSuc_def less_enat_def split: enat.splits)
```
```   395
```
```   396 lemma ile_eSuc [simp]: "n \<le> eSuc n"
```
```   397   by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
```
```   398
```
```   399 lemma not_eSuc_ilei0 [simp]: "\<not> eSuc n \<le> 0"
```
```   400   by (simp add: zero_enat_def eSuc_def less_eq_enat_def split: enat.splits)
```
```   401
```
```   402 lemma i0_iless_eSuc [simp]: "0 < eSuc n"
```
```   403   by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.splits)
```
```   404
```
```   405 lemma iless_eSuc0[simp]: "(n < eSuc 0) = (n = 0)"
```
```   406   by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.split)
```
```   407
```
```   408 lemma ileI1: "m < n \<Longrightarrow> eSuc m \<le> n"
```
```   409   by (simp add: eSuc_def less_eq_enat_def less_enat_def split: enat.splits)
```
```   410
```
```   411 lemma Suc_ile_eq: "enat (Suc m) \<le> n \<longleftrightarrow> enat m < n"
```
```   412   by (cases n) auto
```
```   413
```
```   414 lemma iless_Suc_eq [simp]: "enat m < eSuc n \<longleftrightarrow> enat m \<le> n"
```
```   415   by (auto simp add: eSuc_def less_enat_def split: enat.splits)
```
```   416
```
```   417 lemma imult_infinity: "(0::enat) < n \<Longrightarrow> \<infinity> * n = \<infinity>"
```
```   418   by (simp add: zero_enat_def less_enat_def split: enat.splits)
```
```   419
```
```   420 lemma imult_infinity_right: "(0::enat) < n \<Longrightarrow> n * \<infinity> = \<infinity>"
```
```   421   by (simp add: zero_enat_def less_enat_def split: enat.splits)
```
```   422
```
```   423 lemma enat_0_less_mult_iff: "(0 < (m::enat) * n) = (0 < m \<and> 0 < n)"
```
```   424   by (simp only: i0_less imult_is_0, simp)
```
```   425
```
```   426 lemma mono_eSuc: "mono eSuc"
```
```   427   by (simp add: mono_def)
```
```   428
```
```   429
```
```   430 lemma min_enat_simps [simp]:
```
```   431   "min (enat m) (enat n) = enat (min m n)"
```
```   432   "min q 0 = 0"
```
```   433   "min 0 q = 0"
```
```   434   "min q (\<infinity>::enat) = q"
```
```   435   "min (\<infinity>::enat) q = q"
```
```   436   by (auto simp add: min_def)
```
```   437
```
```   438 lemma max_enat_simps [simp]:
```
```   439   "max (enat m) (enat n) = enat (max m n)"
```
```   440   "max q 0 = q"
```
```   441   "max 0 q = q"
```
```   442   "max q \<infinity> = (\<infinity>::enat)"
```
```   443   "max \<infinity> q = (\<infinity>::enat)"
```
```   444   by (simp_all add: max_def)
```
```   445
```
```   446 lemma enat_ile: "n \<le> enat m \<Longrightarrow> \<exists>k. n = enat k"
```
```   447   by (cases n) simp_all
```
```   448
```
```   449 lemma enat_iless: "n < enat m \<Longrightarrow> \<exists>k. n = enat k"
```
```   450   by (cases n) simp_all
```
```   451
```
```   452 lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. enat k < Y j"
```
```   453 apply (induct_tac k)
```
```   454  apply (simp (no_asm) only: enat_0)
```
```   455  apply (fast intro: le_less_trans [OF i0_lb])
```
```   456 apply (erule exE)
```
```   457 apply (drule spec)
```
```   458 apply (erule exE)
```
```   459 apply (drule ileI1)
```
```   460 apply (rule eSuc_enat [THEN subst])
```
```   461 apply (rule exI)
```
```   462 apply (erule (1) le_less_trans)
```
```   463 done
```
```   464
```
```   465 instantiation enat :: "{bot, top}"
```
```   466 begin
```
```   467
```
```   468 definition bot_enat :: enat where
```
```   469   "bot_enat = 0"
```
```   470
```
```   471 definition top_enat :: enat where
```
```   472   "top_enat = \<infinity>"
```
```   473
```
```   474 instance proof
```
```   475 qed (simp_all add: bot_enat_def top_enat_def)
```
```   476
```
```   477 end
```
```   478
```
```   479 lemma finite_enat_bounded:
```
```   480   assumes le_fin: "\<And>y. y \<in> A \<Longrightarrow> y \<le> enat n"
```
```   481   shows "finite A"
```
```   482 proof (rule finite_subset)
```
```   483   show "finite (enat ` {..n})" by blast
```
```   484
```
```   485   have "A \<subseteq> {..enat n}" using le_fin by fastforce
```
```   486   also have "\<dots> \<subseteq> enat ` {..n}"
```
```   487     by (rule subsetI) (case_tac x, auto)
```
```   488   finally show "A \<subseteq> enat ` {..n}" .
```
```   489 qed
```
```   490
```
```   491
```
```   492 subsection {* Well-ordering *}
```
```   493
```
```   494 lemma less_enatE:
```
```   495   "[| n < enat m; !!k. n = enat k ==> k < m ==> P |] ==> P"
```
```   496 by (induct n) auto
```
```   497
```
```   498 lemma less_infinityE:
```
```   499   "[| n < \<infinity>; !!k. n = enat k ==> P |] ==> P"
```
```   500 by (induct n) auto
```
```   501
```
```   502 lemma enat_less_induct:
```
```   503   assumes prem: "!!n. \<forall>m::enat. m < n --> P m ==> P n" shows "P n"
```
```   504 proof -
```
```   505   have P_enat: "!!k. P (enat k)"
```
```   506     apply (rule nat_less_induct)
```
```   507     apply (rule prem, clarify)
```
```   508     apply (erule less_enatE, simp)
```
```   509     done
```
```   510   show ?thesis
```
```   511   proof (induct n)
```
```   512     fix nat
```
```   513     show "P (enat nat)" by (rule P_enat)
```
```   514   next
```
```   515     show "P \<infinity>"
```
```   516       apply (rule prem, clarify)
```
```   517       apply (erule less_infinityE)
```
```   518       apply (simp add: P_enat)
```
```   519       done
```
```   520   qed
```
```   521 qed
```
```   522
```
```   523 instance enat :: wellorder
```
```   524 proof
```
```   525   fix P and n
```
```   526   assume hyp: "(\<And>n\<Colon>enat. (\<And>m\<Colon>enat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
```
```   527   show "P n" by (blast intro: enat_less_induct hyp)
```
```   528 qed
```
```   529
```
```   530 subsection {* Complete Lattice *}
```
```   531
```
```   532 instantiation enat :: complete_lattice
```
```   533 begin
```
```   534
```
```   535 definition inf_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
```
```   536   "inf_enat \<equiv> min"
```
```   537
```
```   538 definition sup_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
```
```   539   "sup_enat \<equiv> max"
```
```   540
```
```   541 definition Inf_enat :: "enat set \<Rightarrow> enat" where
```
```   542   "Inf_enat A \<equiv> if A = {} then \<infinity> else (LEAST x. x \<in> A)"
```
```   543
```
```   544 definition Sup_enat :: "enat set \<Rightarrow> enat" where
```
```   545   "Sup_enat A \<equiv> if A = {} then 0
```
```   546     else if finite A then Max A
```
```   547                      else \<infinity>"
```
```   548 instance proof
```
```   549   fix x :: "enat" and A :: "enat set"
```
```   550   { assume "x \<in> A" then show "Inf A \<le> x"
```
```   551       unfolding Inf_enat_def by (auto intro: Least_le) }
```
```   552   { assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A"
```
```   553       unfolding Inf_enat_def
```
```   554       by (cases "A = {}") (auto intro: LeastI2_ex) }
```
```   555   { assume "x \<in> A" then show "x \<le> Sup A"
```
```   556       unfolding Sup_enat_def by (cases "finite A") auto }
```
```   557   { assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x"
```
```   558       unfolding Sup_enat_def using finite_enat_bounded by auto }
```
```   559 qed (simp_all add: inf_enat_def sup_enat_def)
```
```   560 end
```
```   561
```
```   562 instance enat :: complete_linorder ..
```
```   563
```
```   564 subsection {* Traditional theorem names *}
```
```   565
```
```   566 lemmas enat_defs = zero_enat_def one_enat_def number_of_enat_def eSuc_def
```
```   567   plus_enat_def less_eq_enat_def less_enat_def
```
```   568
```
```   569 end
```