src/HOL/Library/Extended_Nat.thy
 author haftmann Fri Mar 22 19:18:08 2019 +0000 (3 months ago) changeset 69946 494934c30f38 parent 69861 62e47f06d22c permissions -rw-r--r--
improved code equations taken over from AFP
```     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 section \<open>Extended natural numbers (i.e. with infinity)\<close>
```
```     7
```
```     8 theory Extended_Nat
```
```     9 imports Main Countable Order_Continuity
```
```    10 begin
```
```    11
```
```    12 class infinity =
```
```    13   fixes infinity :: "'a"  ("\<infinity>")
```
```    14
```
```    15 context
```
```    16   fixes f :: "nat \<Rightarrow> 'a::{canonically_ordered_monoid_add, linorder_topology, complete_linorder}"
```
```    17 begin
```
```    18
```
```    19 lemma sums_SUP[simp, intro]: "f sums (SUP n. \<Sum>i<n. f i)"
```
```    20   unfolding sums_def by (intro LIMSEQ_SUP monoI sum_mono2 zero_le) auto
```
```    21
```
```    22 lemma suminf_eq_SUP: "suminf f = (SUP n. \<Sum>i<n. f i)"
```
```    23   using sums_SUP by (rule sums_unique[symmetric])
```
```    24
```
```    25 end
```
```    26
```
```    27 subsection \<open>Type definition\<close>
```
```    28
```
```    29 text \<open>
```
```    30   We extend the standard natural numbers by a special value indicating
```
```    31   infinity.
```
```    32 \<close>
```
```    33
```
```    34 typedef enat = "UNIV :: nat option set" ..
```
```    35
```
```    36 text \<open>TODO: introduce enat as coinductive datatype, enat is just \<^const>\<open>of_nat\<close>\<close>
```
```    37
```
```    38 definition enat :: "nat \<Rightarrow> enat" where
```
```    39   "enat n = Abs_enat (Some n)"
```
```    40
```
```    41 instantiation enat :: infinity
```
```    42 begin
```
```    43
```
```    44 definition "\<infinity> = Abs_enat None"
```
```    45 instance ..
```
```    46
```
```    47 end
```
```    48
```
```    49 instance enat :: countable
```
```    50 proof
```
```    51   show "\<exists>to_nat::enat \<Rightarrow> nat. inj to_nat"
```
```    52     by (rule exI[of _ "to_nat \<circ> Rep_enat"]) (simp add: inj_on_def Rep_enat_inject)
```
```    53 qed
```
```    54
```
```    55 old_rep_datatype enat "\<infinity> :: enat"
```
```    56 proof -
```
```    57   fix P i assume "\<And>j. P (enat j)" "P \<infinity>"
```
```    58   then show "P i"
```
```    59   proof induct
```
```    60     case (Abs_enat y) then show ?case
```
```    61       by (cases y rule: option.exhaust)
```
```    62          (auto simp: enat_def infinity_enat_def)
```
```    63   qed
```
```    64 qed (auto simp add: enat_def infinity_enat_def Abs_enat_inject)
```
```    65
```
```    66 declare [[coercion "enat::nat\<Rightarrow>enat"]]
```
```    67
```
```    68 lemmas enat2_cases = enat.exhaust[case_product enat.exhaust]
```
```    69 lemmas enat3_cases = enat.exhaust[case_product enat.exhaust enat.exhaust]
```
```    70
```
```    71 lemma not_infinity_eq [iff]: "(x \<noteq> \<infinity>) = (\<exists>i. x = enat i)"
```
```    72   by (cases x) auto
```
```    73
```
```    74 lemma not_enat_eq [iff]: "(\<forall>y. x \<noteq> enat y) = (x = \<infinity>)"
```
```    75   by (cases x) auto
```
```    76
```
```    77 lemma enat_ex_split: "(\<exists>c::enat. P c) \<longleftrightarrow> P \<infinity> \<or> (\<exists>c::nat. P c)"
```
```    78   by (metis enat.exhaust)
```
```    79
```
```    80 primrec the_enat :: "enat \<Rightarrow> nat"
```
```    81   where "the_enat (enat n) = n"
```
```    82
```
```    83
```
```    84 subsection \<open>Constructors and numbers\<close>
```
```    85
```
```    86 instantiation enat :: zero_neq_one
```
```    87 begin
```
```    88
```
```    89 definition
```
```    90   "0 = enat 0"
```
```    91
```
```    92 definition
```
```    93   "1 = enat 1"
```
```    94
```
```    95 instance
```
```    96   proof qed (simp add: zero_enat_def one_enat_def)
```
```    97
```
```    98 end
```
```    99
```
```   100 definition eSuc :: "enat \<Rightarrow> enat" where
```
```   101   "eSuc i = (case i of enat n \<Rightarrow> enat (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
```
```   102
```
```   103 lemma enat_0 [code_post]: "enat 0 = 0"
```
```   104   by (simp add: zero_enat_def)
```
```   105
```
```   106 lemma enat_1 [code_post]: "enat 1 = 1"
```
```   107   by (simp add: one_enat_def)
```
```   108
```
```   109 lemma enat_0_iff: "enat x = 0 \<longleftrightarrow> x = 0" "0 = enat x \<longleftrightarrow> x = 0"
```
```   110   by (auto simp add: zero_enat_def)
```
```   111
```
```   112 lemma enat_1_iff: "enat x = 1 \<longleftrightarrow> x = 1" "1 = enat x \<longleftrightarrow> x = 1"
```
```   113   by (auto simp add: one_enat_def)
```
```   114
```
```   115 lemma one_eSuc: "1 = eSuc 0"
```
```   116   by (simp add: zero_enat_def one_enat_def eSuc_def)
```
```   117
```
```   118 lemma infinity_ne_i0 [simp]: "(\<infinity>::enat) \<noteq> 0"
```
```   119   by (simp add: zero_enat_def)
```
```   120
```
```   121 lemma i0_ne_infinity [simp]: "0 \<noteq> (\<infinity>::enat)"
```
```   122   by (simp add: zero_enat_def)
```
```   123
```
```   124 lemma zero_one_enat_neq:
```
```   125   "\<not> 0 = (1::enat)"
```
```   126   "\<not> 1 = (0::enat)"
```
```   127   unfolding zero_enat_def one_enat_def by simp_all
```
```   128
```
```   129 lemma infinity_ne_i1 [simp]: "(\<infinity>::enat) \<noteq> 1"
```
```   130   by (simp add: one_enat_def)
```
```   131
```
```   132 lemma i1_ne_infinity [simp]: "1 \<noteq> (\<infinity>::enat)"
```
```   133   by (simp add: one_enat_def)
```
```   134
```
```   135 lemma eSuc_enat: "eSuc (enat n) = enat (Suc n)"
```
```   136   by (simp add: eSuc_def)
```
```   137
```
```   138 lemma eSuc_infinity [simp]: "eSuc \<infinity> = \<infinity>"
```
```   139   by (simp add: eSuc_def)
```
```   140
```
```   141 lemma eSuc_ne_0 [simp]: "eSuc n \<noteq> 0"
```
```   142   by (simp add: eSuc_def zero_enat_def split: enat.splits)
```
```   143
```
```   144 lemma zero_ne_eSuc [simp]: "0 \<noteq> eSuc n"
```
```   145   by (rule eSuc_ne_0 [symmetric])
```
```   146
```
```   147 lemma eSuc_inject [simp]: "eSuc m = eSuc n \<longleftrightarrow> m = n"
```
```   148   by (simp add: eSuc_def split: enat.splits)
```
```   149
```
```   150 lemma eSuc_enat_iff: "eSuc x = enat y \<longleftrightarrow> (\<exists>n. y = Suc n \<and> x = enat n)"
```
```   151   by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
```
```   152
```
```   153 lemma enat_eSuc_iff: "enat y = eSuc x \<longleftrightarrow> (\<exists>n. y = Suc n \<and> enat n = x)"
```
```   154   by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
```
```   155
```
```   156 subsection \<open>Addition\<close>
```
```   157
```
```   158 instantiation enat :: comm_monoid_add
```
```   159 begin
```
```   160
```
```   161 definition [nitpick_simp]:
```
```   162   "m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | enat m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | enat n \<Rightarrow> enat (m + n)))"
```
```   163
```
```   164 lemma plus_enat_simps [simp, code]:
```
```   165   fixes q :: enat
```
```   166   shows "enat m + enat n = enat (m + n)"
```
```   167     and "\<infinity> + q = \<infinity>"
```
```   168     and "q + \<infinity> = \<infinity>"
```
```   169   by (simp_all add: plus_enat_def split: enat.splits)
```
```   170
```
```   171 instance
```
```   172 proof
```
```   173   fix n m q :: enat
```
```   174   show "n + m + q = n + (m + q)"
```
```   175     by (cases n m q rule: enat3_cases) auto
```
```   176   show "n + m = m + n"
```
```   177     by (cases n m rule: enat2_cases) auto
```
```   178   show "0 + n = n"
```
```   179     by (cases n) (simp_all add: zero_enat_def)
```
```   180 qed
```
```   181
```
```   182 end
```
```   183
```
```   184 lemma eSuc_plus_1:
```
```   185   "eSuc n = n + 1"
```
```   186   by (cases n) (simp_all add: eSuc_enat one_enat_def)
```
```   187
```
```   188 lemma plus_1_eSuc:
```
```   189   "1 + q = eSuc q"
```
```   190   "q + 1 = eSuc q"
```
```   191   by (simp_all add: eSuc_plus_1 ac_simps)
```
```   192
```
```   193 lemma iadd_Suc: "eSuc m + n = eSuc (m + n)"
```
```   194   by (simp_all add: eSuc_plus_1 ac_simps)
```
```   195
```
```   196 lemma iadd_Suc_right: "m + eSuc n = eSuc (m + n)"
```
```   197   by (simp only: add.commute[of m] iadd_Suc)
```
```   198
```
```   199 subsection \<open>Multiplication\<close>
```
```   200
```
```   201 instantiation enat :: "{comm_semiring_1, semiring_no_zero_divisors}"
```
```   202 begin
```
```   203
```
```   204 definition times_enat_def [nitpick_simp]:
```
```   205   "m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | enat m \<Rightarrow>
```
```   206     (case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | enat n \<Rightarrow> enat (m * n)))"
```
```   207
```
```   208 lemma times_enat_simps [simp, code]:
```
```   209   "enat m * enat n = enat (m * n)"
```
```   210   "\<infinity> * \<infinity> = (\<infinity>::enat)"
```
```   211   "\<infinity> * enat n = (if n = 0 then 0 else \<infinity>)"
```
```   212   "enat m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
```
```   213   unfolding times_enat_def zero_enat_def
```
```   214   by (simp_all split: enat.split)
```
```   215
```
```   216 instance
```
```   217 proof
```
```   218   fix a b c :: enat
```
```   219   show "(a * b) * c = a * (b * c)"
```
```   220     unfolding times_enat_def zero_enat_def
```
```   221     by (simp split: enat.split)
```
```   222   show comm: "a * b = b * a"
```
```   223     unfolding times_enat_def zero_enat_def
```
```   224     by (simp split: enat.split)
```
```   225   show "1 * a = a"
```
```   226     unfolding times_enat_def zero_enat_def one_enat_def
```
```   227     by (simp split: enat.split)
```
```   228   show distr: "(a + b) * c = a * c + b * c"
```
```   229     unfolding times_enat_def zero_enat_def
```
```   230     by (simp split: enat.split add: distrib_right)
```
```   231   show "0 * a = 0"
```
```   232     unfolding times_enat_def zero_enat_def
```
```   233     by (simp split: enat.split)
```
```   234   show "a * 0 = 0"
```
```   235     unfolding times_enat_def zero_enat_def
```
```   236     by (simp split: enat.split)
```
```   237   show "a * (b + c) = a * b + a * c"
```
```   238     by (cases a b c rule: enat3_cases) (auto simp: times_enat_def zero_enat_def distrib_left)
```
```   239   show "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
```
```   240     by (cases a b rule: enat2_cases) (auto simp: times_enat_def zero_enat_def)
```
```   241 qed
```
```   242
```
```   243 end
```
```   244
```
```   245 lemma mult_eSuc: "eSuc m * n = n + m * n"
```
```   246   unfolding eSuc_plus_1 by (simp add: algebra_simps)
```
```   247
```
```   248 lemma mult_eSuc_right: "m * eSuc n = m + m * n"
```
```   249   unfolding eSuc_plus_1 by (simp add: algebra_simps)
```
```   250
```
```   251 lemma of_nat_eq_enat: "of_nat n = enat n"
```
```   252   apply (induct n)
```
```   253   apply (simp add: enat_0)
```
```   254   apply (simp add: plus_1_eSuc eSuc_enat)
```
```   255   done
```
```   256
```
```   257 instance enat :: semiring_char_0
```
```   258 proof
```
```   259   have "inj enat" by (rule injI) simp
```
```   260   then show "inj (\<lambda>n. of_nat n :: enat)" by (simp add: of_nat_eq_enat)
```
```   261 qed
```
```   262
```
```   263 lemma imult_is_infinity: "((a::enat) * b = \<infinity>) = (a = \<infinity> \<and> b \<noteq> 0 \<or> b = \<infinity> \<and> a \<noteq> 0)"
```
```   264   by (auto simp add: times_enat_def zero_enat_def split: enat.split)
```
```   265
```
```   266 subsection \<open>Numerals\<close>
```
```   267
```
```   268 lemma numeral_eq_enat:
```
```   269   "numeral k = enat (numeral k)"
```
```   270   using of_nat_eq_enat [of "numeral k"] by simp
```
```   271
```
```   272 lemma enat_numeral [code_abbrev]:
```
```   273   "enat (numeral k) = numeral k"
```
```   274   using numeral_eq_enat ..
```
```   275
```
```   276 lemma infinity_ne_numeral [simp]: "(\<infinity>::enat) \<noteq> numeral k"
```
```   277   by (simp add: numeral_eq_enat)
```
```   278
```
```   279 lemma numeral_ne_infinity [simp]: "numeral k \<noteq> (\<infinity>::enat)"
```
```   280   by (simp add: numeral_eq_enat)
```
```   281
```
```   282 lemma eSuc_numeral [simp]: "eSuc (numeral k) = numeral (k + Num.One)"
```
```   283   by (simp only: eSuc_plus_1 numeral_plus_one)
```
```   284
```
```   285 subsection \<open>Subtraction\<close>
```
```   286
```
```   287 instantiation enat :: minus
```
```   288 begin
```
```   289
```
```   290 definition diff_enat_def:
```
```   291 "a - b = (case a of (enat x) \<Rightarrow> (case b of (enat y) \<Rightarrow> enat (x - y) | \<infinity> \<Rightarrow> 0)
```
```   292           | \<infinity> \<Rightarrow> \<infinity>)"
```
```   293
```
```   294 instance ..
```
```   295
```
```   296 end
```
```   297
```
```   298 lemma idiff_enat_enat [simp, code]: "enat a - enat b = enat (a - b)"
```
```   299   by (simp add: diff_enat_def)
```
```   300
```
```   301 lemma idiff_infinity [simp, code]: "\<infinity> - n = (\<infinity>::enat)"
```
```   302   by (simp add: diff_enat_def)
```
```   303
```
```   304 lemma idiff_infinity_right [simp, code]: "enat a - \<infinity> = 0"
```
```   305   by (simp add: diff_enat_def)
```
```   306
```
```   307 lemma idiff_0 [simp]: "(0::enat) - n = 0"
```
```   308   by (cases n, simp_all add: zero_enat_def)
```
```   309
```
```   310 lemmas idiff_enat_0 [simp] = idiff_0 [unfolded zero_enat_def]
```
```   311
```
```   312 lemma idiff_0_right [simp]: "(n::enat) - 0 = n"
```
```   313   by (cases n) (simp_all add: zero_enat_def)
```
```   314
```
```   315 lemmas idiff_enat_0_right [simp] = idiff_0_right [unfolded zero_enat_def]
```
```   316
```
```   317 lemma idiff_self [simp]: "n \<noteq> \<infinity> \<Longrightarrow> (n::enat) - n = 0"
```
```   318   by (auto simp: zero_enat_def)
```
```   319
```
```   320 lemma eSuc_minus_eSuc [simp]: "eSuc n - eSuc m = n - m"
```
```   321   by (simp add: eSuc_def split: enat.split)
```
```   322
```
```   323 lemma eSuc_minus_1 [simp]: "eSuc n - 1 = n"
```
```   324   by (simp add: one_enat_def flip: eSuc_enat zero_enat_def)
```
```   325
```
```   326 (*lemmas idiff_self_eq_0_enat = idiff_self_eq_0[unfolded zero_enat_def]*)
```
```   327
```
```   328 subsection \<open>Ordering\<close>
```
```   329
```
```   330 instantiation enat :: linordered_ab_semigroup_add
```
```   331 begin
```
```   332
```
```   333 definition [nitpick_simp]:
```
```   334   "m \<le> n = (case n of enat n1 \<Rightarrow> (case m of enat m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
```
```   335     | \<infinity> \<Rightarrow> True)"
```
```   336
```
```   337 definition [nitpick_simp]:
```
```   338   "m < n = (case m of enat m1 \<Rightarrow> (case n of enat n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
```
```   339     | \<infinity> \<Rightarrow> False)"
```
```   340
```
```   341 lemma enat_ord_simps [simp]:
```
```   342   "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
```
```   343   "enat m < enat n \<longleftrightarrow> m < n"
```
```   344   "q \<le> (\<infinity>::enat)"
```
```   345   "q < (\<infinity>::enat) \<longleftrightarrow> q \<noteq> \<infinity>"
```
```   346   "(\<infinity>::enat) \<le> q \<longleftrightarrow> q = \<infinity>"
```
```   347   "(\<infinity>::enat) < q \<longleftrightarrow> False"
```
```   348   by (simp_all add: less_eq_enat_def less_enat_def split: enat.splits)
```
```   349
```
```   350 lemma numeral_le_enat_iff[simp]:
```
```   351   shows "numeral m \<le> enat n \<longleftrightarrow> numeral m \<le> n"
```
```   352 by (auto simp: numeral_eq_enat)
```
```   353
```
```   354 lemma numeral_less_enat_iff[simp]:
```
```   355   shows "numeral m < enat n \<longleftrightarrow> numeral m < n"
```
```   356 by (auto simp: numeral_eq_enat)
```
```   357
```
```   358 lemma enat_ord_code [code]:
```
```   359   "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
```
```   360   "enat m < enat n \<longleftrightarrow> m < n"
```
```   361   "q \<le> (\<infinity>::enat) \<longleftrightarrow> True"
```
```   362   "enat m < \<infinity> \<longleftrightarrow> True"
```
```   363   "\<infinity> \<le> enat n \<longleftrightarrow> False"
```
```   364   "(\<infinity>::enat) < q \<longleftrightarrow> False"
```
```   365   by simp_all
```
```   366
```
```   367 instance
```
```   368   by standard (auto simp add: less_eq_enat_def less_enat_def plus_enat_def split: enat.splits)
```
```   369
```
```   370 end
```
```   371
```
```   372 instance enat :: dioid
```
```   373 proof
```
```   374   fix a b :: enat show "(a \<le> b) = (\<exists>c. b = a + c)"
```
```   375     by (cases a b rule: enat2_cases) (auto simp: le_iff_add enat_ex_split)
```
```   376 qed
```
```   377
```
```   378 instance enat :: "{linordered_nonzero_semiring, strict_ordered_comm_monoid_add}"
```
```   379 proof
```
```   380   fix a b c :: enat
```
```   381   show "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow>c * a \<le> c * b"
```
```   382     unfolding times_enat_def less_eq_enat_def zero_enat_def
```
```   383     by (simp split: enat.splits)
```
```   384   show "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" for a b c d :: enat
```
```   385     by (cases a b c d rule: enat2_cases[case_product enat2_cases]) auto
```
```   386   show "a < b \<Longrightarrow> a + 1 < b + 1"
```
```   387     by (metis add_right_mono eSuc_minus_1 eSuc_plus_1 less_le)
```
```   388 qed (simp add: zero_enat_def one_enat_def)
```
```   389
```
```   390 (* BH: These equations are already proven generally for any type in
```
```   391 class linordered_semidom. However, enat is not in that class because
```
```   392 it does not have the cancellation property. Would it be worthwhile to
```
```   393 a generalize linordered_semidom to a new class that includes enat? *)
```
```   394
```
```   395 lemma add_diff_assoc_enat: "z \<le> y \<Longrightarrow> x + (y - z) = x + y - (z::enat)"
```
```   396 by(cases x)(auto simp add: diff_enat_def split: enat.split)
```
```   397
```
```   398 lemma enat_ord_number [simp]:
```
```   399   "(numeral m :: enat) \<le> numeral n \<longleftrightarrow> (numeral m :: nat) \<le> numeral n"
```
```   400   "(numeral m :: enat) < numeral n \<longleftrightarrow> (numeral m :: nat) < numeral n"
```
```   401   by (simp_all add: numeral_eq_enat)
```
```   402
```
```   403 lemma infinity_ileE [elim!]: "\<infinity> \<le> enat m \<Longrightarrow> R"
```
```   404   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
```
```   405
```
```   406 lemma infinity_ilessE [elim!]: "\<infinity> < enat m \<Longrightarrow> R"
```
```   407   by simp
```
```   408
```
```   409 lemma eSuc_ile_mono [simp]: "eSuc n \<le> eSuc m \<longleftrightarrow> n \<le> m"
```
```   410   by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
```
```   411
```
```   412 lemma eSuc_mono [simp]: "eSuc n < eSuc m \<longleftrightarrow> n < m"
```
```   413   by (simp add: eSuc_def less_enat_def split: enat.splits)
```
```   414
```
```   415 lemma ile_eSuc [simp]: "n \<le> eSuc n"
```
```   416   by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
```
```   417
```
```   418 lemma not_eSuc_ilei0 [simp]: "\<not> eSuc n \<le> 0"
```
```   419   by (simp add: zero_enat_def eSuc_def less_eq_enat_def split: enat.splits)
```
```   420
```
```   421 lemma i0_iless_eSuc [simp]: "0 < eSuc n"
```
```   422   by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.splits)
```
```   423
```
```   424 lemma iless_eSuc0[simp]: "(n < eSuc 0) = (n = 0)"
```
```   425   by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.split)
```
```   426
```
```   427 lemma ileI1: "m < n \<Longrightarrow> eSuc m \<le> n"
```
```   428   by (simp add: eSuc_def less_eq_enat_def less_enat_def split: enat.splits)
```
```   429
```
```   430 lemma Suc_ile_eq: "enat (Suc m) \<le> n \<longleftrightarrow> enat m < n"
```
```   431   by (cases n) auto
```
```   432
```
```   433 lemma iless_Suc_eq [simp]: "enat m < eSuc n \<longleftrightarrow> enat m \<le> n"
```
```   434   by (auto simp add: eSuc_def less_enat_def split: enat.splits)
```
```   435
```
```   436 lemma imult_infinity: "(0::enat) < n \<Longrightarrow> \<infinity> * n = \<infinity>"
```
```   437   by (simp add: zero_enat_def less_enat_def split: enat.splits)
```
```   438
```
```   439 lemma imult_infinity_right: "(0::enat) < n \<Longrightarrow> n * \<infinity> = \<infinity>"
```
```   440   by (simp add: zero_enat_def less_enat_def split: enat.splits)
```
```   441
```
```   442 lemma enat_0_less_mult_iff: "(0 < (m::enat) * n) = (0 < m \<and> 0 < n)"
```
```   443   by (simp only: zero_less_iff_neq_zero mult_eq_0_iff, simp)
```
```   444
```
```   445 lemma mono_eSuc: "mono eSuc"
```
```   446   by (simp add: mono_def)
```
```   447
```
```   448 lemma min_enat_simps [simp]:
```
```   449   "min (enat m) (enat n) = enat (min m n)"
```
```   450   "min q 0 = 0"
```
```   451   "min 0 q = 0"
```
```   452   "min q (\<infinity>::enat) = q"
```
```   453   "min (\<infinity>::enat) q = q"
```
```   454   by (auto simp add: min_def)
```
```   455
```
```   456 lemma max_enat_simps [simp]:
```
```   457   "max (enat m) (enat n) = enat (max m n)"
```
```   458   "max q 0 = q"
```
```   459   "max 0 q = q"
```
```   460   "max q \<infinity> = (\<infinity>::enat)"
```
```   461   "max \<infinity> q = (\<infinity>::enat)"
```
```   462   by (simp_all add: max_def)
```
```   463
```
```   464 lemma enat_ile: "n \<le> enat m \<Longrightarrow> \<exists>k. n = enat k"
```
```   465   by (cases n) simp_all
```
```   466
```
```   467 lemma enat_iless: "n < enat m \<Longrightarrow> \<exists>k. n = enat k"
```
```   468   by (cases n) simp_all
```
```   469
```
```   470 lemma iadd_le_enat_iff:
```
```   471   "x + y \<le> enat n \<longleftrightarrow> (\<exists>y' x'. x = enat x' \<and> y = enat y' \<and> x' + y' \<le> n)"
```
```   472 by(cases x y rule: enat.exhaust[case_product enat.exhaust]) simp_all
```
```   473
```
```   474 lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j \<Longrightarrow> \<exists>j. enat k < Y j"
```
```   475 apply (induct_tac k)
```
```   476  apply (simp (no_asm) only: enat_0)
```
```   477  apply (fast intro: le_less_trans [OF zero_le])
```
```   478 apply (erule exE)
```
```   479 apply (drule spec)
```
```   480 apply (erule exE)
```
```   481 apply (drule ileI1)
```
```   482 apply (rule eSuc_enat [THEN subst])
```
```   483 apply (rule exI)
```
```   484 apply (erule (1) le_less_trans)
```
```   485 done
```
```   486
```
```   487 lemma eSuc_max: "eSuc (max x y) = max (eSuc x) (eSuc y)"
```
```   488   by (simp add: eSuc_def split: enat.split)
```
```   489
```
```   490 lemma eSuc_Max:
```
```   491   assumes "finite A" "A \<noteq> {}"
```
```   492   shows "eSuc (Max A) = Max (eSuc ` A)"
```
```   493 using assms proof induction
```
```   494   case (insert x A)
```
```   495   thus ?case by(cases "A = {}")(simp_all add: eSuc_max)
```
```   496 qed simp
```
```   497
```
```   498 instantiation enat :: "{order_bot, order_top}"
```
```   499 begin
```
```   500
```
```   501 definition bot_enat :: enat where "bot_enat = 0"
```
```   502 definition top_enat :: enat where "top_enat = \<infinity>"
```
```   503
```
```   504 instance
```
```   505   by standard (simp_all add: bot_enat_def top_enat_def)
```
```   506
```
```   507 end
```
```   508
```
```   509 lemma finite_enat_bounded:
```
```   510   assumes le_fin: "\<And>y. y \<in> A \<Longrightarrow> y \<le> enat n"
```
```   511   shows "finite A"
```
```   512 proof (rule finite_subset)
```
```   513   show "finite (enat ` {..n})" by blast
```
```   514   have "A \<subseteq> {..enat n}" using le_fin by fastforce
```
```   515   also have "\<dots> \<subseteq> enat ` {..n}"
```
```   516     apply (rule subsetI)
```
```   517     subgoal for x by (cases x) auto
```
```   518     done
```
```   519   finally show "A \<subseteq> enat ` {..n}" .
```
```   520 qed
```
```   521
```
```   522
```
```   523 subsection \<open>Cancellation simprocs\<close>
```
```   524
```
```   525 lemma add_diff_cancel_enat[simp]: "x \<noteq> \<infinity> \<Longrightarrow> x + y - x = (y::enat)"
```
```   526 by (metis add.commute add.right_neutral add_diff_assoc_enat idiff_self order_refl)
```
```   527
```
```   528 lemma enat_add_left_cancel: "a + b = a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b = c"
```
```   529   unfolding plus_enat_def by (simp split: enat.split)
```
```   530
```
```   531 lemma enat_add_left_cancel_le: "a + b \<le> a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b \<le> c"
```
```   532   unfolding plus_enat_def by (simp split: enat.split)
```
```   533
```
```   534 lemma enat_add_left_cancel_less: "a + b < a + c \<longleftrightarrow> a \<noteq> (\<infinity>::enat) \<and> b < c"
```
```   535   unfolding plus_enat_def by (simp split: enat.split)
```
```   536
```
```   537 lemma plus_eq_infty_iff_enat: "(m::enat) + n = \<infinity> \<longleftrightarrow> m=\<infinity> \<or> n=\<infinity>"
```
```   538 using enat_add_left_cancel by fastforce
```
```   539
```
```   540 ML \<open>
```
```   541 structure Cancel_Enat_Common =
```
```   542 struct
```
```   543   (* copied from src/HOL/Tools/nat_numeral_simprocs.ML *)
```
```   544   fun find_first_t _    _ []         = raise TERM("find_first_t", [])
```
```   545     | find_first_t past u (t::terms) =
```
```   546           if u aconv t then (rev past @ terms)
```
```   547           else find_first_t (t::past) u terms
```
```   548
```
```   549   fun dest_summing (Const (\<^const_name>\<open>Groups.plus\<close>, _) \$ t \$ u, ts) =
```
```   550         dest_summing (t, dest_summing (u, ts))
```
```   551     | dest_summing (t, ts) = t :: ts
```
```   552
```
```   553   val mk_sum = Arith_Data.long_mk_sum
```
```   554   fun dest_sum t = dest_summing (t, [])
```
```   555   val find_first = find_first_t []
```
```   556   val trans_tac = Numeral_Simprocs.trans_tac
```
```   557   val norm_ss =
```
```   558     simpset_of (put_simpset HOL_basic_ss \<^context>
```
```   559       addsimps @{thms ac_simps add_0_left add_0_right})
```
```   560   fun norm_tac ctxt = ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
```
```   561   fun simplify_meta_eq ctxt cancel_th th =
```
```   562     Arith_Data.simplify_meta_eq [] ctxt
```
```   563       ([th, cancel_th] MRS trans)
```
```   564   fun mk_eq (a, b) = HOLogic.mk_Trueprop (HOLogic.mk_eq (a, b))
```
```   565 end
```
```   566
```
```   567 structure Eq_Enat_Cancel = ExtractCommonTermFun
```
```   568 (open Cancel_Enat_Common
```
```   569   val mk_bal = HOLogic.mk_eq
```
```   570   val dest_bal = HOLogic.dest_bin \<^const_name>\<open>HOL.eq\<close> \<^typ>\<open>enat\<close>
```
```   571   fun simp_conv _ _ = SOME @{thm enat_add_left_cancel}
```
```   572 )
```
```   573
```
```   574 structure Le_Enat_Cancel = ExtractCommonTermFun
```
```   575 (open Cancel_Enat_Common
```
```   576   val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Orderings.less_eq\<close>
```
```   577   val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less_eq\<close> \<^typ>\<open>enat\<close>
```
```   578   fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_le}
```
```   579 )
```
```   580
```
```   581 structure Less_Enat_Cancel = ExtractCommonTermFun
```
```   582 (open Cancel_Enat_Common
```
```   583   val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Orderings.less\<close>
```
```   584   val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less\<close> \<^typ>\<open>enat\<close>
```
```   585   fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_less}
```
```   586 )
```
```   587 \<close>
```
```   588
```
```   589 simproc_setup enat_eq_cancel
```
```   590   ("(l::enat) + m = n" | "(l::enat) = m + n") =
```
```   591   \<open>fn phi => fn ctxt => fn ct => Eq_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
```
```   592
```
```   593 simproc_setup enat_le_cancel
```
```   594   ("(l::enat) + m \<le> n" | "(l::enat) \<le> m + n") =
```
```   595   \<open>fn phi => fn ctxt => fn ct => Le_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
```
```   596
```
```   597 simproc_setup enat_less_cancel
```
```   598   ("(l::enat) + m < n" | "(l::enat) < m + n") =
```
```   599   \<open>fn phi => fn ctxt => fn ct => Less_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
```
```   600
```
```   601 text \<open>TODO: add regression tests for these simprocs\<close>
```
```   602
```
```   603 text \<open>TODO: add simprocs for combining and cancelling numerals\<close>
```
```   604
```
```   605 subsection \<open>Well-ordering\<close>
```
```   606
```
```   607 lemma less_enatE:
```
```   608   "[| n < enat m; !!k. n = enat k ==> k < m ==> P |] ==> P"
```
```   609 by (induct n) auto
```
```   610
```
```   611 lemma less_infinityE:
```
```   612   "[| n < \<infinity>; !!k. n = enat k ==> P |] ==> P"
```
```   613 by (induct n) auto
```
```   614
```
```   615 lemma enat_less_induct:
```
```   616   assumes prem: "\<And>n. \<forall>m::enat. m < n \<longrightarrow> P m \<Longrightarrow> P n" shows "P n"
```
```   617 proof -
```
```   618   have P_enat: "\<And>k. P (enat k)"
```
```   619     apply (rule nat_less_induct)
```
```   620     apply (rule prem, clarify)
```
```   621     apply (erule less_enatE, simp)
```
```   622     done
```
```   623   show ?thesis
```
```   624   proof (induct n)
```
```   625     fix nat
```
```   626     show "P (enat nat)" by (rule P_enat)
```
```   627   next
```
```   628     show "P \<infinity>"
```
```   629       apply (rule prem, clarify)
```
```   630       apply (erule less_infinityE)
```
```   631       apply (simp add: P_enat)
```
```   632       done
```
```   633   qed
```
```   634 qed
```
```   635
```
```   636 instance enat :: wellorder
```
```   637 proof
```
```   638   fix P and n
```
```   639   assume hyp: "(\<And>n::enat. (\<And>m::enat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
```
```   640   show "P n" by (blast intro: enat_less_induct hyp)
```
```   641 qed
```
```   642
```
```   643 subsection \<open>Complete Lattice\<close>
```
```   644
```
```   645 instantiation enat :: complete_lattice
```
```   646 begin
```
```   647
```
```   648 definition inf_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
```
```   649   "inf_enat = min"
```
```   650
```
```   651 definition sup_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
```
```   652   "sup_enat = max"
```
```   653
```
```   654 definition Inf_enat :: "enat set \<Rightarrow> enat" where
```
```   655   "Inf_enat A = (if A = {} then \<infinity> else (LEAST x. x \<in> A))"
```
```   656
```
```   657 definition Sup_enat :: "enat set \<Rightarrow> enat" where
```
```   658   "Sup_enat A = (if A = {} then 0 else if finite A then Max A else \<infinity>)"
```
```   659 instance
```
```   660 proof
```
```   661   fix x :: "enat" and A :: "enat set"
```
```   662   { assume "x \<in> A" then show "Inf A \<le> x"
```
```   663       unfolding Inf_enat_def by (auto intro: Least_le) }
```
```   664   { assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A"
```
```   665       unfolding Inf_enat_def
```
```   666       by (cases "A = {}") (auto intro: LeastI2_ex) }
```
```   667   { assume "x \<in> A" then show "x \<le> Sup A"
```
```   668       unfolding Sup_enat_def by (cases "finite A") auto }
```
```   669   { assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x"
```
```   670       unfolding Sup_enat_def using finite_enat_bounded by auto }
```
```   671 qed (simp_all add:
```
```   672  inf_enat_def sup_enat_def bot_enat_def top_enat_def Inf_enat_def Sup_enat_def)
```
```   673 end
```
```   674
```
```   675 instance enat :: complete_linorder ..
```
```   676
```
```   677 lemma eSuc_Sup: "A \<noteq> {} \<Longrightarrow> eSuc (Sup A) = Sup (eSuc ` A)"
```
```   678   by(auto simp add: Sup_enat_def eSuc_Max inj_on_def dest: finite_imageD)
```
```   679
```
```   680 lemma sup_continuous_eSuc: "sup_continuous f \<Longrightarrow> sup_continuous (\<lambda>x. eSuc (f x))"
```
```   681   using eSuc_Sup [of "_ ` UNIV"] by (auto simp: sup_continuous_def image_comp)
```
```   682
```
```   683
```
```   684 subsection \<open>Traditional theorem names\<close>
```
```   685
```
```   686 lemmas enat_defs = zero_enat_def one_enat_def eSuc_def
```
```   687   plus_enat_def less_eq_enat_def less_enat_def
```
```   688
```
```   689 lemma iadd_is_0: "(m + n = (0::enat)) = (m = 0 \<and> n = 0)"
```
```   690   by (rule add_eq_0_iff_both_eq_0)
```
```   691
```
```   692 lemma i0_lb : "(0::enat) \<le> n"
```
```   693   by (rule zero_le)
```
```   694
```
```   695 lemma ile0_eq: "n \<le> (0::enat) \<longleftrightarrow> n = 0"
```
```   696   by (rule le_zero_eq)
```
```   697
```
```   698 lemma not_iless0: "\<not> n < (0::enat)"
```
```   699   by (rule not_less_zero)
```
```   700
```
```   701 lemma i0_less[simp]: "(0::enat) < n \<longleftrightarrow> n \<noteq> 0"
```
```   702   by (rule zero_less_iff_neq_zero)
```
```   703
```
```   704 lemma imult_is_0: "((m::enat) * n = 0) = (m = 0 \<or> n = 0)"
```
```   705   by (rule mult_eq_0_iff)
```
```   706
```
```   707 end
```