src/HOL/Library/Extended_Nat.thy
author paulson <lp15@cam.ac.uk>
Mon Feb 22 14:37:56 2016 +0000 (2016-02-22)
changeset 62379 340738057c8c
parent 62378 85ed00c1fe7c
child 64267 b9a1486e79be
permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
hoelzl@43919
     1
(*  Title:      HOL/Library/Extended_Nat.thy
haftmann@27110
     2
    Author:     David von Oheimb, TU Muenchen;  Florian Haftmann, TU Muenchen
nipkow@41853
     3
    Contributions: David Trachtenherz, TU Muenchen
oheimb@11351
     4
*)
oheimb@11351
     5
wenzelm@60500
     6
section \<open>Extended natural numbers (i.e. with infinity)\<close>
oheimb@11351
     7
hoelzl@43919
     8
theory Extended_Nat
hoelzl@60636
     9
imports Main Countable Order_Continuity
nipkow@15131
    10
begin
oheimb@11351
    11
hoelzl@43921
    12
class infinity =
wenzelm@61384
    13
  fixes infinity :: "'a"  ("\<infinity>")
hoelzl@43921
    14
hoelzl@62378
    15
context
hoelzl@62378
    16
  fixes f :: "nat \<Rightarrow> 'a::{canonically_ordered_monoid_add, linorder_topology, complete_linorder}"
hoelzl@62378
    17
begin
hoelzl@62378
    18
hoelzl@62378
    19
lemma sums_SUP[simp, intro]: "f sums (SUP n. \<Sum>i<n. f i)"
hoelzl@62378
    20
  unfolding sums_def by (intro LIMSEQ_SUP monoI setsum_mono2 zero_le) auto
hoelzl@62378
    21
hoelzl@62378
    22
lemma suminf_eq_SUP: "suminf f = (SUP n. \<Sum>i<n. f i)"
hoelzl@62378
    23
  using sums_SUP by (rule sums_unique[symmetric])
hoelzl@62378
    24
hoelzl@62378
    25
end
hoelzl@62378
    26
wenzelm@60500
    27
subsection \<open>Type definition\<close>
oheimb@11351
    28
wenzelm@60500
    29
text \<open>
wenzelm@11355
    30
  We extend the standard natural numbers by a special value indicating
haftmann@27110
    31
  infinity.
wenzelm@60500
    32
\<close>
oheimb@11351
    33
wenzelm@49834
    34
typedef enat = "UNIV :: nat option set" ..
hoelzl@54415
    35
wenzelm@60500
    36
text \<open>TODO: introduce enat as coinductive datatype, enat is just @{const of_nat}\<close>
hoelzl@54415
    37
hoelzl@43924
    38
definition enat :: "nat \<Rightarrow> enat" where
hoelzl@43924
    39
  "enat n = Abs_enat (Some n)"
hoelzl@62374
    40
hoelzl@43921
    41
instantiation enat :: infinity
hoelzl@43921
    42
begin
wenzelm@60679
    43
wenzelm@60679
    44
definition "\<infinity> = Abs_enat None"
wenzelm@60679
    45
instance ..
wenzelm@60679
    46
hoelzl@43921
    47
end
hoelzl@54415
    48
hoelzl@54415
    49
instance enat :: countable
hoelzl@54415
    50
proof
hoelzl@54415
    51
  show "\<exists>to_nat::enat \<Rightarrow> nat. inj to_nat"
hoelzl@54415
    52
    by (rule exI[of _ "to_nat \<circ> Rep_enat"]) (simp add: inj_on_def Rep_enat_inject)
hoelzl@54415
    53
qed
hoelzl@62374
    54
blanchet@58306
    55
old_rep_datatype enat "\<infinity> :: enat"
hoelzl@43921
    56
proof -
hoelzl@43924
    57
  fix P i assume "\<And>j. P (enat j)" "P \<infinity>"
hoelzl@43921
    58
  then show "P i"
hoelzl@43921
    59
  proof induct
hoelzl@43921
    60
    case (Abs_enat y) then show ?case
hoelzl@43921
    61
      by (cases y rule: option.exhaust)
hoelzl@43924
    62
         (auto simp: enat_def infinity_enat_def)
hoelzl@43921
    63
  qed
hoelzl@43924
    64
qed (auto simp add: enat_def infinity_enat_def Abs_enat_inject)
wenzelm@19736
    65
hoelzl@43924
    66
declare [[coercion "enat::nat\<Rightarrow>enat"]]
wenzelm@19736
    67
noschinl@45934
    68
lemmas enat2_cases = enat.exhaust[case_product enat.exhaust]
noschinl@45934
    69
lemmas enat3_cases = enat.exhaust[case_product enat.exhaust enat.exhaust]
noschinl@45934
    70
hoelzl@54416
    71
lemma not_infinity_eq [iff]: "(x \<noteq> \<infinity>) = (\<exists>i. x = enat i)"
huffman@44019
    72
  by (cases x) auto
nipkow@31084
    73
hoelzl@54416
    74
lemma not_enat_eq [iff]: "(\<forall>y. x \<noteq> enat y) = (x = \<infinity>)"
huffman@44019
    75
  by (cases x) auto
nipkow@31077
    76
hoelzl@62376
    77
lemma enat_ex_split: "(\<exists>c::enat. P c) \<longleftrightarrow> P \<infinity> \<or> (\<exists>c::nat. P c)"
hoelzl@62376
    78
  by (metis enat.exhaust)
hoelzl@62376
    79
hoelzl@43924
    80
primrec the_enat :: "enat \<Rightarrow> nat"
huffman@44019
    81
  where "the_enat (enat n) = n"
nipkow@41855
    82
huffman@47108
    83
wenzelm@60500
    84
subsection \<open>Constructors and numbers\<close>
haftmann@27110
    85
hoelzl@62378
    86
instantiation enat :: zero_neq_one
haftmann@25594
    87
begin
haftmann@25594
    88
haftmann@25594
    89
definition
hoelzl@43924
    90
  "0 = enat 0"
haftmann@25594
    91
haftmann@25594
    92
definition
huffman@47108
    93
  "1 = enat 1"
oheimb@11351
    94
hoelzl@62378
    95
instance
hoelzl@62378
    96
  proof qed (simp add: zero_enat_def one_enat_def)
haftmann@25594
    97
haftmann@25594
    98
end
haftmann@25594
    99
huffman@44019
   100
definition eSuc :: "enat \<Rightarrow> enat" where
huffman@44019
   101
  "eSuc i = (case i of enat n \<Rightarrow> enat (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
oheimb@11351
   102
huffman@47108
   103
lemma enat_0 [code_post]: "enat 0 = 0"
hoelzl@43919
   104
  by (simp add: zero_enat_def)
haftmann@27110
   105
huffman@47108
   106
lemma enat_1 [code_post]: "enat 1 = 1"
hoelzl@43919
   107
  by (simp add: one_enat_def)
haftmann@27110
   108
hoelzl@54416
   109
lemma enat_0_iff: "enat x = 0 \<longleftrightarrow> x = 0" "0 = enat x \<longleftrightarrow> x = 0"
hoelzl@54416
   110
  by (auto simp add: zero_enat_def)
hoelzl@54416
   111
hoelzl@54416
   112
lemma enat_1_iff: "enat x = 1 \<longleftrightarrow> x = 1" "1 = enat x \<longleftrightarrow> x = 1"
hoelzl@54416
   113
  by (auto simp add: one_enat_def)
hoelzl@54416
   114
huffman@44019
   115
lemma one_eSuc: "1 = eSuc 0"
huffman@44019
   116
  by (simp add: zero_enat_def one_enat_def eSuc_def)
oheimb@11351
   117
huffman@44019
   118
lemma infinity_ne_i0 [simp]: "(\<infinity>::enat) \<noteq> 0"
hoelzl@43919
   119
  by (simp add: zero_enat_def)
oheimb@11351
   120
huffman@44019
   121
lemma i0_ne_infinity [simp]: "0 \<noteq> (\<infinity>::enat)"
hoelzl@43919
   122
  by (simp add: zero_enat_def)
haftmann@27110
   123
hoelzl@62378
   124
lemma zero_one_enat_neq:
wenzelm@61076
   125
  "\<not> 0 = (1::enat)"
wenzelm@61076
   126
  "\<not> 1 = (0::enat)"
hoelzl@43919
   127
  unfolding zero_enat_def one_enat_def by simp_all
oheimb@11351
   128
huffman@44019
   129
lemma infinity_ne_i1 [simp]: "(\<infinity>::enat) \<noteq> 1"
hoelzl@43919
   130
  by (simp add: one_enat_def)
haftmann@27110
   131
huffman@44019
   132
lemma i1_ne_infinity [simp]: "1 \<noteq> (\<infinity>::enat)"
hoelzl@43919
   133
  by (simp add: one_enat_def)
haftmann@27110
   134
huffman@44019
   135
lemma eSuc_enat: "eSuc (enat n) = enat (Suc n)"
huffman@44019
   136
  by (simp add: eSuc_def)
haftmann@27110
   137
huffman@44019
   138
lemma eSuc_infinity [simp]: "eSuc \<infinity> = \<infinity>"
huffman@44019
   139
  by (simp add: eSuc_def)
oheimb@11351
   140
huffman@44019
   141
lemma eSuc_ne_0 [simp]: "eSuc n \<noteq> 0"
huffman@44019
   142
  by (simp add: eSuc_def zero_enat_def split: enat.splits)
haftmann@27110
   143
huffman@44019
   144
lemma zero_ne_eSuc [simp]: "0 \<noteq> eSuc n"
huffman@44019
   145
  by (rule eSuc_ne_0 [symmetric])
oheimb@11351
   146
huffman@44019
   147
lemma eSuc_inject [simp]: "eSuc m = eSuc n \<longleftrightarrow> m = n"
huffman@44019
   148
  by (simp add: eSuc_def split: enat.splits)
haftmann@27110
   149
hoelzl@59000
   150
lemma eSuc_enat_iff: "eSuc x = enat y \<longleftrightarrow> (\<exists>n. y = Suc n \<and> x = enat n)"
hoelzl@59000
   151
  by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
hoelzl@59000
   152
hoelzl@59000
   153
lemma enat_eSuc_iff: "enat y = eSuc x \<longleftrightarrow> (\<exists>n. y = Suc n \<and> enat n = x)"
hoelzl@59000
   154
  by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
hoelzl@59000
   155
wenzelm@60500
   156
subsection \<open>Addition\<close>
haftmann@27110
   157
hoelzl@43919
   158
instantiation enat :: comm_monoid_add
haftmann@27110
   159
begin
haftmann@27110
   160
blanchet@38167
   161
definition [nitpick_simp]:
hoelzl@43924
   162
  "m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | enat m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | enat n \<Rightarrow> enat (m + n)))"
oheimb@11351
   163
hoelzl@43919
   164
lemma plus_enat_simps [simp, code]:
hoelzl@43921
   165
  fixes q :: enat
hoelzl@43924
   166
  shows "enat m + enat n = enat (m + n)"
hoelzl@43921
   167
    and "\<infinity> + q = \<infinity>"
hoelzl@43921
   168
    and "q + \<infinity> = \<infinity>"
hoelzl@43919
   169
  by (simp_all add: plus_enat_def split: enat.splits)
haftmann@27110
   170
wenzelm@60679
   171
instance
wenzelm@60679
   172
proof
hoelzl@43919
   173
  fix n m q :: enat
haftmann@27110
   174
  show "n + m + q = n + (m + q)"
noschinl@45934
   175
    by (cases n m q rule: enat3_cases) auto
haftmann@27110
   176
  show "n + m = m + n"
noschinl@45934
   177
    by (cases n m rule: enat2_cases) auto
haftmann@27110
   178
  show "0 + n = n"
hoelzl@43919
   179
    by (cases n) (simp_all add: zero_enat_def)
huffman@26089
   180
qed
huffman@26089
   181
haftmann@27110
   182
end
oheimb@11351
   183
huffman@44019
   184
lemma eSuc_plus_1:
huffman@44019
   185
  "eSuc n = n + 1"
huffman@44019
   186
  by (cases n) (simp_all add: eSuc_enat one_enat_def)
hoelzl@62374
   187
huffman@44019
   188
lemma plus_1_eSuc:
huffman@44019
   189
  "1 + q = eSuc q"
huffman@44019
   190
  "q + 1 = eSuc q"
haftmann@57514
   191
  by (simp_all add: eSuc_plus_1 ac_simps)
nipkow@41853
   192
huffman@44019
   193
lemma iadd_Suc: "eSuc m + n = eSuc (m + n)"
haftmann@57514
   194
  by (simp_all add: eSuc_plus_1 ac_simps)
oheimb@11351
   195
huffman@44019
   196
lemma iadd_Suc_right: "m + eSuc n = eSuc (m + n)"
haftmann@57512
   197
  by (simp only: add.commute[of m] iadd_Suc)
nipkow@41853
   198
wenzelm@60500
   199
subsection \<open>Multiplication\<close>
huffman@29014
   200
hoelzl@62378
   201
instantiation enat :: "{comm_semiring_1, semiring_no_zero_divisors}"
huffman@29014
   202
begin
huffman@29014
   203
hoelzl@43919
   204
definition times_enat_def [nitpick_simp]:
hoelzl@43924
   205
  "m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | enat m \<Rightarrow>
hoelzl@43924
   206
    (case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | enat n \<Rightarrow> enat (m * n)))"
huffman@29014
   207
hoelzl@43919
   208
lemma times_enat_simps [simp, code]:
hoelzl@43924
   209
  "enat m * enat n = enat (m * n)"
hoelzl@43921
   210
  "\<infinity> * \<infinity> = (\<infinity>::enat)"
hoelzl@43924
   211
  "\<infinity> * enat n = (if n = 0 then 0 else \<infinity>)"
hoelzl@43924
   212
  "enat m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
hoelzl@43919
   213
  unfolding times_enat_def zero_enat_def
hoelzl@43919
   214
  by (simp_all split: enat.split)
huffman@29014
   215
wenzelm@60679
   216
instance
wenzelm@60679
   217
proof
hoelzl@43919
   218
  fix a b c :: enat
huffman@29014
   219
  show "(a * b) * c = a * (b * c)"
hoelzl@43919
   220
    unfolding times_enat_def zero_enat_def
hoelzl@43919
   221
    by (simp split: enat.split)
hoelzl@62378
   222
  show comm: "a * b = b * a"
hoelzl@43919
   223
    unfolding times_enat_def zero_enat_def
hoelzl@43919
   224
    by (simp split: enat.split)
huffman@29014
   225
  show "1 * a = a"
hoelzl@43919
   226
    unfolding times_enat_def zero_enat_def one_enat_def
hoelzl@43919
   227
    by (simp split: enat.split)
hoelzl@62378
   228
  show distr: "(a + b) * c = a * c + b * c"
hoelzl@43919
   229
    unfolding times_enat_def zero_enat_def
webertj@49962
   230
    by (simp split: enat.split add: distrib_right)
huffman@29014
   231
  show "0 * a = 0"
hoelzl@43919
   232
    unfolding times_enat_def zero_enat_def
hoelzl@43919
   233
    by (simp split: enat.split)
huffman@29014
   234
  show "a * 0 = 0"
hoelzl@43919
   235
    unfolding times_enat_def zero_enat_def
hoelzl@43919
   236
    by (simp split: enat.split)
hoelzl@62378
   237
  show "a * (b + c) = a * b + a * c"
hoelzl@62378
   238
    by (cases a b c rule: enat3_cases) (auto simp: times_enat_def zero_enat_def distrib_left)
hoelzl@62378
   239
  show "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
hoelzl@62378
   240
    by (cases a b rule: enat2_cases) (auto simp: times_enat_def zero_enat_def)
huffman@29014
   241
qed
huffman@29014
   242
huffman@29014
   243
end
huffman@29014
   244
huffman@44019
   245
lemma mult_eSuc: "eSuc m * n = n + m * n"
huffman@44019
   246
  unfolding eSuc_plus_1 by (simp add: algebra_simps)
huffman@29014
   247
huffman@44019
   248
lemma mult_eSuc_right: "m * eSuc n = m + m * n"
huffman@44019
   249
  unfolding eSuc_plus_1 by (simp add: algebra_simps)
huffman@29014
   250
hoelzl@43924
   251
lemma of_nat_eq_enat: "of_nat n = enat n"
huffman@29023
   252
  apply (induct n)
hoelzl@43924
   253
  apply (simp add: enat_0)
huffman@44019
   254
  apply (simp add: plus_1_eSuc eSuc_enat)
huffman@29023
   255
  done
huffman@29023
   256
wenzelm@60679
   257
instance enat :: semiring_char_0
wenzelm@60679
   258
proof
hoelzl@43924
   259
  have "inj enat" by (rule injI) simp
hoelzl@43924
   260
  then show "inj (\<lambda>n. of_nat n :: enat)" by (simp add: of_nat_eq_enat)
haftmann@38621
   261
qed
huffman@29023
   262
huffman@44019
   263
lemma imult_is_infinity: "((a::enat) * b = \<infinity>) = (a = \<infinity> \<and> b \<noteq> 0 \<or> b = \<infinity> \<and> a \<noteq> 0)"
huffman@44019
   264
  by (auto simp add: times_enat_def zero_enat_def split: enat.split)
nipkow@41853
   265
wenzelm@60500
   266
subsection \<open>Numerals\<close>
huffman@47108
   267
huffman@47108
   268
lemma numeral_eq_enat:
huffman@47108
   269
  "numeral k = enat (numeral k)"
huffman@47108
   270
  using of_nat_eq_enat [of "numeral k"] by simp
huffman@47108
   271
huffman@47108
   272
lemma enat_numeral [code_abbrev]:
huffman@47108
   273
  "enat (numeral k) = numeral k"
huffman@47108
   274
  using numeral_eq_enat ..
huffman@47108
   275
huffman@47108
   276
lemma infinity_ne_numeral [simp]: "(\<infinity>::enat) \<noteq> numeral k"
huffman@47108
   277
  by (simp add: numeral_eq_enat)
huffman@47108
   278
huffman@47108
   279
lemma numeral_ne_infinity [simp]: "numeral k \<noteq> (\<infinity>::enat)"
huffman@47108
   280
  by (simp add: numeral_eq_enat)
huffman@47108
   281
huffman@47108
   282
lemma eSuc_numeral [simp]: "eSuc (numeral k) = numeral (k + Num.One)"
huffman@47108
   283
  by (simp only: eSuc_plus_1 numeral_plus_one)
huffman@47108
   284
wenzelm@60500
   285
subsection \<open>Subtraction\<close>
nipkow@41853
   286
hoelzl@43919
   287
instantiation enat :: minus
nipkow@41853
   288
begin
nipkow@41853
   289
hoelzl@43919
   290
definition diff_enat_def:
hoelzl@43924
   291
"a - b = (case a of (enat x) \<Rightarrow> (case b of (enat y) \<Rightarrow> enat (x - y) | \<infinity> \<Rightarrow> 0)
nipkow@41853
   292
          | \<infinity> \<Rightarrow> \<infinity>)"
nipkow@41853
   293
nipkow@41853
   294
instance ..
nipkow@41853
   295
nipkow@41853
   296
end
nipkow@41853
   297
huffman@47108
   298
lemma idiff_enat_enat [simp, code]: "enat a - enat b = enat (a - b)"
huffman@44019
   299
  by (simp add: diff_enat_def)
nipkow@41853
   300
huffman@47108
   301
lemma idiff_infinity [simp, code]: "\<infinity> - n = (\<infinity>::enat)"
huffman@44019
   302
  by (simp add: diff_enat_def)
nipkow@41853
   303
huffman@47108
   304
lemma idiff_infinity_right [simp, code]: "enat a - \<infinity> = 0"
huffman@44019
   305
  by (simp add: diff_enat_def)
nipkow@41853
   306
huffman@44019
   307
lemma idiff_0 [simp]: "(0::enat) - n = 0"
huffman@44019
   308
  by (cases n, simp_all add: zero_enat_def)
nipkow@41853
   309
huffman@44019
   310
lemmas idiff_enat_0 [simp] = idiff_0 [unfolded zero_enat_def]
nipkow@41853
   311
huffman@44019
   312
lemma idiff_0_right [simp]: "(n::enat) - 0 = n"
huffman@44019
   313
  by (cases n) (simp_all add: zero_enat_def)
nipkow@41853
   314
huffman@44019
   315
lemmas idiff_enat_0_right [simp] = idiff_0_right [unfolded zero_enat_def]
nipkow@41853
   316
huffman@44019
   317
lemma idiff_self [simp]: "n \<noteq> \<infinity> \<Longrightarrow> (n::enat) - n = 0"
huffman@44019
   318
  by (auto simp: zero_enat_def)
nipkow@41853
   319
huffman@44019
   320
lemma eSuc_minus_eSuc [simp]: "eSuc n - eSuc m = n - m"
huffman@44019
   321
  by (simp add: eSuc_def split: enat.split)
nipkow@41855
   322
huffman@44019
   323
lemma eSuc_minus_1 [simp]: "eSuc n - 1 = n"
huffman@44019
   324
  by (simp add: one_enat_def eSuc_enat[symmetric] zero_enat_def[symmetric])
nipkow@41855
   325
hoelzl@43924
   326
(*lemmas idiff_self_eq_0_enat = idiff_self_eq_0[unfolded zero_enat_def]*)
nipkow@41853
   327
wenzelm@60500
   328
subsection \<open>Ordering\<close>
haftmann@27110
   329
hoelzl@43919
   330
instantiation enat :: linordered_ab_semigroup_add
haftmann@27110
   331
begin
oheimb@11351
   332
blanchet@38167
   333
definition [nitpick_simp]:
hoelzl@43924
   334
  "m \<le> n = (case n of enat n1 \<Rightarrow> (case m of enat m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
haftmann@27110
   335
    | \<infinity> \<Rightarrow> True)"
oheimb@11351
   336
blanchet@38167
   337
definition [nitpick_simp]:
hoelzl@43924
   338
  "m < n = (case m of enat m1 \<Rightarrow> (case n of enat n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
haftmann@27110
   339
    | \<infinity> \<Rightarrow> False)"
oheimb@11351
   340
hoelzl@43919
   341
lemma enat_ord_simps [simp]:
hoelzl@43924
   342
  "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
hoelzl@43924
   343
  "enat m < enat n \<longleftrightarrow> m < n"
hoelzl@43921
   344
  "q \<le> (\<infinity>::enat)"
hoelzl@43921
   345
  "q < (\<infinity>::enat) \<longleftrightarrow> q \<noteq> \<infinity>"
hoelzl@43921
   346
  "(\<infinity>::enat) \<le> q \<longleftrightarrow> q = \<infinity>"
hoelzl@43921
   347
  "(\<infinity>::enat) < q \<longleftrightarrow> False"
hoelzl@43919
   348
  by (simp_all add: less_eq_enat_def less_enat_def split: enat.splits)
oheimb@11351
   349
huffman@47108
   350
lemma numeral_le_enat_iff[simp]:
huffman@47108
   351
  shows "numeral m \<le> enat n \<longleftrightarrow> numeral m \<le> n"
huffman@47108
   352
by (auto simp: numeral_eq_enat)
noschinl@45934
   353
huffman@47108
   354
lemma numeral_less_enat_iff[simp]:
huffman@47108
   355
  shows "numeral m < enat n \<longleftrightarrow> numeral m < n"
huffman@47108
   356
by (auto simp: numeral_eq_enat)
noschinl@45934
   357
hoelzl@43919
   358
lemma enat_ord_code [code]:
hoelzl@43924
   359
  "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
hoelzl@43924
   360
  "enat m < enat n \<longleftrightarrow> m < n"
hoelzl@43921
   361
  "q \<le> (\<infinity>::enat) \<longleftrightarrow> True"
hoelzl@43924
   362
  "enat m < \<infinity> \<longleftrightarrow> True"
hoelzl@43924
   363
  "\<infinity> \<le> enat n \<longleftrightarrow> False"
hoelzl@43921
   364
  "(\<infinity>::enat) < q \<longleftrightarrow> False"
haftmann@27110
   365
  by simp_all
oheimb@11351
   366
wenzelm@60679
   367
instance
wenzelm@60679
   368
  by standard (auto simp add: less_eq_enat_def less_enat_def plus_enat_def split: enat.splits)
oheimb@11351
   369
haftmann@27110
   370
end
haftmann@27110
   371
hoelzl@62376
   372
instance enat :: dioid
hoelzl@62376
   373
proof
hoelzl@62376
   374
  fix a b :: enat show "(a \<le> b) = (\<exists>c. b = a + c)"
hoelzl@62376
   375
    by (cases a b rule: enat2_cases) (auto simp: le_iff_add enat_ex_split)
hoelzl@62376
   376
qed
hoelzl@62376
   377
hoelzl@62378
   378
instance enat :: "{linordered_nonzero_semiring, strict_ordered_comm_monoid_add}"
huffman@29014
   379
proof
hoelzl@43919
   380
  fix a b c :: enat
hoelzl@62378
   381
  show "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow>c * a \<le> c * b"
hoelzl@43919
   382
    unfolding times_enat_def less_eq_enat_def zero_enat_def
hoelzl@43919
   383
    by (simp split: enat.splits)
hoelzl@62378
   384
  show "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" for a b c d :: enat
hoelzl@62378
   385
    by (cases a b c d rule: enat2_cases[case_product enat2_cases]) auto
hoelzl@62378
   386
qed (simp add: zero_enat_def one_enat_def)
huffman@29014
   387
huffman@47108
   388
(* BH: These equations are already proven generally for any type in
huffman@47108
   389
class linordered_semidom. However, enat is not in that class because
huffman@47108
   390
it does not have the cancellation property. Would it be worthwhile to
huffman@47108
   391
a generalize linordered_semidom to a new class that includes enat? *)
huffman@47108
   392
hoelzl@43919
   393
lemma enat_ord_number [simp]:
wenzelm@61076
   394
  "(numeral m :: enat) \<le> numeral n \<longleftrightarrow> (numeral m :: nat) \<le> numeral n"
wenzelm@61076
   395
  "(numeral m :: enat) < numeral n \<longleftrightarrow> (numeral m :: nat) < numeral n"
huffman@47108
   396
  by (simp_all add: numeral_eq_enat)
oheimb@11351
   397
huffman@44019
   398
lemma infinity_ileE [elim!]: "\<infinity> \<le> enat m \<Longrightarrow> R"
huffman@44019
   399
  by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
huffman@44019
   400
huffman@44019
   401
lemma infinity_ilessE [elim!]: "\<infinity> < enat m \<Longrightarrow> R"
haftmann@27110
   402
  by simp
oheimb@11351
   403
huffman@44019
   404
lemma eSuc_ile_mono [simp]: "eSuc n \<le> eSuc m \<longleftrightarrow> n \<le> m"
huffman@44019
   405
  by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
hoelzl@62374
   406
huffman@44019
   407
lemma eSuc_mono [simp]: "eSuc n < eSuc m \<longleftrightarrow> n < m"
huffman@44019
   408
  by (simp add: eSuc_def less_enat_def split: enat.splits)
oheimb@11351
   409
huffman@44019
   410
lemma ile_eSuc [simp]: "n \<le> eSuc n"
huffman@44019
   411
  by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
oheimb@11351
   412
huffman@44019
   413
lemma not_eSuc_ilei0 [simp]: "\<not> eSuc n \<le> 0"
huffman@44019
   414
  by (simp add: zero_enat_def eSuc_def less_eq_enat_def split: enat.splits)
haftmann@27110
   415
huffman@44019
   416
lemma i0_iless_eSuc [simp]: "0 < eSuc n"
huffman@44019
   417
  by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.splits)
haftmann@27110
   418
huffman@44019
   419
lemma iless_eSuc0[simp]: "(n < eSuc 0) = (n = 0)"
huffman@44019
   420
  by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.split)
nipkow@41853
   421
huffman@44019
   422
lemma ileI1: "m < n \<Longrightarrow> eSuc m \<le> n"
huffman@44019
   423
  by (simp add: eSuc_def less_eq_enat_def less_enat_def split: enat.splits)
haftmann@27110
   424
hoelzl@43924
   425
lemma Suc_ile_eq: "enat (Suc m) \<le> n \<longleftrightarrow> enat m < n"
haftmann@27110
   426
  by (cases n) auto
haftmann@27110
   427
huffman@44019
   428
lemma iless_Suc_eq [simp]: "enat m < eSuc n \<longleftrightarrow> enat m \<le> n"
huffman@44019
   429
  by (auto simp add: eSuc_def less_enat_def split: enat.splits)
oheimb@11351
   430
huffman@44019
   431
lemma imult_infinity: "(0::enat) < n \<Longrightarrow> \<infinity> * n = \<infinity>"
huffman@44019
   432
  by (simp add: zero_enat_def less_enat_def split: enat.splits)
nipkow@41853
   433
huffman@44019
   434
lemma imult_infinity_right: "(0::enat) < n \<Longrightarrow> n * \<infinity> = \<infinity>"
huffman@44019
   435
  by (simp add: zero_enat_def less_enat_def split: enat.splits)
nipkow@41853
   436
hoelzl@43919
   437
lemma enat_0_less_mult_iff: "(0 < (m::enat) * n) = (0 < m \<and> 0 < n)"
hoelzl@62378
   438
  by (simp only: zero_less_iff_neq_zero mult_eq_0_iff, simp)
nipkow@41853
   439
huffman@44019
   440
lemma mono_eSuc: "mono eSuc"
huffman@44019
   441
  by (simp add: mono_def)
nipkow@41853
   442
hoelzl@43919
   443
lemma min_enat_simps [simp]:
hoelzl@43924
   444
  "min (enat m) (enat n) = enat (min m n)"
haftmann@27110
   445
  "min q 0 = 0"
haftmann@27110
   446
  "min 0 q = 0"
hoelzl@43921
   447
  "min q (\<infinity>::enat) = q"
hoelzl@43921
   448
  "min (\<infinity>::enat) q = q"
haftmann@27110
   449
  by (auto simp add: min_def)
oheimb@11351
   450
hoelzl@43919
   451
lemma max_enat_simps [simp]:
hoelzl@43924
   452
  "max (enat m) (enat n) = enat (max m n)"
haftmann@27110
   453
  "max q 0 = q"
haftmann@27110
   454
  "max 0 q = q"
hoelzl@43921
   455
  "max q \<infinity> = (\<infinity>::enat)"
hoelzl@43921
   456
  "max \<infinity> q = (\<infinity>::enat)"
haftmann@27110
   457
  by (simp_all add: max_def)
haftmann@27110
   458
hoelzl@43924
   459
lemma enat_ile: "n \<le> enat m \<Longrightarrow> \<exists>k. n = enat k"
haftmann@27110
   460
  by (cases n) simp_all
haftmann@27110
   461
hoelzl@43924
   462
lemma enat_iless: "n < enat m \<Longrightarrow> \<exists>k. n = enat k"
haftmann@27110
   463
  by (cases n) simp_all
oheimb@11351
   464
Andreas@61631
   465
lemma iadd_le_enat_iff:
Andreas@61631
   466
  "x + y \<le> enat n \<longleftrightarrow> (\<exists>y' x'. x = enat x' \<and> y = enat y' \<and> x' + y' \<le> n)"
Andreas@61631
   467
by(cases x y rule: enat.exhaust[case_product enat.exhaust]) simp_all
Andreas@61631
   468
hoelzl@62378
   469
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j \<Longrightarrow> \<exists>j. enat k < Y j"
nipkow@25134
   470
apply (induct_tac k)
hoelzl@43924
   471
 apply (simp (no_asm) only: enat_0)
hoelzl@62378
   472
 apply (fast intro: le_less_trans [OF zero_le])
nipkow@25134
   473
apply (erule exE)
nipkow@25134
   474
apply (drule spec)
nipkow@25134
   475
apply (erule exE)
nipkow@25134
   476
apply (drule ileI1)
huffman@44019
   477
apply (rule eSuc_enat [THEN subst])
nipkow@25134
   478
apply (rule exI)
haftmann@27110
   479
apply (erule (1) le_less_trans)
nipkow@25134
   480
done
oheimb@11351
   481
hoelzl@60636
   482
lemma eSuc_max: "eSuc (max x y) = max (eSuc x) (eSuc y)"
hoelzl@60636
   483
  by (simp add: eSuc_def split: enat.split)
hoelzl@60636
   484
hoelzl@62374
   485
lemma eSuc_Max:
hoelzl@60636
   486
  assumes "finite A" "A \<noteq> {}"
hoelzl@60636
   487
  shows "eSuc (Max A) = Max (eSuc ` A)"
hoelzl@60636
   488
using assms proof induction
hoelzl@60636
   489
  case (insert x A)
hoelzl@60636
   490
  thus ?case by(cases "A = {}")(simp_all add: eSuc_max)
hoelzl@60636
   491
qed simp
hoelzl@60636
   492
haftmann@52729
   493
instantiation enat :: "{order_bot, order_top}"
haftmann@29337
   494
begin
haftmann@29337
   495
wenzelm@60679
   496
definition bot_enat :: enat where "bot_enat = 0"
wenzelm@60679
   497
definition top_enat :: enat where "top_enat = \<infinity>"
haftmann@29337
   498
wenzelm@60679
   499
instance
wenzelm@60679
   500
  by standard (simp_all add: bot_enat_def top_enat_def)
haftmann@29337
   501
haftmann@29337
   502
end
haftmann@29337
   503
hoelzl@43924
   504
lemma finite_enat_bounded:
hoelzl@43924
   505
  assumes le_fin: "\<And>y. y \<in> A \<Longrightarrow> y \<le> enat n"
noschinl@42993
   506
  shows "finite A"
noschinl@42993
   507
proof (rule finite_subset)
hoelzl@43924
   508
  show "finite (enat ` {..n})" by blast
nipkow@44890
   509
  have "A \<subseteq> {..enat n}" using le_fin by fastforce
hoelzl@43924
   510
  also have "\<dots> \<subseteq> enat ` {..n}"
wenzelm@60679
   511
    apply (rule subsetI)
wenzelm@60679
   512
    subgoal for x by (cases x) auto
wenzelm@60679
   513
    done
hoelzl@43924
   514
  finally show "A \<subseteq> enat ` {..n}" .
noschinl@42993
   515
qed
noschinl@42993
   516
huffman@26089
   517
wenzelm@60500
   518
subsection \<open>Cancellation simprocs\<close>
huffman@45775
   519
huffman@45775
   520
lemma enat_add_left_cancel: "a + b = a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b = c"
huffman@45775
   521
  unfolding plus_enat_def by (simp split: enat.split)
huffman@45775
   522
huffman@45775
   523
lemma enat_add_left_cancel_le: "a + b \<le> a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b \<le> c"
huffman@45775
   524
  unfolding plus_enat_def by (simp split: enat.split)
huffman@45775
   525
huffman@45775
   526
lemma enat_add_left_cancel_less: "a + b < a + c \<longleftrightarrow> a \<noteq> (\<infinity>::enat) \<and> b < c"
huffman@45775
   527
  unfolding plus_enat_def by (simp split: enat.split)
huffman@45775
   528
wenzelm@60500
   529
ML \<open>
huffman@45775
   530
structure Cancel_Enat_Common =
huffman@45775
   531
struct
huffman@45775
   532
  (* copied from src/HOL/Tools/nat_numeral_simprocs.ML *)
huffman@45775
   533
  fun find_first_t _    _ []         = raise TERM("find_first_t", [])
huffman@45775
   534
    | find_first_t past u (t::terms) =
huffman@45775
   535
          if u aconv t then (rev past @ terms)
huffman@45775
   536
          else find_first_t (t::past) u terms
huffman@45775
   537
huffman@51366
   538
  fun dest_summing (Const (@{const_name Groups.plus}, _) $ t $ u, ts) =
huffman@51366
   539
        dest_summing (t, dest_summing (u, ts))
huffman@51366
   540
    | dest_summing (t, ts) = t :: ts
huffman@51366
   541
huffman@45775
   542
  val mk_sum = Arith_Data.long_mk_sum
huffman@51366
   543
  fun dest_sum t = dest_summing (t, [])
huffman@45775
   544
  val find_first = find_first_t []
huffman@45775
   545
  val trans_tac = Numeral_Simprocs.trans_tac
wenzelm@51717
   546
  val norm_ss =
wenzelm@51717
   547
    simpset_of (put_simpset HOL_basic_ss @{context}
haftmann@57514
   548
      addsimps @{thms ac_simps add_0_left add_0_right})
wenzelm@51717
   549
  fun norm_tac ctxt = ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
wenzelm@51717
   550
  fun simplify_meta_eq ctxt cancel_th th =
wenzelm@51717
   551
    Arith_Data.simplify_meta_eq [] ctxt
huffman@45775
   552
      ([th, cancel_th] MRS trans)
huffman@45775
   553
  fun mk_eq (a, b) = HOLogic.mk_Trueprop (HOLogic.mk_eq (a, b))
huffman@45775
   554
end
huffman@45775
   555
huffman@45775
   556
structure Eq_Enat_Cancel = ExtractCommonTermFun
huffman@45775
   557
(open Cancel_Enat_Common
huffman@45775
   558
  val mk_bal = HOLogic.mk_eq
huffman@45775
   559
  val dest_bal = HOLogic.dest_bin @{const_name HOL.eq} @{typ enat}
huffman@45775
   560
  fun simp_conv _ _ = SOME @{thm enat_add_left_cancel}
huffman@45775
   561
)
huffman@45775
   562
huffman@45775
   563
structure Le_Enat_Cancel = ExtractCommonTermFun
huffman@45775
   564
(open Cancel_Enat_Common
huffman@45775
   565
  val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less_eq}
huffman@45775
   566
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} @{typ enat}
huffman@45775
   567
  fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_le}
huffman@45775
   568
)
huffman@45775
   569
huffman@45775
   570
structure Less_Enat_Cancel = ExtractCommonTermFun
huffman@45775
   571
(open Cancel_Enat_Common
huffman@45775
   572
  val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less}
huffman@45775
   573
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} @{typ enat}
huffman@45775
   574
  fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_less}
huffman@45775
   575
)
wenzelm@60500
   576
\<close>
huffman@45775
   577
huffman@45775
   578
simproc_setup enat_eq_cancel
huffman@45775
   579
  ("(l::enat) + m = n" | "(l::enat) = m + n") =
wenzelm@60500
   580
  \<open>fn phi => fn ctxt => fn ct => Eq_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
huffman@45775
   581
huffman@45775
   582
simproc_setup enat_le_cancel
huffman@45775
   583
  ("(l::enat) + m \<le> n" | "(l::enat) \<le> m + n") =
wenzelm@60500
   584
  \<open>fn phi => fn ctxt => fn ct => Le_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
huffman@45775
   585
huffman@45775
   586
simproc_setup enat_less_cancel
huffman@45775
   587
  ("(l::enat) + m < n" | "(l::enat) < m + n") =
wenzelm@60500
   588
  \<open>fn phi => fn ctxt => fn ct => Less_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
huffman@45775
   589
wenzelm@60500
   590
text \<open>TODO: add regression tests for these simprocs\<close>
huffman@45775
   591
wenzelm@60500
   592
text \<open>TODO: add simprocs for combining and cancelling numerals\<close>
huffman@45775
   593
wenzelm@60500
   594
subsection \<open>Well-ordering\<close>
huffman@26089
   595
hoelzl@43924
   596
lemma less_enatE:
hoelzl@43924
   597
  "[| n < enat m; !!k. n = enat k ==> k < m ==> P |] ==> P"
huffman@26089
   598
by (induct n) auto
huffman@26089
   599
huffman@44019
   600
lemma less_infinityE:
hoelzl@43924
   601
  "[| n < \<infinity>; !!k. n = enat k ==> P |] ==> P"
huffman@26089
   602
by (induct n) auto
huffman@26089
   603
hoelzl@43919
   604
lemma enat_less_induct:
hoelzl@43919
   605
  assumes prem: "!!n. \<forall>m::enat. m < n --> P m ==> P n" shows "P n"
huffman@26089
   606
proof -
hoelzl@43924
   607
  have P_enat: "!!k. P (enat k)"
huffman@26089
   608
    apply (rule nat_less_induct)
huffman@26089
   609
    apply (rule prem, clarify)
hoelzl@43924
   610
    apply (erule less_enatE, simp)
huffman@26089
   611
    done
huffman@26089
   612
  show ?thesis
huffman@26089
   613
  proof (induct n)
huffman@26089
   614
    fix nat
hoelzl@43924
   615
    show "P (enat nat)" by (rule P_enat)
huffman@26089
   616
  next
hoelzl@43921
   617
    show "P \<infinity>"
huffman@26089
   618
      apply (rule prem, clarify)
huffman@44019
   619
      apply (erule less_infinityE)
hoelzl@43924
   620
      apply (simp add: P_enat)
huffman@26089
   621
      done
huffman@26089
   622
  qed
huffman@26089
   623
qed
huffman@26089
   624
hoelzl@43919
   625
instance enat :: wellorder
huffman@26089
   626
proof
haftmann@27823
   627
  fix P and n
wenzelm@61076
   628
  assume hyp: "(\<And>n::enat. (\<And>m::enat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
hoelzl@43919
   629
  show "P n" by (blast intro: enat_less_induct hyp)
huffman@26089
   630
qed
huffman@26089
   631
wenzelm@60500
   632
subsection \<open>Complete Lattice\<close>
noschinl@42993
   633
hoelzl@43919
   634
instantiation enat :: complete_lattice
noschinl@42993
   635
begin
noschinl@42993
   636
hoelzl@43919
   637
definition inf_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
wenzelm@56777
   638
  "inf_enat = min"
noschinl@42993
   639
hoelzl@43919
   640
definition sup_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
wenzelm@56777
   641
  "sup_enat = max"
noschinl@42993
   642
hoelzl@43919
   643
definition Inf_enat :: "enat set \<Rightarrow> enat" where
wenzelm@56777
   644
  "Inf_enat A = (if A = {} then \<infinity> else (LEAST x. x \<in> A))"
noschinl@42993
   645
hoelzl@43919
   646
definition Sup_enat :: "enat set \<Rightarrow> enat" where
wenzelm@56777
   647
  "Sup_enat A = (if A = {} then 0 else if finite A then Max A else \<infinity>)"
wenzelm@56777
   648
instance
wenzelm@56777
   649
proof
hoelzl@43919
   650
  fix x :: "enat" and A :: "enat set"
noschinl@42993
   651
  { assume "x \<in> A" then show "Inf A \<le> x"
hoelzl@43919
   652
      unfolding Inf_enat_def by (auto intro: Least_le) }
noschinl@42993
   653
  { assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A"
hoelzl@43919
   654
      unfolding Inf_enat_def
noschinl@42993
   655
      by (cases "A = {}") (auto intro: LeastI2_ex) }
noschinl@42993
   656
  { assume "x \<in> A" then show "x \<le> Sup A"
hoelzl@43919
   657
      unfolding Sup_enat_def by (cases "finite A") auto }
noschinl@42993
   658
  { assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x"
hoelzl@43924
   659
      unfolding Sup_enat_def using finite_enat_bounded by auto }
haftmann@52729
   660
qed (simp_all add:
haftmann@52729
   661
 inf_enat_def sup_enat_def bot_enat_def top_enat_def Inf_enat_def Sup_enat_def)
noschinl@42993
   662
end
noschinl@42993
   663
hoelzl@43978
   664
instance enat :: complete_linorder ..
haftmann@27110
   665
hoelzl@60636
   666
lemma eSuc_Sup: "A \<noteq> {} \<Longrightarrow> eSuc (Sup A) = Sup (eSuc ` A)"
hoelzl@60636
   667
  by(auto simp add: Sup_enat_def eSuc_Max inj_on_def dest: finite_imageD)
hoelzl@60636
   668
hoelzl@60636
   669
lemma sup_continuous_eSuc: "sup_continuous f \<Longrightarrow> sup_continuous (\<lambda>x. eSuc (f x))"
hoelzl@60636
   670
  using  eSuc_Sup[of "_ ` UNIV"] by (auto simp: sup_continuous_def)
hoelzl@60636
   671
wenzelm@60500
   672
subsection \<open>Traditional theorem names\<close>
haftmann@27110
   673
huffman@47108
   674
lemmas enat_defs = zero_enat_def one_enat_def eSuc_def
hoelzl@43919
   675
  plus_enat_def less_eq_enat_def less_enat_def
haftmann@27110
   676
hoelzl@62378
   677
lemma iadd_is_0: "(m + n = (0::enat)) = (m = 0 \<and> n = 0)"
hoelzl@62378
   678
  by (rule add_eq_0_iff_both_eq_0)
hoelzl@62378
   679
hoelzl@62378
   680
lemma i0_lb : "(0::enat) \<le> n"
hoelzl@62378
   681
  by (rule zero_le)
hoelzl@62378
   682
hoelzl@62378
   683
lemma ile0_eq: "n \<le> (0::enat) \<longleftrightarrow> n = 0"
hoelzl@62378
   684
  by (rule le_zero_eq)
hoelzl@62378
   685
hoelzl@62378
   686
lemma not_iless0: "\<not> n < (0::enat)"
hoelzl@62378
   687
  by (rule not_less_zero)
hoelzl@62378
   688
hoelzl@62378
   689
lemma i0_less[simp]: "(0::enat) < n \<longleftrightarrow> n \<noteq> 0"
hoelzl@62378
   690
  by (rule zero_less_iff_neq_zero)
hoelzl@62378
   691
hoelzl@62378
   692
lemma imult_is_0: "((m::enat) * n = 0) = (m = 0 \<or> n = 0)"
hoelzl@62378
   693
  by (rule mult_eq_0_iff)
hoelzl@62378
   694
oheimb@11351
   695
end