src/HOL/Lattices.thy
author haftmann
Tue Jul 14 15:54:19 2009 +0200 (2009-07-14)
changeset 32064 53ca12ff305d
parent 32063 2aab4f2af536
child 32204 b330aa4d59cb
permissions -rw-r--r--
refinement of lattice classes
haftmann@21249
     1
(*  Title:      HOL/Lattices.thy
haftmann@21249
     2
    Author:     Tobias Nipkow
haftmann@21249
     3
*)
haftmann@21249
     4
haftmann@22454
     5
header {* Abstract lattices *}
haftmann@21249
     6
haftmann@21249
     7
theory Lattices
haftmann@30302
     8
imports Orderings
haftmann@21249
     9
begin
haftmann@21249
    10
haftmann@28562
    11
subsection {* Lattices *}
haftmann@21249
    12
haftmann@25206
    13
notation
wenzelm@25382
    14
  less_eq  (infix "\<sqsubseteq>" 50) and
wenzelm@25382
    15
  less  (infix "\<sqsubset>" 50)
haftmann@25206
    16
haftmann@22422
    17
class lower_semilattice = order +
haftmann@21249
    18
  fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
haftmann@22737
    19
  assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
haftmann@22737
    20
  and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
nipkow@21733
    21
  and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
haftmann@21249
    22
haftmann@22422
    23
class upper_semilattice = order +
haftmann@21249
    24
  fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
haftmann@22737
    25
  assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
haftmann@22737
    26
  and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
nipkow@21733
    27
  and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
haftmann@26014
    28
begin
haftmann@26014
    29
haftmann@26014
    30
text {* Dual lattice *}
haftmann@26014
    31
haftmann@31991
    32
lemma dual_semilattice:
haftmann@26014
    33
  "lower_semilattice (op \<ge>) (op >) sup"
haftmann@27682
    34
by (rule lower_semilattice.intro, rule dual_order)
haftmann@27682
    35
  (unfold_locales, simp_all add: sup_least)
haftmann@26014
    36
haftmann@26014
    37
end
haftmann@21249
    38
haftmann@22422
    39
class lattice = lower_semilattice + upper_semilattice
haftmann@21249
    40
wenzelm@25382
    41
haftmann@28562
    42
subsubsection {* Intro and elim rules*}
nipkow@21733
    43
nipkow@21733
    44
context lower_semilattice
nipkow@21733
    45
begin
haftmann@21249
    46
haftmann@32064
    47
lemma le_infI1:
haftmann@32064
    48
  "a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
haftmann@32064
    49
  by (rule order_trans) auto
haftmann@21249
    50
haftmann@32064
    51
lemma le_infI2:
haftmann@32064
    52
  "b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
haftmann@32064
    53
  by (rule order_trans) auto
nipkow@21733
    54
haftmann@32064
    55
lemma le_infI: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
haftmann@32064
    56
  by (blast intro: inf_greatest)
haftmann@21249
    57
haftmann@32064
    58
lemma le_infE: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@32064
    59
  by (blast intro: order_trans le_infI1 le_infI2)
haftmann@21249
    60
nipkow@21734
    61
lemma le_inf_iff [simp]:
haftmann@32064
    62
  "x \<sqsubseteq> y \<sqinter> z \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
haftmann@32064
    63
  by (blast intro: le_infI elim: le_infE)
nipkow@21733
    64
haftmann@32064
    65
lemma le_iff_inf:
haftmann@32064
    66
  "x \<sqsubseteq> y \<longleftrightarrow> x \<sqinter> y = x"
haftmann@32064
    67
  by (auto intro: le_infI1 antisym dest: eq_iff [THEN iffD1])
haftmann@21249
    68
haftmann@25206
    69
lemma mono_inf:
haftmann@25206
    70
  fixes f :: "'a \<Rightarrow> 'b\<Colon>lower_semilattice"
haftmann@25206
    71
  shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<le> f A \<sqinter> f B"
haftmann@25206
    72
  by (auto simp add: mono_def intro: Lattices.inf_greatest)
nipkow@21733
    73
haftmann@25206
    74
end
nipkow@21733
    75
nipkow@21733
    76
context upper_semilattice
nipkow@21733
    77
begin
haftmann@21249
    78
haftmann@32064
    79
lemma le_supI1:
haftmann@32064
    80
  "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
haftmann@25062
    81
  by (rule order_trans) auto
haftmann@21249
    82
haftmann@32064
    83
lemma le_supI2:
haftmann@32064
    84
  "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
haftmann@25062
    85
  by (rule order_trans) auto 
nipkow@21733
    86
haftmann@32064
    87
lemma le_supI:
haftmann@32064
    88
  "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
haftmann@26014
    89
  by (blast intro: sup_least)
haftmann@21249
    90
haftmann@32064
    91
lemma le_supE:
haftmann@32064
    92
  "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@32064
    93
  by (blast intro: le_supI1 le_supI2 order_trans)
haftmann@22422
    94
haftmann@32064
    95
lemma le_sup_iff [simp]:
haftmann@32064
    96
  "x \<squnion> y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<and> y \<sqsubseteq> z"
haftmann@32064
    97
  by (blast intro: le_supI elim: le_supE)
nipkow@21733
    98
haftmann@32064
    99
lemma le_iff_sup:
haftmann@32064
   100
  "x \<sqsubseteq> y \<longleftrightarrow> x \<squnion> y = y"
haftmann@32064
   101
  by (auto intro: le_supI2 antisym dest: eq_iff [THEN iffD1])
nipkow@21734
   102
haftmann@25206
   103
lemma mono_sup:
haftmann@25206
   104
  fixes f :: "'a \<Rightarrow> 'b\<Colon>upper_semilattice"
haftmann@25206
   105
  shows "mono f \<Longrightarrow> f A \<squnion> f B \<le> f (A \<squnion> B)"
haftmann@25206
   106
  by (auto simp add: mono_def intro: Lattices.sup_least)
nipkow@21733
   107
haftmann@25206
   108
end
haftmann@23878
   109
nipkow@21733
   110
haftmann@32064
   111
subsubsection {* Equational laws *}
haftmann@21249
   112
nipkow@21733
   113
context lower_semilattice
nipkow@21733
   114
begin
nipkow@21733
   115
nipkow@21733
   116
lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
haftmann@32064
   117
  by (rule antisym) auto
nipkow@21733
   118
nipkow@21733
   119
lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
haftmann@32064
   120
  by (rule antisym) (auto intro: le_infI1 le_infI2)
nipkow@21733
   121
nipkow@21733
   122
lemma inf_idem[simp]: "x \<sqinter> x = x"
haftmann@32064
   123
  by (rule antisym) auto
nipkow@21733
   124
nipkow@21733
   125
lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
haftmann@32064
   126
  by (rule antisym) (auto intro: le_infI2)
nipkow@21733
   127
nipkow@21733
   128
lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
haftmann@32064
   129
  by (rule antisym) auto
nipkow@21733
   130
nipkow@21733
   131
lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
haftmann@32064
   132
  by (rule antisym) auto
nipkow@21733
   133
nipkow@21733
   134
lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
haftmann@32064
   135
  by (rule mk_left_commute [of inf]) (fact inf_assoc inf_commute)+
haftmann@32064
   136
  
haftmann@32064
   137
lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem
nipkow@21733
   138
nipkow@21733
   139
end
nipkow@21733
   140
nipkow@21733
   141
context upper_semilattice
nipkow@21733
   142
begin
haftmann@21249
   143
nipkow@21733
   144
lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
haftmann@32064
   145
  by (rule antisym) auto
nipkow@21733
   146
nipkow@21733
   147
lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
haftmann@32064
   148
  by (rule antisym) (auto intro: le_supI1 le_supI2)
nipkow@21733
   149
nipkow@21733
   150
lemma sup_idem[simp]: "x \<squnion> x = x"
haftmann@32064
   151
  by (rule antisym) auto
nipkow@21733
   152
nipkow@21733
   153
lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
haftmann@32064
   154
  by (rule antisym) (auto intro: le_supI2)
nipkow@21733
   155
nipkow@21733
   156
lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
haftmann@32064
   157
  by (rule antisym) auto
nipkow@21733
   158
nipkow@21733
   159
lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
haftmann@32064
   160
  by (rule antisym) auto
haftmann@21249
   161
nipkow@21733
   162
lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
haftmann@32064
   163
  by (rule mk_left_commute [of sup]) (fact sup_assoc sup_commute)+
nipkow@21733
   164
haftmann@32064
   165
lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem
nipkow@21733
   166
nipkow@21733
   167
end
haftmann@21249
   168
nipkow@21733
   169
context lattice
nipkow@21733
   170
begin
nipkow@21733
   171
haftmann@31991
   172
lemma dual_lattice:
haftmann@31991
   173
  "lattice (op \<ge>) (op >) sup inf"
haftmann@31991
   174
  by (rule lattice.intro, rule dual_semilattice, rule upper_semilattice.intro, rule dual_order)
haftmann@31991
   175
    (unfold_locales, auto)
haftmann@31991
   176
nipkow@21733
   177
lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
haftmann@25102
   178
  by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
nipkow@21733
   179
nipkow@21733
   180
lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
haftmann@25102
   181
  by (blast intro: antisym sup_ge1 sup_least inf_le1)
nipkow@21733
   182
haftmann@32064
   183
lemmas inf_sup_aci = inf_aci sup_aci
nipkow@21734
   184
haftmann@22454
   185
lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
haftmann@22454
   186
nipkow@21734
   187
text{* Towards distributivity *}
haftmann@21249
   188
nipkow@21734
   189
lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
haftmann@32064
   190
  by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
nipkow@21734
   191
nipkow@21734
   192
lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
haftmann@32064
   193
  by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
nipkow@21734
   194
nipkow@21734
   195
text{* If you have one of them, you have them all. *}
haftmann@21249
   196
nipkow@21733
   197
lemma distrib_imp1:
haftmann@21249
   198
assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
haftmann@21249
   199
shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
haftmann@21249
   200
proof-
haftmann@21249
   201
  have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
haftmann@21249
   202
  also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
haftmann@21249
   203
  also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
haftmann@21249
   204
    by(simp add:inf_sup_absorb inf_commute)
haftmann@21249
   205
  also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
haftmann@21249
   206
  finally show ?thesis .
haftmann@21249
   207
qed
haftmann@21249
   208
nipkow@21733
   209
lemma distrib_imp2:
haftmann@21249
   210
assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
haftmann@21249
   211
shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
haftmann@21249
   212
proof-
haftmann@21249
   213
  have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
haftmann@21249
   214
  also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
haftmann@21249
   215
  also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
haftmann@21249
   216
    by(simp add:sup_inf_absorb sup_commute)
haftmann@21249
   217
  also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
haftmann@21249
   218
  finally show ?thesis .
haftmann@21249
   219
qed
haftmann@21249
   220
nipkow@21733
   221
end
haftmann@21249
   222
haftmann@21249
   223
haftmann@24164
   224
subsection {* Distributive lattices *}
haftmann@21249
   225
haftmann@22454
   226
class distrib_lattice = lattice +
haftmann@21249
   227
  assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
haftmann@21249
   228
nipkow@21733
   229
context distrib_lattice
nipkow@21733
   230
begin
nipkow@21733
   231
nipkow@21733
   232
lemma sup_inf_distrib2:
haftmann@21249
   233
 "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
haftmann@32064
   234
by(simp add: inf_sup_aci sup_inf_distrib1)
haftmann@21249
   235
nipkow@21733
   236
lemma inf_sup_distrib1:
haftmann@21249
   237
 "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
haftmann@21249
   238
by(rule distrib_imp2[OF sup_inf_distrib1])
haftmann@21249
   239
nipkow@21733
   240
lemma inf_sup_distrib2:
haftmann@21249
   241
 "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
haftmann@32064
   242
by(simp add: inf_sup_aci inf_sup_distrib1)
haftmann@21249
   243
haftmann@31991
   244
lemma dual_distrib_lattice:
haftmann@31991
   245
  "distrib_lattice (op \<ge>) (op >) sup inf"
haftmann@31991
   246
  by (rule distrib_lattice.intro, rule dual_lattice)
haftmann@31991
   247
    (unfold_locales, fact inf_sup_distrib1)
haftmann@31991
   248
nipkow@21733
   249
lemmas distrib =
haftmann@21249
   250
  sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
haftmann@21249
   251
nipkow@21733
   252
end
nipkow@21733
   253
haftmann@21249
   254
haftmann@31991
   255
subsection {* Boolean algebras *}
haftmann@31991
   256
haftmann@31991
   257
class boolean_algebra = distrib_lattice + top + bot + minus + uminus +
haftmann@31991
   258
  assumes inf_compl_bot: "x \<sqinter> - x = bot"
haftmann@31991
   259
    and sup_compl_top: "x \<squnion> - x = top"
haftmann@31991
   260
  assumes diff_eq: "x - y = x \<sqinter> - y"
haftmann@31991
   261
begin
haftmann@31991
   262
haftmann@31991
   263
lemma dual_boolean_algebra:
haftmann@31991
   264
  "boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) top bot"
haftmann@31991
   265
  by (rule boolean_algebra.intro, rule dual_distrib_lattice)
haftmann@31991
   266
    (unfold_locales,
haftmann@31991
   267
      auto simp add: inf_compl_bot sup_compl_top diff_eq less_le_not_le)
haftmann@31991
   268
haftmann@31991
   269
lemma compl_inf_bot:
haftmann@31991
   270
  "- x \<sqinter> x = bot"
haftmann@31991
   271
  by (simp add: inf_commute inf_compl_bot)
haftmann@31991
   272
haftmann@31991
   273
lemma compl_sup_top:
haftmann@31991
   274
  "- x \<squnion> x = top"
haftmann@31991
   275
  by (simp add: sup_commute sup_compl_top)
haftmann@31991
   276
haftmann@31991
   277
lemma inf_bot_left [simp]:
haftmann@31991
   278
  "bot \<sqinter> x = bot"
haftmann@31991
   279
  by (rule inf_absorb1) simp
haftmann@31991
   280
haftmann@31991
   281
lemma inf_bot_right [simp]:
haftmann@31991
   282
  "x \<sqinter> bot = bot"
haftmann@31991
   283
  by (rule inf_absorb2) simp
haftmann@31991
   284
haftmann@31991
   285
lemma sup_top_left [simp]:
haftmann@31991
   286
  "top \<squnion> x = top"
haftmann@31991
   287
  by (rule sup_absorb1) simp
haftmann@31991
   288
haftmann@31991
   289
lemma sup_top_right [simp]:
haftmann@31991
   290
  "x \<squnion> top = top"
haftmann@31991
   291
  by (rule sup_absorb2) simp
haftmann@31991
   292
haftmann@31991
   293
lemma inf_top_left [simp]:
haftmann@31991
   294
  "top \<sqinter> x = x"
haftmann@31991
   295
  by (rule inf_absorb2) simp
haftmann@31991
   296
haftmann@31991
   297
lemma inf_top_right [simp]:
haftmann@31991
   298
  "x \<sqinter> top = x"
haftmann@31991
   299
  by (rule inf_absorb1) simp
haftmann@31991
   300
haftmann@31991
   301
lemma sup_bot_left [simp]:
haftmann@31991
   302
  "bot \<squnion> x = x"
haftmann@31991
   303
  by (rule sup_absorb2) simp
haftmann@31991
   304
haftmann@31991
   305
lemma sup_bot_right [simp]:
haftmann@31991
   306
  "x \<squnion> bot = x"
haftmann@31991
   307
  by (rule sup_absorb1) simp
haftmann@31991
   308
haftmann@31991
   309
lemma compl_unique:
haftmann@31991
   310
  assumes "x \<sqinter> y = bot"
haftmann@31991
   311
    and "x \<squnion> y = top"
haftmann@31991
   312
  shows "- x = y"
haftmann@31991
   313
proof -
haftmann@31991
   314
  have "(x \<sqinter> - x) \<squnion> (- x \<sqinter> y) = (x \<sqinter> y) \<squnion> (- x \<sqinter> y)"
haftmann@31991
   315
    using inf_compl_bot assms(1) by simp
haftmann@31991
   316
  then have "(- x \<sqinter> x) \<squnion> (- x \<sqinter> y) = (y \<sqinter> x) \<squnion> (y \<sqinter> - x)"
haftmann@31991
   317
    by (simp add: inf_commute)
haftmann@31991
   318
  then have "- x \<sqinter> (x \<squnion> y) = y \<sqinter> (x \<squnion> - x)"
haftmann@31991
   319
    by (simp add: inf_sup_distrib1)
haftmann@31991
   320
  then have "- x \<sqinter> top = y \<sqinter> top"
haftmann@31991
   321
    using sup_compl_top assms(2) by simp
haftmann@31991
   322
  then show "- x = y" by (simp add: inf_top_right)
haftmann@31991
   323
qed
haftmann@31991
   324
haftmann@31991
   325
lemma double_compl [simp]:
haftmann@31991
   326
  "- (- x) = x"
haftmann@31991
   327
  using compl_inf_bot compl_sup_top by (rule compl_unique)
haftmann@31991
   328
haftmann@31991
   329
lemma compl_eq_compl_iff [simp]:
haftmann@31991
   330
  "- x = - y \<longleftrightarrow> x = y"
haftmann@31991
   331
proof
haftmann@31991
   332
  assume "- x = - y"
haftmann@31991
   333
  then have "- x \<sqinter> y = bot"
haftmann@31991
   334
    and "- x \<squnion> y = top"
haftmann@31991
   335
    by (simp_all add: compl_inf_bot compl_sup_top)
haftmann@31991
   336
  then have "- (- x) = y" by (rule compl_unique)
haftmann@31991
   337
  then show "x = y" by simp
haftmann@31991
   338
next
haftmann@31991
   339
  assume "x = y"
haftmann@31991
   340
  then show "- x = - y" by simp
haftmann@31991
   341
qed
haftmann@31991
   342
haftmann@31991
   343
lemma compl_bot_eq [simp]:
haftmann@31991
   344
  "- bot = top"
haftmann@31991
   345
proof -
haftmann@31991
   346
  from sup_compl_top have "bot \<squnion> - bot = top" .
haftmann@31991
   347
  then show ?thesis by simp
haftmann@31991
   348
qed
haftmann@31991
   349
haftmann@31991
   350
lemma compl_top_eq [simp]:
haftmann@31991
   351
  "- top = bot"
haftmann@31991
   352
proof -
haftmann@31991
   353
  from inf_compl_bot have "top \<sqinter> - top = bot" .
haftmann@31991
   354
  then show ?thesis by simp
haftmann@31991
   355
qed
haftmann@31991
   356
haftmann@31991
   357
lemma compl_inf [simp]:
haftmann@31991
   358
  "- (x \<sqinter> y) = - x \<squnion> - y"
haftmann@31991
   359
proof (rule compl_unique)
haftmann@31991
   360
  have "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = ((x \<sqinter> y) \<sqinter> - x) \<squnion> ((x \<sqinter> y) \<sqinter> - y)"
haftmann@31991
   361
    by (rule inf_sup_distrib1)
haftmann@31991
   362
  also have "... = (y \<sqinter> (x \<sqinter> - x)) \<squnion> (x \<sqinter> (y \<sqinter> - y))"
haftmann@31991
   363
    by (simp only: inf_commute inf_assoc inf_left_commute)
haftmann@31991
   364
  finally show "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = bot"
haftmann@31991
   365
    by (simp add: inf_compl_bot)
haftmann@31991
   366
next
haftmann@31991
   367
  have "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = (x \<squnion> (- x \<squnion> - y)) \<sqinter> (y \<squnion> (- x \<squnion> - y))"
haftmann@31991
   368
    by (rule sup_inf_distrib2)
haftmann@31991
   369
  also have "... = (- y \<squnion> (x \<squnion> - x)) \<sqinter> (- x \<squnion> (y \<squnion> - y))"
haftmann@31991
   370
    by (simp only: sup_commute sup_assoc sup_left_commute)
haftmann@31991
   371
  finally show "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = top"
haftmann@31991
   372
    by (simp add: sup_compl_top)
haftmann@31991
   373
qed
haftmann@31991
   374
haftmann@31991
   375
lemma compl_sup [simp]:
haftmann@31991
   376
  "- (x \<squnion> y) = - x \<sqinter> - y"
haftmann@31991
   377
proof -
haftmann@31991
   378
  interpret boolean_algebra "\<lambda>x y. x \<squnion> - y" uminus "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" top bot
haftmann@31991
   379
    by (rule dual_boolean_algebra)
haftmann@31991
   380
  then show ?thesis by simp
haftmann@31991
   381
qed
haftmann@31991
   382
haftmann@31991
   383
end
haftmann@31991
   384
haftmann@31991
   385
haftmann@22454
   386
subsection {* Uniqueness of inf and sup *}
haftmann@22454
   387
haftmann@22737
   388
lemma (in lower_semilattice) inf_unique:
haftmann@22454
   389
  fixes f (infixl "\<triangle>" 70)
haftmann@25062
   390
  assumes le1: "\<And>x y. x \<triangle> y \<le> x" and le2: "\<And>x y. x \<triangle> y \<le> y"
haftmann@25062
   391
  and greatest: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z"
haftmann@22737
   392
  shows "x \<sqinter> y = x \<triangle> y"
haftmann@22454
   393
proof (rule antisym)
haftmann@25062
   394
  show "x \<triangle> y \<le> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2)
haftmann@22454
   395
next
haftmann@25062
   396
  have leI: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" by (blast intro: greatest)
haftmann@25062
   397
  show "x \<sqinter> y \<le> x \<triangle> y" by (rule leI) simp_all
haftmann@22454
   398
qed
haftmann@22454
   399
haftmann@22737
   400
lemma (in upper_semilattice) sup_unique:
haftmann@22454
   401
  fixes f (infixl "\<nabla>" 70)
haftmann@25062
   402
  assumes ge1 [simp]: "\<And>x y. x \<le> x \<nabla> y" and ge2: "\<And>x y. y \<le> x \<nabla> y"
haftmann@25062
   403
  and least: "\<And>x y z. y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<nabla> z \<le> x"
haftmann@22737
   404
  shows "x \<squnion> y = x \<nabla> y"
haftmann@22454
   405
proof (rule antisym)
haftmann@25062
   406
  show "x \<squnion> y \<le> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2)
haftmann@22454
   407
next
haftmann@25062
   408
  have leI: "\<And>x y z. x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<nabla> y \<le> z" by (blast intro: least)
haftmann@25062
   409
  show "x \<nabla> y \<le> x \<squnion> y" by (rule leI) simp_all
haftmann@22454
   410
qed
haftmann@22454
   411
  
haftmann@22454
   412
haftmann@22916
   413
subsection {* @{const min}/@{const max} on linear orders as
haftmann@22916
   414
  special case of @{const inf}/@{const sup} *}
haftmann@22916
   415
haftmann@22916
   416
lemma (in linorder) distrib_lattice_min_max:
haftmann@25062
   417
  "distrib_lattice (op \<le>) (op <) min max"
haftmann@28823
   418
proof
haftmann@25062
   419
  have aux: "\<And>x y \<Colon> 'a. x < y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
haftmann@22916
   420
    by (auto simp add: less_le antisym)
haftmann@22916
   421
  fix x y z
haftmann@22916
   422
  show "max x (min y z) = min (max x y) (max x z)"
haftmann@22916
   423
  unfolding min_def max_def
ballarin@24640
   424
  by auto
haftmann@22916
   425
qed (auto simp add: min_def max_def not_le less_imp_le)
haftmann@21249
   426
wenzelm@30729
   427
interpretation min_max: distrib_lattice "op \<le> :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max
haftmann@23948
   428
  by (rule distrib_lattice_min_max)
haftmann@21249
   429
haftmann@22454
   430
lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
haftmann@25102
   431
  by (rule ext)+ (auto intro: antisym)
nipkow@21733
   432
haftmann@22454
   433
lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
haftmann@25102
   434
  by (rule ext)+ (auto intro: antisym)
nipkow@21733
   435
haftmann@21249
   436
lemmas le_maxI1 = min_max.sup_ge1
haftmann@21249
   437
lemmas le_maxI2 = min_max.sup_ge2
haftmann@21381
   438
 
haftmann@21249
   439
lemmas max_ac = min_max.sup_assoc min_max.sup_commute
haftmann@22422
   440
  mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
haftmann@21249
   441
haftmann@21249
   442
lemmas min_ac = min_max.inf_assoc min_max.inf_commute
haftmann@22422
   443
  mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
haftmann@21249
   444
haftmann@22454
   445
haftmann@22454
   446
subsection {* Bool as lattice *}
haftmann@22454
   447
haftmann@31991
   448
instantiation bool :: boolean_algebra
haftmann@25510
   449
begin
haftmann@25510
   450
haftmann@25510
   451
definition
haftmann@31991
   452
  bool_Compl_def: "uminus = Not"
haftmann@31991
   453
haftmann@31991
   454
definition
haftmann@31991
   455
  bool_diff_def: "A - B \<longleftrightarrow> A \<and> \<not> B"
haftmann@31991
   456
haftmann@31991
   457
definition
haftmann@25510
   458
  inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
haftmann@25510
   459
haftmann@25510
   460
definition
haftmann@25510
   461
  sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
haftmann@25510
   462
haftmann@31991
   463
instance proof
haftmann@31991
   464
qed (simp_all add: inf_bool_eq sup_bool_eq le_bool_def
haftmann@31991
   465
  bot_bool_eq top_bool_eq bool_Compl_def bool_diff_def, auto)
haftmann@22454
   466
haftmann@25510
   467
end
haftmann@25510
   468
haftmann@23878
   469
haftmann@23878
   470
subsection {* Fun as lattice *}
haftmann@23878
   471
haftmann@25510
   472
instantiation "fun" :: (type, lattice) lattice
haftmann@25510
   473
begin
haftmann@25510
   474
haftmann@25510
   475
definition
haftmann@28562
   476
  inf_fun_eq [code del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
haftmann@25510
   477
haftmann@25510
   478
definition
haftmann@28562
   479
  sup_fun_eq [code del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
haftmann@25510
   480
haftmann@25510
   481
instance
haftmann@23878
   482
apply intro_classes
haftmann@23878
   483
unfolding inf_fun_eq sup_fun_eq
haftmann@23878
   484
apply (auto intro: le_funI)
haftmann@23878
   485
apply (rule le_funI)
haftmann@23878
   486
apply (auto dest: le_funD)
haftmann@23878
   487
apply (rule le_funI)
haftmann@23878
   488
apply (auto dest: le_funD)
haftmann@23878
   489
done
haftmann@23878
   490
haftmann@25510
   491
end
haftmann@23878
   492
haftmann@23878
   493
instance "fun" :: (type, distrib_lattice) distrib_lattice
haftmann@31991
   494
proof
haftmann@31991
   495
qed (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
haftmann@31991
   496
haftmann@31991
   497
instantiation "fun" :: (type, uminus) uminus
haftmann@31991
   498
begin
haftmann@31991
   499
haftmann@31991
   500
definition
haftmann@31991
   501
  fun_Compl_def: "- A = (\<lambda>x. - A x)"
haftmann@31991
   502
haftmann@31991
   503
instance ..
haftmann@31991
   504
haftmann@31991
   505
end
haftmann@31991
   506
haftmann@31991
   507
instantiation "fun" :: (type, minus) minus
haftmann@31991
   508
begin
haftmann@31991
   509
haftmann@31991
   510
definition
haftmann@31991
   511
  fun_diff_def: "A - B = (\<lambda>x. A x - B x)"
haftmann@31991
   512
haftmann@31991
   513
instance ..
haftmann@31991
   514
haftmann@31991
   515
end
haftmann@31991
   516
haftmann@31991
   517
instance "fun" :: (type, boolean_algebra) boolean_algebra
haftmann@31991
   518
proof
haftmann@31991
   519
qed (simp_all add: inf_fun_eq sup_fun_eq bot_fun_eq top_fun_eq fun_Compl_def fun_diff_def
haftmann@31991
   520
  inf_compl_bot sup_compl_top diff_eq)
haftmann@23878
   521
berghofe@26794
   522
haftmann@23878
   523
text {* redundant bindings *}
haftmann@22454
   524
haftmann@22454
   525
haftmann@25062
   526
no_notation
wenzelm@25382
   527
  less_eq  (infix "\<sqsubseteq>" 50) and
wenzelm@25382
   528
  less (infix "\<sqsubset>" 50) and
wenzelm@25382
   529
  inf  (infixl "\<sqinter>" 70) and
haftmann@30302
   530
  sup  (infixl "\<squnion>" 65)
haftmann@25062
   531
haftmann@21249
   532
end