src/HOL/Lattices.thy
 author haftmann Fri Jun 11 17:14:02 2010 +0200 (2010-06-11) changeset 37407 61dd8c145da7 parent 36673 6d25e8dab1e3 child 37767 a2b7a20d6ea3 permissions -rw-r--r--
declare lex_prod_def [code del]
```     1 (*  Title:      HOL/Lattices.thy
```
```     2     Author:     Tobias Nipkow
```
```     3 *)
```
```     4
```
```     5 header {* Abstract lattices *}
```
```     6
```
```     7 theory Lattices
```
```     8 imports Orderings Groups
```
```     9 begin
```
```    10
```
```    11 subsection {* Abstract semilattice *}
```
```    12
```
```    13 text {*
```
```    14   This locales provide a basic structure for interpretation into
```
```    15   bigger structures;  extensions require careful thinking, otherwise
```
```    16   undesired effects may occur due to interpretation.
```
```    17 *}
```
```    18
```
```    19 locale semilattice = abel_semigroup +
```
```    20   assumes idem [simp]: "f a a = a"
```
```    21 begin
```
```    22
```
```    23 lemma left_idem [simp]:
```
```    24   "f a (f a b) = f a b"
```
```    25   by (simp add: assoc [symmetric])
```
```    26
```
```    27 end
```
```    28
```
```    29
```
```    30 subsection {* Idempotent semigroup *}
```
```    31
```
```    32 class ab_semigroup_idem_mult = ab_semigroup_mult +
```
```    33   assumes mult_idem: "x * x = x"
```
```    34
```
```    35 sublocale ab_semigroup_idem_mult < times!: semilattice times proof
```
```    36 qed (fact mult_idem)
```
```    37
```
```    38 context ab_semigroup_idem_mult
```
```    39 begin
```
```    40
```
```    41 lemmas mult_left_idem = times.left_idem
```
```    42
```
```    43 end
```
```    44
```
```    45
```
```    46 subsection {* Concrete lattices *}
```
```    47
```
```    48 notation
```
```    49   less_eq  (infix "\<sqsubseteq>" 50) and
```
```    50   less  (infix "\<sqsubset>" 50) and
```
```    51   top ("\<top>") and
```
```    52   bot ("\<bottom>")
```
```    53
```
```    54 class semilattice_inf = order +
```
```    55   fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
```
```    56   assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
```
```    57   and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
```
```    58   and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
```
```    59
```
```    60 class semilattice_sup = order +
```
```    61   fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
```
```    62   assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
```
```    63   and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
```
```    64   and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
```
```    65 begin
```
```    66
```
```    67 text {* Dual lattice *}
```
```    68
```
```    69 lemma dual_semilattice:
```
```    70   "class.semilattice_inf (op \<ge>) (op >) sup"
```
```    71 by (rule class.semilattice_inf.intro, rule dual_order)
```
```    72   (unfold_locales, simp_all add: sup_least)
```
```    73
```
```    74 end
```
```    75
```
```    76 class lattice = semilattice_inf + semilattice_sup
```
```    77
```
```    78
```
```    79 subsubsection {* Intro and elim rules*}
```
```    80
```
```    81 context semilattice_inf
```
```    82 begin
```
```    83
```
```    84 lemma le_infI1:
```
```    85   "a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
```
```    86   by (rule order_trans) auto
```
```    87
```
```    88 lemma le_infI2:
```
```    89   "b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
```
```    90   by (rule order_trans) auto
```
```    91
```
```    92 lemma le_infI: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
```
```    93   by (rule inf_greatest) (* FIXME: duplicate lemma *)
```
```    94
```
```    95 lemma le_infE: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"
```
```    96   by (blast intro: order_trans inf_le1 inf_le2)
```
```    97
```
```    98 lemma le_inf_iff [simp]:
```
```    99   "x \<sqsubseteq> y \<sqinter> z \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
```
```   100   by (blast intro: le_infI elim: le_infE)
```
```   101
```
```   102 lemma le_iff_inf:
```
```   103   "x \<sqsubseteq> y \<longleftrightarrow> x \<sqinter> y = x"
```
```   104   by (auto intro: le_infI1 antisym dest: eq_iff [THEN iffD1])
```
```   105
```
```   106 lemma inf_mono: "a \<sqsubseteq> c \<Longrightarrow> b \<le> d \<Longrightarrow> a \<sqinter> b \<sqsubseteq> c \<sqinter> d"
```
```   107   by (fast intro: inf_greatest le_infI1 le_infI2)
```
```   108
```
```   109 lemma mono_inf:
```
```   110   fixes f :: "'a \<Rightarrow> 'b\<Colon>semilattice_inf"
```
```   111   shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<sqsubseteq> f A \<sqinter> f B"
```
```   112   by (auto simp add: mono_def intro: Lattices.inf_greatest)
```
```   113
```
```   114 end
```
```   115
```
```   116 context semilattice_sup
```
```   117 begin
```
```   118
```
```   119 lemma le_supI1:
```
```   120   "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
```
```   121   by (rule order_trans) auto
```
```   122
```
```   123 lemma le_supI2:
```
```   124   "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
```
```   125   by (rule order_trans) auto
```
```   126
```
```   127 lemma le_supI:
```
```   128   "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
```
```   129   by (rule sup_least) (* FIXME: duplicate lemma *)
```
```   130
```
```   131 lemma le_supE:
```
```   132   "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   133   by (blast intro: order_trans sup_ge1 sup_ge2)
```
```   134
```
```   135 lemma le_sup_iff [simp]:
```
```   136   "x \<squnion> y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<and> y \<sqsubseteq> z"
```
```   137   by (blast intro: le_supI elim: le_supE)
```
```   138
```
```   139 lemma le_iff_sup:
```
```   140   "x \<sqsubseteq> y \<longleftrightarrow> x \<squnion> y = y"
```
```   141   by (auto intro: le_supI2 antisym dest: eq_iff [THEN iffD1])
```
```   142
```
```   143 lemma sup_mono: "a \<sqsubseteq> c \<Longrightarrow> b \<le> d \<Longrightarrow> a \<squnion> b \<sqsubseteq> c \<squnion> d"
```
```   144   by (fast intro: sup_least le_supI1 le_supI2)
```
```   145
```
```   146 lemma mono_sup:
```
```   147   fixes f :: "'a \<Rightarrow> 'b\<Colon>semilattice_sup"
```
```   148   shows "mono f \<Longrightarrow> f A \<squnion> f B \<sqsubseteq> f (A \<squnion> B)"
```
```   149   by (auto simp add: mono_def intro: Lattices.sup_least)
```
```   150
```
```   151 end
```
```   152
```
```   153
```
```   154 subsubsection {* Equational laws *}
```
```   155
```
```   156 sublocale semilattice_inf < inf!: semilattice inf
```
```   157 proof
```
```   158   fix a b c
```
```   159   show "(a \<sqinter> b) \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
```
```   160     by (rule antisym) (auto intro: le_infI1 le_infI2)
```
```   161   show "a \<sqinter> b = b \<sqinter> a"
```
```   162     by (rule antisym) auto
```
```   163   show "a \<sqinter> a = a"
```
```   164     by (rule antisym) auto
```
```   165 qed
```
```   166
```
```   167 context semilattice_inf
```
```   168 begin
```
```   169
```
```   170 lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
```
```   171   by (fact inf.assoc)
```
```   172
```
```   173 lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
```
```   174   by (fact inf.commute)
```
```   175
```
```   176 lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
```
```   177   by (fact inf.left_commute)
```
```   178
```
```   179 lemma inf_idem: "x \<sqinter> x = x"
```
```   180   by (fact inf.idem)
```
```   181
```
```   182 lemma inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
```
```   183   by (fact inf.left_idem)
```
```   184
```
```   185 lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
```
```   186   by (rule antisym) auto
```
```   187
```
```   188 lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
```
```   189   by (rule antisym) auto
```
```   190
```
```   191 lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem
```
```   192
```
```   193 end
```
```   194
```
```   195 sublocale semilattice_sup < sup!: semilattice sup
```
```   196 proof
```
```   197   fix a b c
```
```   198   show "(a \<squnion> b) \<squnion> c = a \<squnion> (b \<squnion> c)"
```
```   199     by (rule antisym) (auto intro: le_supI1 le_supI2)
```
```   200   show "a \<squnion> b = b \<squnion> a"
```
```   201     by (rule antisym) auto
```
```   202   show "a \<squnion> a = a"
```
```   203     by (rule antisym) auto
```
```   204 qed
```
```   205
```
```   206 context semilattice_sup
```
```   207 begin
```
```   208
```
```   209 lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
```
```   210   by (fact sup.assoc)
```
```   211
```
```   212 lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
```
```   213   by (fact sup.commute)
```
```   214
```
```   215 lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
```
```   216   by (fact sup.left_commute)
```
```   217
```
```   218 lemma sup_idem: "x \<squnion> x = x"
```
```   219   by (fact sup.idem)
```
```   220
```
```   221 lemma sup_left_idem: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
```
```   222   by (fact sup.left_idem)
```
```   223
```
```   224 lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
```
```   225   by (rule antisym) auto
```
```   226
```
```   227 lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
```
```   228   by (rule antisym) auto
```
```   229
```
```   230 lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem
```
```   231
```
```   232 end
```
```   233
```
```   234 context lattice
```
```   235 begin
```
```   236
```
```   237 lemma dual_lattice:
```
```   238   "class.lattice (op \<ge>) (op >) sup inf"
```
```   239   by (rule class.lattice.intro, rule dual_semilattice, rule class.semilattice_sup.intro, rule dual_order)
```
```   240     (unfold_locales, auto)
```
```   241
```
```   242 lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
```
```   243   by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
```
```   244
```
```   245 lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
```
```   246   by (blast intro: antisym sup_ge1 sup_least inf_le1)
```
```   247
```
```   248 lemmas inf_sup_aci = inf_aci sup_aci
```
```   249
```
```   250 lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
```
```   251
```
```   252 text{* Towards distributivity *}
```
```   253
```
```   254 lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   255   by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
```
```   256
```
```   257 lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
```
```   258   by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
```
```   259
```
```   260 text{* If you have one of them, you have them all. *}
```
```   261
```
```   262 lemma distrib_imp1:
```
```   263 assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   264 shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   265 proof-
```
```   266   have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
```
```   267   also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
```
```   268   also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
```
```   269     by(simp add:inf_sup_absorb inf_commute)
```
```   270   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
```
```   271   finally show ?thesis .
```
```   272 qed
```
```   273
```
```   274 lemma distrib_imp2:
```
```   275 assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   276 shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   277 proof-
```
```   278   have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
```
```   279   also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
```
```   280   also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
```
```   281     by(simp add:sup_inf_absorb sup_commute)
```
```   282   also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
```
```   283   finally show ?thesis .
```
```   284 qed
```
```   285
```
```   286 end
```
```   287
```
```   288 subsubsection {* Strict order *}
```
```   289
```
```   290 context semilattice_inf
```
```   291 begin
```
```   292
```
```   293 lemma less_infI1:
```
```   294   "a \<sqsubset> x \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
```
```   295   by (auto simp add: less_le inf_absorb1 intro: le_infI1)
```
```   296
```
```   297 lemma less_infI2:
```
```   298   "b \<sqsubset> x \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
```
```   299   by (auto simp add: less_le inf_absorb2 intro: le_infI2)
```
```   300
```
```   301 end
```
```   302
```
```   303 context semilattice_sup
```
```   304 begin
```
```   305
```
```   306 lemma less_supI1:
```
```   307   "x \<sqsubset> a \<Longrightarrow> x \<sqsubset> a \<squnion> b"
```
```   308 proof -
```
```   309   interpret dual: semilattice_inf "op \<ge>" "op >" sup
```
```   310     by (fact dual_semilattice)
```
```   311   assume "x \<sqsubset> a"
```
```   312   then show "x \<sqsubset> a \<squnion> b"
```
```   313     by (fact dual.less_infI1)
```
```   314 qed
```
```   315
```
```   316 lemma less_supI2:
```
```   317   "x \<sqsubset> b \<Longrightarrow> x \<sqsubset> a \<squnion> b"
```
```   318 proof -
```
```   319   interpret dual: semilattice_inf "op \<ge>" "op >" sup
```
```   320     by (fact dual_semilattice)
```
```   321   assume "x \<sqsubset> b"
```
```   322   then show "x \<sqsubset> a \<squnion> b"
```
```   323     by (fact dual.less_infI2)
```
```   324 qed
```
```   325
```
```   326 end
```
```   327
```
```   328
```
```   329 subsection {* Distributive lattices *}
```
```   330
```
```   331 class distrib_lattice = lattice +
```
```   332   assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   333
```
```   334 context distrib_lattice
```
```   335 begin
```
```   336
```
```   337 lemma sup_inf_distrib2:
```
```   338  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
```
```   339 by(simp add: inf_sup_aci sup_inf_distrib1)
```
```   340
```
```   341 lemma inf_sup_distrib1:
```
```   342  "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   343 by(rule distrib_imp2[OF sup_inf_distrib1])
```
```   344
```
```   345 lemma inf_sup_distrib2:
```
```   346  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
```
```   347 by(simp add: inf_sup_aci inf_sup_distrib1)
```
```   348
```
```   349 lemma dual_distrib_lattice:
```
```   350   "class.distrib_lattice (op \<ge>) (op >) sup inf"
```
```   351   by (rule class.distrib_lattice.intro, rule dual_lattice)
```
```   352     (unfold_locales, fact inf_sup_distrib1)
```
```   353
```
```   354 lemmas sup_inf_distrib =
```
```   355   sup_inf_distrib1 sup_inf_distrib2
```
```   356
```
```   357 lemmas inf_sup_distrib =
```
```   358   inf_sup_distrib1 inf_sup_distrib2
```
```   359
```
```   360 lemmas distrib =
```
```   361   sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
```
```   362
```
```   363 end
```
```   364
```
```   365
```
```   366 subsection {* Bounded lattices and boolean algebras *}
```
```   367
```
```   368 class bounded_lattice_bot = lattice + bot
```
```   369 begin
```
```   370
```
```   371 lemma inf_bot_left [simp]:
```
```   372   "\<bottom> \<sqinter> x = \<bottom>"
```
```   373   by (rule inf_absorb1) simp
```
```   374
```
```   375 lemma inf_bot_right [simp]:
```
```   376   "x \<sqinter> \<bottom> = \<bottom>"
```
```   377   by (rule inf_absorb2) simp
```
```   378
```
```   379 lemma sup_bot_left [simp]:
```
```   380   "\<bottom> \<squnion> x = x"
```
```   381   by (rule sup_absorb2) simp
```
```   382
```
```   383 lemma sup_bot_right [simp]:
```
```   384   "x \<squnion> \<bottom> = x"
```
```   385   by (rule sup_absorb1) simp
```
```   386
```
```   387 lemma sup_eq_bot_iff [simp]:
```
```   388   "x \<squnion> y = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
```
```   389   by (simp add: eq_iff)
```
```   390
```
```   391 end
```
```   392
```
```   393 class bounded_lattice_top = lattice + top
```
```   394 begin
```
```   395
```
```   396 lemma sup_top_left [simp]:
```
```   397   "\<top> \<squnion> x = \<top>"
```
```   398   by (rule sup_absorb1) simp
```
```   399
```
```   400 lemma sup_top_right [simp]:
```
```   401   "x \<squnion> \<top> = \<top>"
```
```   402   by (rule sup_absorb2) simp
```
```   403
```
```   404 lemma inf_top_left [simp]:
```
```   405   "\<top> \<sqinter> x = x"
```
```   406   by (rule inf_absorb2) simp
```
```   407
```
```   408 lemma inf_top_right [simp]:
```
```   409   "x \<sqinter> \<top> = x"
```
```   410   by (rule inf_absorb1) simp
```
```   411
```
```   412 lemma inf_eq_top_iff [simp]:
```
```   413   "x \<sqinter> y = \<top> \<longleftrightarrow> x = \<top> \<and> y = \<top>"
```
```   414   by (simp add: eq_iff)
```
```   415
```
```   416 end
```
```   417
```
```   418 class bounded_lattice = bounded_lattice_bot + bounded_lattice_top
```
```   419 begin
```
```   420
```
```   421 lemma dual_bounded_lattice:
```
```   422   "class.bounded_lattice (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
```
```   423   by unfold_locales (auto simp add: less_le_not_le)
```
```   424
```
```   425 end
```
```   426
```
```   427 class boolean_algebra = distrib_lattice + bounded_lattice + minus + uminus +
```
```   428   assumes inf_compl_bot: "x \<sqinter> - x = \<bottom>"
```
```   429     and sup_compl_top: "x \<squnion> - x = \<top>"
```
```   430   assumes diff_eq: "x - y = x \<sqinter> - y"
```
```   431 begin
```
```   432
```
```   433 lemma dual_boolean_algebra:
```
```   434   "class.boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
```
```   435   by (rule class.boolean_algebra.intro, rule dual_bounded_lattice, rule dual_distrib_lattice)
```
```   436     (unfold_locales, auto simp add: inf_compl_bot sup_compl_top diff_eq)
```
```   437
```
```   438 lemma compl_inf_bot:
```
```   439   "- x \<sqinter> x = \<bottom>"
```
```   440   by (simp add: inf_commute inf_compl_bot)
```
```   441
```
```   442 lemma compl_sup_top:
```
```   443   "- x \<squnion> x = \<top>"
```
```   444   by (simp add: sup_commute sup_compl_top)
```
```   445
```
```   446 lemma compl_unique:
```
```   447   assumes "x \<sqinter> y = \<bottom>"
```
```   448     and "x \<squnion> y = \<top>"
```
```   449   shows "- x = y"
```
```   450 proof -
```
```   451   have "(x \<sqinter> - x) \<squnion> (- x \<sqinter> y) = (x \<sqinter> y) \<squnion> (- x \<sqinter> y)"
```
```   452     using inf_compl_bot assms(1) by simp
```
```   453   then have "(- x \<sqinter> x) \<squnion> (- x \<sqinter> y) = (y \<sqinter> x) \<squnion> (y \<sqinter> - x)"
```
```   454     by (simp add: inf_commute)
```
```   455   then have "- x \<sqinter> (x \<squnion> y) = y \<sqinter> (x \<squnion> - x)"
```
```   456     by (simp add: inf_sup_distrib1)
```
```   457   then have "- x \<sqinter> \<top> = y \<sqinter> \<top>"
```
```   458     using sup_compl_top assms(2) by simp
```
```   459   then show "- x = y" by simp
```
```   460 qed
```
```   461
```
```   462 lemma double_compl [simp]:
```
```   463   "- (- x) = x"
```
```   464   using compl_inf_bot compl_sup_top by (rule compl_unique)
```
```   465
```
```   466 lemma compl_eq_compl_iff [simp]:
```
```   467   "- x = - y \<longleftrightarrow> x = y"
```
```   468 proof
```
```   469   assume "- x = - y"
```
```   470   then have "- (- x) = - (- y)" by (rule arg_cong)
```
```   471   then show "x = y" by simp
```
```   472 next
```
```   473   assume "x = y"
```
```   474   then show "- x = - y" by simp
```
```   475 qed
```
```   476
```
```   477 lemma compl_bot_eq [simp]:
```
```   478   "- \<bottom> = \<top>"
```
```   479 proof -
```
```   480   from sup_compl_top have "\<bottom> \<squnion> - \<bottom> = \<top>" .
```
```   481   then show ?thesis by simp
```
```   482 qed
```
```   483
```
```   484 lemma compl_top_eq [simp]:
```
```   485   "- \<top> = \<bottom>"
```
```   486 proof -
```
```   487   from inf_compl_bot have "\<top> \<sqinter> - \<top> = \<bottom>" .
```
```   488   then show ?thesis by simp
```
```   489 qed
```
```   490
```
```   491 lemma compl_inf [simp]:
```
```   492   "- (x \<sqinter> y) = - x \<squnion> - y"
```
```   493 proof (rule compl_unique)
```
```   494   have "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = (y \<sqinter> (x \<sqinter> - x)) \<squnion> (x \<sqinter> (y \<sqinter> - y))"
```
```   495     by (simp only: inf_sup_distrib inf_aci)
```
```   496   then show "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = \<bottom>"
```
```   497     by (simp add: inf_compl_bot)
```
```   498 next
```
```   499   have "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = (- y \<squnion> (x \<squnion> - x)) \<sqinter> (- x \<squnion> (y \<squnion> - y))"
```
```   500     by (simp only: sup_inf_distrib sup_aci)
```
```   501   then show "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = \<top>"
```
```   502     by (simp add: sup_compl_top)
```
```   503 qed
```
```   504
```
```   505 lemma compl_sup [simp]:
```
```   506   "- (x \<squnion> y) = - x \<sqinter> - y"
```
```   507 proof -
```
```   508   interpret boolean_algebra "\<lambda>x y. x \<squnion> - y" uminus "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" \<top> \<bottom>
```
```   509     by (rule dual_boolean_algebra)
```
```   510   then show ?thesis by simp
```
```   511 qed
```
```   512
```
```   513 lemma compl_mono:
```
```   514   "x \<sqsubseteq> y \<Longrightarrow> - y \<sqsubseteq> - x"
```
```   515 proof -
```
```   516   assume "x \<sqsubseteq> y"
```
```   517   then have "x \<squnion> y = y" by (simp only: le_iff_sup)
```
```   518   then have "- (x \<squnion> y) = - y" by simp
```
```   519   then have "- x \<sqinter> - y = - y" by simp
```
```   520   then have "- y \<sqinter> - x = - y" by (simp only: inf_commute)
```
```   521   then show "- y \<sqsubseteq> - x" by (simp only: le_iff_inf)
```
```   522 qed
```
```   523
```
```   524 lemma compl_le_compl_iff: (* TODO: declare [simp] ? *)
```
```   525   "- x \<le> - y \<longleftrightarrow> y \<le> x"
```
```   526 by (auto dest: compl_mono)
```
```   527
```
```   528 end
```
```   529
```
```   530
```
```   531 subsection {* Uniqueness of inf and sup *}
```
```   532
```
```   533 lemma (in semilattice_inf) inf_unique:
```
```   534   fixes f (infixl "\<triangle>" 70)
```
```   535   assumes le1: "\<And>x y. x \<triangle> y \<sqsubseteq> x" and le2: "\<And>x y. x \<triangle> y \<sqsubseteq> y"
```
```   536   and greatest: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<triangle> z"
```
```   537   shows "x \<sqinter> y = x \<triangle> y"
```
```   538 proof (rule antisym)
```
```   539   show "x \<triangle> y \<sqsubseteq> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2)
```
```   540 next
```
```   541   have leI: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<triangle> z" by (blast intro: greatest)
```
```   542   show "x \<sqinter> y \<sqsubseteq> x \<triangle> y" by (rule leI) simp_all
```
```   543 qed
```
```   544
```
```   545 lemma (in semilattice_sup) sup_unique:
```
```   546   fixes f (infixl "\<nabla>" 70)
```
```   547   assumes ge1 [simp]: "\<And>x y. x \<sqsubseteq> x \<nabla> y" and ge2: "\<And>x y. y \<sqsubseteq> x \<nabla> y"
```
```   548   and least: "\<And>x y z. y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<nabla> z \<sqsubseteq> x"
```
```   549   shows "x \<squnion> y = x \<nabla> y"
```
```   550 proof (rule antisym)
```
```   551   show "x \<squnion> y \<sqsubseteq> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2)
```
```   552 next
```
```   553   have leI: "\<And>x y z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<nabla> y \<sqsubseteq> z" by (blast intro: least)
```
```   554   show "x \<nabla> y \<sqsubseteq> x \<squnion> y" by (rule leI) simp_all
```
```   555 qed
```
```   556
```
```   557
```
```   558 subsection {* @{const min}/@{const max} on linear orders as
```
```   559   special case of @{const inf}/@{const sup} *}
```
```   560
```
```   561 sublocale linorder < min_max!: distrib_lattice less_eq less min max
```
```   562 proof
```
```   563   fix x y z
```
```   564   show "max x (min y z) = min (max x y) (max x z)"
```
```   565     by (auto simp add: min_def max_def)
```
```   566 qed (auto simp add: min_def max_def not_le less_imp_le)
```
```   567
```
```   568 lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{semilattice_inf, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```   569   by (rule ext)+ (auto intro: antisym)
```
```   570
```
```   571 lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{semilattice_sup, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```   572   by (rule ext)+ (auto intro: antisym)
```
```   573
```
```   574 lemmas le_maxI1 = min_max.sup_ge1
```
```   575 lemmas le_maxI2 = min_max.sup_ge2
```
```   576
```
```   577 lemmas min_ac = min_max.inf_assoc min_max.inf_commute
```
```   578   min_max.inf.left_commute
```
```   579
```
```   580 lemmas max_ac = min_max.sup_assoc min_max.sup_commute
```
```   581   min_max.sup.left_commute
```
```   582
```
```   583
```
```   584 subsection {* Bool as lattice *}
```
```   585
```
```   586 instantiation bool :: boolean_algebra
```
```   587 begin
```
```   588
```
```   589 definition
```
```   590   bool_Compl_def: "uminus = Not"
```
```   591
```
```   592 definition
```
```   593   bool_diff_def: "A - B \<longleftrightarrow> A \<and> \<not> B"
```
```   594
```
```   595 definition
```
```   596   inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
```
```   597
```
```   598 definition
```
```   599   sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
```
```   600
```
```   601 instance proof
```
```   602 qed (simp_all add: inf_bool_eq sup_bool_eq le_bool_def
```
```   603   bot_bool_eq top_bool_eq bool_Compl_def bool_diff_def, auto)
```
```   604
```
```   605 end
```
```   606
```
```   607 lemma sup_boolI1:
```
```   608   "P \<Longrightarrow> P \<squnion> Q"
```
```   609   by (simp add: sup_bool_eq)
```
```   610
```
```   611 lemma sup_boolI2:
```
```   612   "Q \<Longrightarrow> P \<squnion> Q"
```
```   613   by (simp add: sup_bool_eq)
```
```   614
```
```   615 lemma sup_boolE:
```
```   616   "P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
```
```   617   by (auto simp add: sup_bool_eq)
```
```   618
```
```   619
```
```   620 subsection {* Fun as lattice *}
```
```   621
```
```   622 instantiation "fun" :: (type, lattice) lattice
```
```   623 begin
```
```   624
```
```   625 definition
```
```   626   inf_fun_eq [code del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
```
```   627
```
```   628 definition
```
```   629   sup_fun_eq [code del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
```
```   630
```
```   631 instance proof
```
```   632 qed (simp_all add: le_fun_def inf_fun_eq sup_fun_eq)
```
```   633
```
```   634 end
```
```   635
```
```   636 instance "fun" :: (type, distrib_lattice) distrib_lattice
```
```   637 proof
```
```   638 qed (simp_all add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
```
```   639
```
```   640 instance "fun" :: (type, bounded_lattice) bounded_lattice ..
```
```   641
```
```   642 instantiation "fun" :: (type, uminus) uminus
```
```   643 begin
```
```   644
```
```   645 definition
```
```   646   fun_Compl_def: "- A = (\<lambda>x. - A x)"
```
```   647
```
```   648 instance ..
```
```   649
```
```   650 end
```
```   651
```
```   652 instantiation "fun" :: (type, minus) minus
```
```   653 begin
```
```   654
```
```   655 definition
```
```   656   fun_diff_def: "A - B = (\<lambda>x. A x - B x)"
```
```   657
```
```   658 instance ..
```
```   659
```
```   660 end
```
```   661
```
```   662 instance "fun" :: (type, boolean_algebra) boolean_algebra
```
```   663 proof
```
```   664 qed (simp_all add: inf_fun_eq sup_fun_eq bot_fun_eq top_fun_eq fun_Compl_def fun_diff_def
```
```   665   inf_compl_bot sup_compl_top diff_eq)
```
```   666
```
```   667
```
```   668 no_notation
```
```   669   less_eq  (infix "\<sqsubseteq>" 50) and
```
```   670   less (infix "\<sqsubset>" 50) and
```
```   671   inf  (infixl "\<sqinter>" 70) and
```
```   672   sup  (infixl "\<squnion>" 65) and
```
```   673   top ("\<top>") and
```
```   674   bot ("\<bottom>")
```
```   675
```
```   676 end
```