src/HOL/Lattices.thy
 author haftmann Wed Jan 30 10:57:46 2008 +0100 (2008-01-30) changeset 26014 00c2c3525bef parent 25510 38c15efe603b child 26233 3751b3dbb67c permissions -rw-r--r--
dual orders and dual lattices
```     1 (*  Title:      HOL/Lattices.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow
```
```     4 *)
```
```     5
```
```     6 header {* Abstract lattices *}
```
```     7
```
```     8 theory Lattices
```
```     9 imports Orderings
```
```    10 begin
```
```    11
```
```    12 subsection{* Lattices *}
```
```    13
```
```    14 notation
```
```    15   less_eq  (infix "\<sqsubseteq>" 50) and
```
```    16   less  (infix "\<sqsubset>" 50)
```
```    17
```
```    18 class lower_semilattice = order +
```
```    19   fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
```
```    20   assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
```
```    21   and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
```
```    22   and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
```
```    23
```
```    24 class upper_semilattice = order +
```
```    25   fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
```
```    26   assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
```
```    27   and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
```
```    28   and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
```
```    29 begin
```
```    30
```
```    31 text {* Dual lattice *}
```
```    32
```
```    33 lemma dual_lattice:
```
```    34   "lower_semilattice (op \<ge>) (op >) sup"
```
```    35 by unfold_locales
```
```    36   (auto simp add: sup_least)
```
```    37
```
```    38 end
```
```    39
```
```    40 class lattice = lower_semilattice + upper_semilattice
```
```    41
```
```    42
```
```    43 subsubsection{* Intro and elim rules*}
```
```    44
```
```    45 context lower_semilattice
```
```    46 begin
```
```    47
```
```    48 lemma le_infI1[intro]:
```
```    49   assumes "a \<sqsubseteq> x"
```
```    50   shows "a \<sqinter> b \<sqsubseteq> x"
```
```    51 proof (rule order_trans)
```
```    52   from assms show "a \<sqsubseteq> x" .
```
```    53   show "a \<sqinter> b \<sqsubseteq> a" by simp
```
```    54 qed
```
```    55 lemmas (in -) [rule del] = le_infI1
```
```    56
```
```    57 lemma le_infI2[intro]:
```
```    58   assumes "b \<sqsubseteq> x"
```
```    59   shows "a \<sqinter> b \<sqsubseteq> x"
```
```    60 proof (rule order_trans)
```
```    61   from assms show "b \<sqsubseteq> x" .
```
```    62   show "a \<sqinter> b \<sqsubseteq> b" by simp
```
```    63 qed
```
```    64 lemmas (in -) [rule del] = le_infI2
```
```    65
```
```    66 lemma le_infI[intro!]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
```
```    67 by(blast intro: inf_greatest)
```
```    68 lemmas (in -) [rule del] = le_infI
```
```    69
```
```    70 lemma le_infE [elim!]: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"
```
```    71   by (blast intro: order_trans)
```
```    72 lemmas (in -) [rule del] = le_infE
```
```    73
```
```    74 lemma le_inf_iff [simp]:
```
```    75   "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
```
```    76 by blast
```
```    77
```
```    78 lemma le_iff_inf: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
```
```    79   by (blast intro: antisym dest: eq_iff [THEN iffD1])
```
```    80
```
```    81 lemma mono_inf:
```
```    82   fixes f :: "'a \<Rightarrow> 'b\<Colon>lower_semilattice"
```
```    83   shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<le> f A \<sqinter> f B"
```
```    84   by (auto simp add: mono_def intro: Lattices.inf_greatest)
```
```    85
```
```    86 end
```
```    87
```
```    88 context upper_semilattice
```
```    89 begin
```
```    90
```
```    91 lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
```
```    92   by (rule order_trans) auto
```
```    93 lemmas (in -) [rule del] = le_supI1
```
```    94
```
```    95 lemma le_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
```
```    96   by (rule order_trans) auto
```
```    97 lemmas (in -) [rule del] = le_supI2
```
```    98
```
```    99 lemma le_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
```
```   100   by (blast intro: sup_least)
```
```   101 lemmas (in -) [rule del] = le_supI
```
```   102
```
```   103 lemma le_supE[elim!]: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   104   by (blast intro: order_trans)
```
```   105 lemmas (in -) [rule del] = le_supE
```
```   106
```
```   107 lemma ge_sup_conv[simp]:
```
```   108   "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
```
```   109 by blast
```
```   110
```
```   111 lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
```
```   112   by (blast intro: antisym dest: eq_iff [THEN iffD1])
```
```   113
```
```   114 lemma mono_sup:
```
```   115   fixes f :: "'a \<Rightarrow> 'b\<Colon>upper_semilattice"
```
```   116   shows "mono f \<Longrightarrow> f A \<squnion> f B \<le> f (A \<squnion> B)"
```
```   117   by (auto simp add: mono_def intro: Lattices.sup_least)
```
```   118
```
```   119 end
```
```   120
```
```   121
```
```   122 subsubsection{* Equational laws *}
```
```   123
```
```   124 context lower_semilattice
```
```   125 begin
```
```   126
```
```   127 lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
```
```   128   by (blast intro: antisym)
```
```   129
```
```   130 lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
```
```   131   by (blast intro: antisym)
```
```   132
```
```   133 lemma inf_idem[simp]: "x \<sqinter> x = x"
```
```   134   by (blast intro: antisym)
```
```   135
```
```   136 lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
```
```   137   by (blast intro: antisym)
```
```   138
```
```   139 lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
```
```   140   by (blast intro: antisym)
```
```   141
```
```   142 lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
```
```   143   by (blast intro: antisym)
```
```   144
```
```   145 lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
```
```   146   by (blast intro: antisym)
```
```   147
```
```   148 lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem
```
```   149
```
```   150 end
```
```   151
```
```   152
```
```   153 context upper_semilattice
```
```   154 begin
```
```   155
```
```   156 lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
```
```   157   by (blast intro: antisym)
```
```   158
```
```   159 lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
```
```   160   by (blast intro: antisym)
```
```   161
```
```   162 lemma sup_idem[simp]: "x \<squnion> x = x"
```
```   163   by (blast intro: antisym)
```
```   164
```
```   165 lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
```
```   166   by (blast intro: antisym)
```
```   167
```
```   168 lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
```
```   169   by (blast intro: antisym)
```
```   170
```
```   171 lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
```
```   172   by (blast intro: antisym)
```
```   173
```
```   174 lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
```
```   175   by (blast intro: antisym)
```
```   176
```
```   177 lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem
```
```   178
```
```   179 end
```
```   180
```
```   181 context lattice
```
```   182 begin
```
```   183
```
```   184 lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
```
```   185   by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
```
```   186
```
```   187 lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
```
```   188   by (blast intro: antisym sup_ge1 sup_least inf_le1)
```
```   189
```
```   190 lemmas ACI = inf_ACI sup_ACI
```
```   191
```
```   192 lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
```
```   193
```
```   194 text{* Towards distributivity *}
```
```   195
```
```   196 lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   197   by blast
```
```   198
```
```   199 lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
```
```   200   by blast
```
```   201
```
```   202
```
```   203 text{* If you have one of them, you have them all. *}
```
```   204
```
```   205 lemma distrib_imp1:
```
```   206 assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   207 shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   208 proof-
```
```   209   have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
```
```   210   also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
```
```   211   also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
```
```   212     by(simp add:inf_sup_absorb inf_commute)
```
```   213   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
```
```   214   finally show ?thesis .
```
```   215 qed
```
```   216
```
```   217 lemma distrib_imp2:
```
```   218 assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   219 shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   220 proof-
```
```   221   have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
```
```   222   also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
```
```   223   also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
```
```   224     by(simp add:sup_inf_absorb sup_commute)
```
```   225   also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
```
```   226   finally show ?thesis .
```
```   227 qed
```
```   228
```
```   229 (* seems unused *)
```
```   230 lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z"
```
```   231 by blast
```
```   232
```
```   233 end
```
```   234
```
```   235
```
```   236 subsection {* Distributive lattices *}
```
```   237
```
```   238 class distrib_lattice = lattice +
```
```   239   assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   240
```
```   241 context distrib_lattice
```
```   242 begin
```
```   243
```
```   244 lemma sup_inf_distrib2:
```
```   245  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
```
```   246 by(simp add:ACI sup_inf_distrib1)
```
```   247
```
```   248 lemma inf_sup_distrib1:
```
```   249  "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   250 by(rule distrib_imp2[OF sup_inf_distrib1])
```
```   251
```
```   252 lemma inf_sup_distrib2:
```
```   253  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
```
```   254 by(simp add:ACI inf_sup_distrib1)
```
```   255
```
```   256 lemmas distrib =
```
```   257   sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
```
```   258
```
```   259 end
```
```   260
```
```   261
```
```   262 subsection {* Uniqueness of inf and sup *}
```
```   263
```
```   264 lemma (in lower_semilattice) inf_unique:
```
```   265   fixes f (infixl "\<triangle>" 70)
```
```   266   assumes le1: "\<And>x y. x \<triangle> y \<le> x" and le2: "\<And>x y. x \<triangle> y \<le> y"
```
```   267   and greatest: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z"
```
```   268   shows "x \<sqinter> y = x \<triangle> y"
```
```   269 proof (rule antisym)
```
```   270   show "x \<triangle> y \<le> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2)
```
```   271 next
```
```   272   have leI: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" by (blast intro: greatest)
```
```   273   show "x \<sqinter> y \<le> x \<triangle> y" by (rule leI) simp_all
```
```   274 qed
```
```   275
```
```   276 lemma (in upper_semilattice) sup_unique:
```
```   277   fixes f (infixl "\<nabla>" 70)
```
```   278   assumes ge1 [simp]: "\<And>x y. x \<le> x \<nabla> y" and ge2: "\<And>x y. y \<le> x \<nabla> y"
```
```   279   and least: "\<And>x y z. y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<nabla> z \<le> x"
```
```   280   shows "x \<squnion> y = x \<nabla> y"
```
```   281 proof (rule antisym)
```
```   282   show "x \<squnion> y \<le> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2)
```
```   283 next
```
```   284   have leI: "\<And>x y z. x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<nabla> y \<le> z" by (blast intro: least)
```
```   285   show "x \<nabla> y \<le> x \<squnion> y" by (rule leI) simp_all
```
```   286 qed
```
```   287
```
```   288
```
```   289 subsection {* @{const min}/@{const max} on linear orders as
```
```   290   special case of @{const inf}/@{const sup} *}
```
```   291
```
```   292 lemma (in linorder) distrib_lattice_min_max:
```
```   293   "distrib_lattice (op \<le>) (op <) min max"
```
```   294 proof unfold_locales
```
```   295   have aux: "\<And>x y \<Colon> 'a. x < y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
```
```   296     by (auto simp add: less_le antisym)
```
```   297   fix x y z
```
```   298   show "max x (min y z) = min (max x y) (max x z)"
```
```   299   unfolding min_def max_def
```
```   300   by auto
```
```   301 qed (auto simp add: min_def max_def not_le less_imp_le)
```
```   302
```
```   303 interpretation min_max:
```
```   304   distrib_lattice ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max]
```
```   305   by (rule distrib_lattice_min_max)
```
```   306
```
```   307 lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```   308   by (rule ext)+ (auto intro: antisym)
```
```   309
```
```   310 lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```   311   by (rule ext)+ (auto intro: antisym)
```
```   312
```
```   313 lemmas le_maxI1 = min_max.sup_ge1
```
```   314 lemmas le_maxI2 = min_max.sup_ge2
```
```   315
```
```   316 lemmas max_ac = min_max.sup_assoc min_max.sup_commute
```
```   317   mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
```
```   318
```
```   319 lemmas min_ac = min_max.inf_assoc min_max.inf_commute
```
```   320   mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
```
```   321
```
```   322 text {*
```
```   323   Now we have inherited antisymmetry as an intro-rule on all
```
```   324   linear orders. This is a problem because it applies to bool, which is
```
```   325   undesirable.
```
```   326 *}
```
```   327
```
```   328 lemmas [rule del] = min_max.le_infI min_max.le_supI
```
```   329   min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
```
```   330   min_max.le_infI1 min_max.le_infI2
```
```   331
```
```   332
```
```   333 subsection {* Complete lattices *}
```
```   334
```
```   335 class complete_lattice = lattice +
```
```   336   fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
```
```   337     and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
```
```   338   assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
```
```   339      and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
```
```   340   assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
```
```   341      and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
```
```   342 begin
```
```   343
```
```   344 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
```
```   345   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   346
```
```   347 lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
```
```   348   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   349
```
```   350 lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
```
```   351   unfolding Sup_Inf by auto
```
```   352
```
```   353 lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
```
```   354   unfolding Inf_Sup by auto
```
```   355
```
```   356 lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
```
```   357   apply (rule antisym)
```
```   358   apply (rule le_infI)
```
```   359   apply (rule Inf_lower)
```
```   360   apply simp
```
```   361   apply (rule Inf_greatest)
```
```   362   apply (rule Inf_lower)
```
```   363   apply simp
```
```   364   apply (rule Inf_greatest)
```
```   365   apply (erule insertE)
```
```   366   apply (rule le_infI1)
```
```   367   apply simp
```
```   368   apply (rule le_infI2)
```
```   369   apply (erule Inf_lower)
```
```   370   done
```
```   371
```
```   372 lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
```
```   373   apply (rule antisym)
```
```   374   apply (rule Sup_least)
```
```   375   apply (erule insertE)
```
```   376   apply (rule le_supI1)
```
```   377   apply simp
```
```   378   apply (rule le_supI2)
```
```   379   apply (erule Sup_upper)
```
```   380   apply (rule le_supI)
```
```   381   apply (rule Sup_upper)
```
```   382   apply simp
```
```   383   apply (rule Sup_least)
```
```   384   apply (rule Sup_upper)
```
```   385   apply simp
```
```   386   done
```
```   387
```
```   388 lemma Inf_singleton [simp]:
```
```   389   "\<Sqinter>{a} = a"
```
```   390   by (auto intro: antisym Inf_lower Inf_greatest)
```
```   391
```
```   392 lemma Sup_singleton [simp]:
```
```   393   "\<Squnion>{a} = a"
```
```   394   by (auto intro: antisym Sup_upper Sup_least)
```
```   395
```
```   396 lemma Inf_insert_simp:
```
```   397   "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
```
```   398   by (cases "A = {}") (simp_all, simp add: Inf_insert)
```
```   399
```
```   400 lemma Sup_insert_simp:
```
```   401   "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
```
```   402   by (cases "A = {}") (simp_all, simp add: Sup_insert)
```
```   403
```
```   404 lemma Inf_binary:
```
```   405   "\<Sqinter>{a, b} = a \<sqinter> b"
```
```   406   by (simp add: Inf_insert_simp)
```
```   407
```
```   408 lemma Sup_binary:
```
```   409   "\<Squnion>{a, b} = a \<squnion> b"
```
```   410   by (simp add: Sup_insert_simp)
```
```   411
```
```   412 definition
```
```   413   top :: 'a where
```
```   414   "top = \<Sqinter>{}"
```
```   415
```
```   416 definition
```
```   417   bot :: 'a where
```
```   418   "bot = \<Squnion>{}"
```
```   419
```
```   420 lemma top_greatest [simp]: "x \<le> top"
```
```   421   by (unfold top_def, rule Inf_greatest, simp)
```
```   422
```
```   423 lemma bot_least [simp]: "bot \<le> x"
```
```   424   by (unfold bot_def, rule Sup_least, simp)
```
```   425
```
```   426 definition
```
```   427   SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
```
```   428 where
```
```   429   "SUPR A f == \<Squnion> (f ` A)"
```
```   430
```
```   431 definition
```
```   432   INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
```
```   433 where
```
```   434   "INFI A f == \<Sqinter> (f ` A)"
```
```   435
```
```   436 end
```
```   437
```
```   438 syntax
```
```   439   "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
```
```   440   "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
```
```   441   "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
```
```   442   "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
```
```   443
```
```   444 translations
```
```   445   "SUP x y. B"   == "SUP x. SUP y. B"
```
```   446   "SUP x. B"     == "CONST SUPR UNIV (%x. B)"
```
```   447   "SUP x. B"     == "SUP x:UNIV. B"
```
```   448   "SUP x:A. B"   == "CONST SUPR A (%x. B)"
```
```   449   "INF x y. B"   == "INF x. INF y. B"
```
```   450   "INF x. B"     == "CONST INFI UNIV (%x. B)"
```
```   451   "INF x. B"     == "INF x:UNIV. B"
```
```   452   "INF x:A. B"   == "CONST INFI A (%x. B)"
```
```   453
```
```   454 (* To avoid eta-contraction of body: *)
```
```   455 print_translation {*
```
```   456 let
```
```   457   fun btr' syn (A :: Abs abs :: ts) =
```
```   458     let val (x,t) = atomic_abs_tr' abs
```
```   459     in list_comb (Syntax.const syn \$ x \$ A \$ t, ts) end
```
```   460   val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
```
```   461 in
```
```   462 [(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
```
```   463 end
```
```   464 *}
```
```   465
```
```   466 context complete_lattice
```
```   467 begin
```
```   468
```
```   469 lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
```
```   470   by (auto simp add: SUPR_def intro: Sup_upper)
```
```   471
```
```   472 lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
```
```   473   by (auto simp add: SUPR_def intro: Sup_least)
```
```   474
```
```   475 lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
```
```   476   by (auto simp add: INFI_def intro: Inf_lower)
```
```   477
```
```   478 lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
```
```   479   by (auto simp add: INFI_def intro: Inf_greatest)
```
```   480
```
```   481 lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
```
```   482   by (auto intro: antisym SUP_leI le_SUPI)
```
```   483
```
```   484 lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
```
```   485   by (auto intro: antisym INF_leI le_INFI)
```
```   486
```
```   487 end
```
```   488
```
```   489
```
```   490 subsection {* Bool as lattice *}
```
```   491
```
```   492 instantiation bool :: distrib_lattice
```
```   493 begin
```
```   494
```
```   495 definition
```
```   496   inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
```
```   497
```
```   498 definition
```
```   499   sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
```
```   500
```
```   501 instance
```
```   502   by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
```
```   503
```
```   504 end
```
```   505
```
```   506 instantiation bool :: complete_lattice
```
```   507 begin
```
```   508
```
```   509 definition
```
```   510   Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
```
```   511
```
```   512 definition
```
```   513   Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
```
```   514
```
```   515 instance
```
```   516   by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
```
```   517
```
```   518 end
```
```   519
```
```   520 lemma Inf_empty_bool [simp]:
```
```   521   "\<Sqinter>{}"
```
```   522   unfolding Inf_bool_def by auto
```
```   523
```
```   524 lemma not_Sup_empty_bool [simp]:
```
```   525   "\<not> Sup {}"
```
```   526   unfolding Sup_bool_def by auto
```
```   527
```
```   528 lemma top_bool_eq: "top = True"
```
```   529   by (iprover intro!: order_antisym le_boolI top_greatest)
```
```   530
```
```   531 lemma bot_bool_eq: "bot = False"
```
```   532   by (iprover intro!: order_antisym le_boolI bot_least)
```
```   533
```
```   534
```
```   535 subsection {* Set as lattice *}
```
```   536
```
```   537 instantiation set :: (type) distrib_lattice
```
```   538 begin
```
```   539
```
```   540 definition
```
```   541   inf_set_eq [code func del]: "A \<sqinter> B = A \<inter> B"
```
```   542
```
```   543 definition
```
```   544   sup_set_eq [code func del]: "A \<squnion> B = A \<union> B"
```
```   545
```
```   546 instance
```
```   547   by intro_classes (auto simp add: inf_set_eq sup_set_eq)
```
```   548
```
```   549 end
```
```   550
```
```   551 lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
```
```   552   apply (fold inf_set_eq sup_set_eq)
```
```   553   apply (erule mono_inf)
```
```   554   done
```
```   555
```
```   556 lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
```
```   557   apply (fold inf_set_eq sup_set_eq)
```
```   558   apply (erule mono_sup)
```
```   559   done
```
```   560
```
```   561 instantiation set :: (type) complete_lattice
```
```   562 begin
```
```   563
```
```   564 definition
```
```   565   Inf_set_def [code func del]: "\<Sqinter>S \<equiv> \<Inter>S"
```
```   566
```
```   567 definition
```
```   568   Sup_set_def [code func del]: "\<Squnion>S \<equiv> \<Union>S"
```
```   569
```
```   570 instance
```
```   571   by intro_classes (auto simp add: Inf_set_def Sup_set_def)
```
```   572
```
```   573 end
```
```   574
```
```   575 lemma top_set_eq: "top = UNIV"
```
```   576   by (iprover intro!: subset_antisym subset_UNIV top_greatest)
```
```   577
```
```   578 lemma bot_set_eq: "bot = {}"
```
```   579   by (iprover intro!: subset_antisym empty_subsetI bot_least)
```
```   580
```
```   581
```
```   582 subsection {* Fun as lattice *}
```
```   583
```
```   584 instantiation "fun" :: (type, lattice) lattice
```
```   585 begin
```
```   586
```
```   587 definition
```
```   588   inf_fun_eq [code func del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
```
```   589
```
```   590 definition
```
```   591   sup_fun_eq [code func del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
```
```   592
```
```   593 instance
```
```   594 apply intro_classes
```
```   595 unfolding inf_fun_eq sup_fun_eq
```
```   596 apply (auto intro: le_funI)
```
```   597 apply (rule le_funI)
```
```   598 apply (auto dest: le_funD)
```
```   599 apply (rule le_funI)
```
```   600 apply (auto dest: le_funD)
```
```   601 done
```
```   602
```
```   603 end
```
```   604
```
```   605 instance "fun" :: (type, distrib_lattice) distrib_lattice
```
```   606   by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
```
```   607
```
```   608 instantiation "fun" :: (type, complete_lattice) complete_lattice
```
```   609 begin
```
```   610
```
```   611 definition
```
```   612   Inf_fun_def [code func del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
```
```   613
```
```   614 definition
```
```   615   Sup_fun_def [code func del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
```
```   616
```
```   617 instance
```
```   618   by intro_classes
```
```   619     (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
```
```   620       intro: Inf_lower Sup_upper Inf_greatest Sup_least)
```
```   621
```
```   622 end
```
```   623
```
```   624 lemma Inf_empty_fun:
```
```   625   "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
```
```   626   by rule (auto simp add: Inf_fun_def)
```
```   627
```
```   628 lemma Sup_empty_fun:
```
```   629   "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
```
```   630   by rule (auto simp add: Sup_fun_def)
```
```   631
```
```   632 lemma top_fun_eq: "top = (\<lambda>x. top)"
```
```   633   by (iprover intro!: order_antisym le_funI top_greatest)
```
```   634
```
```   635 lemma bot_fun_eq: "bot = (\<lambda>x. bot)"
```
```   636   by (iprover intro!: order_antisym le_funI bot_least)
```
```   637
```
```   638
```
```   639 text {* redundant bindings *}
```
```   640
```
```   641 lemmas inf_aci = inf_ACI
```
```   642 lemmas sup_aci = sup_ACI
```
```   643
```
```   644 no_notation
```
```   645   less_eq  (infix "\<sqsubseteq>" 50) and
```
```   646   less (infix "\<sqsubset>" 50) and
```
```   647   inf  (infixl "\<sqinter>" 70) and
```
```   648   sup  (infixl "\<squnion>" 65) and
```
```   649   Inf  ("\<Sqinter>_" [900] 900) and
```
```   650   Sup  ("\<Squnion>_" [900] 900)
```
```   651
```
```   652 end
```