src/HOL/Lattices.thy
 author haftmann Wed Nov 28 09:01:37 2007 +0100 (2007-11-28) changeset 25482 4ed49eccb1eb parent 25382 72cfe89f7b21 child 25510 38c15efe603b permissions -rw-r--r--
dropped implicit assumption proof
```     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
```
```    30 class lattice = lower_semilattice + upper_semilattice
```
```    31
```
```    32
```
```    33 subsubsection{* Intro and elim rules*}
```
```    34
```
```    35 context lower_semilattice
```
```    36 begin
```
```    37
```
```    38 lemma le_infI1[intro]:
```
```    39   assumes "a \<sqsubseteq> x"
```
```    40   shows "a \<sqinter> b \<sqsubseteq> x"
```
```    41 proof (rule order_trans)
```
```    42   from assms show "a \<sqsubseteq> x" .
```
```    43   show "a \<sqinter> b \<sqsubseteq> a" by simp
```
```    44 qed
```
```    45 lemmas (in -) [rule del] = le_infI1
```
```    46
```
```    47 lemma le_infI2[intro]:
```
```    48   assumes "b \<sqsubseteq> x"
```
```    49   shows "a \<sqinter> b \<sqsubseteq> x"
```
```    50 proof (rule order_trans)
```
```    51   from assms show "b \<sqsubseteq> x" .
```
```    52   show "a \<sqinter> b \<sqsubseteq> b" by simp
```
```    53 qed
```
```    54 lemmas (in -) [rule del] = le_infI2
```
```    55
```
```    56 lemma le_infI[intro!]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
```
```    57 by(blast intro: inf_greatest)
```
```    58 lemmas (in -) [rule del] = le_infI
```
```    59
```
```    60 lemma le_infE [elim!]: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"
```
```    61   by (blast intro: order_trans)
```
```    62 lemmas (in -) [rule del] = le_infE
```
```    63
```
```    64 lemma le_inf_iff [simp]:
```
```    65   "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
```
```    66 by blast
```
```    67
```
```    68 lemma le_iff_inf: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
```
```    69   by (blast intro: antisym dest: eq_iff [THEN iffD1])
```
```    70
```
```    71 lemma mono_inf:
```
```    72   fixes f :: "'a \<Rightarrow> 'b\<Colon>lower_semilattice"
```
```    73   shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<le> f A \<sqinter> f B"
```
```    74   by (auto simp add: mono_def intro: Lattices.inf_greatest)
```
```    75
```
```    76 end
```
```    77
```
```    78 context upper_semilattice
```
```    79 begin
```
```    80
```
```    81 lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
```
```    82   by (rule order_trans) auto
```
```    83 lemmas (in -) [rule del] = le_supI1
```
```    84
```
```    85 lemma le_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
```
```    86   by (rule order_trans) auto
```
```    87 lemmas (in -) [rule del] = le_supI2
```
```    88
```
```    89 lemma le_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
```
```    90 by(blast intro: sup_least)
```
```    91 lemmas (in -) [rule del] = le_supI
```
```    92
```
```    93 lemma le_supE[elim!]: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
```
```    94   by (blast intro: order_trans)
```
```    95 lemmas (in -) [rule del] = le_supE
```
```    96
```
```    97 lemma ge_sup_conv[simp]:
```
```    98   "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
```
```    99 by blast
```
```   100
```
```   101 lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
```
```   102   by (blast intro: antisym dest: eq_iff [THEN iffD1])
```
```   103
```
```   104 lemma mono_sup:
```
```   105   fixes f :: "'a \<Rightarrow> 'b\<Colon>upper_semilattice"
```
```   106   shows "mono f \<Longrightarrow> f A \<squnion> f B \<le> f (A \<squnion> B)"
```
```   107   by (auto simp add: mono_def intro: Lattices.sup_least)
```
```   108
```
```   109 end
```
```   110
```
```   111
```
```   112 subsubsection{* Equational laws *}
```
```   113
```
```   114 context lower_semilattice
```
```   115 begin
```
```   116
```
```   117 lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
```
```   118   by (blast intro: antisym)
```
```   119
```
```   120 lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
```
```   121   by (blast intro: antisym)
```
```   122
```
```   123 lemma inf_idem[simp]: "x \<sqinter> x = x"
```
```   124   by (blast intro: antisym)
```
```   125
```
```   126 lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
```
```   127   by (blast intro: antisym)
```
```   128
```
```   129 lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
```
```   130   by (blast intro: antisym)
```
```   131
```
```   132 lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
```
```   133   by (blast intro: antisym)
```
```   134
```
```   135 lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
```
```   136   by (blast intro: antisym)
```
```   137
```
```   138 lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem
```
```   139
```
```   140 end
```
```   141
```
```   142
```
```   143 context upper_semilattice
```
```   144 begin
```
```   145
```
```   146 lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
```
```   147   by (blast intro: antisym)
```
```   148
```
```   149 lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
```
```   150   by (blast intro: antisym)
```
```   151
```
```   152 lemma sup_idem[simp]: "x \<squnion> x = x"
```
```   153   by (blast intro: antisym)
```
```   154
```
```   155 lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
```
```   156   by (blast intro: antisym)
```
```   157
```
```   158 lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
```
```   159   by (blast intro: antisym)
```
```   160
```
```   161 lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
```
```   162   by (blast intro: antisym)
```
```   163
```
```   164 lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
```
```   165   by (blast intro: antisym)
```
```   166
```
```   167 lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem
```
```   168
```
```   169 end
```
```   170
```
```   171 context lattice
```
```   172 begin
```
```   173
```
```   174 lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
```
```   175   by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
```
```   176
```
```   177 lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
```
```   178   by (blast intro: antisym sup_ge1 sup_least inf_le1)
```
```   179
```
```   180 lemmas ACI = inf_ACI sup_ACI
```
```   181
```
```   182 lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
```
```   183
```
```   184 text{* Towards distributivity *}
```
```   185
```
```   186 lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   187   by blast
```
```   188
```
```   189 lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
```
```   190   by blast
```
```   191
```
```   192
```
```   193 text{* If you have one of them, you have them all. *}
```
```   194
```
```   195 lemma distrib_imp1:
```
```   196 assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   197 shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   198 proof-
```
```   199   have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
```
```   200   also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
```
```   201   also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
```
```   202     by(simp add:inf_sup_absorb inf_commute)
```
```   203   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
```
```   204   finally show ?thesis .
```
```   205 qed
```
```   206
```
```   207 lemma distrib_imp2:
```
```   208 assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   209 shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   210 proof-
```
```   211   have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
```
```   212   also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
```
```   213   also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
```
```   214     by(simp add:sup_inf_absorb sup_commute)
```
```   215   also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
```
```   216   finally show ?thesis .
```
```   217 qed
```
```   218
```
```   219 (* seems unused *)
```
```   220 lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z"
```
```   221 by blast
```
```   222
```
```   223 end
```
```   224
```
```   225
```
```   226 subsection {* Distributive lattices *}
```
```   227
```
```   228 class distrib_lattice = lattice +
```
```   229   assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   230
```
```   231 context distrib_lattice
```
```   232 begin
```
```   233
```
```   234 lemma sup_inf_distrib2:
```
```   235  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
```
```   236 by(simp add:ACI sup_inf_distrib1)
```
```   237
```
```   238 lemma inf_sup_distrib1:
```
```   239  "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   240 by(rule distrib_imp2[OF sup_inf_distrib1])
```
```   241
```
```   242 lemma inf_sup_distrib2:
```
```   243  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
```
```   244 by(simp add:ACI inf_sup_distrib1)
```
```   245
```
```   246 lemmas distrib =
```
```   247   sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
```
```   248
```
```   249 end
```
```   250
```
```   251
```
```   252 subsection {* Uniqueness of inf and sup *}
```
```   253
```
```   254 lemma (in lower_semilattice) inf_unique:
```
```   255   fixes f (infixl "\<triangle>" 70)
```
```   256   assumes le1: "\<And>x y. x \<triangle> y \<le> x" and le2: "\<And>x y. x \<triangle> y \<le> y"
```
```   257   and greatest: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z"
```
```   258   shows "x \<sqinter> y = x \<triangle> y"
```
```   259 proof (rule antisym)
```
```   260   show "x \<triangle> y \<le> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2)
```
```   261 next
```
```   262   have leI: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" by (blast intro: greatest)
```
```   263   show "x \<sqinter> y \<le> x \<triangle> y" by (rule leI) simp_all
```
```   264 qed
```
```   265
```
```   266 lemma (in upper_semilattice) sup_unique:
```
```   267   fixes f (infixl "\<nabla>" 70)
```
```   268   assumes ge1 [simp]: "\<And>x y. x \<le> x \<nabla> y" and ge2: "\<And>x y. y \<le> x \<nabla> y"
```
```   269   and least: "\<And>x y z. y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<nabla> z \<le> x"
```
```   270   shows "x \<squnion> y = x \<nabla> y"
```
```   271 proof (rule antisym)
```
```   272   show "x \<squnion> y \<le> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2)
```
```   273 next
```
```   274   have leI: "\<And>x y z. x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<nabla> y \<le> z" by (blast intro: least)
```
```   275   show "x \<nabla> y \<le> x \<squnion> y" by (rule leI) simp_all
```
```   276 qed
```
```   277
```
```   278
```
```   279 subsection {* @{const min}/@{const max} on linear orders as
```
```   280   special case of @{const inf}/@{const sup} *}
```
```   281
```
```   282 lemma (in linorder) distrib_lattice_min_max:
```
```   283   "distrib_lattice (op \<le>) (op <) min max"
```
```   284 proof unfold_locales
```
```   285   have aux: "\<And>x y \<Colon> 'a. x < y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
```
```   286     by (auto simp add: less_le antisym)
```
```   287   fix x y z
```
```   288   show "max x (min y z) = min (max x y) (max x z)"
```
```   289   unfolding min_def max_def
```
```   290   by auto
```
```   291 qed (auto simp add: min_def max_def not_le less_imp_le)
```
```   292
```
```   293 interpretation min_max:
```
```   294   distrib_lattice ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max]
```
```   295   by (rule distrib_lattice_min_max)
```
```   296
```
```   297 lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```   298   by (rule ext)+ (auto intro: antisym)
```
```   299
```
```   300 lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```   301   by (rule ext)+ (auto intro: antisym)
```
```   302
```
```   303 lemmas le_maxI1 = min_max.sup_ge1
```
```   304 lemmas le_maxI2 = min_max.sup_ge2
```
```   305
```
```   306 lemmas max_ac = min_max.sup_assoc min_max.sup_commute
```
```   307   mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
```
```   308
```
```   309 lemmas min_ac = min_max.inf_assoc min_max.inf_commute
```
```   310   mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
```
```   311
```
```   312 text {*
```
```   313   Now we have inherited antisymmetry as an intro-rule on all
```
```   314   linear orders. This is a problem because it applies to bool, which is
```
```   315   undesirable.
```
```   316 *}
```
```   317
```
```   318 lemmas [rule del] = min_max.le_infI min_max.le_supI
```
```   319   min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
```
```   320   min_max.le_infI1 min_max.le_infI2
```
```   321
```
```   322
```
```   323 subsection {* Complete lattices *}
```
```   324
```
```   325 class complete_lattice = lattice +
```
```   326   fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
```
```   327     and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
```
```   328   assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
```
```   329      and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
```
```   330   assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
```
```   331      and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
```
```   332 begin
```
```   333
```
```   334 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
```
```   335   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   336
```
```   337 lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
```
```   338   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
```
```   339
```
```   340 lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
```
```   341   unfolding Sup_Inf by auto
```
```   342
```
```   343 lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
```
```   344   unfolding Inf_Sup by auto
```
```   345
```
```   346 lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
```
```   347   apply (rule antisym)
```
```   348   apply (rule le_infI)
```
```   349   apply (rule Inf_lower)
```
```   350   apply simp
```
```   351   apply (rule Inf_greatest)
```
```   352   apply (rule Inf_lower)
```
```   353   apply simp
```
```   354   apply (rule Inf_greatest)
```
```   355   apply (erule insertE)
```
```   356   apply (rule le_infI1)
```
```   357   apply simp
```
```   358   apply (rule le_infI2)
```
```   359   apply (erule Inf_lower)
```
```   360   done
```
```   361
```
```   362 lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
```
```   363   apply (rule antisym)
```
```   364   apply (rule Sup_least)
```
```   365   apply (erule insertE)
```
```   366   apply (rule le_supI1)
```
```   367   apply simp
```
```   368   apply (rule le_supI2)
```
```   369   apply (erule Sup_upper)
```
```   370   apply (rule le_supI)
```
```   371   apply (rule Sup_upper)
```
```   372   apply simp
```
```   373   apply (rule Sup_least)
```
```   374   apply (rule Sup_upper)
```
```   375   apply simp
```
```   376   done
```
```   377
```
```   378 lemma Inf_singleton [simp]:
```
```   379   "\<Sqinter>{a} = a"
```
```   380   by (auto intro: antisym Inf_lower Inf_greatest)
```
```   381
```
```   382 lemma Sup_singleton [simp]:
```
```   383   "\<Squnion>{a} = a"
```
```   384   by (auto intro: antisym Sup_upper Sup_least)
```
```   385
```
```   386 lemma Inf_insert_simp:
```
```   387   "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
```
```   388   by (cases "A = {}") (simp_all, simp add: Inf_insert)
```
```   389
```
```   390 lemma Sup_insert_simp:
```
```   391   "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
```
```   392   by (cases "A = {}") (simp_all, simp add: Sup_insert)
```
```   393
```
```   394 lemma Inf_binary:
```
```   395   "\<Sqinter>{a, b} = a \<sqinter> b"
```
```   396   by (simp add: Inf_insert_simp)
```
```   397
```
```   398 lemma Sup_binary:
```
```   399   "\<Squnion>{a, b} = a \<squnion> b"
```
```   400   by (simp add: Sup_insert_simp)
```
```   401
```
```   402 definition
```
```   403   top :: 'a where
```
```   404   "top = \<Sqinter>{}"
```
```   405
```
```   406 definition
```
```   407   bot :: 'a where
```
```   408   "bot = \<Squnion>{}"
```
```   409
```
```   410 lemma top_greatest [simp]: "x \<le> top"
```
```   411   by (unfold top_def, rule Inf_greatest, simp)
```
```   412
```
```   413 lemma bot_least [simp]: "bot \<le> x"
```
```   414   by (unfold bot_def, rule Sup_least, simp)
```
```   415
```
```   416 definition
```
```   417   SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
```
```   418 where
```
```   419   "SUPR A f == \<Squnion> (f ` A)"
```
```   420
```
```   421 definition
```
```   422   INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
```
```   423 where
```
```   424   "INFI A f == \<Sqinter> (f ` A)"
```
```   425
```
```   426 end
```
```   427
```
```   428 syntax
```
```   429   "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
```
```   430   "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
```
```   431   "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
```
```   432   "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
```
```   433
```
```   434 translations
```
```   435   "SUP x y. B"   == "SUP x. SUP y. B"
```
```   436   "SUP x. B"     == "CONST SUPR UNIV (%x. B)"
```
```   437   "SUP x. B"     == "SUP x:UNIV. B"
```
```   438   "SUP x:A. B"   == "CONST SUPR A (%x. B)"
```
```   439   "INF x y. B"   == "INF x. INF y. B"
```
```   440   "INF x. B"     == "CONST INFI UNIV (%x. B)"
```
```   441   "INF x. B"     == "INF x:UNIV. B"
```
```   442   "INF x:A. B"   == "CONST INFI A (%x. B)"
```
```   443
```
```   444 (* To avoid eta-contraction of body: *)
```
```   445 print_translation {*
```
```   446 let
```
```   447   fun btr' syn (A :: Abs abs :: ts) =
```
```   448     let val (x,t) = atomic_abs_tr' abs
```
```   449     in list_comb (Syntax.const syn \$ x \$ A \$ t, ts) end
```
```   450   val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
```
```   451 in
```
```   452 [(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
```
```   453 end
```
```   454 *}
```
```   455
```
```   456 context complete_lattice
```
```   457 begin
```
```   458
```
```   459 lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
```
```   460   by (auto simp add: SUPR_def intro: Sup_upper)
```
```   461
```
```   462 lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
```
```   463   by (auto simp add: SUPR_def intro: Sup_least)
```
```   464
```
```   465 lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
```
```   466   by (auto simp add: INFI_def intro: Inf_lower)
```
```   467
```
```   468 lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
```
```   469   by (auto simp add: INFI_def intro: Inf_greatest)
```
```   470
```
```   471 lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
```
```   472   by (auto intro: antisym SUP_leI le_SUPI)
```
```   473
```
```   474 lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
```
```   475   by (auto intro: antisym INF_leI le_INFI)
```
```   476
```
```   477 end
```
```   478
```
```   479
```
```   480 subsection {* Bool as lattice *}
```
```   481
```
```   482 instance bool :: distrib_lattice
```
```   483   inf_bool_eq: "P \<sqinter> Q \<equiv> P \<and> Q"
```
```   484   sup_bool_eq: "P \<squnion> Q \<equiv> P \<or> Q"
```
```   485   by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
```
```   486
```
```   487 instance bool :: complete_lattice
```
```   488   Inf_bool_def: "\<Sqinter>A \<equiv> \<forall>x\<in>A. x"
```
```   489   Sup_bool_def: "\<Squnion>A \<equiv> \<exists>x\<in>A. x"
```
```   490   by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
```
```   491
```
```   492 lemma Inf_empty_bool [simp]:
```
```   493   "\<Sqinter>{}"
```
```   494   unfolding Inf_bool_def by auto
```
```   495
```
```   496 lemma not_Sup_empty_bool [simp]:
```
```   497   "\<not> Sup {}"
```
```   498   unfolding Sup_bool_def by auto
```
```   499
```
```   500 lemma top_bool_eq: "top = True"
```
```   501   by (iprover intro!: order_antisym le_boolI top_greatest)
```
```   502
```
```   503 lemma bot_bool_eq: "bot = False"
```
```   504   by (iprover intro!: order_antisym le_boolI bot_least)
```
```   505
```
```   506
```
```   507 subsection {* Set as lattice *}
```
```   508
```
```   509 instance set :: (type) distrib_lattice
```
```   510   inf_set_eq: "A \<sqinter> B \<equiv> A \<inter> B"
```
```   511   sup_set_eq: "A \<squnion> B \<equiv> A \<union> B"
```
```   512   by intro_classes (auto simp add: inf_set_eq sup_set_eq)
```
```   513
```
```   514 lemmas [code func del] = inf_set_eq sup_set_eq
```
```   515
```
```   516 lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
```
```   517   apply (fold inf_set_eq sup_set_eq)
```
```   518   apply (erule mono_inf)
```
```   519   done
```
```   520
```
```   521 lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
```
```   522   apply (fold inf_set_eq sup_set_eq)
```
```   523   apply (erule mono_sup)
```
```   524   done
```
```   525
```
```   526 instance set :: (type) complete_lattice
```
```   527   Inf_set_def: "\<Sqinter>S \<equiv> \<Inter>S"
```
```   528   Sup_set_def: "\<Squnion>S \<equiv> \<Union>S"
```
```   529   by intro_classes (auto simp add: Inf_set_def Sup_set_def)
```
```   530
```
```   531 lemmas [code func del] = Inf_set_def Sup_set_def
```
```   532
```
```   533 lemma top_set_eq: "top = UNIV"
```
```   534   by (iprover intro!: subset_antisym subset_UNIV top_greatest)
```
```   535
```
```   536 lemma bot_set_eq: "bot = {}"
```
```   537   by (iprover intro!: subset_antisym empty_subsetI bot_least)
```
```   538
```
```   539
```
```   540 subsection {* Fun as lattice *}
```
```   541
```
```   542 instance "fun" :: (type, lattice) lattice
```
```   543   inf_fun_eq: "f \<sqinter> g \<equiv> (\<lambda>x. f x \<sqinter> g x)"
```
```   544   sup_fun_eq: "f \<squnion> g \<equiv> (\<lambda>x. f x \<squnion> g x)"
```
```   545 apply intro_classes
```
```   546 unfolding inf_fun_eq sup_fun_eq
```
```   547 apply (auto intro: le_funI)
```
```   548 apply (rule le_funI)
```
```   549 apply (auto dest: le_funD)
```
```   550 apply (rule le_funI)
```
```   551 apply (auto dest: le_funD)
```
```   552 done
```
```   553
```
```   554 lemmas [code func del] = inf_fun_eq sup_fun_eq
```
```   555
```
```   556 instance "fun" :: (type, distrib_lattice) distrib_lattice
```
```   557   by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
```
```   558
```
```   559 instance "fun" :: (type, complete_lattice) complete_lattice
```
```   560   Inf_fun_def: "\<Sqinter>A \<equiv> (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
```
```   561   Sup_fun_def: "\<Squnion>A \<equiv> (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
```
```   562   by intro_classes
```
```   563     (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
```
```   564       intro: Inf_lower Sup_upper Inf_greatest Sup_least)
```
```   565
```
```   566 lemmas [code func del] = Inf_fun_def Sup_fun_def
```
```   567
```
```   568 lemma Inf_empty_fun:
```
```   569   "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
```
```   570   by rule (auto simp add: Inf_fun_def)
```
```   571
```
```   572 lemma Sup_empty_fun:
```
```   573   "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
```
```   574   by rule (auto simp add: Sup_fun_def)
```
```   575
```
```   576 lemma top_fun_eq: "top = (\<lambda>x. top)"
```
```   577   by (iprover intro!: order_antisym le_funI top_greatest)
```
```   578
```
```   579 lemma bot_fun_eq: "bot = (\<lambda>x. bot)"
```
```   580   by (iprover intro!: order_antisym le_funI bot_least)
```
```   581
```
```   582
```
```   583 text {* redundant bindings *}
```
```   584
```
```   585 lemmas inf_aci = inf_ACI
```
```   586 lemmas sup_aci = sup_ACI
```
```   587
```
```   588 no_notation
```
```   589   less_eq  (infix "\<sqsubseteq>" 50) and
```
```   590   less (infix "\<sqsubset>" 50) and
```
```   591   inf  (infixl "\<sqinter>" 70) and
```
```   592   sup  (infixl "\<squnion>" 65) and
```
```   593   Inf  ("\<Sqinter>_" [900] 900) and
```
```   594   Sup  ("\<Squnion>_" [900] 900)
```
```   595
```
```   596 end
```