src/HOL/Lattices.thy
 author haftmann Fri Mar 02 15:43:15 2007 +0100 (2007-03-02) changeset 22384 33a46e6c7f04 parent 22168 627e7aee1b82 child 22422 ee19cdb07528 permissions -rw-r--r--
prefix of class interpretation not mandatory any longer
```     1 (*  Title:      HOL/Lattices.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow
```
```     4 *)
```
```     5
```
```     6 header {* Lattices via Locales *}
```
```     7
```
```     8 theory Lattices
```
```     9 imports Orderings
```
```    10 begin
```
```    11
```
```    12 subsection{* Lattices *}
```
```    13
```
```    14 text{* This theory of lattice locales only defines binary sup and inf
```
```    15 operations. The extension to finite sets is done in theory @{text
```
```    16 Finite_Set}. In the longer term it may be better to define arbitrary
```
```    17 sups and infs via @{text THE}. *}
```
```    18
```
```    19 locale lower_semilattice = order +
```
```    20   fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
```
```    21   assumes inf_le1[simp]: "x \<sqinter> y \<sqsubseteq> x" 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 locale upper_semilattice = order +
```
```    25   fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
```
```    26   assumes sup_ge1[simp]: "x \<sqsubseteq> x \<squnion> y" and sup_ge2[simp]: "y \<sqsubseteq> x \<squnion> y"
```
```    27   and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
```
```    28
```
```    29 locale lattice = lower_semilattice + upper_semilattice
```
```    30
```
```    31 subsubsection{* Intro and elim rules*}
```
```    32
```
```    33 context lower_semilattice
```
```    34 begin
```
```    35
```
```    36 lemmas antisym_intro[intro!] = antisym
```
```    37
```
```    38 lemma le_infI1[intro]: "a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
```
```    39 apply(subgoal_tac "a \<sqinter> b \<sqsubseteq> a")
```
```    40  apply(blast intro: order_trans)
```
```    41 apply simp
```
```    42 done
```
```    43
```
```    44 lemma le_infI2[intro]: "b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
```
```    45 apply(subgoal_tac "a \<sqinter> b \<sqsubseteq> b")
```
```    46  apply(blast intro: order_trans)
```
```    47 apply simp
```
```    48 done
```
```    49
```
```    50 lemma le_infI[intro!]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
```
```    51 by(blast intro: inf_greatest)
```
```    52
```
```    53 lemma le_infE[elim!]: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"
```
```    54 by(blast intro: order_trans)
```
```    55
```
```    56 lemma le_inf_iff [simp]:
```
```    57  "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
```
```    58 by blast
```
```    59
```
```    60 lemma le_iff_inf: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
```
```    61 by(blast dest:eq_iff[THEN iffD1])
```
```    62
```
```    63 end
```
```    64
```
```    65
```
```    66 context upper_semilattice
```
```    67 begin
```
```    68
```
```    69 lemmas antisym_intro[intro!] = antisym
```
```    70
```
```    71 lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
```
```    72 apply(subgoal_tac "a \<sqsubseteq> a \<squnion> b")
```
```    73  apply(blast intro: order_trans)
```
```    74 apply simp
```
```    75 done
```
```    76
```
```    77 lemma le_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
```
```    78 apply(subgoal_tac "b \<sqsubseteq> a \<squnion> b")
```
```    79  apply(blast intro: order_trans)
```
```    80 apply simp
```
```    81 done
```
```    82
```
```    83 lemma le_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
```
```    84 by(blast intro: sup_least)
```
```    85
```
```    86 lemma le_supE[elim!]: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
```
```    87 by(blast intro: order_trans)
```
```    88
```
```    89 lemma ge_sup_conv[simp]:
```
```    90  "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
```
```    91 by blast
```
```    92
```
```    93 lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
```
```    94 by(blast dest:eq_iff[THEN iffD1])
```
```    95
```
```    96 end
```
```    97
```
```    98
```
```    99 subsubsection{* Equational laws *}
```
```   100
```
```   101
```
```   102 context lower_semilattice
```
```   103 begin
```
```   104
```
```   105 lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
```
```   106 by blast
```
```   107
```
```   108 lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
```
```   109 by blast
```
```   110
```
```   111 lemma inf_idem[simp]: "x \<sqinter> x = x"
```
```   112 by blast
```
```   113
```
```   114 lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
```
```   115 by blast
```
```   116
```
```   117 lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
```
```   118 by blast
```
```   119
```
```   120 lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
```
```   121 by blast
```
```   122
```
```   123 lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
```
```   124 by blast
```
```   125
```
```   126 lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem
```
```   127
```
```   128 end
```
```   129
```
```   130
```
```   131 context upper_semilattice
```
```   132 begin
```
```   133
```
```   134 lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
```
```   135 by blast
```
```   136
```
```   137 lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
```
```   138 by blast
```
```   139
```
```   140 lemma sup_idem[simp]: "x \<squnion> x = x"
```
```   141 by blast
```
```   142
```
```   143 lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
```
```   144 by blast
```
```   145
```
```   146 lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
```
```   147 by blast
```
```   148
```
```   149 lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
```
```   150 by blast
```
```   151
```
```   152 lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
```
```   153 by blast
```
```   154
```
```   155 lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem
```
```   156
```
```   157 end
```
```   158
```
```   159 context lattice
```
```   160 begin
```
```   161
```
```   162 lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
```
```   163 by(blast intro: antisym inf_le1 inf_greatest sup_ge1)
```
```   164
```
```   165 lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
```
```   166 by(blast intro: antisym sup_ge1 sup_least inf_le1)
```
```   167
```
```   168 lemmas ACI = inf_ACI sup_ACI
```
```   169
```
```   170 text{* Towards distributivity *}
```
```   171
```
```   172 lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   173 by blast
```
```   174
```
```   175 lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
```
```   176 by blast
```
```   177
```
```   178
```
```   179 text{* If you have one of them, you have them all. *}
```
```   180
```
```   181 lemma distrib_imp1:
```
```   182 assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   183 shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   184 proof-
```
```   185   have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
```
```   186   also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
```
```   187   also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
```
```   188     by(simp add:inf_sup_absorb inf_commute)
```
```   189   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
```
```   190   finally show ?thesis .
```
```   191 qed
```
```   192
```
```   193 lemma distrib_imp2:
```
```   194 assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   195 shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   196 proof-
```
```   197   have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
```
```   198   also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
```
```   199   also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
```
```   200     by(simp add:sup_inf_absorb sup_commute)
```
```   201   also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
```
```   202   finally show ?thesis .
```
```   203 qed
```
```   204
```
```   205 (* seems unused *)
```
```   206 lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z"
```
```   207 by blast
```
```   208
```
```   209 end
```
```   210
```
```   211
```
```   212 subsection{* Distributive lattices *}
```
```   213
```
```   214 locale distrib_lattice = lattice +
```
```   215   assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   216
```
```   217 context distrib_lattice
```
```   218 begin
```
```   219
```
```   220 lemma sup_inf_distrib2:
```
```   221  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
```
```   222 by(simp add:ACI sup_inf_distrib1)
```
```   223
```
```   224 lemma inf_sup_distrib1:
```
```   225  "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   226 by(rule distrib_imp2[OF sup_inf_distrib1])
```
```   227
```
```   228 lemma inf_sup_distrib2:
```
```   229  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
```
```   230 by(simp add:ACI inf_sup_distrib1)
```
```   231
```
```   232 lemmas distrib =
```
```   233   sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
```
```   234
```
```   235 end
```
```   236
```
```   237
```
```   238 subsection {* min/max on linear orders as special case of inf/sup *}
```
```   239
```
```   240 interpretation min_max:
```
```   241   distrib_lattice ["op \<le>" "op <" "min \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]
```
```   242 apply unfold_locales
```
```   243 apply (simp add: min_def linorder_not_le order_less_imp_le)
```
```   244 apply (simp add: min_def linorder_not_le order_less_imp_le)
```
```   245 apply (simp add: min_def linorder_not_le order_less_imp_le)
```
```   246 apply (simp add: max_def linorder_not_le order_less_imp_le)
```
```   247 apply (simp add: max_def linorder_not_le order_less_imp_le)
```
```   248 unfolding min_def max_def by auto
```
```   249
```
```   250 text{* Now we have inherited antisymmetry as an intro-rule on all
```
```   251 linear orders. This is a problem because it applies to bool, which is
```
```   252 undesirable. *}
```
```   253
```
```   254 declare
```
```   255  min_max.antisym_intro[rule del]
```
```   256  min_max.le_infI[rule del] min_max.le_supI[rule del]
```
```   257  min_max.le_supE[rule del] min_max.le_infE[rule del]
```
```   258  min_max.le_supI1[rule del] min_max.le_supI2[rule del]
```
```   259  min_max.le_infI1[rule del] min_max.le_infI2[rule del]
```
```   260
```
```   261 lemmas le_maxI1 = min_max.sup_ge1
```
```   262 lemmas le_maxI2 = min_max.sup_ge2
```
```   263
```
```   264 lemmas max_ac = min_max.sup_assoc min_max.sup_commute
```
```   265                mk_left_commute[of max,OF min_max.sup_assoc min_max.sup_commute]
```
```   266
```
```   267 lemmas min_ac = min_max.inf_assoc min_max.inf_commute
```
```   268                mk_left_commute[of min,OF min_max.inf_assoc min_max.inf_commute]
```
```   269
```
```   270 text {* ML legacy bindings *}
```
```   271
```
```   272 ML {*
```
```   273 val Least_def = @{thm Least_def}
```
```   274 val Least_equality = @{thm Least_equality}
```
```   275 val min_def = @{thm min_def}
```
```   276 val min_of_mono = @{thm min_of_mono}
```
```   277 val max_def = @{thm max_def}
```
```   278 val max_of_mono = @{thm max_of_mono}
```
```   279 val min_leastL = @{thm min_leastL}
```
```   280 val max_leastL = @{thm max_leastL}
```
```   281 val min_leastR = @{thm min_leastR}
```
```   282 val max_leastR = @{thm max_leastR}
```
```   283 val le_max_iff_disj = @{thm le_max_iff_disj}
```
```   284 val le_maxI1 = @{thm le_maxI1}
```
```   285 val le_maxI2 = @{thm le_maxI2}
```
```   286 val less_max_iff_disj = @{thm less_max_iff_disj}
```
```   287 val max_less_iff_conj = @{thm max_less_iff_conj}
```
```   288 val min_less_iff_conj = @{thm min_less_iff_conj}
```
```   289 val min_le_iff_disj = @{thm min_le_iff_disj}
```
```   290 val min_less_iff_disj = @{thm min_less_iff_disj}
```
```   291 val split_min = @{thm split_min}
```
```   292 val split_max = @{thm split_max}
```
```   293 *}
```
```   294
```
```   295 end
```