src/HOL/Lattices.thy
 author nipkow Sun Dec 10 13:14:43 2006 +0100 (2006-12-10) changeset 21734 283461c15fa7 parent 21733 131dd2a27137 child 22068 00bed5ac9884 permissions -rw-r--r--
renaming
```     1 (*  Title:      HOL/Lattices.thy
```
```     2     ID:         \$Id\$
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```     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
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```    14 text{* This theory of lattice locales only defines binary sup and inf
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```    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
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```    17 sups and infs via @{text THE}. *}
```
```    18
```
```    19 locale lower_semilattice = partial_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 = partial_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*}
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```    32
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```    33 context lower_semilattice
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```    34 begin
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```    35
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```    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: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: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: 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 apply rule
```
```    62  apply(simp add: antisym)
```
```    63 apply(subgoal_tac "x \<sqinter> y \<sqsubseteq> y")
```
```    64  apply(simp)
```
```    65 apply(simp (no_asm))
```
```    66 done
```
```    67
```
```    68 end
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```    69
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```    70
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```    71 context upper_semilattice
```
```    72 begin
```
```    73
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```    74 lemmas antisym_intro[intro!] = antisym
```
```    75
```
```    76 lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
```
```    77 apply(subgoal_tac "a \<sqsubseteq> a \<squnion> b")
```
```    78  apply(blast intro:trans)
```
```    79 apply simp
```
```    80 done
```
```    81
```
```    82 lemma le_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
```
```    83 apply(subgoal_tac "b \<sqsubseteq> a \<squnion> b")
```
```    84  apply(blast intro:trans)
```
```    85 apply simp
```
```    86 done
```
```    87
```
```    88 lemma le_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
```
```    89 by(blast intro: sup_least)
```
```    90
```
```    91 lemma le_supE[elim!]: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
```
```    92 by(blast intro: trans)
```
```    93
```
```    94 lemma ge_sup_conv[simp]:
```
```    95  "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
```
```    96 by blast
```
```    97
```
```    98 lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
```
```    99 apply rule
```
```   100  apply(simp add: antisym)
```
```   101 apply(subgoal_tac "x \<sqsubseteq> x \<squnion> y")
```
```   102 apply(simp)
```
```   103  apply(simp (no_asm))
```
```   104 done
```
```   105
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```   106 end
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```   107
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```   108
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```   109 subsubsection{* Equational laws *}
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```   110
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```   111
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```   112 context lower_semilattice
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```   113 begin
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```   114
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```   115 lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
```
```   116 by blast
```
```   117
```
```   118 lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
```
```   119 by blast
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```   120
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```   121 lemma inf_idem[simp]: "x \<sqinter> x = x"
```
```   122 by blast
```
```   123
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```   124 lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
```
```   125 by blast
```
```   126
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```   127 lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
```
```   128 by blast
```
```   129
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```   130 lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
```
```   131 by blast
```
```   132
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```   133 lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
```
```   134 by blast
```
```   135
```
```   136 lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem
```
```   137
```
```   138 end
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```   139
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```   140
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```   141 context upper_semilattice
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```   142 begin
```
```   143
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```   144 lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
```
```   145 by blast
```
```   146
```
```   147 lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
```
```   148 by blast
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```   149
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```   150 lemma sup_idem[simp]: "x \<squnion> x = x"
```
```   151 by blast
```
```   152
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```   153 lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
```
```   154 by blast
```
```   155
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```   156 lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
```
```   157 by blast
```
```   158
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```   159 lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
```
```   160 by blast
```
```   161
```
```   162 lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
```
```   163 by blast
```
```   164
```
```   165 lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem
```
```   166
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```   167 end
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```   168
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```   169 context lattice
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```   170 begin
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```   171
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```   172 lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
```
```   173 by(blast intro: antisym inf_le1 inf_greatest sup_ge1)
```
```   174
```
```   175 lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
```
```   176 by(blast intro: antisym sup_ge1 sup_least inf_le1)
```
```   177
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```   178 lemmas ACI = inf_ACI sup_ACI
```
```   179
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```   180 text{* Towards distributivity *}
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```   181
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```   182 lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   183 by blast
```
```   184
```
```   185 lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
```
```   186 by blast
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```   187
```
```   188
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```   189 text{* If you have one of them, you have them all. *}
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```   190
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```   191 lemma distrib_imp1:
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```   192 assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   193 shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   194 proof-
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```   195   have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
```
```   196   also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
```
```   197   also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
```
```   198     by(simp add:inf_sup_absorb inf_commute)
```
```   199   also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
```
```   200   finally show ?thesis .
```
```   201 qed
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```   202
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```   203 lemma distrib_imp2:
```
```   204 assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   205 shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   206 proof-
```
```   207   have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
```
```   208   also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
```
```   209   also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
```
```   210     by(simp add:sup_inf_absorb sup_commute)
```
```   211   also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
```
```   212   finally show ?thesis .
```
```   213 qed
```
```   214
```
```   215 (* seems unused *)
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```   216 lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z"
```
```   217 by blast
```
```   218
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```   219 end
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```   220
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```   221
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```   222 subsection{* Distributive lattices *}
```
```   223
```
```   224 locale distrib_lattice = lattice +
```
```   225   assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
```
```   226
```
```   227 context distrib_lattice
```
```   228 begin
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```   229
```
```   230 lemma sup_inf_distrib2:
```
```   231  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
```
```   232 by(simp add:ACI sup_inf_distrib1)
```
```   233
```
```   234 lemma inf_sup_distrib1:
```
```   235  "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
```
```   236 by(rule distrib_imp2[OF sup_inf_distrib1])
```
```   237
```
```   238 lemma inf_sup_distrib2:
```
```   239  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
```
```   240 by(simp add:ACI inf_sup_distrib1)
```
```   241
```
```   242 lemmas distrib =
```
```   243   sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
```
```   244
```
```   245 end
```
```   246
```
```   247
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```   248 subsection {* min/max on linear orders as special case of inf/sup *}
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```   249
```
```   250 interpretation min_max:
```
```   251   distrib_lattice ["op \<le>" "op <" "min \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]
```
```   252 apply unfold_locales
```
```   253 apply (simp add: min_def linorder_not_le order_less_imp_le)
```
```   254 apply (simp add: min_def linorder_not_le order_less_imp_le)
```
```   255 apply (simp add: min_def linorder_not_le order_less_imp_le)
```
```   256 apply (simp add: max_def linorder_not_le order_less_imp_le)
```
```   257 apply (simp add: max_def linorder_not_le order_less_imp_le)
```
```   258 unfolding min_def max_def by auto
```
```   259
```
```   260 text{* Now we have inherited antisymmetry as an intro-rule on all
```
```   261 linear orders. This is a problem because it applies to bool, which is
```
```   262 undesirable. *}
```
```   263
```
```   264 declare
```
```   265  min_max.antisym_intro[rule del]
```
```   266  min_max.le_infI[rule del] min_max.le_supI[rule del]
```
```   267  min_max.le_supE[rule del] min_max.le_infE[rule del]
```
```   268  min_max.le_supI1[rule del] min_max.le_supI2[rule del]
```
```   269  min_max.le_infI1[rule del] min_max.le_infI2[rule del]
```
```   270
```
```   271 lemmas le_maxI1 = min_max.sup_ge1
```
```   272 lemmas le_maxI2 = min_max.sup_ge2
```
```   273
```
```   274 lemmas max_ac = min_max.sup_assoc min_max.sup_commute
```
```   275                mk_left_commute[of max,OF min_max.sup_assoc min_max.sup_commute]
```
```   276
```
```   277 lemmas min_ac = min_max.inf_assoc min_max.inf_commute
```
```   278                mk_left_commute[of min,OF min_max.inf_assoc min_max.inf_commute]
```
```   279
```
```   280 text {* ML legacy bindings *}
```
```   281
```
```   282 ML {*
```
```   283 val Least_def = thm "Least_def";
```
```   284 val Least_equality = thm "Least_equality";
```
```   285 val min_def = thm "min_def";
```
```   286 val min_of_mono = thm "min_of_mono";
```
```   287 val max_def = thm "max_def";
```
```   288 val max_of_mono = thm "max_of_mono";
```
```   289 val min_leastL = thm "min_leastL";
```
```   290 val max_leastL = thm "max_leastL";
```
```   291 val min_leastR = thm "min_leastR";
```
```   292 val max_leastR = thm "max_leastR";
```
```   293 val le_max_iff_disj = thm "le_max_iff_disj";
```
```   294 val le_maxI1 = thm "le_maxI1";
```
```   295 val le_maxI2 = thm "le_maxI2";
```
```   296 val less_max_iff_disj = thm "less_max_iff_disj";
```
```   297 val max_less_iff_conj = thm "max_less_iff_conj";
```
```   298 val min_less_iff_conj = thm "min_less_iff_conj";
```
```   299 val min_le_iff_disj = thm "min_le_iff_disj";
```
```   300 val min_less_iff_disj = thm "min_less_iff_disj";
```
```   301 val split_min = thm "split_min";
```
```   302 val split_max = thm "split_max";
```
```   303 *}
```
```   304
```
```   305 end
```