21249

1 
(* Title: HOL/Lattices.thy


2 
ID: $Id$


3 
Author: Tobias Nipkow


4 
*)


5 

21733

6 
header {* Lattices via Locales *}

21249

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 = partial_order +


20 
fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)

21312

21 
assumes inf_le1[simp]: "x \<sqinter> y \<sqsubseteq> x" and inf_le2[simp]: "x \<sqinter> y \<sqsubseteq> y"

21733

22 
and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"

21249

23 


24 
locale upper_semilattice = partial_order +


25 
fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)

21312

26 
assumes sup_ge1[simp]: "x \<sqsubseteq> x \<squnion> y" and sup_ge2[simp]: "y \<sqsubseteq> x \<squnion> y"

21733

27 
and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"

21249

28 


29 
locale lattice = lower_semilattice + upper_semilattice


30 

21733

31 
subsubsection{* Intro and elim rules*}


32 


33 
context lower_semilattice


34 
begin

21249

35 

21733

36 
lemmas antisym_intro[intro!] = antisym

21249

37 

21733

38 
lemma less_eq_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

21249

43 

21733

44 
lemma less_eq_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 less_eq_infI[intro!]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"


51 
by(blast intro: inf_greatest)

21249

52 

21733

53 
lemma less_eq_infE[elim!]: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"


54 
by(blast intro: trans)

21249

55 

21733

56 
lemma less_eq_inf_conv [simp]:


57 
"x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"


58 
by blast


59 


60 
lemmas below_inf_conv = less_eq_inf_conv


61 
 {* a duplicate for backward compatibility *}

21249

62 

21733

63 
end


64 


65 


66 
context upper_semilattice


67 
begin

21249

68 

21733

69 
lemmas antisym_intro[intro!] = antisym

21249

70 

21733

71 
lemma less_eq_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"


72 
apply(subgoal_tac "a \<sqsubseteq> a \<squnion> b")


73 
apply(blast intro:trans)


74 
apply simp


75 
done

21249

76 

21733

77 
lemma less_eq_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"


78 
apply(subgoal_tac "b \<sqsubseteq> a \<squnion> b")


79 
apply(blast intro:trans)


80 
apply simp


81 
done


82 


83 
lemma less_eq_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"


84 
by(blast intro: sup_least)

21249

85 

21733

86 
lemma less_eq_supE[elim!]: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"


87 
by(blast intro: trans)

21249

88 

21733

89 
lemma above_sup_conv[simp]:


90 
"x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"


91 
by blast


92 


93 
end


94 


95 


96 
subsubsection{* Equational laws *}

21249

97 


98 

21733

99 
context lower_semilattice


100 
begin


101 


102 
lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"


103 
by blast


104 


105 
lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"


106 
by blast


107 


108 
lemma inf_idem[simp]: "x \<sqinter> x = x"


109 
by blast


110 


111 
lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"


112 
by blast


113 


114 
lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"


115 
by blast


116 


117 
lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"


118 
by blast


119 


120 
lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"


121 
by blast


122 


123 
lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem


124 


125 
end


126 


127 


128 
context upper_semilattice


129 
begin

21249

130 

21733

131 
lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"


132 
by blast


133 


134 
lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"


135 
by blast


136 


137 
lemma sup_idem[simp]: "x \<squnion> x = x"


138 
by blast


139 


140 
lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"


141 
by blast


142 


143 
lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"


144 
by blast


145 


146 
lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"


147 
by blast

21249

148 

21733

149 
lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"


150 
by blast


151 


152 
lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem


153 


154 
end

21249

155 

21733

156 
context lattice


157 
begin


158 


159 
lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"


160 
by(blast intro: antisym inf_le1 inf_greatest sup_ge1)


161 


162 
lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"


163 
by(blast intro: antisym sup_ge1 sup_least inf_le1)


164 


165 
lemmas (in lattice) ACI = inf_ACI sup_ACI

21249

166 


167 
text{* Towards distributivity: if you have one of them, you have them all. *}


168 

21733

169 
lemma distrib_imp1:

21249

170 
assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"


171 
shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"


172 
proof


173 
have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)


174 
also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)


175 
also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"


176 
by(simp add:inf_sup_absorb inf_commute)


177 
also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)


178 
finally show ?thesis .


179 
qed


180 

21733

181 
lemma distrib_imp2:

21249

182 
assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"


183 
shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"


184 
proof


185 
have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)


186 
also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)


187 
also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"


188 
by(simp add:sup_inf_absorb sup_commute)


189 
also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)


190 
finally show ?thesis .


191 
qed


192 

21733

193 
end

21249

194 


195 


196 
subsection{* Distributive lattices *}


197 


198 
locale distrib_lattice = lattice +


199 
assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"


200 

21733

201 
context distrib_lattice


202 
begin


203 


204 
lemma sup_inf_distrib2:

21249

205 
"(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"


206 
by(simp add:ACI sup_inf_distrib1)


207 

21733

208 
lemma inf_sup_distrib1:

21249

209 
"x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"


210 
by(rule distrib_imp2[OF sup_inf_distrib1])


211 

21733

212 
lemma inf_sup_distrib2:

21249

213 
"(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"


214 
by(simp add:ACI inf_sup_distrib1)


215 

21733

216 
lemmas distrib =

21249

217 
sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2


218 

21733

219 
end


220 

21249

221 

21381

222 
subsection {* min/max on linear orders as special case of inf/sup *}

21249

223 


224 
interpretation min_max:

21381

225 
distrib_lattice ["op \<le>" "op <" "min \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]

21249

226 
apply unfold_locales

21381

227 
apply (simp add: min_def linorder_not_le order_less_imp_le)


228 
apply (simp add: min_def linorder_not_le order_less_imp_le)


229 
apply (simp add: min_def linorder_not_le order_less_imp_le)


230 
apply (simp add: max_def linorder_not_le order_less_imp_le)


231 
apply (simp add: max_def linorder_not_le order_less_imp_le)


232 
unfolding min_def max_def by auto

21249

233 

21733

234 
text{* Now we have inherited antisymmetry as an introrule on all


235 
linear orders. This is a problem because it applies to bool, which is


236 
undesirable. *}


237 


238 
declare


239 
min_max.antisym_intro[rule del]


240 
min_max.less_eq_infI[rule del] min_max.less_eq_supI[rule del]


241 
min_max.less_eq_supE[rule del] min_max.less_eq_infE[rule del]


242 
min_max.less_eq_supI1[rule del] min_max.less_eq_supI2[rule del]


243 
min_max.less_eq_infI1[rule del] min_max.less_eq_infI2[rule del]


244 

21249

245 
lemmas le_maxI1 = min_max.sup_ge1


246 
lemmas le_maxI2 = min_max.sup_ge2

21381

247 

21249

248 
lemmas max_ac = min_max.sup_assoc min_max.sup_commute


249 
mk_left_commute[of max,OF min_max.sup_assoc min_max.sup_commute]


250 


251 
lemmas min_ac = min_max.inf_assoc min_max.inf_commute


252 
mk_left_commute[of min,OF min_max.inf_assoc min_max.inf_commute]


253 

21733

254 
text {* ML legacy bindings *}


255 


256 
ML {*


257 
val Least_def = thm "Least_def";


258 
val Least_equality = thm "Least_equality";


259 
val min_def = thm "min_def";


260 
val min_of_mono = thm "min_of_mono";


261 
val max_def = thm "max_def";


262 
val max_of_mono = thm "max_of_mono";


263 
val min_leastL = thm "min_leastL";


264 
val max_leastL = thm "max_leastL";


265 
val min_leastR = thm "min_leastR";


266 
val max_leastR = thm "max_leastR";


267 
val le_max_iff_disj = thm "le_max_iff_disj";


268 
val le_maxI1 = thm "le_maxI1";


269 
val le_maxI2 = thm "le_maxI2";


270 
val less_max_iff_disj = thm "less_max_iff_disj";


271 
val max_less_iff_conj = thm "max_less_iff_conj";


272 
val min_less_iff_conj = thm "min_less_iff_conj";


273 
val min_le_iff_disj = thm "min_le_iff_disj";


274 
val min_less_iff_disj = thm "min_less_iff_disj";


275 
val split_min = thm "split_min";


276 
val split_max = thm "split_max";


277 
*}


278 

21249

279 
end
