added dual for complete lattice
authorhaftmann
Thu Sep 24 18:29:28 2009 +0200 (2009-09-24)
changeset 32678de1f7d4da21a
parent 32674 b629fbcc5313
child 32679 096306d7391d
added dual for complete lattice
src/HOL/Complete_Lattice.thy
     1.1 --- a/src/HOL/Complete_Lattice.thy	Thu Sep 24 08:28:27 2009 +0200
     1.2 +++ b/src/HOL/Complete_Lattice.thy	Thu Sep 24 18:29:28 2009 +0200
     1.3 @@ -10,7 +10,9 @@
     1.4    less_eq  (infix "\<sqsubseteq>" 50) and
     1.5    less (infix "\<sqsubset>" 50) and
     1.6    inf  (infixl "\<sqinter>" 70) and
     1.7 -  sup  (infixl "\<squnion>" 65)
     1.8 +  sup  (infixl "\<squnion>" 65) and
     1.9 +  top ("\<top>") and
    1.10 +  bot ("\<bottom>")
    1.11  
    1.12  
    1.13  subsection {* Abstract complete lattices *}
    1.14 @@ -24,6 +26,15 @@
    1.15       and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
    1.16  begin
    1.17  
    1.18 +term complete_lattice
    1.19 +
    1.20 +lemma dual_complete_lattice:
    1.21 +  "complete_lattice (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom> Sup Inf"
    1.22 +  by (auto intro!: complete_lattice.intro dual_lattice
    1.23 +    bot.intro top.intro dual_preorder, unfold_locales)
    1.24 +      (fact bot_least top_greatest
    1.25 +        Sup_upper Sup_least Inf_lower Inf_greatest)+
    1.26 +
    1.27  lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
    1.28    by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
    1.29  
    1.30 @@ -784,7 +795,9 @@
    1.31    inf  (infixl "\<sqinter>" 70) and
    1.32    sup  (infixl "\<squnion>" 65) and
    1.33    Inf  ("\<Sqinter>_" [900] 900) and
    1.34 -  Sup  ("\<Squnion>_" [900] 900)
    1.35 +  Sup  ("\<Squnion>_" [900] 900) and
    1.36 +  top ("\<top>") and
    1.37 +  bot ("\<bottom>")
    1.38  
    1.39  lemmas mem_simps =
    1.40    insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff