src/HOL/Algebra/Group.thy
 changeset 22063 717425609192 parent 21041 60e418260b4d child 23350 50c5b0912a0c
     1.1 --- a/src/HOL/Algebra/Group.thy	Fri Jan 12 09:58:31 2007 +0100
1.2 +++ b/src/HOL/Algebra/Group.thy	Fri Jan 12 15:37:21 2007 +0100
1.3 @@ -685,7 +685,7 @@
1.4  text_raw {* \label{sec:subgroup-lattice} *}
1.5
1.6  theorem (in group) subgroups_partial_order:
1.7 -  "partial_order {H. subgroup H G} (op \<subseteq>)"
1.8 +  "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
1.9    by (rule partial_order.intro) simp_all
1.10
1.11  lemma (in group) subgroup_self:
1.12 @@ -730,23 +730,23 @@
1.13  qed
1.14
1.15  theorem (in group) subgroups_complete_lattice:
1.16 -  "complete_lattice {H. subgroup H G} (op \<subseteq>)"
1.17 -    (is "complete_lattice ?car ?le")
1.18 +  "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
1.19 +    (is "complete_lattice ?L")
1.20  proof (rule partial_order.complete_lattice_criterion1)
1.21 -  show "partial_order ?car ?le" by (rule subgroups_partial_order)
1.22 +  show "partial_order ?L" by (rule subgroups_partial_order)
1.23  next
1.24 -  have "order_syntax.greatest ?car ?le (carrier G) ?car"
1.25 -    by (unfold order_syntax.greatest_def)
1.26 +  have "greatest ?L (carrier G) (carrier ?L)"
1.27 +    by (unfold greatest_def)
1.29 -  then show "\<exists>G. order_syntax.greatest ?car ?le G ?car" ..
1.30 +  then show "\<exists>G. greatest ?L G (carrier ?L)" ..
1.31  next
1.32    fix A
1.33 -  assume L: "A \<subseteq> ?car" and non_empty: "A ~= {}"
1.34 +  assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
1.35    then have Int_subgroup: "subgroup (\<Inter>A) G"
1.36      by (fastsimp intro: subgroups_Inter)
1.37 -  have "order_syntax.greatest ?car ?le (\<Inter>A) (order_syntax.Lower ?car ?le A)"
1.38 -    (is "order_syntax.greatest _ _ ?Int _")
1.39 -  proof (rule order_syntax.greatest_LowerI)
1.40 +  have "greatest ?L (\<Inter>A) (Lower ?L A)"
1.41 +    (is "greatest _ ?Int _")
1.42 +  proof (rule greatest_LowerI)
1.43      fix H
1.44      assume H: "H \<in> A"
1.45      with L have subgroupH: "subgroup H G" by auto
1.46 @@ -755,18 +755,18 @@
1.47      from groupH have monoidH: "monoid ?H"
1.48        by (rule group.is_monoid)
1.49      from H have Int_subset: "?Int \<subseteq> H" by fastsimp
1.50 -    then show "?le ?Int H" by simp
1.51 +    then show "le ?L ?Int H" by simp
1.52    next
1.53      fix H
1.54 -    assume H: "H \<in> order_syntax.Lower ?car ?le A"
1.55 -    with L Int_subgroup show "?le H ?Int"
1.56 -      by (fastsimp simp: order_syntax.Lower_def intro: Inter_greatest)
1.57 +    assume H: "H \<in> Lower ?L A"
1.58 +    with L Int_subgroup show "le ?L H ?Int"
1.59 +      by (fastsimp simp: Lower_def intro: Inter_greatest)
1.60    next
1.61 -    show "A \<subseteq> ?car" by (rule L)
1.62 +    show "A \<subseteq> carrier ?L" by (rule L)
1.63    next
1.64 -    show "?Int \<in> ?car" by simp (rule Int_subgroup)
1.65 +    show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
1.66    qed
1.67 -  then show "\<exists>I. order_syntax.greatest ?car ?le I (order_syntax.Lower ?car ?le A)" ..
1.68 +  then show "\<exists>I. greatest ?L I (Lower ?L A)" ..
1.69  qed
1.70
1.71  end