# HG changeset patch
# User ballarin
# Date 1168612641 -3600
# Node ID 71742560919279baeaa6eade105eb574ef2c9762
# Parent  f4cfc4101c8f9040d7f4816e53b1ca60b29a2f42
Reverted to structure representation with records.

diff -r f4cfc4101c8f -r 717425609192 src/HOL/Algebra/Group.thy
--- a/src/HOL/Algebra/Group.thy	Fri Jan 12 09:58:31 2007 +0100
+++ b/src/HOL/Algebra/Group.thy	Fri Jan 12 15:37:21 2007 +0100
@@ -685,7 +685,7 @@
 text_raw {* \label{sec:subgroup-lattice} *}
 
 theorem (in group) subgroups_partial_order:
-  "partial_order {H. subgroup H G} (op \<subseteq>)"
+  "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
   by (rule partial_order.intro) simp_all
 
 lemma (in group) subgroup_self:
@@ -730,23 +730,23 @@
 qed
 
 theorem (in group) subgroups_complete_lattice:
-  "complete_lattice {H. subgroup H G} (op \<subseteq>)"
-    (is "complete_lattice ?car ?le")
+  "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
+    (is "complete_lattice ?L")
 proof (rule partial_order.complete_lattice_criterion1)
-  show "partial_order ?car ?le" by (rule subgroups_partial_order)
+  show "partial_order ?L" by (rule subgroups_partial_order)
 next
-  have "order_syntax.greatest ?car ?le (carrier G) ?car"
-    by (unfold order_syntax.greatest_def)
+  have "greatest ?L (carrier G) (carrier ?L)"
+    by (unfold greatest_def)
       (simp add: subgroup.subset subgroup_self)
-  then show "\<exists>G. order_syntax.greatest ?car ?le G ?car" ..
+  then show "\<exists>G. greatest ?L G (carrier ?L)" ..
 next
   fix A
-  assume L: "A \<subseteq> ?car" and non_empty: "A ~= {}"
+  assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
   then have Int_subgroup: "subgroup (\<Inter>A) G"
     by (fastsimp intro: subgroups_Inter)
-  have "order_syntax.greatest ?car ?le (\<Inter>A) (order_syntax.Lower ?car ?le A)"
-    (is "order_syntax.greatest _ _ ?Int _")
-  proof (rule order_syntax.greatest_LowerI)
+  have "greatest ?L (\<Inter>A) (Lower ?L A)"
+    (is "greatest _ ?Int _")
+  proof (rule greatest_LowerI)
     fix H
     assume H: "H \<in> A"
     with L have subgroupH: "subgroup H G" by auto
@@ -755,18 +755,18 @@
     from groupH have monoidH: "monoid ?H"
       by (rule group.is_monoid)
     from H have Int_subset: "?Int \<subseteq> H" by fastsimp
-    then show "?le ?Int H" by simp
+    then show "le ?L ?Int H" by simp
   next
     fix H
-    assume H: "H \<in> order_syntax.Lower ?car ?le A"
-    with L Int_subgroup show "?le H ?Int"
-      by (fastsimp simp: order_syntax.Lower_def intro: Inter_greatest)
+    assume H: "H \<in> Lower ?L A"
+    with L Int_subgroup show "le ?L H ?Int"
+      by (fastsimp simp: Lower_def intro: Inter_greatest)
   next
-    show "A \<subseteq> ?car" by (rule L)
+    show "A \<subseteq> carrier ?L" by (rule L)
   next
-    show "?Int \<in> ?car" by simp (rule Int_subgroup)
+    show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
   qed
-  then show "\<exists>I. order_syntax.greatest ?car ?le I (order_syntax.Lower ?car ?le A)" ..
+  then show "\<exists>I. greatest ?L I (Lower ?L A)" ..
 qed
 
 end
diff -r f4cfc4101c8f -r 717425609192 src/HOL/Algebra/IntRing.thy
--- a/src/HOL/Algebra/IntRing.thy	Fri Jan 12 09:58:31 2007 +0100
+++ b/src/HOL/Algebra/IntRing.thy	Fri Jan 12 15:37:21 2007 +0100
@@ -96,54 +96,60 @@
 interpretation "domain" ["\<Z>"] by (rule int_is_domain)
 
 lemma int_le_total_order:
-  "total_order (UNIV::int set) (op \<le>)"
-apply (rule partial_order.total_orderI)
- apply (rule partial_order.intro, simp+)
-apply clarsimp
-done
+  "total_order (| carrier = UNIV::int set, le = op \<le> |)"
+  apply (rule partial_order.total_orderI)
+   apply (rule partial_order.intro, simp+)
+  apply clarsimp
+  done
 
 lemma less_int:
-  "order_syntax.less (op \<le>::[int, int] => bool) = (op <)"
-  by (auto simp add: expand_fun_eq order_syntax.less_def)
+  "lless (| carrier = UNIV::int set, le = op \<le> |) = (op <)"
+  by (auto simp add: expand_fun_eq lless_def)
 
 lemma join_int:
-  "order_syntax.join (UNIV::int set) (op \<le>) = max"
+  "join (| carrier = UNIV::int set, le = op \<le> |) = max"
   apply (simp add: expand_fun_eq max_def)
   apply (rule+)
   apply (rule lattice.joinI)
   apply (simp add: int_le_total_order total_order.axioms)
-  apply (simp add: order_syntax.least_def order_syntax.Upper_def)
+  apply (simp add: least_def Upper_def)
   apply arith
   apply simp apply simp
   apply (rule lattice.joinI)
   apply (simp add: int_le_total_order total_order.axioms)
-  apply (simp add: order_syntax.least_def order_syntax.Upper_def)
+  apply (simp add: least_def Upper_def)
   apply arith
   apply simp apply simp
   done
 
 lemma meet_int:
-  "order_syntax.meet (UNIV::int set) (op \<le>) = min"
+  "meet (| carrier = UNIV::int set, le = op \<le> |) = min"
   apply (simp add: expand_fun_eq min_def)
   apply (rule+)
   apply (rule lattice.meetI)
   apply (simp add: int_le_total_order total_order.axioms)
-  apply (simp add: order_syntax.greatest_def order_syntax.Lower_def)
+  apply (simp add: greatest_def Lower_def)
   apply arith
   apply simp apply simp
   apply (rule lattice.meetI)
   apply (simp add: int_le_total_order total_order.axioms)
-  apply (simp add: order_syntax.greatest_def order_syntax.Lower_def)
+  apply (simp add: greatest_def Lower_def)
   apply arith
   apply simp apply simp
   done
 
-text {* Interpretation unfolding @{text less}, @{text join} and @{text
+lemma carrier_int:
+  "carrier (| carrier = UNIV::int set, le = op \<le> |) = UNIV"
+  apply simp
+  done
+
+text {* Interpretation unfolding @{text lless}, @{text join} and @{text
   meet} since they have natural representations in @{typ int}. *}
 
 interpretation 
-  int [unfolded less_int join_int meet_int]:
-  total_order ["UNIV::int set" "op \<le>"] by (rule int_le_total_order)
+  int [unfolded less_int join_int meet_int carrier_int]:
+  total_order ["(| carrier = UNIV::int set, le = op \<le> |)"]
+  by (rule int_le_total_order)
 
 
 subsubsection {* Generated Ideals of @{text "\<Z>"} *}
diff -r f4cfc4101c8f -r 717425609192 src/HOL/Algebra/Lattice.thy
--- a/src/HOL/Algebra/Lattice.thy	Fri Jan 12 09:58:31 2007 +0100
+++ b/src/HOL/Algebra/Lattice.thy	Fri Jan 12 15:37:21 2007 +0100
@@ -18,163 +18,157 @@
 
 subsection {* Partial Orders *}
 
-text {* Locale @{text order_syntax} is required since we want to refer
-  to definitions (and their derived theorems) outside of @{text partial_order}.
-  *}
-
-locale order_syntax =
-  fixes L :: "'a set" and le :: "['a, 'a] => bool" (infix "\<sqsubseteq>" 50)
-
-text {* Note that the type constraints above are necessary, because the
-  definition command cannot specialise the types. *}
-
-definition (in order_syntax)
-  less (infixl "\<sqsubset>" 50) where "x \<sqsubset> y == x \<sqsubseteq> y & x ~= y"
-
-text {* Upper and lower bounds of a set. *}
-
-definition (in order_syntax)
-  Upper :: "'a set => 'a set" where
-  "Upper A == {u. (ALL x. x \<in> A \<inter> L --> x \<sqsubseteq> u)} \<inter> L"
-
-definition (in order_syntax)
-  Lower :: "'a set => 'a set" where
-  "Lower A == {l. (ALL x. x \<in> A \<inter> L --> l \<sqsubseteq> x)} \<inter> L"
-
-text {* Least and greatest, as predicate. *}
+record 'a order = "'a partial_object" +
+  le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50)
 
-definition (in order_syntax)
-  least :: "['a, 'a set] => bool" where
-  "least l A == A \<subseteq> L & l \<in> A & (ALL x : A. l \<sqsubseteq> x)"
-
-definition (in order_syntax)
-  greatest :: "['a, 'a set] => bool" where
-  "greatest g A == A \<subseteq> L & g \<in> A & (ALL x : A. x \<sqsubseteq> g)"
-
-text {* Supremum and infimum *}
-
-definition (in order_syntax)
-  sup :: "'a set => 'a" ("\<Squnion>_" [90] 90) where
-  "\<Squnion>A == THE x. least x (Upper A)"
-
-definition (in order_syntax)
-  inf :: "'a set => 'a" ("\<Sqinter>_" [90] 90) where
-  "\<Sqinter>A == THE x. greatest x (Lower A)"
-
-definition (in order_syntax)
-  join :: "['a, 'a] => 'a" (infixl "\<squnion>" 65) where
-  "x \<squnion> y == sup {x, y}"
-
-definition (in order_syntax)
-  meet :: "['a, 'a] => 'a" (infixl "\<sqinter>" 70) where
-  "x \<sqinter> y == inf {x, y}"
-
-locale partial_order = order_syntax +
+locale partial_order =
+  fixes L (structure)
   assumes refl [intro, simp]:
-                  "x \<in> L ==> x \<sqsubseteq> x"
+                  "x \<in> carrier L ==> x \<sqsubseteq> x"
     and anti_sym [intro]:
-                  "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> L; y \<in> L |] ==> x = y"
+                  "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x = y"
     and trans [trans]:
                   "[| x \<sqsubseteq> y; y \<sqsubseteq> z;
-                   x \<in> L; y \<in> L; z \<in> L |] ==> x \<sqsubseteq> z"
+                   x \<in> carrier L; y \<in> carrier L; z \<in> carrier L |] ==> x \<sqsubseteq> z"
+
+constdefs (structure L)
+  lless :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50)
+  "x \<sqsubset> y == x \<sqsubseteq> y & x ~= y"
+
+  -- {* Upper and lower bounds of a set. *}
+  Upper :: "[_, 'a set] => 'a set"
+  "Upper L A == {u. (ALL x. x \<in> A \<inter> carrier L --> x \<sqsubseteq> u)} \<inter>
+                carrier L"
+
+  Lower :: "[_, 'a set] => 'a set"
+  "Lower L A == {l. (ALL x. x \<in> A \<inter> carrier L --> l \<sqsubseteq> x)} \<inter>
+                carrier L"
+
+  -- {* Least and greatest, as predicate. *}
+  least :: "[_, 'a, 'a set] => bool"
+  "least L l A == A \<subseteq> carrier L & l \<in> A & (ALL x : A. l \<sqsubseteq> x)"
+
+  greatest :: "[_, 'a, 'a set] => bool"
+  "greatest L g A == A \<subseteq> carrier L & g \<in> A & (ALL x : A. x \<sqsubseteq> g)"
+
+  -- {* Supremum and infimum *}
+  sup :: "[_, 'a set] => 'a" ("\<Squnion>\<index>_" [90] 90)
+  "\<Squnion>A == THE x. least L x (Upper L A)"
+
+  inf :: "[_, 'a set] => 'a" ("\<Sqinter>\<index>_" [90] 90)
+  "\<Sqinter>A == THE x. greatest L x (Lower L A)"
+
+  join :: "[_, 'a, 'a] => 'a" (infixl "\<squnion>\<index>" 65)
+  "x \<squnion> y == sup L {x, y}"
+
+  meet :: "[_, 'a, 'a] => 'a" (infixl "\<sqinter>\<index>" 70)
+  "x \<sqinter> y == inf L {x, y}"
 
 
 subsubsection {* Upper *}
 
-lemma (in order_syntax) Upper_closed [intro, simp]:
-  "Upper A \<subseteq> L"
+lemma Upper_closed [intro, simp]:
+  "Upper L A \<subseteq> carrier L"
   by (unfold Upper_def) clarify
 
-lemma (in order_syntax) UpperD [dest]:
-  "[| u \<in> Upper A; x \<in> A; A \<subseteq> L |] ==> x \<sqsubseteq> u"
+lemma UpperD [dest]:
+  fixes L (structure)
+  shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u"
   by (unfold Upper_def) blast
 
-lemma (in order_syntax) Upper_memI:
-  "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> L |] ==> x \<in> Upper A"
+lemma Upper_memI:
+  fixes L (structure)
+  shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A"
   by (unfold Upper_def) blast
 
-lemma (in order_syntax) Upper_antimono:
-  "A \<subseteq> B ==> Upper B \<subseteq> Upper A"
+lemma Upper_antimono:
+  "A \<subseteq> B ==> Upper L B \<subseteq> Upper L A"
   by (unfold Upper_def) blast
 
 
 subsubsection {* Lower *}
 
-lemma (in order_syntax) Lower_closed [intro, simp]:
-  "Lower A \<subseteq> L"
+lemma Lower_closed [intro, simp]:
+  "Lower L A \<subseteq> carrier L"
   by (unfold Lower_def) clarify
 
-lemma (in order_syntax) LowerD [dest]:
-  "[| l \<in> Lower A; x \<in> A; A \<subseteq> L |] ==> l \<sqsubseteq> x"
+lemma LowerD [dest]:
+  fixes L (structure)
+  shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x"
   by (unfold Lower_def) blast
 
-lemma (in order_syntax) Lower_memI:
-  "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> L |] ==> x \<in> Lower A"
+lemma Lower_memI:
+  fixes L (structure)
+  shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A"
   by (unfold Lower_def) blast
 
-lemma (in order_syntax) Lower_antimono:
-  "A \<subseteq> B ==> Lower B \<subseteq> Lower A"
+lemma Lower_antimono:
+  "A \<subseteq> B ==> Lower L B \<subseteq> Lower L A"
   by (unfold Lower_def) blast
 
 
 subsubsection {* least *}
 
-lemma (in order_syntax) least_closed [intro, simp]:
-  "least l A ==> l \<in> L"
+lemma least_carrier [intro, simp]:
+  shows "least L l A ==> l \<in> carrier L"
   by (unfold least_def) fast
 
-lemma (in order_syntax) least_mem:
-  "least l A ==> l \<in> A"
+lemma least_mem:
+  "least L l A ==> l \<in> A"
   by (unfold least_def) fast
 
 lemma (in partial_order) least_unique:
-  "[| least x A; least y A |] ==> x = y"
+  "[| least L x A; least L y A |] ==> x = y"
   by (unfold least_def) blast
 
-lemma (in order_syntax) least_le:
-  "[| least x A; a \<in> A |] ==> x \<sqsubseteq> a"
+lemma least_le:
+  fixes L (structure)
+  shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a"
   by (unfold least_def) fast
 
-lemma (in order_syntax) least_UpperI:
+lemma least_UpperI:
+  fixes L (structure)
   assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s"
-    and below: "!! y. y \<in> Upper A ==> s \<sqsubseteq> y"
-    and L: "A \<subseteq> L"  "s \<in> L"
-  shows "least s (Upper A)"
+    and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y"
+    and L: "A \<subseteq> carrier L"  "s \<in> carrier L"
+  shows "least L s (Upper L A)"
 proof -
-  have "Upper A \<subseteq> L" by simp
-  moreover from above L have "s \<in> Upper A" by (simp add: Upper_def)
-  moreover from below have "ALL x : Upper A. s \<sqsubseteq> x" by fast
+  have "Upper L A \<subseteq> carrier L" by simp
+  moreover from above L have "s \<in> Upper L A" by (simp add: Upper_def)
+  moreover from below have "ALL x : Upper L A. s \<sqsubseteq> x" by fast
   ultimately show ?thesis by (simp add: least_def)
 qed
 
 
 subsubsection {* greatest *}
 
-lemma (in order_syntax) greatest_closed [intro, simp]:
-  "greatest l A ==> l \<in> L"
+lemma greatest_carrier [intro, simp]:
+  shows "greatest L l A ==> l \<in> carrier L"
   by (unfold greatest_def) fast
 
-lemma (in order_syntax) greatest_mem:
-  "greatest l A ==> l \<in> A"
+lemma greatest_mem:
+  "greatest L l A ==> l \<in> A"
   by (unfold greatest_def) fast
 
 lemma (in partial_order) greatest_unique:
-  "[| greatest x A; greatest y A |] ==> x = y"
+  "[| greatest L x A; greatest L y A |] ==> x = y"
   by (unfold greatest_def) blast
 
-lemma (in order_syntax) greatest_le:
-  "[| greatest x A; a \<in> A |] ==> a \<sqsubseteq> x"
+lemma greatest_le:
+  fixes L (structure)
+  shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x"
   by (unfold greatest_def) fast
 
-lemma (in order_syntax) greatest_LowerI:
+lemma greatest_LowerI:
+  fixes L (structure)
   assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x"
-    and above: "!! y. y \<in> Lower A ==> y \<sqsubseteq> i"
-    and L: "A \<subseteq> L"  "i \<in> L"
-  shows "greatest i (Lower A)"
+    and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i"
+    and L: "A \<subseteq> carrier L"  "i \<in> carrier L"
+  shows "greatest L i (Lower L A)"
 proof -
-  have "Lower A \<subseteq> L" by simp
-  moreover from below L have "i \<in> Lower A" by (simp add: Lower_def)
-  moreover from above have "ALL x : Lower A. x \<sqsubseteq> i" by fast
+  have "Lower L A \<subseteq> carrier L" by simp
+  moreover from below L have "i \<in> Lower L A" by (simp add: Lower_def)
+  moreover from above have "ALL x : Lower L A. x \<sqsubseteq> i" by fast
   ultimately show ?thesis by (simp add: greatest_def)
 qed
 
@@ -183,61 +177,63 @@
 
 locale lattice = partial_order +
   assumes sup_of_two_exists:
-    "[| x \<in> L; y \<in> L |] ==> EX s. order_syntax.least L le s (order_syntax.Upper L le {x, y})"
+    "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. least L s (Upper L {x, y})"
     and inf_of_two_exists:
-    "[| x \<in> L; y \<in> L |] ==> EX s. order_syntax.greatest L le s (order_syntax.Lower L le {x, y})"
+    "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})"
 
-lemma (in order_syntax) least_Upper_above:
-  "[| least s (Upper A); x \<in> A; A \<subseteq> L |] ==> x \<sqsubseteq> s"
+lemma least_Upper_above:
+  fixes L (structure)
+  shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s"
   by (unfold least_def) blast
 
-lemma (in order_syntax) greatest_Lower_above:
-  "[| greatest i (Lower A); x \<in> A; A \<subseteq> L |] ==> i \<sqsubseteq> x"
+lemma greatest_Lower_above:
+  fixes L (structure)
+  shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x"
   by (unfold greatest_def) blast
 
 
 subsubsection {* Supremum *}
 
 lemma (in lattice) joinI:
-  "[| !!l. least l (Upper {x, y}) ==> P l; x \<in> L; y \<in> L |]
+  "[| !!l. least L l (Upper L {x, y}) ==> P l; x \<in> carrier L; y \<in> carrier L |]
   ==> P (x \<squnion> y)"
 proof (unfold join_def sup_def)
-  assume L: "x \<in> L"  "y \<in> L"
-    and P: "!!l. least l (Upper {x, y}) ==> P l"
-  with sup_of_two_exists obtain s where "least s (Upper {x, y})" by fast
-  with L show "P (THE l. least l (Upper {x, y}))"
+  assume L: "x \<in> carrier L"  "y \<in> carrier L"
+    and P: "!!l. least L l (Upper L {x, y}) ==> P l"
+  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
+  with L show "P (THE l. least L l (Upper L {x, y}))"
     by (fast intro: theI2 least_unique P)
 qed
 
 lemma (in lattice) join_closed [simp]:
-  "[| x \<in> L; y \<in> L |] ==> x \<squnion> y \<in> L"
-  by (rule joinI) (rule least_closed)
+  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<squnion> y \<in> carrier L"
+  by (rule joinI) (rule least_carrier)
 
-lemma (in partial_order) sup_of_singletonI:     (* only reflexivity needed ? *)
-  "x \<in> L ==> least x (Upper {x})"
+lemma (in partial_order) sup_of_singletonI:      (* only reflexivity needed ? *)
+  "x \<in> carrier L ==> least L x (Upper L {x})"
   by (rule least_UpperI) fast+
 
 lemma (in partial_order) sup_of_singleton [simp]:
-  "x \<in> L ==> \<Squnion>{x} = x"
+  "x \<in> carrier L ==> \<Squnion>{x} = x"
   by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI)
 
 
 text {* Condition on @{text A}: supremum exists. *}
 
 lemma (in lattice) sup_insertI:
-  "[| !!s. least s (Upper (insert x A)) ==> P s;
-  least a (Upper A); x \<in> L; A \<subseteq> L |]
+  "[| !!s. least L s (Upper L (insert x A)) ==> P s;
+  least L a (Upper L A); x \<in> carrier L; A \<subseteq> carrier L |]
   ==> P (\<Squnion>(insert x A))"
 proof (unfold sup_def)
-  assume L: "x \<in> L"  "A \<subseteq> L"
-    and P: "!!l. least l (Upper (insert x A)) ==> P l"
-    and least_a: "least a (Upper A)"
-  from least_a have La: "a \<in> L" by simp
+  assume L: "x \<in> carrier L"  "A \<subseteq> carrier L"
+    and P: "!!l. least L l (Upper L (insert x A)) ==> P l"
+    and least_a: "least L a (Upper L A)"
+  from L least_a have La: "a \<in> carrier L" by simp
   from L sup_of_two_exists least_a
-  obtain s where least_s: "least s (Upper {a, x})" by blast
-  show "P (THE l. least l (Upper (insert x A)))"
+  obtain s where least_s: "least L s (Upper L {a, x})" by blast
+  show "P (THE l. least L l (Upper L (insert x A)))"
   proof (rule theI2)
-    show "least s (Upper (insert x A))"
+    show "least L s (Upper L (insert x A))"
     proof (rule least_UpperI)
       fix z
       assume "z \<in> insert x A"
@@ -252,15 +248,15 @@
       qed
     next
       fix y
-      assume y: "y \<in> Upper (insert x A)"
+      assume y: "y \<in> Upper L (insert x A)"
       show "s \<sqsubseteq> y"
       proof (rule least_le [OF least_s], rule Upper_memI)
 	fix z
 	assume z: "z \<in> {a, x}"
 	then show "z \<sqsubseteq> y"
 	proof
-          have y': "y \<in> Upper A"
-            apply (rule subsetD [where A = "Upper (insert x A)"])
+          have y': "y \<in> Upper L A"
+            apply (rule subsetD [where A = "Upper L (insert x A)"])
             apply (rule Upper_antimono) apply clarify apply assumption
             done
           assume "z = a"
@@ -271,15 +267,15 @@
 	qed
       qed (rule Upper_closed [THEN subsetD])
     next
-      from L show "insert x A \<subseteq> L" by simp
-      from least_s show "s \<in> L" by simp
+      from L show "insert x A \<subseteq> carrier L" by simp
+      from least_s show "s \<in> carrier L" by simp
     qed
   next
     fix l
-    assume least_l: "least l (Upper (insert x A))"
+    assume least_l: "least L l (Upper L (insert x A))"
     show "l = s"
     proof (rule least_unique)
-      show "least s (Upper (insert x A))"
+      show "least L s (Upper L (insert x A))"
       proof (rule least_UpperI)
         fix z
         assume "z \<in> insert x A"
@@ -294,15 +290,15 @@
 	qed
       next
         fix y
-        assume y: "y \<in> Upper (insert x A)"
+        assume y: "y \<in> Upper L (insert x A)"
         show "s \<sqsubseteq> y"
         proof (rule least_le [OF least_s], rule Upper_memI)
           fix z
           assume z: "z \<in> {a, x}"
           then show "z \<sqsubseteq> y"
           proof
-            have y': "y \<in> Upper A"
-	      apply (rule subsetD [where A = "Upper (insert x A)"])
+            have y': "y \<in> Upper L A"
+	      apply (rule subsetD [where A = "Upper L (insert x A)"])
 	      apply (rule Upper_antimono) apply clarify apply assumption
 	      done
             assume "z = a"
@@ -313,15 +309,15 @@
           qed
         qed (rule Upper_closed [THEN subsetD])
       next
-        from L show "insert x A \<subseteq> L" by simp
-        from least_s show "s \<in> L" by simp
+        from L show "insert x A \<subseteq> carrier L" by simp
+        from least_s show "s \<in> carrier L" by simp
       qed
     qed
   qed
 qed
 
 lemma (in lattice) finite_sup_least:
-  "[| finite A; A \<subseteq> L; A ~= {} |] ==> least (\<Squnion>A) (Upper A)"
+  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion>A) (Upper L A)"
 proof (induct set: Finites)
   case empty
   then show ?case by simp
@@ -333,15 +329,15 @@
     with insert show ?thesis by (simp add: sup_of_singletonI)
   next
     case False
-    with insert have "least (\<Squnion>A) (Upper A)" by simp
+    with insert have "least L (\<Squnion>A) (Upper L A)" by simp
     with _ show ?thesis
       by (rule sup_insertI) (simp_all add: insert [simplified])
   qed
 qed
 
 lemma (in lattice) finite_sup_insertI:
-  assumes P: "!!l. least l (Upper (insert x A)) ==> P l"
-    and xA: "finite A"  "x \<in> L"  "A \<subseteq> L"
+  assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l"
+    and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
   shows "P (\<Squnion> (insert x A))"
 proof (cases "A = {}")
   case True with P and xA show ?thesis
@@ -352,7 +348,7 @@
 qed
 
 lemma (in lattice) finite_sup_closed:
-  "[| finite A; A \<subseteq> L; A ~= {} |] ==> \<Squnion>A \<in> L"
+  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion>A \<in> carrier L"
 proof (induct set: Finites)
   case empty then show ?case by simp
 next
@@ -361,39 +357,39 @@
 qed
 
 lemma (in lattice) join_left:
-  "[| x \<in> L; y \<in> L |] ==> x \<sqsubseteq> x \<squnion> y"
+  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
   by (rule joinI [folded join_def]) (blast dest: least_mem)
 
 lemma (in lattice) join_right:
-  "[| x \<in> L; y \<in> L |] ==> y \<sqsubseteq> x \<squnion> y"
+  "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
   by (rule joinI [folded join_def]) (blast dest: least_mem)
 
 lemma (in lattice) sup_of_two_least:
-  "[| x \<in> L; y \<in> L |] ==> least (\<Squnion>{x, y}) (Upper {x, y})"
+  "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion>{x, y}) (Upper L {x, y})"
 proof (unfold sup_def)
-  assume L: "x \<in> L"  "y \<in> L"
-  with sup_of_two_exists obtain s where "least s (Upper {x, y})" by fast
-  with L show "least (THE xa. least xa (Upper {x, y})) (Upper {x, y})"
+  assume L: "x \<in> carrier L"  "y \<in> carrier L"
+  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
+  with L show "least L (THE xa. least L xa (Upper L {x, y})) (Upper L {x, y})"
   by (fast intro: theI2 least_unique)  (* blast fails *)
 qed
 
 lemma (in lattice) join_le:
   assumes sub: "x \<sqsubseteq> z"  "y \<sqsubseteq> z"
-    and L: "x \<in> L"  "y \<in> L"  "z \<in> L"
+    and L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
   shows "x \<squnion> y \<sqsubseteq> z"
 proof (rule joinI)
   fix s
-  assume "least s (Upper {x, y})"
+  assume "least L s (Upper L {x, y})"
   with sub L show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI)
 qed
 
 lemma (in lattice) join_assoc_lemma:
-  assumes L: "x \<in> L"  "y \<in> L"  "z \<in> L"
+  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
   shows "x \<squnion> (y \<squnion> z) = \<Squnion>{x, y, z}"
 proof (rule finite_sup_insertI)
   -- {* The textbook argument in Jacobson I, p 457 *}
   fix s
-  assume sup: "least s (Upper {x, y, z})"
+  assume sup: "least L s (Upper L {x, y, z})"
   show "x \<squnion> (y \<squnion> z) = s"
   proof (rule anti_sym)
     from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s"
@@ -402,15 +398,16 @@
     from sup L show "s \<sqsubseteq> x \<squnion> (y \<squnion> z)"
     by (erule_tac least_le)
       (blast intro!: Upper_memI intro: trans join_left join_right join_closed)
-  qed (simp_all add: L least_closed [OF sup])
+  qed (simp_all add: L least_carrier [OF sup])
 qed (simp_all add: L)
 
-lemma (in order_syntax) join_comm:
-  "x \<squnion> y = y \<squnion> x"
+lemma join_comm:
+  fixes L (structure)
+  shows "x \<squnion> y = y \<squnion> x"
   by (unfold join_def) (simp add: insert_commute)
 
 lemma (in lattice) join_assoc:
-  assumes L: "x \<in> L"  "y \<in> L"  "z \<in> L"
+  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
   shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
 proof -
   have "(x \<squnion> y) \<squnion> z = z \<squnion> (x \<squnion> y)" by (simp only: join_comm)
@@ -424,44 +421,45 @@
 subsubsection {* Infimum *}
 
 lemma (in lattice) meetI:
-  "[| !!i. greatest i (Lower {x, y}) ==> P i; x \<in> L; y \<in> L |]
+  "[| !!i. greatest L i (Lower L {x, y}) ==> P i;
+  x \<in> carrier L; y \<in> carrier L |]
   ==> P (x \<sqinter> y)"
 proof (unfold meet_def inf_def)
-  assume L: "x \<in> L"  "y \<in> L"
-    and P: "!!g. greatest g (Lower {x, y}) ==> P g"
-  with inf_of_two_exists obtain i where "greatest i (Lower {x, y})" by fast
-  with L show "P (THE g. greatest g (Lower {x, y}))"
+  assume L: "x \<in> carrier L"  "y \<in> carrier L"
+    and P: "!!g. greatest L g (Lower L {x, y}) ==> P g"
+  with inf_of_two_exists obtain i where "greatest L i (Lower L {x, y})" by fast
+  with L show "P (THE g. greatest L g (Lower L {x, y}))"
   by (fast intro: theI2 greatest_unique P)
 qed
 
 lemma (in lattice) meet_closed [simp]:
-  "[| x \<in> L; y \<in> L |] ==> x \<sqinter> y \<in> L"
-  by (rule meetI) (rule greatest_closed)
+  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<in> carrier L"
+  by (rule meetI) (rule greatest_carrier)
 
 lemma (in partial_order) inf_of_singletonI:      (* only reflexivity needed ? *)
-  "x \<in> L ==> greatest x (Lower {x})"
+  "x \<in> carrier L ==> greatest L x (Lower L {x})"
   by (rule greatest_LowerI) fast+
 
 lemma (in partial_order) inf_of_singleton [simp]:
-  "x \<in> L ==> \<Sqinter> {x} = x"
+  "x \<in> carrier L ==> \<Sqinter> {x} = x"
   by (unfold inf_def) (blast intro: greatest_unique greatest_LowerI inf_of_singletonI)
 
 text {* Condition on A: infimum exists. *}
 
 lemma (in lattice) inf_insertI:
-  "[| !!i. greatest i (Lower (insert x A)) ==> P i;
-  greatest a (Lower A); x \<in> L; A \<subseteq> L |]
+  "[| !!i. greatest L i (Lower L (insert x A)) ==> P i;
+  greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |]
   ==> P (\<Sqinter>(insert x A))"
 proof (unfold inf_def)
-  assume L: "x \<in> L"  "A \<subseteq> L"
-    and P: "!!g. greatest g (Lower (insert x A)) ==> P g"
-    and greatest_a: "greatest a (Lower A)"
-  from greatest_a have La: "a \<in> L" by simp
+  assume L: "x \<in> carrier L"  "A \<subseteq> carrier L"
+    and P: "!!g. greatest L g (Lower L (insert x A)) ==> P g"
+    and greatest_a: "greatest L a (Lower L A)"
+  from L greatest_a have La: "a \<in> carrier L" by simp
   from L inf_of_two_exists greatest_a
-  obtain i where greatest_i: "greatest i (Lower {a, x})" by blast
-  show "P (THE g. greatest g (Lower (insert x A)))"
+  obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast
+  show "P (THE g. greatest L g (Lower L (insert x A)))"
   proof (rule theI2)
-    show "greatest i (Lower (insert x A))"
+    show "greatest L i (Lower L (insert x A))"
     proof (rule greatest_LowerI)
       fix z
       assume "z \<in> insert x A"
@@ -476,15 +474,15 @@
       qed
     next
       fix y
-      assume y: "y \<in> Lower (insert x A)"
+      assume y: "y \<in> Lower L (insert x A)"
       show "y \<sqsubseteq> i"
       proof (rule greatest_le [OF greatest_i], rule Lower_memI)
 	fix z
 	assume z: "z \<in> {a, x}"
 	then show "y \<sqsubseteq> z"
 	proof
-          have y': "y \<in> Lower A"
-            apply (rule subsetD [where A = "Lower (insert x A)"])
+          have y': "y \<in> Lower L A"
+            apply (rule subsetD [where A = "Lower L (insert x A)"])
             apply (rule Lower_antimono) apply clarify apply assumption
             done
           assume "z = a"
@@ -495,15 +493,15 @@
 	qed
       qed (rule Lower_closed [THEN subsetD])
     next
-      from L show "insert x A \<subseteq> L" by simp
-      from greatest_i show "i \<in> L" by simp
+      from L show "insert x A \<subseteq> carrier L" by simp
+      from greatest_i show "i \<in> carrier L" by simp
     qed
   next
     fix g
-    assume greatest_g: "greatest g (Lower (insert x A))"
+    assume greatest_g: "greatest L g (Lower L (insert x A))"
     show "g = i"
     proof (rule greatest_unique)
-      show "greatest i (Lower (insert x A))"
+      show "greatest L i (Lower L (insert x A))"
       proof (rule greatest_LowerI)
         fix z
         assume "z \<in> insert x A"
@@ -518,15 +516,15 @@
         qed
       next
         fix y
-        assume y: "y \<in> Lower (insert x A)"
+        assume y: "y \<in> Lower L (insert x A)"
         show "y \<sqsubseteq> i"
         proof (rule greatest_le [OF greatest_i], rule Lower_memI)
           fix z
           assume z: "z \<in> {a, x}"
           then show "y \<sqsubseteq> z"
           proof
-            have y': "y \<in> Lower A"
-	      apply (rule subsetD [where A = "Lower (insert x A)"])
+            have y': "y \<in> Lower L A"
+	      apply (rule subsetD [where A = "Lower L (insert x A)"])
 	      apply (rule Lower_antimono) apply clarify apply assumption
 	      done
             assume "z = a"
@@ -537,15 +535,15 @@
 	  qed
         qed (rule Lower_closed [THEN subsetD])
       next
-        from L show "insert x A \<subseteq> L" by simp
-        from greatest_i show "i \<in> L" by simp
+        from L show "insert x A \<subseteq> carrier L" by simp
+        from greatest_i show "i \<in> carrier L" by simp
       qed
     qed
   qed
 qed
 
 lemma (in lattice) finite_inf_greatest:
-  "[| finite A; A \<subseteq> L; A ~= {} |] ==> greatest (\<Sqinter>A) (Lower A)"
+  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter>A) (Lower L A)"
 proof (induct set: Finites)
   case empty then show ?case by simp
 next
@@ -558,14 +556,14 @@
     case False
     from insert show ?thesis
     proof (rule_tac inf_insertI)
-      from False insert show "greatest (\<Sqinter>A) (Lower A)" by simp
+      from False insert show "greatest L (\<Sqinter>A) (Lower L A)" by simp
     qed simp_all
   qed
 qed
 
 lemma (in lattice) finite_inf_insertI:
-  assumes P: "!!i. greatest i (Lower (insert x A)) ==> P i"
-    and xA: "finite A"  "x \<in> L"  "A \<subseteq> L"
+  assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i"
+    and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
   shows "P (\<Sqinter> (insert x A))"
 proof (cases "A = {}")
   case True with P and xA show ?thesis
@@ -576,7 +574,7 @@
 qed
 
 lemma (in lattice) finite_inf_closed:
-  "[| finite A; A \<subseteq> L; A ~= {} |] ==> \<Sqinter>A \<in> L"
+  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter>A \<in> carrier L"
 proof (induct set: Finites)
   case empty then show ?case by simp
 next
@@ -585,40 +583,41 @@
 qed
 
 lemma (in lattice) meet_left:
-  "[| x \<in> L; y \<in> L |] ==> x \<sqinter> y \<sqsubseteq> x"
+  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
   by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
 
 lemma (in lattice) meet_right:
-  "[| x \<in> L; y \<in> L |] ==> x \<sqinter> y \<sqsubseteq> y"
+  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y"
   by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
 
 lemma (in lattice) inf_of_two_greatest:
-  "[| x \<in> L; y \<in> L |] ==> greatest (\<Sqinter> {x, y}) (Lower {x, y})"
+  "[| x \<in> carrier L; y \<in> carrier L |] ==>
+  greatest L (\<Sqinter> {x, y}) (Lower L {x, y})"
 proof (unfold inf_def)
-  assume L: "x \<in> L"  "y \<in> L"
-  with inf_of_two_exists obtain s where "greatest s (Lower {x, y})" by fast
+  assume L: "x \<in> carrier L"  "y \<in> carrier L"
+  with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast
   with L
-  show "greatest (THE xa. greatest xa (Lower {x, y})) (Lower {x, y})"
+  show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {x, y})"
   by (fast intro: theI2 greatest_unique)  (* blast fails *)
 qed
 
 lemma (in lattice) meet_le:
   assumes sub: "z \<sqsubseteq> x"  "z \<sqsubseteq> y"
-    and L: "x \<in> L"  "y \<in> L"  "z \<in> L"
+    and L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
   shows "z \<sqsubseteq> x \<sqinter> y"
 proof (rule meetI)
   fix i
-  assume "greatest i (Lower {x, y})"
+  assume "greatest L i (Lower L {x, y})"
   with sub L show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI)
 qed
 
 lemma (in lattice) meet_assoc_lemma:
-  assumes L: "x \<in> L"  "y \<in> L"  "z \<in> L"
+  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
   shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{x, y, z}"
 proof (rule finite_inf_insertI)
   txt {* The textbook argument in Jacobson I, p 457 *}
   fix i
-  assume inf: "greatest i (Lower {x, y, z})"
+  assume inf: "greatest L i (Lower L {x, y, z})"
   show "x \<sqinter> (y \<sqinter> z) = i"
   proof (rule anti_sym)
     from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"
@@ -627,15 +626,16 @@
     from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> i"
     by (erule_tac greatest_le)
       (blast intro!: Lower_memI intro: trans meet_left meet_right meet_closed)
-  qed (simp_all add: L greatest_closed [OF inf])
+  qed (simp_all add: L greatest_carrier [OF inf])
 qed (simp_all add: L)
 
-lemma (in order_syntax) meet_comm:
-  "x \<sqinter> y = y \<sqinter> x"
+lemma meet_comm:
+  fixes L (structure)
+  shows "x \<sqinter> y = y \<sqinter> x"
   by (unfold meet_def) (simp add: insert_commute)
 
 lemma (in lattice) meet_assoc:
-  assumes L: "x \<in> L"  "y \<in> L"  "z \<in> L"
+  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
   shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
 proof -
   have "(x \<sqinter> y) \<sqinter> z = z \<sqinter> (x \<sqinter> y)" by (simp only: meet_comm)
@@ -649,52 +649,51 @@
 subsection {* Total Orders *}
 
 locale total_order = lattice +
-  assumes total: "[| x \<in> L; y \<in> L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
-
+  assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
 
 text {* Introduction rule: the usual definition of total order *}
 
 lemma (in partial_order) total_orderI:
-  assumes total: "!!x y. [| x \<in> L; y \<in> L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
-  shows "total_order L le"
+  assumes total: "!!x y. [| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
+  shows "total_order L"
 proof intro_locales
-  show "lattice_axioms L le"
+  show "lattice_axioms L"
   proof (rule lattice_axioms.intro)
     fix x y
-    assume L: "x \<in> L"  "y \<in> L"
-    show "EX s. least s (Upper {x, y})"
+    assume L: "x \<in> carrier L"  "y \<in> carrier L"
+    show "EX s. least L s (Upper L {x, y})"
     proof -
       note total L
       moreover
       {
         assume "x \<sqsubseteq> y"
-        with L have "least y (Upper {x, y})"
+        with L have "least L y (Upper L {x, y})"
           by (rule_tac least_UpperI) auto
       }
       moreover
       {
         assume "y \<sqsubseteq> x"
-        with L have "least x (Upper {x, y})"
+        with L have "least L x (Upper L {x, y})"
           by (rule_tac least_UpperI) auto
       }
       ultimately show ?thesis by blast
     qed
   next
     fix x y
-    assume L: "x \<in> L"  "y \<in> L"
-    show "EX i. greatest i (Lower {x, y})"
+    assume L: "x \<in> carrier L"  "y \<in> carrier L"
+    show "EX i. greatest L i (Lower L {x, y})"
     proof -
       note total L
       moreover
       {
         assume "y \<sqsubseteq> x"
-        with L have "greatest y (Lower {x, y})"
+        with L have "greatest L y (Lower L {x, y})"
           by (rule_tac greatest_LowerI) auto
       }
       moreover
       {
         assume "x \<sqsubseteq> y"
-        with L have "greatest x (Lower {x, y})"
+        with L have "greatest L x (Lower L {x, y})"
           by (rule_tac greatest_LowerI) auto
       }
       ultimately show ?thesis by blast
@@ -707,98 +706,97 @@
 
 locale complete_lattice = lattice +
   assumes sup_exists:
-    "[| A \<subseteq> L |] ==> EX s. order_syntax.least L le s (order_syntax.Upper L le A)"
+    "[| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
     and inf_exists:
-    "[| A \<subseteq> L |] ==> EX i. order_syntax.greatest L le i (order_syntax.Lower L le A)"
+    "[| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
 
 text {* Introduction rule: the usual definition of complete lattice *}
 
 lemma (in partial_order) complete_latticeI:
   assumes sup_exists:
-    "!!A. [| A \<subseteq> L |] ==> EX s. least s (Upper A)"
+    "!!A. [| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
     and inf_exists:
-    "!!A. [| A \<subseteq> L |] ==> EX i. greatest i (Lower A)"
-  shows "complete_lattice L le"
+    "!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
+  shows "complete_lattice L"
 proof intro_locales
-  show "lattice_axioms L le"
+  show "lattice_axioms L"
     by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
 qed (assumption | rule complete_lattice_axioms.intro)+
 
-definition (in order_syntax)
-  top ("\<top>") where
-  "\<top> == sup L"
+constdefs (structure L)
+  top :: "_ => 'a" ("\<top>\<index>")
+  "\<top> == sup L (carrier L)"
 
-definition (in order_syntax)
-  bottom ("\<bottom>") where
-  "\<bottom> == inf L"
+  bottom :: "_ => 'a" ("\<bottom>\<index>")
+  "\<bottom> == inf L (carrier L)"
 
 
 lemma (in complete_lattice) supI:
-  "[| !!l. least l (Upper A) ==> P l; A \<subseteq> L |]
+  "[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |]
   ==> P (\<Squnion>A)"
 proof (unfold sup_def)
-  assume L: "A \<subseteq> L"
-    and P: "!!l. least l (Upper A) ==> P l"
-  with sup_exists obtain s where "least s (Upper A)" by blast
-  with L show "P (THE l. least l (Upper A))"
+  assume L: "A \<subseteq> carrier L"
+    and P: "!!l. least L l (Upper L A) ==> P l"
+  with sup_exists obtain s where "least L s (Upper L A)" by blast
+  with L show "P (THE l. least L l (Upper L A))"
   by (fast intro: theI2 least_unique P)
 qed
 
 lemma (in complete_lattice) sup_closed [simp]:
-  "A \<subseteq> L ==> \<Squnion>A \<in> L"
+  "A \<subseteq> carrier L ==> \<Squnion>A \<in> carrier L"
   by (rule supI) simp_all
 
 lemma (in complete_lattice) top_closed [simp, intro]:
-  "\<top> \<in> L"
+  "\<top> \<in> carrier L"
   by (unfold top_def) simp
 
 lemma (in complete_lattice) infI:
-  "[| !!i. greatest i (Lower A) ==> P i; A \<subseteq> L |]
+  "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |]
   ==> P (\<Sqinter>A)"
 proof (unfold inf_def)
-  assume L: "A \<subseteq> L"
-    and P: "!!l. greatest l (Lower A) ==> P l"
-  with inf_exists obtain s where "greatest s (Lower A)" by blast
-  with L show "P (THE l. greatest l (Lower A))"
+  assume L: "A \<subseteq> carrier L"
+    and P: "!!l. greatest L l (Lower L A) ==> P l"
+  with inf_exists obtain s where "greatest L s (Lower L A)" by blast
+  with L show "P (THE l. greatest L l (Lower L A))"
   by (fast intro: theI2 greatest_unique P)
 qed
 
 lemma (in complete_lattice) inf_closed [simp]:
-  "A \<subseteq> L ==> \<Sqinter>A \<in> L"
+  "A \<subseteq> carrier L ==> \<Sqinter>A \<in> carrier L"
   by (rule infI) simp_all
 
 lemma (in complete_lattice) bottom_closed [simp, intro]:
-  "\<bottom> \<in> L"
+  "\<bottom> \<in> carrier L"
   by (unfold bottom_def) simp
 
 text {* Jacobson: Theorem 8.1 *}
 
-lemma (in order_syntax) Lower_empty [simp]:
-  "Lower {} = L"
+lemma Lower_empty [simp]:
+  "Lower L {} = carrier L"
   by (unfold Lower_def) simp
 
-lemma (in order_syntax) Upper_empty [simp]:
-  "Upper {} = L"
+lemma Upper_empty [simp]:
+  "Upper L {} = carrier L"
   by (unfold Upper_def) simp
 
 theorem (in partial_order) complete_lattice_criterion1:
-  assumes top_exists: "EX g. greatest g L"
+  assumes top_exists: "EX g. greatest L g (carrier L)"
     and inf_exists:
-      "!!A. [| A \<subseteq> L; A ~= {} |] ==> EX i. greatest i (Lower A)"
-  shows "complete_lattice L le"
+      "!!A. [| A \<subseteq> carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)"
+  shows "complete_lattice L"
 proof (rule complete_latticeI)
-  from top_exists obtain top where top: "greatest top L" ..
+  from top_exists obtain top where top: "greatest L top (carrier L)" ..
   fix A
-  assume L: "A \<subseteq> L"
-  let ?B = "Upper A"
+  assume L: "A \<subseteq> carrier L"
+  let ?B = "Upper L A"
   from L top have "top \<in> ?B" by (fast intro!: Upper_memI intro: greatest_le)
   then have B_non_empty: "?B ~= {}" by fast
-  have B_L: "?B \<subseteq> L" by simp
+  have B_L: "?B \<subseteq> carrier L" by simp
   from inf_exists [OF B_L B_non_empty]
-  obtain b where b_inf_B: "greatest b (Lower ?B)" ..
-  have "least b (Upper A)"
+  obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
+  have "least L b (Upper L A)"
 apply (rule least_UpperI)
-   apply (rule greatest_le [where A = "Lower ?B"])
+   apply (rule greatest_le [where A = "Lower L ?B"])
     apply (rule b_inf_B)
    apply (rule Lower_memI)
     apply (erule UpperD)
@@ -808,13 +806,13 @@
   apply (erule greatest_Lower_above [OF b_inf_B])
   apply simp
  apply (rule L)
-apply (rule greatest_closed [OF b_inf_B]) (* rename rule: _closed *)
+apply (rule greatest_carrier [OF b_inf_B]) (* rename rule: _closed *)
 done
-  then show "EX s. least s (Upper A)" ..
+  then show "EX s. least L s (Upper L A)" ..
 next
   fix A
-  assume L: "A \<subseteq> L"
-  show "EX i. greatest i (Lower A)"
+  assume L: "A \<subseteq> carrier L"
+  show "EX i. greatest L i (Lower L A)"
   proof (cases "A = {}")
     case True then show ?thesis
       by (simp add: top_exists)
@@ -832,25 +830,25 @@
 subsubsection {* Powerset of a Set is a Complete Lattice *}
 
 theorem powerset_is_complete_lattice:
-  "complete_lattice (Pow A) (op \<subseteq>)"
-  (is "complete_lattice ?L ?le")
+  "complete_lattice (| carrier = Pow A, le = op \<subseteq> |)"
+  (is "complete_lattice ?L")
 proof (rule partial_order.complete_latticeI)
-  show "partial_order ?L ?le"
+  show "partial_order ?L"
     by (rule partial_order.intro) auto
 next
   fix B
-  assume "B \<subseteq> ?L"
-  then have "order_syntax.least ?L ?le (\<Union> B) (order_syntax.Upper ?L ?le B)"
-    by (fastsimp intro!: order_syntax.least_UpperI simp: order_syntax.Upper_def)
-  then show "EX s. order_syntax.least ?L ?le s (order_syntax.Upper ?L ?le B)" ..
+  assume "B \<subseteq> carrier ?L"
+  then have "least ?L (\<Union> B) (Upper ?L B)"
+    by (fastsimp intro!: least_UpperI simp: Upper_def)
+  then show "EX s. least ?L s (Upper ?L B)" ..
 next
   fix B
-  assume "B \<subseteq> ?L"
-  then have "order_syntax.greatest ?L ?le (\<Inter> B \<inter> A) (order_syntax.Lower ?L ?le B)"
+  assume "B \<subseteq> carrier ?L"
+  then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L B)"
     txt {* @{term "\<Inter> B"} is not the infimum of @{term B}:
       @{term "\<Inter> {} = UNIV"} which is in general bigger than @{term "A"}! *}
-    by (fastsimp intro!: order_syntax.greatest_LowerI simp: order_syntax.Lower_def)
-  then show "EX i. order_syntax.greatest ?L ?le i (order_syntax.Lower ?L ?le B)" ..
+    by (fastsimp intro!: greatest_LowerI simp: Lower_def)
+  then show "EX i. greatest ?L i (Lower ?L B)" ..
 qed
 
 text {* An other example, that of the lattice of subgroups of a group,