src/HOL/Algebra/Lattice.thy

 subsection {* Partial Orders *}
 
 record 'a order = "'a partial_object" +
   le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50)
 
-locale order_syntax = struct L
-
-locale partial_order = order_syntax +
+locale partial_order = struct L +
   assumes refl [intro, simp]:
                   "x \<in> carrier L ==> x \<sqsubseteq> x"
     and anti_sym [intro]:
                   "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x = y"
     and trans [trans]:

   less :: "[_, '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 --> le L x u)} \<inter>
+  "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 --> le L l x)} \<inter>
+  "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. le L l x)"
+  "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. le L x g)"
+  "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)"
 

 lemma Upper_closed [intro, simp]:
   "Upper L A \<subseteq> carrier L"
   by (unfold Upper_def) clarify
 
 lemma UpperD [dest]:
-  includes order_syntax
+  includes struct L
   shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u"
-  by (unfold Upper_def) blast 
+  by (unfold Upper_def) blast
 
 lemma Upper_memI:
-  includes order_syntax
+  includes struct L
   shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A"
-  by (unfold Upper_def) blast 
+  by (unfold Upper_def) blast
 
 lemma Upper_antimono:
   "A \<subseteq> B ==> Upper L B \<subseteq> Upper L A"
   by (unfold Upper_def) blast
 

 lemma Lower_closed [intro, simp]:
   "Lower L A \<subseteq> carrier L"
   by (unfold Lower_def) clarify
 
 lemma LowerD [dest]:
-  includes order_syntax
+  includes struct L
   shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x"
-  by (unfold Lower_def) blast 
+  by (unfold Lower_def) blast
 
 lemma Lower_memI:
-  includes order_syntax
+  includes struct L
   shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A"
-  by (unfold Lower_def) blast 
+  by (unfold Lower_def) blast
 
 lemma Lower_antimono:
   "A \<subseteq> B ==> Lower L B \<subseteq> Lower L A"
   by (unfold Lower_def) blast
 

 lemma (in partial_order) least_unique:
   "[| least L x A; least L y A |] ==> x = y"
   by (unfold least_def) blast
 
 lemma least_le:
-  includes order_syntax
+  includes struct L
   shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a"
   by (unfold least_def) fast
 
 lemma least_UpperI:
-  includes order_syntax
+  includes struct L
   assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s"
     and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y"
-    and L: "A \<subseteq> carrier L" "s \<in> carrier L"
+    and L: "A \<subseteq> carrier L"  "s \<in> carrier L"
   shows "least L s (Upper L A)"
-proof (unfold least_def, intro conjI)
-  show "Upper L A \<subseteq> carrier L" by simp
-next
-  from above L show "s \<in> Upper L A" by (simp add: Upper_def)
-next
-  from below show "ALL x : Upper L A. s \<sqsubseteq> x" by fast
+proof -
+  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 partial_order) greatest_unique:
   "[| greatest L x A; greatest L y A |] ==> x = y"
   by (unfold greatest_def) blast
 
 lemma greatest_le:
-  includes order_syntax
+  includes struct L
   shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x"
   by (unfold greatest_def) fast
 
 lemma greatest_LowerI:
-  includes order_syntax
+  includes struct L
   assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x"
     and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i"
-    and L: "A \<subseteq> carrier L" "i \<in> carrier L"
+    and L: "A \<subseteq> carrier L"  "i \<in> carrier L"
   shows "greatest L i (Lower L A)"
-proof (unfold greatest_def, intro conjI)
-  show "Lower L A \<subseteq> carrier L" by simp
-next
-  from below L show "i \<in> Lower L A" by (simp add: Lower_def)
-next
-  from above show "ALL x : Lower L A. x \<sqsubseteq> i" by fast
-qed
+proof -
+  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
+
 
 subsection {* Lattices *}
 
 locale lattice = partial_order +
   assumes sup_of_two_exists:
     "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. least L s (Upper L {x, y})"
     and inf_of_two_exists:
     "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})"
 
 lemma least_Upper_above:
-  includes order_syntax
+  includes struct L
   shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s"
   by (unfold least_def) blast
 
 lemma greatest_Lower_above:
-  includes order_syntax
+  includes struct L
   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 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> carrier L" "y \<in> carrier L"
+  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)
+    by (fast intro: theI2 least_unique P)
 qed
 
 lemma (in lattice) join_closed [simp]:
   "[| 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> carrier L ==> least L x (Upper L {x})"
   by (rule least_UpperI) fast+
 
 lemma (in partial_order) sup_of_singleton [simp]:
-  includes order_syntax
-  shows "x \<in> carrier L ==> \<Squnion> {x} = x"
+  includes struct L
+  shows "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 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))"
+  ==> P (\<Squnion>(insert x A))"
 proof (unfold sup_def)
-  assume L: "x \<in> carrier L" "A \<subseteq> carrier L"
+  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 L s (Upper L {a, x})" by blast
   show "P (THE l. least L l (Upper L (insert x A)))"
-  proof (rule theI2 [where a = s])
+  proof (rule theI2)
     show "least L s (Upper L (insert x A))"
     proof (rule least_UpperI)
       fix z
-      assume xA: "z \<in> insert x A"
-      show "z \<sqsubseteq> s"
-      proof -
-	{
-	  assume "z = x" then have ?thesis
-	    by (simp add: least_Upper_above [OF least_s] L La)
-        }
-	moreover
-        {
-	  assume "z \<in> A"
-          with L least_s least_a have ?thesis
-	    by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
-        }
-      moreover note xA
-      ultimately show ?thesis by blast
+      assume "z \<in> insert x A"
+      then show "z \<sqsubseteq> s"
+      proof
+        assume "z = x" then show ?thesis
+          by (simp add: least_Upper_above [OF least_s] L La)
+      next
+        assume "z \<in> A"
+        with L least_s least_a show ?thesis
+          by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
+      qed
+    next
+      fix y
+      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 L A"
+            apply (rule subsetD [where A = "Upper L (insert x A)"])
+            apply (rule Upper_antimono) apply clarify apply assumption
+            done
+          assume "z = a"
+          with y' least_a show ?thesis by (fast dest: least_le)
+	next
+	  assume "z \<in> {x}"  (* FIXME "z = x"; declare specific elim rule for "insert x {}" (!?) *)
+          with y L show ?thesis by blast
+	qed
+      qed (rule Upper_closed [THEN subsetD])
+    next
+      from L show "insert x A \<subseteq> carrier L" by simp
+      from least_s show "s \<in> carrier L" by simp
     qed
   next
-    fix y
-    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}"
-      show "z \<sqsubseteq> y"
-      proof -
-	{
-          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"
-	  with y' least_a have ?thesis by (fast dest: least_le)
-        }
-	moreover
-	{
-           assume "z = x"
-           with y L have ?thesis by blast
-        }
-        moreover note z
-        ultimately show ?thesis by blast
-      qed
-    qed (rule Upper_closed [THEN subsetD])
-  next
-    from L show "insert x A \<subseteq> carrier L" by simp
-  next
-    from least_s show "s \<in> carrier L" by simp
-  qed
-next
     fix l
     assume least_l: "least L l (Upper L (insert x A))"
     show "l = s"
     proof (rule least_unique)
       show "least L s (Upper L (insert x A))"
       proof (rule least_UpperI)
-	fix z
-	assume xA: "z \<in> insert x A"
-	show "z \<sqsubseteq> s"
-      proof -
-	{
-	  assume "z = x" then have ?thesis
-	    by (simp add: least_Upper_above [OF least_s] L La)
-        }
-	moreover
-        {
-	  assume "z \<in> A"
-          with L least_s least_a have ?thesis
-	    by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
-        }
-	  moreover note xA
-	  ultimately show ?thesis by blast
+        fix z
+        assume "z \<in> insert x A"
+        then show "z \<sqsubseteq> s"
+	proof
+          assume "z = x" then show ?thesis
+            by (simp add: least_Upper_above [OF least_s] L La)
+	next
+          assume "z \<in> A"
+          with L least_s least_a show ?thesis
+            by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
 	qed
       next
-	fix y
-	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}"
-	  show "z \<sqsubseteq> y"
-	  proof -
-	    {
-          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"
-	  with y' least_a have ?thesis by (fast dest: least_le)
-        }
-	moreover
-	{
-           assume "z = x"
-           with y L have ?thesis by blast
-            }
-            moreover note z
-            ultimately show ?thesis by blast
-	  qed
-	qed (rule Upper_closed [THEN subsetD])
+        fix y
+        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 L A"
+	      apply (rule subsetD [where A = "Upper L (insert x A)"])
+	      apply (rule Upper_antimono) apply clarify apply assumption
+	      done
+            assume "z = a"
+            with y' least_a show ?thesis by (fast dest: least_le)
+	  next
+            assume "z \<in> {x}"
+            with y L show ?thesis by blast
+          qed
+        qed (rule Upper_closed [THEN subsetD])
       next
-	from L show "insert x A \<subseteq> carrier L" by simp
-      next
-	from least_s show "s \<in> carrier 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> carrier L; A ~= {} |] ==> least L (\<Squnion> A) (Upper L 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
+  case empty
+  then show ?case by simp
 next
   case (insert A x)
   show ?case
   proof (cases "A = {}")
     case True
     with insert show ?thesis by (simp add: sup_of_singletonI)
   next
     case False
-    from insert show ?thesis
-    proof (rule_tac sup_insertI)
-      from False insert show "least L (\<Squnion> A) (Upper L A)" by simp
-    qed simp_all
+    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 l (Upper L (insert x A)) ==> P l"
-    and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier 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
     by (simp add: sup_of_singletonI)
 next
   case False with P and xA show ?thesis
     by (simp add: sup_insertI finite_sup_least)
 qed
 
 lemma (in lattice) finite_sup_closed:
-  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion> A \<in> carrier 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
   case (insert A x) then show ?case
-    by (rule_tac finite_sup_insertI) (simp_all)
+    by - (rule finite_sup_insertI, simp_all)
 qed
 
 lemma (in lattice) join_left:
   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
-  by (rule joinI [folded join_def]) (blast dest: least_mem )
+  by (rule joinI [folded join_def]) (blast dest: least_mem)
 
 lemma (in lattice) join_right:
   "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
-  by (rule joinI [folded join_def]) (blast dest: least_mem )
+  by (rule joinI [folded join_def]) (blast dest: least_mem)
 
 lemma (in lattice) sup_of_two_least:
-  "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion> {x, y}) (Upper L {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> carrier L" "y \<in> carrier L"
+  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> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+  assumes sub: "x \<sqsubseteq> z"  "y \<sqsubseteq> z"
+    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 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> carrier L" "y \<in> carrier L" "z \<in> carrier L"
-  shows "x \<squnion> (y \<squnion> z) = \<Squnion> {x, y, z}"
+  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 L s (Upper L {x, y, z})"
   show "x \<squnion> (y \<squnion> z) = s"

       (blast intro!: Upper_memI intro: trans join_left join_right join_closed)
   qed (simp_all add: L least_carrier [OF sup])
 qed (simp_all add: L)
 
 lemma join_comm:
-  includes order_syntax
+  includes struct L
   shows "x \<squnion> y = y \<squnion> x"
   by (unfold join_def) (simp add: insert_commute)
 
 lemma (in lattice) join_assoc:
-  assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier 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)
-  also from L have "... = \<Squnion> {z, x, y}" by (simp add: join_assoc_lemma)
-  also from L have "... = \<Squnion> {x, y, z}" by (simp add: insert_commute)
+  also from L have "... = \<Squnion>{z, x, y}" by (simp add: join_assoc_lemma)
+  also from L have "... = \<Squnion>{x, y, z}" by (simp add: insert_commute)
   also from L have "... = x \<squnion> (y \<squnion> z)" by (simp add: join_assoc_lemma)
   finally show ?thesis .
 qed
+
 
 subsubsection {* Infimum *}
 
 lemma (in lattice) meetI:
   "[| !!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> carrier L" "y \<in> carrier L"
+  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 partial_order) inf_of_singletonI:      (* only reflexivity needed ? *)
   "x \<in> carrier L ==> greatest L x (Lower L {x})"
   by (rule greatest_LowerI) fast+
 
 lemma (in partial_order) inf_of_singleton [simp]:
-  includes order_syntax
+  includes struct L
   shows "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 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))"
+  ==> P (\<Sqinter>(insert x A))"
 proof (unfold inf_def)
-  assume L: "x \<in> carrier L" "A \<subseteq> carrier L"
+  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 L i (Lower L {a, x})" by blast
   show "P (THE g. greatest L g (Lower L (insert x A)))"
-  proof (rule theI2 [where a = i])
+  proof (rule theI2)
     show "greatest L i (Lower L (insert x A))"
     proof (rule greatest_LowerI)
       fix z
-      assume xA: "z \<in> insert x A"
-      show "i \<sqsubseteq> z"
-      proof -
-	{
-	  assume "z = x" then have ?thesis
-	    by (simp add: greatest_Lower_above [OF greatest_i] L La)
-        }
-	moreover
-        {
-	  assume "z \<in> A"
-          with L greatest_i greatest_a have ?thesis
-	    by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
-        }
-      moreover note xA
-      ultimately show ?thesis by blast
+      assume "z \<in> insert x A"
+      then show "i \<sqsubseteq> z"
+      proof
+        assume "z = x" then show ?thesis
+          by (simp add: greatest_Lower_above [OF greatest_i] L La)
+      next
+        assume "z \<in> A"
+        with L greatest_i greatest_a show ?thesis
+          by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
+      qed
+    next
+      fix y
+      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 L A"
+            apply (rule subsetD [where A = "Lower L (insert x A)"])
+            apply (rule Lower_antimono) apply clarify apply assumption
+            done
+          assume "z = a"
+          with y' greatest_a show ?thesis by (fast dest: greatest_le)
+	next
+          assume "z \<in> {x}"
+          with y L show ?thesis by blast
+	qed
+      qed (rule Lower_closed [THEN subsetD])
+    next
+      from L show "insert x A \<subseteq> carrier L" by simp
+      from greatest_i show "i \<in> carrier L" by simp
     qed
   next
-    fix y
-    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}"
-      show "y \<sqsubseteq> z"
-      proof -
-	{
-          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"
-	  with y' greatest_a have ?thesis by (fast dest: greatest_le)
-        }
-	moreover
-	{
-           assume "z = x"
-           with y L have ?thesis by blast
-        }
-        moreover note z
-        ultimately show ?thesis by blast
-      qed
-    qed (rule Lower_closed [THEN subsetD])
-  next
-    from L show "insert x A \<subseteq> carrier L" by simp
-  next
-    from greatest_i show "i \<in> carrier L" by simp
-  qed
-next
     fix g
     assume greatest_g: "greatest L g (Lower L (insert x A))"
     show "g = i"
     proof (rule greatest_unique)
       show "greatest L i (Lower L (insert x A))"
       proof (rule greatest_LowerI)
-	fix z
-	assume xA: "z \<in> insert x A"
-	show "i \<sqsubseteq> z"
-      proof -
-	{
-	  assume "z = x" then have ?thesis
-	    by (simp add: greatest_Lower_above [OF greatest_i] L La)
-        }
-	moreover
-        {
-	  assume "z \<in> A"
-          with L greatest_i greatest_a have ?thesis
-	    by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
-        }
-	  moreover note xA
-	  ultimately show ?thesis by blast
-	qed
+        fix z
+        assume "z \<in> insert x A"
+        then show "i \<sqsubseteq> z"
+	proof
+          assume "z = x" then show ?thesis
+            by (simp add: greatest_Lower_above [OF greatest_i] L La)
+	next
+          assume "z \<in> A"
+          with L greatest_i greatest_a show ?thesis
+            by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
+        qed
       next
-	fix y
-	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}"
-	  show "y \<sqsubseteq> z"
-	  proof -
-	    {
-          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"
-	  with y' greatest_a have ?thesis by (fast dest: greatest_le)
-        }
-	moreover
-	{
-           assume "z = x"
-           with y L have ?thesis by blast
-            }
-            moreover note z
-            ultimately show ?thesis by blast
+        fix y
+        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 L A"
+	      apply (rule subsetD [where A = "Lower L (insert x A)"])
+	      apply (rule Lower_antimono) apply clarify apply assumption
+	      done
+            assume "z = a"
+            with y' greatest_a show ?thesis by (fast dest: greatest_le)
+	  next
+            assume "z \<in> {x}"
+            with y L show ?thesis by blast
 	  qed
-	qed (rule Lower_closed [THEN subsetD])
+        qed (rule Lower_closed [THEN subsetD])
       next
-	from L show "insert x A \<subseteq> carrier L" by simp
-      next
-	from greatest_i show "i \<in> carrier 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> carrier L; A ~= {} |] ==> greatest L (\<Sqinter> A) (Lower L 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
   case (insert A x)
   show ?case

     with insert show ?thesis by (simp add: inf_of_singletonI)
   next
     case False
     from insert show ?thesis
     proof (rule_tac inf_insertI)
-      from False insert show "greatest L (\<Sqinter> A) (Lower L 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 L i (Lower L (insert x A)) ==> P i"
-    and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L"
+    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
     by (simp add: inf_of_singletonI)
 next
   case False with P and xA show ?thesis
     by (simp add: inf_insertI finite_inf_greatest)
 qed
 
 lemma (in lattice) finite_inf_closed:
-  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter> A \<in> carrier 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
   case (insert A x) then show ?case
     by (rule_tac finite_inf_insertI) (simp_all)
 qed
 
 lemma (in lattice) meet_left:
   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
-  by (rule meetI [folded meet_def]) (blast dest: greatest_mem )
+  by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
 
 lemma (in lattice) meet_right:
   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y"
-  by (rule meetI [folded meet_def]) (blast dest: greatest_mem )
+  by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
 
 lemma (in lattice) inf_of_two_greatest:
   "[| 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> carrier L" "y \<in> carrier L"
+  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 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> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+  assumes sub: "z \<sqsubseteq> x"  "z \<sqsubseteq> y"
+    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 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> carrier L" "y \<in> carrier L" "z \<in> carrier L"
-  shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter> {x, y, z}"
+  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 L i (Lower L {x, y, z})"
   show "x \<sqinter> (y \<sqinter> z) = i"

       (blast intro!: Lower_memI intro: trans meet_left meet_right meet_closed)
   qed (simp_all add: L greatest_carrier [OF inf])
 qed (simp_all add: L)
 
 lemma meet_comm:
-  includes order_syntax
+  includes struct L
   shows "x \<sqinter> y = y \<sqinter> x"
   by (unfold meet_def) (simp add: insert_commute)
 
 lemma (in lattice) meet_assoc:
-  assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier 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)
   also from L have "... = \<Sqinter> {z, x, y}" by (simp add: meet_assoc_lemma)
   also from L have "... = \<Sqinter> {x, y, z}" by (simp add: insert_commute)
   also from L have "... = x \<sqinter> (y \<sqinter> z)" by (simp add: meet_assoc_lemma)
   finally show ?thesis .
 qed
 
+
 subsection {* Total Orders *}
 
 locale total_order = lattice +
   assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
 

   shows "total_order L"
 proof (rule total_order.intro)
   show "lattice_axioms L"
   proof (rule lattice_axioms.intro)
     fix x y
-    assume L: "x \<in> carrier L" "y \<in> carrier L"
+    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"
+        assume "x \<sqsubseteq> y"
         with L have "least L y (Upper L {x, y})"
-	  by (rule_tac least_UpperI) auto
+          by (rule_tac least_UpperI) auto
       }
       moreover
       {
-	assume "y \<sqsubseteq> x"
+        assume "y \<sqsubseteq> x"
         with L have "least L x (Upper L {x, y})"
-	  by (rule_tac least_UpperI) auto
+          by (rule_tac least_UpperI) auto
       }
       ultimately show ?thesis by blast
     qed
   next
     fix x y
-    assume L: "x \<in> carrier L" "y \<in> carrier L"
+    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"
+        assume "y \<sqsubseteq> x"
         with L have "greatest L y (Lower L {x, y})"
-	  by (rule_tac greatest_LowerI) auto
+          by (rule_tac greatest_LowerI) auto
       }
       moreover
       {
-	assume "x \<sqsubseteq> y"
+        assume "x \<sqsubseteq> y"
         with L have "greatest L x (Lower L {x, y})"
-	  by (rule_tac greatest_LowerI) auto
+          by (rule_tac greatest_LowerI) auto
       }
       ultimately show ?thesis by blast
     qed
   qed
 qed (assumption | rule total_order_axioms.intro)+
+
 
 subsection {* Complete lattices *}
 
 locale complete_lattice = lattice +
   assumes sup_exists:

     and inf_exists:
     "!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
   shows "complete_lattice L"
 proof (rule complete_lattice.intro)
   show "lattice_axioms L"
-  by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
+    by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
 qed (assumption | rule complete_lattice_axioms.intro)+
 
 constdefs (structure L)
   top :: "_ => 'a" ("\<top>\<index>")
   "\<top> == sup L (carrier L)"

   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> carrier L ==> \<Squnion> A \<in> carrier 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> carrier L"
   by (unfold top_def) simp
 
 lemma (in complete_lattice) infI:
   "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |]
-  ==> P (\<Sqinter> A)"
+  ==> P (\<Sqinter>A)"
 proof (unfold inf_def)
   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> carrier L ==> \<Sqinter> A \<in> carrier 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> carrier L"
   by (unfold bottom_def) 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 L b (Lower L ?B)" ..
   have "least L b (Upper L A)"
 apply (rule least_UpperI)
-   apply (rule greatest_le [where A = "Lower L ?B"]) 
+   apply (rule greatest_le [where A = "Lower L ?B"])
     apply (rule b_inf_B)
    apply (rule Lower_memI)
     apply (erule UpperD)
      apply assumption
     apply (rule L)

 next
   fix x assume "x \<in> \<Inter>A"
   with subgr [THEN subgroup.m_inv_closed]
   show "inv x \<in> \<Inter>A" by blast
 next
-  fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
+  fix x y assume "x \<in> \<Inter>A"  "y \<in> \<Inter>A"
   with subgr [THEN subgroup.m_closed]
   show "x \<otimes> y \<in> \<Inter>A" by blast
 qed
 
 theorem (in group) subgroups_complete_lattice: