# HG changeset patch
# User wenzelm
# Date 1026838346 -7200
# Node ID 041d78bf9403715015acc245960a450621bbe5da
# Parent  b37764a46b16990ad77cd79d7182071adfea3e13
adapted locales;

diff -r b37764a46b16 -r 041d78bf9403 src/HOL/ex/Locales.thy
--- a/src/HOL/ex/Locales.thy	Tue Jul 16 18:46:59 2002 +0200
+++ b/src/HOL/ex/Locales.thy	Tue Jul 16 18:52:26 2002 +0200
@@ -231,11 +231,14 @@
   contexts is rather light-weight and convenient to use for abstract
   reasoning.  Here the ``components'' (the group operations) have been
   exhibited directly as context parameters; logically this corresponds
-  to a curried predicate definition.  Occasionally, this
-  ``externalized'' version of the informal idea of classes of tuple
-  structures may cause some inconveniences, especially in
-  meta-theoretical studies (involving functors from groups to groups,
-  for example).
+  to a curried predicate definition:
+
+  @{thm [display] group_context_axioms_def [no_vars]}
+
+  Occasionally, this ``externalized'' version of the informal idea of
+  classes of tuple structures may cause some inconveniences,
+  especially in meta-theoretical studies (involving functors from
+  groups to groups, for example).
 
   Another minor drawback of the naive approach above is that concrete
   syntax will get lost on any kind of operation on the locale itself
diff -r b37764a46b16 -r 041d78bf9403 src/HOL/ex/Tarski.thy
--- a/src/HOL/ex/Tarski.thy	Tue Jul 16 18:46:59 2002 +0200
+++ b/src/HOL/ex/Tarski.thy	Tue Jul 16 18:52:26 2002 +0200
@@ -1,25 +1,29 @@
-(*  Title:      HOL/ex/Tarski
+(*  Title:      HOL/ex/Tarski.thy
     ID:         $Id$
-    Author:     Florian Kammueller, Cambridge University Computer Laboratory
+    Author:     Florian Kammüller, Cambridge University Computer Laboratory
     Copyright   1999  University of Cambridge
+*)
 
-Minimal version of lattice theory plus the full theorem of Tarski:
-   The fixedpoints of a complete lattice themselves form a complete lattice.
-
-Illustrates first-class theories, using the Sigma representation of structures
-
-Tidied and converted to Isar by lcp
-*)
+header {* The full theorem of Tarski *}
 
 theory Tarski = Main:
 
-record 'a potype = 
+text {*
+  Minimal version of lattice theory plus the full theorem of Tarski:
+  The fixedpoints of a complete lattice themselves form a complete
+  lattice.
+
+  Illustrates first-class theories, using the Sigma representation of
+  structures.  Tidied and converted to Isar by lcp.
+*}
+
+record 'a potype =
   pset  :: "'a set"
   order :: "('a * 'a) set"
 
 syntax
   "@pset"  :: "'a potype => 'a set"            ("_ .<A>"  [90] 90)
-  "@order" :: "'a potype => ('a *'a)set"       ("_ .<r>"  [90] 90) 
+  "@order" :: "'a potype => ('a *'a)set"       ("_ .<r>"  [90] 90)
 
 translations
   "po.<A>" == "pset po"
@@ -27,69 +31,67 @@
 
 constdefs
   monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
-    "monotone f A r == \<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r"
+  "monotone f A r == \<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r"
 
   least :: "['a => bool, 'a potype] => 'a"
-   "least P po == @ x. x: po.<A> & P x &
+  "least P po == @ x. x: po.<A> & P x &
                        (\<forall>y \<in> po.<A>. P y --> (x,y): po.<r>)"
 
   greatest :: "['a => bool, 'a potype] => 'a"
-   "greatest P po == @ x. x: po.<A> & P x &
+  "greatest P po == @ x. x: po.<A> & P x &
                           (\<forall>y \<in> po.<A>. P y --> (y,x): po.<r>)"
 
   lub  :: "['a set, 'a potype] => 'a"
-   "lub S po == least (%x. \<forall>y\<in>S. (y,x): po.<r>) po"
+  "lub S po == least (%x. \<forall>y\<in>S. (y,x): po.<r>) po"
 
   glb  :: "['a set, 'a potype] => 'a"
-   "glb S po == greatest (%x. \<forall>y\<in>S. (x,y): po.<r>) po"
+  "glb S po == greatest (%x. \<forall>y\<in>S. (x,y): po.<r>) po"
 
   isLub :: "['a set, 'a potype, 'a] => bool"
-   "isLub S po == %L. (L: po.<A> & (\<forall>y\<in>S. (y,L): po.<r>) &
-                      (\<forall>z\<in>po.<A>. (\<forall>y\<in>S. (y,z): po.<r>) --> (L,z): po.<r>))"
+  "isLub S po == %L. (L: po.<A> & (\<forall>y\<in>S. (y,L): po.<r>) &
+                     (\<forall>z\<in>po.<A>. (\<forall>y\<in>S. (y,z): po.<r>) --> (L,z): po.<r>))"
 
   isGlb :: "['a set, 'a potype, 'a] => bool"
-   "isGlb S po == %G. (G: po.<A> & (\<forall>y\<in>S. (G,y): po.<r>) &
+  "isGlb S po == %G. (G: po.<A> & (\<forall>y\<in>S. (G,y): po.<r>) &
                      (\<forall>z \<in> po.<A>. (\<forall>y\<in>S. (z,y): po.<r>) --> (z,G): po.<r>))"
 
   "fix"    :: "[('a => 'a), 'a set] => 'a set"
-   "fix f A  == {x. x: A & f x = x}"
+  "fix f A  == {x. x: A & f x = x}"
 
   interval :: "[('a*'a) set,'a, 'a ] => 'a set"
-   "interval r a b == {x. (a,x): r & (x,b): r}"
+  "interval r a b == {x. (a,x): r & (x,b): r}"
 
 
 constdefs
   Bot :: "'a potype => 'a"
-   "Bot po == least (%x. True) po"
+  "Bot po == least (%x. True) po"
 
   Top :: "'a potype => 'a"
-   "Top po == greatest (%x. True) po"
+  "Top po == greatest (%x. True) po"
 
   PartialOrder :: "('a potype) set"
-   "PartialOrder == {P. refl (P.<A>) (P.<r>) & antisym (P.<r>) &
-		        trans (P.<r>)}"
+  "PartialOrder == {P. refl (P.<A>) (P.<r>) & antisym (P.<r>) &
+                       trans (P.<r>)}"
 
   CompleteLattice :: "('a potype) set"
-   "CompleteLattice == {cl. cl: PartialOrder & 
-			(\<forall>S. S <= cl.<A> --> (\<exists>L. isLub S cl L)) &
-			(\<forall>S. S <= cl.<A> --> (\<exists>G. isGlb S cl G))}"
+  "CompleteLattice == {cl. cl: PartialOrder &
+                        (\<forall>S. S <= cl.<A> --> (\<exists>L. isLub S cl L)) &
+                        (\<forall>S. S <= cl.<A> --> (\<exists>G. isGlb S cl G))}"
 
   CLF :: "('a potype * ('a => 'a)) set"
-   "CLF == SIGMA cl: CompleteLattice.
-             {f. f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)}"
-  
+  "CLF == SIGMA cl: CompleteLattice.
+            {f. f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)}"
+
   induced :: "['a set, ('a * 'a) set] => ('a *'a)set"
-   "induced A r == {(a,b). a : A & b: A & (a,b): r}"
-
-
+  "induced A r == {(a,b). a : A & b: A & (a,b): r}"
 
 
 constdefs
   sublattice :: "('a potype * 'a set)set"
-   "sublattice == 
+  "sublattice ==
       SIGMA cl: CompleteLattice.
           {S. S <= cl.<A> &
-	   (| pset = S, order = induced S (cl.<r>) |): CompleteLattice }"
+           (| pset = S, order = induced S (cl.<r>) |): CompleteLattice }"
 
 syntax
   "@SL"  :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
@@ -99,9 +101,9 @@
 
 constdefs
   dual :: "'a potype => 'a potype"
-   "dual po == (| pset = po.<A>, order = converse (po.<r>) |)"
+  "dual po == (| pset = po.<A>, order = converse (po.<r>) |)"
 
-locale PO = 
+locale (open) PO =
   fixes cl :: "'a potype"
     and A  :: "'a set"
     and r  :: "('a * 'a) set"
@@ -109,17 +111,17 @@
   defines A_def: "A == cl.<A>"
      and  r_def: "r == cl.<r>"
 
-locale CL = PO +
+locale (open) CL = PO +
   assumes cl_co:  "cl : CompleteLattice"
 
-locale CLF = CL +
+locale (open) CLF = CL +
   fixes f :: "'a => 'a"
     and P :: "'a set"
   assumes f_cl:  "(cl,f) : CLF" (*was the equivalent "f : CLF``{cl}"*)
   defines P_def: "P == fix f A"
 
 
-locale Tarski = CLF + 
+locale (open) Tarski = CLF +
   fixes Y     :: "'a set"
     and intY1 :: "'a set"
     and v     :: "'a"
@@ -127,41 +129,40 @@
     Y_ss: "Y <= P"
   defines
     intY1_def: "intY1 == interval r (lub Y cl) (Top cl)"
-    and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r & 
+    and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r &
                              x: intY1}
-		      (| pset=intY1, order=induced intY1 r|)"
+                      (| pset=intY1, order=induced intY1 r|)"
 
 
-
-(* Partial Order *)
+subsubsection {* Partial Order *}
 
 lemma (in PO) PO_imp_refl: "refl A r"
-apply (insert cl_po) 
+apply (insert cl_po)
 apply (simp add: PartialOrder_def A_def r_def)
 done
 
 lemma (in PO) PO_imp_sym: "antisym r"
-apply (insert cl_po) 
+apply (insert cl_po)
 apply (simp add: PartialOrder_def A_def r_def)
 done
 
 lemma (in PO) PO_imp_trans: "trans r"
-apply (insert cl_po) 
+apply (insert cl_po)
 apply (simp add: PartialOrder_def A_def r_def)
 done
 
 lemma (in PO) reflE: "[| refl A r; x \<in> A|] ==> (x, x) \<in> r"
-apply (insert cl_po) 
+apply (insert cl_po)
 apply (simp add: PartialOrder_def refl_def)
 done
 
 lemma (in PO) antisymE: "[| antisym r; (a, b) \<in> r; (b, a) \<in> r |] ==> a = b"
-apply (insert cl_po) 
+apply (insert cl_po)
 apply (simp add: PartialOrder_def antisym_def)
 done
 
 lemma (in PO) transE: "[| trans r; (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r"
-apply (insert cl_po) 
+apply (insert cl_po)
 apply (simp add: PartialOrder_def)
 apply (unfold trans_def, fast)
 done
@@ -174,13 +175,13 @@
      "S <= A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
 apply (simp (no_asm) add: PartialOrder_def)
 apply auto
-(* refl *)
+-- {* refl *}
 apply (simp add: refl_def induced_def)
 apply (blast intro: PO_imp_refl [THEN reflE])
-(* antisym *)
+-- {* antisym *}
 apply (simp add: antisym_def induced_def)
 apply (blast intro: PO_imp_sym [THEN antisymE])
-(* trans *)
+-- {* trans *}
 apply (simp add: trans_def induced_def)
 apply (blast intro: PO_imp_trans [THEN transE])
 done
@@ -191,8 +192,8 @@
 lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r"
 by (simp add: add: induced_def)
 
-lemma (in CL) CL_imp_ex_isLub: "S <= A ==> \<exists>L. isLub S cl L" 
-apply (insert cl_co) 
+lemma (in CL) CL_imp_ex_isLub: "S <= A ==> \<exists>L. isLub S cl L"
+apply (insert cl_co)
 apply (simp add: CompleteLattice_def A_def)
 done
 
@@ -211,13 +212,13 @@
 by (simp add: isLub_def isGlb_def dual_def converse_def)
 
 lemma (in PO) dualPO: "dual cl \<in> PartialOrder"
-apply (insert cl_po) 
-apply (simp add: PartialOrder_def dual_def refl_converse 
+apply (insert cl_po)
+apply (simp add: PartialOrder_def dual_def refl_converse
                  trans_converse antisym_converse)
 done
 
 lemma Rdual:
-     "\<forall>S. (S <= A -->( \<exists>L. isLub S (| pset = A, order = r|) L))  
+     "\<forall>S. (S <= A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
       ==> \<forall>S. (S <= A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"
 apply safe
 apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)}
@@ -253,17 +254,17 @@
 by (rule PO_imp_trans)
 
 lemma CompleteLatticeI:
-     "[| po \<in> PartialOrder; (\<forall>S. S <= po.<A> --> (\<exists>L. isLub S po L)); 
-         (\<forall>S. S <= po.<A> --> (\<exists>G. isGlb S po G))|] 
+     "[| po \<in> PartialOrder; (\<forall>S. S <= po.<A> --> (\<exists>L. isLub S po L));
+         (\<forall>S. S <= po.<A> --> (\<exists>G. isGlb S po G))|]
       ==> po \<in> CompleteLattice"
-apply (unfold CompleteLattice_def, blast) 
+apply (unfold CompleteLattice_def, blast)
 done
 
 lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
-apply (insert cl_co) 
+apply (insert cl_co)
 apply (simp add: CompleteLattice_def dual_def)
-apply (fold dual_def) 
-apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric] 
+apply (fold dual_def)
+apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric]
                  dualPO)
 done
 
@@ -307,17 +308,21 @@
 apply (simp add: PO_imp_refl [THEN reflE])
 done
 
-(* sublattice *)
+
+subsubsection {* sublattice *}
+
 lemma (in PO) sublattice_imp_CL:
      "S <<= cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
 by (simp add: sublattice_def CompleteLattice_def A_def r_def)
 
 lemma (in CL) sublatticeI:
-     "[| S <= A; (| pset = S, order = induced S r |) \<in> CompleteLattice |] 
+     "[| S <= A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
       ==> S <<= cl"
 by (simp add: sublattice_def A_def r_def)
 
-(* lub *)
+
+subsubsection {* lub *}
+
 lemma (in CL) lub_unique: "[| S <= A; isLub S cl x; isLub S cl L|] ==> x = L"
 apply (rule antisymE)
 apply (rule CO_antisym)
@@ -329,7 +334,7 @@
 apply (unfold lub_def least_def)
 apply (rule some_equality [THEN ssubst])
   apply (simp add: isLub_def)
- apply (simp add: lub_unique A_def isLub_def) 
+ apply (simp add: lub_unique A_def isLub_def)
 apply (simp add: isLub_def r_def)
 done
 
@@ -339,7 +344,7 @@
 apply (unfold lub_def least_def)
 apply (rule_tac s=x in some_equality [THEN ssubst])
   apply (simp add: isLub_def)
- apply (simp add: lub_unique A_def isLub_def) 
+ apply (simp add: lub_unique A_def isLub_def)
 apply (simp add: isLub_def r_def A_def)
 done
 
@@ -349,11 +354,11 @@
 apply (subst some_equality)
 apply (simp add: isLub_def)
 prefer 2 apply (simp add: isLub_def A_def)
-apply (simp add: lub_unique A_def isLub_def) 
+apply (simp add: lub_unique A_def isLub_def)
 done
 
 lemma (in CL) lubI:
-     "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r; 
+     "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
          \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
 apply (rule lub_unique, assumption)
 apply (simp add: isLub_def A_def r_def)
@@ -378,17 +383,19 @@
 by (simp add: isLub_def A_def r_def)
 
 lemma (in CL) isLubI:
-     "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;  
+     "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
          (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
 by (simp add: isLub_def A_def r_def)
 
-(* glb *)
+
+subsubsection {* glb *}
+
 lemma (in CL) glb_in_lattice: "S <= A ==> glb S cl \<in> A"
 apply (subst glb_dual_lub)
 apply (simp add: A_def)
 apply (rule dualA_iff [THEN subst])
 apply (rule Tarski.lub_in_lattice)
-apply (rule dualPO) 
+apply (rule dualPO)
 apply (rule CL_dualCL)
 apply (simp add: dualA_iff)
 done
@@ -398,18 +405,20 @@
 apply (simp add: r_def)
 apply (rule dualr_iff [THEN subst])
 apply (rule Tarski.lub_upper [rule_format])
-apply (rule dualPO) 
+apply (rule dualPO)
 apply (rule CL_dualCL)
 apply (simp add: dualA_iff A_def, assumption)
 done
 
-(* Reduce the sublattice property by using substructural properties*)
-(* abandoned see Tarski_4.ML *)
+text {*
+  Reduce the sublattice property by using substructural properties;
+  abandoned see @{text "Tarski_4.ML"}.
+*}
 
 lemma (in CLF) [simp]:
     "f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)"
-apply (insert f_cl) 
-apply (simp add: CLF_def) 
+apply (insert f_cl)
+apply (simp add: CLF_def)
 done
 
 declare (in CLF) f_cl [simp]
@@ -426,7 +435,9 @@
 apply (simp add: dualA_iff)
 done
 
-(* fixed points *)
+
+subsubsection {* fixed points *}
+
 lemma fix_subset: "fix f A <= A"
 by (simp add: fix_def, fast)
 
@@ -435,22 +446,24 @@
 
 lemma fixf_subset:
      "[| A <= B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
-apply (simp add: fix_def, auto) 
+apply (simp add: fix_def, auto)
 done
 
-(* lemmas for Tarski, lub *)
+
+subsubsection {* lemmas for Tarski, lub *}
 lemma (in CLF) lubH_le_flubH:
      "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
 apply (rule lub_least, fast)
 apply (rule f_in_funcset [THEN funcset_mem])
 apply (rule lub_in_lattice, fast)
-(* \<forall>x:H. (x, f (lub H r)) \<in> r *)
+-- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *}
 apply (rule ballI)
 apply (rule transE)
 apply (rule CO_trans)
-(* instantiates (x, ???z) \<in> cl.<r> to (x, f x), because of the def of H *)
+-- {* instantiates @{text "(x, ???z) \<in> cl.<r> to (x, f x)"}, *}
+-- {* because of the def of @{text H} *}
 apply fast
-(* so it remains to show (f x, f (lub H cl)) \<in> r *)
+-- {* so it remains to show @{text "(f x, f (lub H cl)) \<in> r"} *}
 apply (rule_tac f = "f" in monotoneE)
 apply (rule monotone_f, fast)
 apply (rule lub_in_lattice, fast)
@@ -469,8 +482,8 @@
 prefer 2 apply fast
 apply (rule_tac f = "f" in monotoneE)
 apply (rule monotone_f)
-  apply (blast intro: lub_in_lattice) 
- apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem]) 
+  apply (blast intro: lub_in_lattice)
+ apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])
 apply (simp add: lubH_le_flubH)
 done
 
@@ -487,19 +500,19 @@
 
 lemma (in CLF) fix_in_H:
      "[| H = {x. (x, f x) \<in> r & x \<in> A};  x \<in> P |] ==> x \<in> H"
-by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl 
-                    fix_subset [of f A, THEN subsetD]) 
+by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl
+                    fix_subset [of f A, THEN subsetD])
 
 lemma (in CLF) fixf_le_lubH:
      "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
 apply (rule ballI)
 apply (rule lub_upper, fast)
 apply (rule fix_in_H)
-apply (simp_all add: P_def) 
+apply (simp_all add: P_def)
 done
 
 lemma (in CLF) lubH_least_fixf:
-     "H = {x. (x, f x) \<in> r & x \<in> A} 
+     "H = {x. (x, f x) \<in> r & x \<in> A}
       ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
 apply (rule allI)
 apply (rule impI)
@@ -507,10 +520,10 @@
 apply (rule lubH_is_fixp, assumption)
 done
 
-(* Tarski fixpoint theorem 1, first part *)
+subsubsection {* Tarski fixpoint theorem 1, first part *}
 lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
 apply (rule sym)
-apply (simp add: P_def) 
+apply (simp add: P_def)
 apply (rule lubI)
 apply (rule fix_subset)
 apply (rule lub_in_lattice, fast)
@@ -518,12 +531,12 @@
 apply (simp add: lubH_least_fixf)
 done
 
-(* Tarski for glb *)
 lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
+  -- {* Tarski for glb *}
 apply (simp add: glb_dual_lub P_def A_def r_def)
 apply (rule dualA_iff [THEN subst])
 apply (rule Tarski.lubH_is_fixp)
-apply (rule dualPO) 
+apply (rule dualPO)
 apply (rule CL_dualCL)
 apply (rule f_cl [THEN CLF_dual])
 apply (simp add: dualr_iff dualA_iff)
@@ -532,14 +545,15 @@
 lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
 apply (simp add: glb_dual_lub P_def A_def r_def)
 apply (rule dualA_iff [THEN subst])
-apply (simp add: Tarski.T_thm_1_lub [of _ f, OF dualPO CL_dualCL] 
+apply (simp add: Tarski.T_thm_1_lub [of _ f, OF dualPO CL_dualCL]
                  dualPO CL_dualCL CLF_dual dualr_iff)
 done
 
-(* interval *)
+subsubsection {* interval *}
+
 lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
-apply (insert CO_refl) 
-apply (simp add: refl_def, blast) 
+apply (insert CO_refl)
+apply (simp add: refl_def, blast)
 done
 
 lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b <= A"
@@ -563,12 +577,12 @@
 done
 
 lemma (in CLF) a_less_lub:
-     "[| S <= A; S \<noteq> {}; 
+     "[| S <= A; S \<noteq> {};
          \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
 by (blast intro: transE PO_imp_trans)
 
 lemma (in CLF) glb_less_b:
-     "[| S <= A; S \<noteq> {}; 
+     "[| S <= A; S \<noteq> {};
          \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
 by (blast intro: transE PO_imp_trans)
 
@@ -577,7 +591,7 @@
 by (simp add: subset_trans [OF _ interval_subset])
 
 lemma (in CLF) L_in_interval:
-     "[| a \<in> A; b \<in> A; S <= interval r a b;  
+     "[| a \<in> A; b \<in> A; S <= interval r a b;
          S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
 apply (rule intervalI)
 apply (rule a_less_lub)
@@ -586,7 +600,7 @@
 apply (rule ballI)
 apply (simp add: interval_lemma1)
 apply (simp add: isLub_upper)
-(* (L, b) \<in> r *)
+-- {* @{text "(L, b) \<in> r"} *}
 apply (simp add: isLub_least interval_lemma2)
 done
 
@@ -594,12 +608,12 @@
      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S <= interval r a b; isGlb S cl G;
          S \<noteq> {} |] ==> G \<in> interval r a b"
 apply (simp add: interval_dual)
-apply (simp add: Tarski.L_in_interval [of _ f] 
+apply (simp add: Tarski.L_in_interval [of _ f]
                  dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub)
 done
 
 lemma (in CLF) intervalPO:
-     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] 
+     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
       ==> (| pset = interval r a b, order = induced (interval r a b) r |)
           \<in> PartialOrder"
 apply (rule po_subset_po)
@@ -607,41 +621,40 @@
 done
 
 lemma (in CLF) intv_CL_lub:
- "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] 
-  ==> \<forall>S. S <= interval r a b --> 
-          (\<exists>L. isLub S (| pset = interval r a b, 
+ "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
+  ==> \<forall>S. S <= interval r a b -->
+          (\<exists>L. isLub S (| pset = interval r a b,
                           order = induced (interval r a b) r |)  L)"
 apply (intro strip)
 apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
 prefer 2 apply assumption
 apply assumption
 apply (erule exE)
-(* define the lub for the interval as *)
+-- {* define the lub for the interval as *}
 apply (rule_tac x = "if S = {} then a else L" in exI)
 apply (simp (no_asm_simp) add: isLub_def split del: split_if)
-apply (intro impI conjI) 
-(* (if S = {} then a else L) \<in> interval r a b *)
+apply (intro impI conjI)
+-- {* @{text "(if S = {} then a else L) \<in> interval r a b"} *}
 apply (simp add: CL_imp_PO L_in_interval)
 apply (simp add: left_in_interval)
-(* lub prop 1 *)
+-- {* lub prop 1 *}
 apply (case_tac "S = {}")
-(* S = {}, y \<in> S = False => everything *)
+-- {* @{text "S = {}, y \<in> S = False => everything"} *}
 apply fast
-(* S \<noteq> {} *)
+-- {* @{text "S \<noteq> {}"} *}
 apply simp
-(* \<forall>y:S. (y, L) \<in> induced (interval r a b) r *)
+-- {* @{text "\<forall>y:S. (y, L) \<in> induced (interval r a b) r"} *}
 apply (rule ballI)
 apply (simp add: induced_def  L_in_interval)
 apply (rule conjI)
 apply (rule subsetD)
 apply (simp add: S_intv_cl, assumption)
 apply (simp add: isLub_upper)
-(* \<forall>z:interval r a b. (\<forall>y:S. (y, z) \<in> induced (interval r a b) r -->
-      (if S = {} then a else L, z) \<in> induced (interval r a b) r *)
+-- {* @{text "\<forall>z:interval r a b. (\<forall>y:S. (y, z) \<in> induced (interval r a b) r \<longrightarrow> (if S = {} then a else L, z) \<in> induced (interval r a b) r"} *}
 apply (rule ballI)
 apply (rule impI)
 apply (case_tac "S = {}")
-(* S = {} *)
+-- {* @{text "S = {}"} *}
 apply simp
 apply (simp add: induced_def  interval_def)
 apply (rule conjI)
@@ -650,7 +663,7 @@
 apply (rule interval_not_empty)
 apply (rule CO_trans)
 apply (simp add: interval_def)
-(* S \<noteq> {} *)
+-- {* @{text "S \<noteq> {}"} *}
 apply simp
 apply (simp add: induced_def  L_in_interval)
 apply (rule isLub_least, assumption)
@@ -662,7 +675,7 @@
 lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
 
 lemma (in CLF) interval_is_sublattice:
-     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] 
+     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
         ==> interval r a b <<= cl"
 apply (rule sublatticeI)
 apply (simp add: interval_subset)
@@ -672,10 +685,11 @@
 apply (simp add: intv_CL_glb)
 done
 
-lemmas (in CLF) interv_is_compl_latt = 
+lemmas (in CLF) interv_is_compl_latt =
     interval_is_sublattice [THEN sublattice_imp_CL]
 
-(* Top and Bottom *)
+
+subsubsection {* Top and Bottom *}
 lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
 by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
 
@@ -696,8 +710,8 @@
 
 lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
 apply (simp add: Top_dual_Bot A_def)
-apply (rule dualA_iff [THEN subst])  
-apply (blast intro!: Tarski.Bot_in_lattice dualPO CL_dualCL CLF_dual f_cl) 
+apply (rule dualA_iff [THEN subst])
+apply (blast intro!: Tarski.Bot_in_lattice dualPO CL_dualCL CLF_dual f_cl)
 done
 
 lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
@@ -713,7 +727,7 @@
 lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
 apply (simp add: Bot_dual_Top r_def)
 apply (rule dualr_iff [THEN subst])
-apply (simp add: Tarski.Top_prop [of _ f] 
+apply (simp add: Tarski.Top_prop [of _ f]
                  dualA_iff A_def dualPO CL_dualCL CLF_dual)
 done
 
@@ -732,11 +746,12 @@
 apply (rule dualA_iff [THEN subst])
 apply (blast intro!: Tarski.Top_in_lattice
                  f_cl dualPO CL_dualCL CLF_dual)
-apply (simp add: Tarski.Top_intv_not_empty [of _ f] 
+apply (simp add: Tarski.Top_intv_not_empty [of _ f]
                  dualA_iff A_def dualPO CL_dualCL CLF_dual)
 done
 
-(* fixed points form a partial order *)
+subsubsection {* fixed points form a partial order *}
+
 lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
 by (simp add: P_def fix_subset po_subset_po)
 
@@ -753,11 +768,11 @@
 apply (rule Y_subset_A)
 apply (rule f_in_funcset [THEN funcset_mem])
 apply (rule lubY_in_A)
-(* Y <= P ==> f x = x *)
+-- {* @{text "Y <= P ==> f x = x"} *}
 apply (rule ballI)
 apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
 apply (erule Y_ss [simplified P_def, THEN subsetD])
-(* reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r by monotonicity *)
+-- {* @{text "reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r"} by monotonicity *}
 apply (rule_tac f = "f" in monotoneE)
 apply (rule monotone_f)
 apply (simp add: Y_subset_A [THEN subsetD])
@@ -780,13 +795,13 @@
 apply (rule transE)
 apply (rule CO_trans)
 apply (rule lubY_le_flubY)
-(* (f (lub Y cl), f x) \<in> r *)
+-- {* @{text "(f (lub Y cl), f x) \<in> r"} *}
 apply (rule_tac f=f in monotoneE)
 apply (rule monotone_f)
 apply (rule lubY_in_A)
 apply (simp add: intY1_def interval_def  intY1_elem)
 apply (simp add: intY1_def  interval_def)
-(* (f x, Top cl) \<in> r *)
+-- {* @{text "(f x, Top cl) \<in> r"} *}
 apply (rule Top_prop)
 apply (rule f_in_funcset [THEN funcset_mem])
 apply (simp add: intY1_def interval_def  intY1_elem)
@@ -803,7 +818,7 @@
 apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
 done
 
-lemma (in Tarski) intY1_is_cl: 
+lemma (in Tarski) intY1_is_cl:
     "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
 apply (unfold intY1_def)
 apply (rule interv_is_compl_latt)
@@ -821,11 +836,11 @@
                  v_def CL_imp_PO intY1_is_cl CLF_def intY1_func intY1_mono)
 done
 
-lemma (in Tarski) z_in_interval: 
+lemma (in Tarski) z_in_interval:
      "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
 apply (unfold intY1_def P_def)
 apply (rule intervalI)
-prefer 2 
+prefer 2
  apply (erule fix_subset [THEN subsetD, THEN Top_prop])
 apply (rule lub_least)
 apply (rule Y_subset_A)
@@ -833,10 +848,10 @@
 apply (simp add: induced_def)
 done
 
-lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] 
+lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
       ==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"
 apply (simp add: induced_def  intY1_f_closed z_in_interval P_def)
-apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD] 
+apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD]
                  CO_refl [THEN reflE])
 done
 
@@ -844,11 +859,11 @@
      "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
 apply (rule_tac x = "v" in exI)
 apply (simp add: isLub_def)
-(* v \<in> P *)
+-- {* @{text "v \<in> P"} *}
 apply (simp add: v_in_P)
 apply (rule conjI)
-(* v is lub *)
-(*  1. \<forall>y:Y. (y, v) \<in> induced P r *)
+-- {* @{text v} is lub *}
+-- {* @{text "1. \<forall>y:Y. (y, v) \<in> induced P r"} *}
 apply (rule ballI)
 apply (simp add: induced_def subsetD v_in_P)
 apply (rule conjI)
@@ -862,7 +877,7 @@
 apply (fold intY1_def)
 apply (rule Tarski.glb_in_lattice [OF _ intY1_is_cl, simplified])
  apply (simp add: CL_imp_PO intY1_is_cl, force)
-(* v is LEAST ub *)
+-- {* @{text v} is LEAST ub *}
 apply clarify
 apply (rule indI)
   prefer 3 apply assumption
@@ -871,18 +886,17 @@
 apply (rule indE)
 apply (rule_tac [2] intY1_subset)
 apply (rule Tarski.glb_lower [OF _ intY1_is_cl, simplified])
-  apply (simp add: CL_imp_PO intY1_is_cl) 
+  apply (simp add: CL_imp_PO intY1_is_cl)
  apply force
 apply (simp add: induced_def intY1_f_closed z_in_interval)
-apply (simp add: P_def fix_imp_eq [of _ f A] 
-                 fix_subset [of f A, THEN subsetD] 
+apply (simp add: P_def fix_imp_eq [of _ f A]
+                 fix_subset [of f A, THEN subsetD]
                  CO_refl [THEN reflE])
 done
 
-
 lemma CompleteLatticeI_simp:
-     "[| (| pset = A, order = r |) \<in> PartialOrder; 
-         \<forall>S. S <= A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]  
+     "[| (| pset = A, order = r |) \<in> PartialOrder;
+         \<forall>S. S <= A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
     ==> (| pset = A, order = r |) \<in> CompleteLattice"
 by (simp add: CompleteLatticeI Rdual)
 
@@ -890,8 +904,8 @@
      "(| pset = P, order = induced P r|) \<in> CompleteLattice"
 apply (rule CompleteLatticeI_simp)
 apply (rule fixf_po, clarify)
-apply (simp add: P_def A_def r_def) 
-apply (blast intro!: Tarski.tarski_full_lemma cl_po cl_co f_cl) 
+apply (simp add: P_def A_def r_def)
+apply (blast intro!: Tarski.tarski_full_lemma cl_po cl_co f_cl)
 done
 
 end