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diff --git a/isar/KnasterTarski.thy b/isar/KnasterTarski.thy
deleted file mode 100644
index 5d0bbce0..00000000
--- a/isar/KnasterTarski.thy
+++ /dev/null
@@ -1,113 +0,0 @@
-(********** This file is from the Isabelle distribution **********)
-
-(*  Title:      HOL/Isar_examples/KnasterTarski.thy
-    Author:     Markus Wenzel, TU Muenchen
-
-Typical textbook proof example.
-*)
-
-header {* Textbook-style reasoning: the Knaster-Tarski Theorem *}
-
-theory KnasterTarski
-imports Main Lattice_Syntax
-begin
-
-
-subsection {* Prose version *}
-
-text {*
-  According to the textbook \cite[pages 93--94]{davey-priestley}, the
-  Knaster-Tarski fixpoint theorem is as follows.\footnote{We have
-  dualized the argument, and tuned the notation a little bit.}
-
-  \textbf{The Knaster-Tarski Fixpoint Theorem.}  Let @{text L} be a
-  complete lattice and @{text "f: L \<rightarrow> L"} an order-preserving map.
-  Then @{text "\<Sqinter>{x \<in> L | f(x) \<le> x}"} is a fixpoint of @{text f}.
-
-  \textbf{Proof.} Let @{text "H = {x \<in> L | f(x) \<le> x}"} and @{text "a =
-  \<Sqinter>H"}.  For all @{text "x \<in> H"} we have @{text "a \<le> x"}, so @{text
-  "f(a) \<le> f(x) \<le> x"}.  Thus @{text "f(a)"} is a lower bound of @{text
-  H}, whence @{text "f(a) \<le> a"}.  We now use this inequality to prove
-  the reverse one (!) and thereby complete the proof that @{text a} is
-  a fixpoint.  Since @{text f} is order-preserving, @{text "f(f(a)) \<le>
-  f(a)"}.  This says @{text "f(a) \<in> H"}, so @{text "a \<le> f(a)"}.
-*}
-
-
-subsection {* Formal versions *}
-
-text {*
-  The Isar proof below closely follows the original presentation.
-  Virtually all of the prose narration has been rephrased in terms of
-  formal Isar language elements.  Just as many textbook-style proofs,
-  there is a strong bias towards forward proof, and several bends in
-  the course of reasoning.
-*}
-
-theorem Knaster_Tarski:
-  fixes f :: "'a::complete_lattice \<Rightarrow> 'a"
-  assumes "mono f"
-  shows "\<exists>a. f a = a"
-proof
-  let ?H = "{u. f u \<le> u}"
-  let ?a = "\<Sqinter>?H"
-  show "f ?a = ?a"
-  proof -
-    {
-      fix x
-      assume "x \<in> ?H"
-      then have "?a \<le> x" by (rule Inf_lower)
-      with `mono f` have "f ?a \<le> f x" ..
-      also from `x \<in> ?H` have "\<dots> \<le> x" ..
-      finally have "f ?a \<le> x" .
-    }
-    then have "f ?a \<le> ?a" by (rule Inf_greatest)
-    {
-      also presume "\<dots> \<le> f ?a"
-      finally (order_antisym) show ?thesis .
-    }
-    from `mono f` and `f ?a \<le> ?a` have "f (f ?a) \<le> f ?a" ..
-    then have "f ?a \<in> ?H" ..
-    then show "?a \<le> f ?a" by (rule Inf_lower)
-  qed
-qed
-
-text {*
-  Above we have used several advanced Isar language elements, such as
-  explicit block structure and weak assumptions.  Thus we have
-  mimicked the particular way of reasoning of the original text.
-
-  In the subsequent version the order of reasoning is changed to
-  achieve structured top-down decomposition of the problem at the
-  outer level, while only the inner steps of reasoning are done in a
-  forward manner.  We are certainly more at ease here, requiring only
-  the most basic features of the Isar language.
-*}
-
-theorem Knaster_Tarski':
-  fixes f :: "'a::complete_lattice \<Rightarrow> 'a"
-  assumes "mono f"
-  shows "\<exists>a. f a = a"
-proof
-  let ?H = "{u. f u \<le> u}"
-  let ?a = "\<Sqinter>?H"
-  show "f ?a = ?a"
-  proof (rule order_antisym)
-    show "f ?a \<le> ?a"
-    proof (rule Inf_greatest)
-      fix x
-      assume "x \<in> ?H"
-      then have "?a \<le> x" by (rule Inf_lower)
-      with `mono f` have "f ?a \<le> f x" ..
-      also from `x \<in> ?H` have "\<dots> \<le> x" ..
-      finally show "f ?a \<le> x" .
-    qed
-    show "?a \<le> f ?a"
-    proof (rule Inf_lower)
-      from `mono f` and `f ?a \<le> ?a` have "f (f ?a) \<le> f ?a" ..
-      then show "f ?a \<in> ?H" ..
-    qed
-  qed
-qed
-
-end
diff --git a/isar/Root2_Isar.thy b/isar/Root2_Isar.thy
deleted file mode 100644
index 7a0123dd..00000000
--- a/isar/Root2_Isar.thy
+++ /dev/null
@@ -1,153 +0,0 @@
-(* Example proof by Markus Wenzel; see http://www.cs.kun.nl/~freek/comparison/ 
-   Taken from Isabelle2005 distribution. *)
-
-
-(*  Title:      HOL/Hyperreal/ex/Sqrt.thy
-    ID:         $Id$
-    Author:     Markus Wenzel, TU Muenchen
-
-*)
-
-header {*  Square roots of primes are irrational *}
-
-theory Root2_Isar
-imports Primes Complex_Main
-begin
-
-subsection {* Preliminaries *}
-
-text {*
-  The set of rational numbers, including the key representation
-  theorem.
-*}
-
-constdefs
-  rationals :: "real set"    ("\<rat>")
-  "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
-
-theorem rationals_rep: "x \<in> \<rat> \<Longrightarrow>
-  \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real m / real n \<and> gcd (m, n) = 1"
-proof -
-  assume "x \<in> \<rat>"
-  then obtain m n :: nat where
-      n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n"
-    by (unfold rationals_def) blast
-  let ?gcd = "gcd (m, n)"
-  from n have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero)
-  let ?k = "m div ?gcd"
-  let ?l = "n div ?gcd"
-  let ?gcd' = "gcd (?k, ?l)"
-  have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
-    by (rule dvd_mult_div_cancel)
-  have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
-    by (rule dvd_mult_div_cancel)
-
-  from n and gcd_l have "?l \<noteq> 0"
-    by (auto iff del: neq0_conv)
-  moreover
-  have "\<bar>x\<bar> = real ?k / real ?l"
-  proof -
-    from gcd have "real ?k / real ?l =
-        real (?gcd * ?k) / real (?gcd * ?l)"
-      by (simp add: mult_divide_cancel_left)
-    also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
-    also from x_rat have "\<dots> = \<bar>x\<bar>" ..
-    finally show ?thesis ..
-  qed
-  moreover
-  have "?gcd' = 1"
-  proof -
-    have "?gcd * ?gcd' = gcd (?gcd * ?k, ?gcd * ?l)"
-      by (rule gcd_mult_distrib2)
-    with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
-    with gcd show ?thesis by simp
-  qed
-  ultimately show ?thesis by blast
-qed
-
-lemma [elim?]: "r \<in> \<rat> \<Longrightarrow>
-  (\<And>m n. n \<noteq> 0 \<Longrightarrow> \<bar>r\<bar> = real m / real n \<Longrightarrow> gcd (m, n) = 1 \<Longrightarrow> C)
-    \<Longrightarrow> C"
-  using rationals_rep by blast
-
-
-subsection {* Main theorem *}
-
-text {*
-  The square root of any prime number (including @{text 2}) is
-  irrational.
-*}
-
-theorem sqrt_prime_irrational: "prime p \<Longrightarrow> sqrt (real p) \<notin> \<rat>"
-proof
-  assume p_prime: "prime p"
-  then have p: "1 < p" by (simp add: prime_def)
-  assume "sqrt (real p) \<in> \<rat>"
-  then obtain m n where
-      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
-    and gcd: "gcd (m, n) = 1" ..
-  have eq: "m\<twosuperior> = p * n\<twosuperior>"
-  proof -
-    from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
-    then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
-      by (auto simp add: power2_eq_square)
-    also have "(sqrt (real p))\<twosuperior> = real p" by simp
-    also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
-    finally show ?thesis ..
-  qed
-  have "p dvd m \<and> p dvd n"
-  proof
-    from eq have "p dvd m\<twosuperior>" ..
-    with p_prime show "p dvd m" by (rule prime_dvd_power_two)
-    then obtain k where "m = p * k" ..
-    with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
-    with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
-    then have "p dvd n\<twosuperior>" ..
-    with p_prime show "p dvd n" by (rule prime_dvd_power_two)
-  qed
-  then have "p dvd gcd (m, n)" ..
-  with gcd have "p dvd 1" by simp
-  then have "p \<le> 1" by (simp add: dvd_imp_le)
-  with p show False by simp
-qed
-
-corollary "sqrt (real (2::nat)) \<notin> \<rat>"
-  by (rule sqrt_prime_irrational) (rule two_is_prime)
-
-
-subsection {* Variations *}
-
-text {*
-  Here is an alternative version of the main proof, using mostly
-  linear forward-reasoning.  While this results in less top-down
-  structure, it is probably closer to proofs seen in mathematics.
-*}
-
-theorem "prime p \<Longrightarrow> sqrt (real p) \<notin> \<rat>"
-proof
-  assume p_prime: "prime p"
-  then have p: "1 < p" by (simp add: prime_def)
-  assume "sqrt (real p) \<in> \<rat>"
-  then obtain m n where
-      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
-    and gcd: "gcd (m, n) = 1" ..
-  from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
-  then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
-    by (auto simp add: power2_eq_square)
-  also have "(sqrt (real p))\<twosuperior> = real p" by simp
-  also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
-  finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
-  then have "p dvd m\<twosuperior>" ..
-  with p_prime have dvd_m: "p dvd m" by (rule prime_dvd_power_two)
-  then obtain k where "m = p * k" ..
-  with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
-  with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
-  then have "p dvd n\<twosuperior>" ..
-  with p_prime have "p dvd n" by (rule prime_dvd_power_two)
-  with dvd_m have "p dvd gcd (m, n)" by (rule gcd_greatest)
-  with gcd have "p dvd 1" by simp
-  then have "p \<le> 1" by (simp add: dvd_imp_le)
-  with p show False by simp
-qed
-
-end
diff --git a/isar/Root2_Tactic.thy b/isar/Root2_Tactic.thy
deleted file mode 100644
index 7c0620c5..00000000
--- a/isar/Root2_Tactic.thy
+++ /dev/null
@@ -1,83 +0,0 @@
-(* Example proof by Larry Paulson; see http://www.cs.kun.nl/~freek/comparison/ 
-   Taken from Isabelle2005 distribution. *)
-
-
-(*  Title:      HOL/Hyperreal/ex/Sqrt_Script.thy
-    ID:         $Id$
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   2001  University of Cambridge
-*)
-
-header {* Square roots of primes are irrational (script version) *}
-
-theory Root2_Tactic
-imports Primes Complex_Main
-begin
-
-text {*
-  \medskip Contrast this linear Isabelle/Isar script with Markus
-  Wenzel's more mathematical version.
-*}
-
-subsection {* Preliminaries *}
-
-lemma prime_nonzero:  "prime p \<Longrightarrow> p \<noteq> 0"
-  by (force simp add: prime_def)
-
-lemma prime_dvd_other_side:
-    "n * n = p * (k * k) \<Longrightarrow> prime p \<Longrightarrow> p dvd n"
-  apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult)
-  apply (rule_tac j = "k * k" in dvd_mult_left, simp)
-  done
-
-lemma reduction: "prime p \<Longrightarrow>
-    0 < k \<Longrightarrow> k * k = p * (j * j) \<Longrightarrow> k < p * j \<and> 0 < j"
-  apply (rule ccontr)
-  apply (simp add: linorder_not_less)
-  apply (erule disjE)
-   apply (frule mult_le_mono, assumption)
-   apply auto
-  apply (force simp add: prime_def)
-  done
-
-lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)"
-  by (simp add: mult_ac)
-
-lemma prime_not_square:
-    "prime p \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))"
-  apply (induct m rule: nat_less_induct)
-  apply clarify
-  apply (frule prime_dvd_other_side, assumption)
-  apply (erule dvdE)
-  apply (simp add: nat_mult_eq_cancel_disj prime_nonzero)
-  apply (blast dest: rearrange reduction)
-  done
-
-
-subsection {* The set of rational numbers *}
-
-constdefs
-  rationals :: "real set"    ("\<rat>")
-  "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
-
-
-subsection {* Main theorem *}
-
-text {*
-  The square root of any prime number (including @{text 2}) is
-  irrational.
-*}
-
-theorem prime_sqrt_irrational:
-    "prime p \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>"
-  apply (simp add: rationals_def real_abs_def)
-  apply clarify
-  apply (erule_tac P = "real m / real n * ?x = ?y" in rev_mp)
-  apply (simp del: real_of_nat_mult
-              add: divide_eq_eq prime_not_square real_of_nat_mult [symmetric])
-  done
-
-lemmas two_sqrt_irrational =
-  prime_sqrt_irrational [OF two_is_prime]
-
-end
diff --git a/isar/Tarski.thy b/isar/Tarski.thy
deleted file mode 100644
index e68006e5..00000000
--- a/isar/Tarski.thy
+++ /dev/null
@@ -1,904 +0,0 @@
-(********** This file is from the Isabelle distribution **********)
-
-(*  Title:      HOL/ex/Tarski.thy
-    ID:         Id: Tarski.thy,v 1.10 2002/09/26 08:51:32 paulson Exp 
-    Author:     Florian Kammüller, Cambridge University Computer Laboratory
-*)
-
-header {* The Full Theorem of Tarski *}
-
-theory Tarski imports Main FuncSet begin
-
-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"
-
-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"
-
-  least :: "['a => bool, 'a potype] => 'a"
-  "least P po == @ x. x: pset po & P x &
-                       (\<forall>y \<in> pset po. P y --> (x,y): order po)"
-
-  greatest :: "['a => bool, 'a potype] => 'a"
-  "greatest P po == @ x. x: pset po & P x &
-                          (\<forall>y \<in> pset po. P y --> (y,x): order po)"
-
-  lub  :: "['a set, 'a potype] => 'a"
-  "lub S po == least (%x. \<forall>y\<in>S. (y,x): order po) po"
-
-  glb  :: "['a set, 'a potype] => 'a"
-  "glb S po == greatest (%x. \<forall>y\<in>S. (x,y): order po) po"
-
-  isLub :: "['a set, 'a potype, 'a] => bool"
-  "isLub S po == %L. (L: pset po & (\<forall>y\<in>S. (y,L): order po) &
-                   (\<forall>z\<in>pset po. (\<forall>y\<in>S. (y,z): order po) --> (L,z): order po))"
-
-  isGlb :: "['a set, 'a potype, 'a] => bool"
-  "isGlb S po == %G. (G: pset po & (\<forall>y\<in>S. (G,y): order po) &
-                 (\<forall>z \<in> pset po. (\<forall>y\<in>S. (z,y): order po) --> (z,G): order po))"
-
-  "fix"    :: "[('a => 'a), 'a set] => 'a set"
-  "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}"
-
-
-constdefs
-  Bot :: "'a potype => 'a"
-  "Bot po == least (%x. True) po"
-
-  Top :: "'a potype => 'a"
-  "Top po == greatest (%x. True) po"
-
-  PartialOrder :: "('a potype) set"
-  "PartialOrder == {P. refl (pset P) (order P) & antisym (order P) &
-                       trans (order P)}"
-
-  CompleteLattice :: "('a potype) set"
-  "CompleteLattice == {cl. cl: PartialOrder &
-                        (\<forall>S. S <= pset cl --> (\<exists>L. isLub S cl L)) &
-                        (\<forall>S. S <= pset cl --> (\<exists>G. isGlb S cl G))}"
-
-  CLF :: "('a potype * ('a => 'a)) set"
-  "CLF == SIGMA cl: CompleteLattice.
-            {f. f: pset cl -> pset cl & monotone f (pset cl) (order cl)}"
-
-  induced :: "['a set, ('a * 'a) set] => ('a *'a)set"
-  "induced A r == {(a,b). a : A & b: A & (a,b): r}"
-
-
-constdefs
-  sublattice :: "('a potype * 'a set)set"
-  "sublattice ==
-      SIGMA cl: CompleteLattice.
-          {S. S <= pset cl &
-           (| pset = S, order = induced S (order cl) |): CompleteLattice }"
-
-syntax
-  "@SL"  :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
-
-translations
-  "S <<= cl" == "S : sublattice `` {cl}"
-
-constdefs
-  dual :: "'a potype => 'a potype"
-  "dual po == (| pset = pset po, order = converse (order po) |)"
-
-locale (open) PO =
-  fixes cl :: "'a potype"
-    and A  :: "'a set"
-    and r  :: "('a * 'a) set"
-  assumes cl_po:  "cl : PartialOrder"
-  defines A_def: "A == pset cl"
-     and  r_def: "r == order cl"
-
-locale (open) CL = PO +
-  assumes cl_co:  "cl : CompleteLattice"
-
-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 (open) Tarski = CLF +
-  fixes Y     :: "'a set"
-    and intY1 :: "'a set"
-    and v     :: "'a"
-  assumes
-    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 &
-                             x: intY1}
-                      (| pset=intY1, order=induced intY1 r|)"
-
-
-subsubsection {* Partial Order *}
-
-lemma (in PO) PO_imp_refl: "refl A r"
-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 (simp add: PartialOrder_def A_def r_def)
-done
-
-lemma (in PO) PO_imp_trans: "trans r"
-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 (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 (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 (simp add: PartialOrder_def)
-apply (unfold trans_def, fast)
-done
-
-lemma (in PO) monotoneE:
-     "[| monotone f A r;  x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r"
-by (simp add: monotone_def)
-
-lemma (in PO) po_subset_po:
-     "S <= A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
-apply (simp (no_asm) add: PartialOrder_def)
-apply auto
--- {* refl *}
-apply (simp add: refl_def induced_def)
-apply (blast intro: PO_imp_refl [THEN reflE])
--- {* antisym *}
-apply (simp add: antisym_def induced_def)
-apply (blast intro: PO_imp_sym [THEN antisymE])
--- {* trans *}
-apply (simp add: trans_def induced_def)
-apply (blast intro: PO_imp_trans [THEN transE])
-done
-
-lemma (in PO) indE: "[| (x, y) \<in> induced S r; S <= A |] ==> (x, y) \<in> r"
-by (simp add: add: induced_def)
-
-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)
-apply (simp add: CompleteLattice_def A_def)
-done
-
-declare (in CL) cl_co [simp]
-
-lemma isLub_lub: "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)"
-by (simp add: lub_def least_def isLub_def some_eq_ex [symmetric])
-
-lemma isGlb_glb: "(\<exists>G. isGlb S cl G) = isGlb S cl (glb S cl)"
-by (simp add: glb_def greatest_def isGlb_def some_eq_ex [symmetric])
-
-lemma isGlb_dual_isLub: "isGlb S cl = isLub S (dual cl)"
-by (simp add: isLub_def isGlb_def dual_def converse_def)
-
-lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)"
-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
-                 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>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)}
-                      (|pset = A, order = r|) " in exI)
-apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> r) }" in spec)
-apply (drule mp, fast)
-apply (simp add: isLub_lub isGlb_def)
-apply (simp add: isLub_def, blast)
-done
-
-lemma lub_dual_glb: "lub S cl = glb S (dual cl)"
-by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
-
-lemma glb_dual_lub: "glb S cl = lub S (dual cl)"
-by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
-
-lemma CL_subset_PO: "CompleteLattice <= PartialOrder"
-by (simp add: PartialOrder_def CompleteLattice_def, fast)
-
-lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
-
-declare CL_imp_PO [THEN Tarski.PO_imp_refl, simp]
-declare CL_imp_PO [THEN Tarski.PO_imp_sym, simp]
-declare CL_imp_PO [THEN Tarski.PO_imp_trans, simp]
-
-lemma (in CL) CO_refl: "refl A r"
-by (rule PO_imp_refl)
-
-lemma (in CL) CO_antisym: "antisym r"
-by (rule PO_imp_sym)
-
-lemma (in CL) CO_trans: "trans r"
-by (rule PO_imp_trans)
-
-lemma CompleteLatticeI:
-     "[| po \<in> PartialOrder; (\<forall>S. S <= pset po --> (\<exists>L. isLub S po L));
-         (\<forall>S. S <= pset po --> (\<exists>G. isGlb S po G))|]
-      ==> po \<in> CompleteLattice"
-apply (unfold CompleteLattice_def, blast)
-done
-
-lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
-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]
-                 dualPO)
-done
-
-lemma (in PO) dualA_iff: "pset (dual cl) = pset cl"
-by (simp add: dual_def)
-
-lemma (in PO) dualr_iff: "((x, y) \<in> (order(dual cl))) = ((y, x) \<in> order cl)"
-by (simp add: dual_def)
-
-lemma (in PO) monotone_dual:
-     "monotone f (pset cl) (order cl) 
-     ==> monotone f (pset (dual cl)) (order(dual cl))"
-by (simp add: monotone_def dualA_iff dualr_iff)
-
-lemma (in PO) interval_dual:
-     "[| x \<in> A; y \<in> A|] ==> interval r x y = interval (order(dual cl)) y x"
-apply (simp add: interval_def dualr_iff)
-apply (fold r_def, fast)
-done
-
-lemma (in PO) interval_not_empty:
-     "[| trans r; interval r a b \<noteq> {} |] ==> (a, b) \<in> r"
-apply (simp add: interval_def)
-apply (unfold trans_def, blast)
-done
-
-lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r"
-by (simp add: interval_def)
-
-lemma (in PO) left_in_interval:
-     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b"
-apply (simp (no_asm_simp) add: interval_def)
-apply (simp add: PO_imp_trans interval_not_empty)
-apply (simp add: PO_imp_refl [THEN reflE])
-done
-
-lemma (in PO) right_in_interval:
-     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b"
-apply (simp (no_asm_simp) add: interval_def)
-apply (simp add: PO_imp_trans interval_not_empty)
-apply (simp add: PO_imp_refl [THEN reflE])
-done
-
-
-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 <<= cl"
-by (simp add: sublattice_def A_def r_def)
-
-
-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)
-apply (auto simp add: isLub_def r_def)
-done
-
-lemma (in CL) lub_upper: "[|S <= A; x \<in> S|] ==> (x, lub S cl) \<in> r"
-apply (rule CL_imp_ex_isLub [THEN exE], assumption)
-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: isLub_def r_def)
-done
-
-lemma (in CL) lub_least:
-     "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
-apply (rule CL_imp_ex_isLub [THEN exE], assumption)
-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: isLub_def r_def A_def)
-done
-
-lemma (in CL) lub_in_lattice: "S <= A ==> lub S cl \<in> A"
-apply (rule CL_imp_ex_isLub [THEN exE], assumption)
-apply (unfold lub_def least_def)
-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)
-done
-
-lemma (in CL) lubI:
-     "[| 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)
-apply (unfold isLub_def)
-apply (rule conjI)
-apply (fold A_def r_def)
-apply (rule lub_in_lattice, assumption)
-apply (simp add: lub_upper lub_least)
-done
-
-lemma (in CL) lubIa: "[| S <= A; isLub S cl L |] ==> L = lub S cl"
-by (simp add: lubI isLub_def A_def r_def)
-
-lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
-by (simp add: isLub_def  A_def)
-
-lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
-by (simp add: isLub_def r_def)
-
-lemma (in CL) isLub_least:
-     "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
-by (simp add: isLub_def A_def r_def)
-
-lemma (in CL) isLubI:
-     "[| 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)
-
-
-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 CL_dualCL)
-apply (simp add: dualA_iff)
-done
-
-lemma (in CL) glb_lower: "[|S <= A; x \<in> S|] ==> (glb S cl, x) \<in> r"
-apply (subst glb_dual_lub)
-apply (simp add: r_def)
-apply (rule dualr_iff [THEN subst])
-apply (rule Tarski.lub_upper [rule_format])
-apply (rule dualPO)
-apply (rule CL_dualCL)
-apply (simp add: dualA_iff A_def, assumption)
-done
-
-text {*
-  Reduce the sublattice property by using substructural properties;
-  abandoned see @{text "Tarski_4.ML"}.
-*}
-
-lemma (in CLF) [simp]:
-    "f: pset cl -> pset cl & monotone f (pset cl) (order cl)"
-apply (insert f_cl)
-apply (simp add: CLF_def)
-done
-
-declare (in CLF) f_cl [simp]
-
-
-lemma (in CLF) f_in_funcset: "f \<in> A -> A"
-by (simp add: A_def)
-
-lemma (in CLF) monotone_f: "monotone f A r"
-by (simp add: A_def r_def)
-
-lemma (in CLF) CLF_dual: "(cl,f) \<in> CLF ==> (dual cl, f) \<in> CLF"
-apply (simp add: CLF_def  CL_dualCL monotone_dual)
-apply (simp add: dualA_iff)
-done
-
-
-subsubsection {* fixed points *}
-
-lemma fix_subset: "fix f A <= A"
-by (simp add: fix_def, fast)
-
-lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
-by (simp add: fix_def)
-
-lemma fixf_subset:
-     "[| A <= B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
-apply (simp add: fix_def, auto)
-done
-
-
-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)
--- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *}
-apply (rule ballI)
-apply (rule transE)
-apply (rule CO_trans)
--- {* instantiates @{text "(x, ???z) \<in> order cl to (x, f x)"}, *}
--- {* because of the def of @{text H} *}
-apply fast
--- {* 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)
-apply (rule lub_upper, fast)
-apply assumption
-done
-
-lemma (in CLF) flubH_le_lubH:
-     "[|  H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
-apply (rule lub_upper, fast)
-apply (rule_tac t = "H" in ssubst, assumption)
-apply (rule CollectI)
-apply (rule conjI)
-apply (rule_tac [2] f_in_funcset [THEN funcset_mem])
-apply (rule_tac [2] lub_in_lattice)
-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 (simp add: lubH_le_flubH)
-done
-
-lemma (in CLF) lubH_is_fixp:
-     "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
-apply (simp add: fix_def)
-apply (rule conjI)
-apply (rule lub_in_lattice, fast)
-apply (rule antisymE)
-apply (rule CO_antisym)
-apply (simp add: flubH_le_lubH)
-apply (simp add: lubH_le_flubH)
-done
-
-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])
-
-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)
-done
-
-lemma (in CLF) lubH_least_fixf:
-     "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)
-apply (erule bspec)
-apply (rule lubH_is_fixp, assumption)
-done
-
-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 (rule lubI)
-apply (rule fix_subset)
-apply (rule lub_in_lattice, fast)
-apply (simp add: fixf_le_lubH)
-apply (simp add: lubH_least_fixf)
-done
-
-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 CL_dualCL)
-apply (rule f_cl [THEN CLF_dual])
-apply (simp add: dualr_iff dualA_iff)
-done
-
-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]
-                 dualPO CL_dualCL CLF_dual dualr_iff)
-done
-
-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)
-done
-
-lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b <= A"
-apply (simp add: interval_def)
-apply (blast intro: rel_imp_elem)
-done
-
-lemma (in CLF) intervalI:
-     "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
-apply (simp add: interval_def)
-done
-
-lemma (in CLF) interval_lemma1:
-     "[| S <= interval r a b; x \<in> S |] ==> (a, x) \<in> r"
-apply (unfold interval_def, fast)
-done
-
-lemma (in CLF) interval_lemma2:
-     "[| S <= interval r a b; x \<in> S |] ==> (x, b) \<in> r"
-apply (unfold interval_def, fast)
-done
-
-lemma (in CLF) a_less_lub:
-     "[| 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> {};
-         \<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)
-
-lemma (in CLF) S_intv_cl:
-     "[| a \<in> A; b \<in> A; S <= interval r a b |]==> S <= A"
-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;
-         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)
-prefer 2 apply assumption
-apply (simp add: S_intv_cl)
-apply (rule ballI)
-apply (simp add: interval_lemma1)
-apply (simp add: isLub_upper)
--- {* @{text "(L, b) \<in> r"} *}
-apply (simp add: isLub_least interval_lemma2)
-done
-
-lemma (in CLF) G_in_interval:
-     "[| 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]
-                 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> {} |]
-      ==> (| pset = interval r a b, order = induced (interval r a b) r |)
-          \<in> PartialOrder"
-apply (rule po_subset_po)
-apply (simp add: interval_subset)
-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,
-                          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 *}
-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)
--- {* @{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 *}
-apply (case_tac "S = {}")
--- {* @{text "S = {}, y \<in> S = False => everything"} *}
-apply fast
--- {* @{text "S \<noteq> {}"} *}
-apply simp
--- {* @{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)
--- {* @{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 = {}")
--- {* @{text "S = {}"} *}
-apply simp
-apply (simp add: induced_def  interval_def)
-apply (rule conjI)
-apply (rule reflE)
-apply (rule CO_refl, assumption)
-apply (rule interval_not_empty)
-apply (rule CO_trans)
-apply (simp add: interval_def)
--- {* @{text "S \<noteq> {}"} *}
-apply simp
-apply (simp add: induced_def  L_in_interval)
-apply (rule isLub_least, assumption)
-apply (rule subsetD)
-prefer 2 apply assumption
-apply (simp add: S_intv_cl, fast)
-done
-
-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> {} |]
-        ==> interval r a b <<= cl"
-apply (rule sublatticeI)
-apply (simp add: interval_subset)
-apply (rule CompleteLatticeI)
-apply (simp add: intervalPO)
- apply (simp add: intv_CL_lub)
-apply (simp add: intv_CL_glb)
-done
-
-lemmas (in CLF) interv_is_compl_latt =
-    interval_is_sublattice [THEN sublattice_imp_CL]
-
-
-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)
-
-lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
-by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
-
-lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
-apply (simp add: Bot_def least_def)
-apply (rule someI2)
-apply (fold A_def)
-apply (erule_tac [2] conjunct1)
-apply (rule conjI)
-apply (rule glb_in_lattice)
-apply (rule subset_refl)
-apply (fold r_def)
-apply (simp add: glb_lower)
-done
-
-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)
-done
-
-lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
-apply (simp add: Top_def greatest_def)
-apply (rule someI2)
-apply (fold r_def  A_def)
-prefer 2 apply fast
-apply (intro conjI ballI)
-apply (rule_tac [2] lub_upper)
-apply (auto simp add: lub_in_lattice)
-done
-
-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]
-                 dualA_iff A_def dualPO CL_dualCL CLF_dual)
-done
-
-lemma (in CLF) Top_intv_not_empty: "x \<in> A  ==> interval r x (Top cl) \<noteq> {}"
-apply (rule notI)
-apply (drule_tac a = "Top cl" in equals0D)
-apply (simp add: interval_def)
-apply (simp add: refl_def Top_in_lattice Top_prop)
-done
-
-lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
-apply (simp add: Bot_dual_Top)
-apply (subst interval_dual)
-prefer 2 apply assumption
-apply (simp add: A_def)
-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]
-                 dualA_iff A_def dualPO CL_dualCL CLF_dual)
-done
-
-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)
-
-lemma (in Tarski) Y_subset_A: "Y <= A"
-apply (rule subset_trans [OF _ fix_subset])
-apply (rule Y_ss [simplified P_def])
-done
-
-lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
-by (simp add: Y_subset_A [THEN lub_in_lattice])
-
-lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
-apply (rule lub_least)
-apply (rule Y_subset_A)
-apply (rule f_in_funcset [THEN funcset_mem])
-apply (rule lubY_in_A)
--- {* @{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])
--- {* @{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])
-apply (rule lubY_in_A)
-apply (simp add: lub_upper Y_subset_A)
-done
-
-lemma (in Tarski) intY1_subset: "intY1 <= A"
-apply (unfold intY1_def)
-apply (rule interval_subset)
-apply (rule lubY_in_A)
-apply (rule Top_in_lattice)
-done
-
-lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
-
-lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
-apply (simp add: intY1_def  interval_def)
-apply (rule conjI)
-apply (rule transE)
-apply (rule CO_trans)
-apply (rule lubY_le_flubY)
--- {* @{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)
--- {* @{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)
-done
-
-lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 -> intY1"
-apply (rule restrictI)
-apply (erule intY1_f_closed)
-done
-
-lemma (in Tarski) intY1_mono:
-     "monotone (%x: intY1. f x) intY1 (induced intY1 r)"
-apply (auto simp add: monotone_def induced_def intY1_f_closed)
-apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
-done
-
-lemma (in Tarski) intY1_is_cl:
-    "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
-apply (unfold intY1_def)
-apply (rule interv_is_compl_latt)
-apply (rule lubY_in_A)
-apply (rule Top_in_lattice)
-apply (rule Top_intv_not_empty)
-apply (rule lubY_in_A)
-done
-
-lemma (in Tarski) v_in_P: "v \<in> P"
-apply (unfold P_def)
-apply (rule_tac A = "intY1" in fixf_subset)
-apply (rule intY1_subset)
-apply (simp add: Tarski.glbH_is_fixp [OF _ intY1_is_cl, simplified]
-                 v_def CL_imp_PO intY1_is_cl CLF_def intY1_func intY1_mono)
-done
-
-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
- apply (erule fix_subset [THEN subsetD, THEN Top_prop])
-apply (rule lub_least)
-apply (rule Y_subset_A)
-apply (fast elim!: fix_subset [THEN subsetD])
-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 |]
-      ==> ((%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]
-                 CO_refl [THEN reflE])
-done
-
-lemma (in Tarski) tarski_full_lemma:
-     "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
-apply (rule_tac x = "v" in exI)
-apply (simp add: isLub_def)
--- {* @{text "v \<in> P"} *}
-apply (simp add: v_in_P)
-apply (rule conjI)
--- {* @{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)
-apply (erule Y_ss [THEN subsetD])
-apply (rule_tac b = "lub Y cl" in transE)
-apply (rule CO_trans)
-apply (rule lub_upper)
-apply (rule Y_subset_A, assumption)
-apply (rule_tac b = "Top cl" in interval_imp_mem)
-apply (simp add: v_def)
-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)
--- {* @{text v} is LEAST ub *}
-apply clarify
-apply (rule indI)
-  prefer 3 apply assumption
- prefer 2 apply (simp add: v_in_P)
-apply (unfold v_def)
-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 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]
-                 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> CompleteLattice"
-by (simp add: CompleteLatticeI Rdual)
-
-theorem (in CLF) Tarski_full:
-     "(| 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)
-done
-
-end