src/HOL/ex/Tarski.thy
author wenzelm
Wed Jun 22 10:09:20 2016 +0200 (2016-06-22)
changeset 63343 fb5d8a50c641
parent 62390 842917225d56
child 64915 2bb0152d82cf
permissions -rw-r--r--
bundle lifting_syntax;
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(*  Title:      HOL/ex/Tarski.thy
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    Author:     Florian Kammüller, Cambridge University Computer Laboratory
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*)
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section \<open>The Full Theorem of Tarski\<close>
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theory Tarski
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imports Main "~~/src/HOL/Library/FuncSet"
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begin
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text \<open>
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  Minimal version of lattice theory plus the full theorem of Tarski:
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  The fixedpoints of a complete lattice themselves form a complete
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  lattice.
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  Illustrates first-class theories, using the Sigma representation of
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  structures.  Tidied and converted to Isar by lcp.
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\<close>
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record 'a potype =
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  pset  :: "'a set"
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  order :: "('a * 'a) set"
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definition
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  monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool" where
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  "monotone f A r = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r)"
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definition
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  least :: "['a => bool, 'a potype] => 'a" where
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  "least P po = (SOME x. x: pset po & P x &
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                       (\<forall>y \<in> pset po. P y --> (x,y): order po))"
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definition
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  greatest :: "['a => bool, 'a potype] => 'a" where
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  "greatest P po = (SOME x. x: pset po & P x &
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                          (\<forall>y \<in> pset po. P y --> (y,x): order po))"
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definition
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  lub  :: "['a set, 'a potype] => 'a" where
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  "lub S po = least (%x. \<forall>y\<in>S. (y,x): order po) po"
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definition
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  glb  :: "['a set, 'a potype] => 'a" where
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  "glb S po = greatest (%x. \<forall>y\<in>S. (x,y): order po) po"
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definition
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  isLub :: "['a set, 'a potype, 'a] => bool" where
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  "isLub S po = (%L. (L: pset po & (\<forall>y\<in>S. (y,L): order po) &
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                   (\<forall>z\<in>pset po. (\<forall>y\<in>S. (y,z): order po) --> (L,z): order po)))"
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definition
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  isGlb :: "['a set, 'a potype, 'a] => bool" where
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  "isGlb S po = (%G. (G: pset po & (\<forall>y\<in>S. (G,y): order po) &
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                 (\<forall>z \<in> pset po. (\<forall>y\<in>S. (z,y): order po) --> (z,G): order po)))"
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definition
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  "fix"    :: "[('a => 'a), 'a set] => 'a set" where
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  "fix f A  = {x. x: A & f x = x}"
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definition
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  interval :: "[('a*'a) set,'a, 'a ] => 'a set" where
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  "interval r a b = {x. (a,x): r & (x,b): r}"
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definition
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  Bot :: "'a potype => 'a" where
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  "Bot po = least (%x. True) po"
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definition
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  Top :: "'a potype => 'a" where
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  "Top po = greatest (%x. True) po"
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definition
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  PartialOrder :: "('a potype) set" where
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  "PartialOrder = {P. refl_on (pset P) (order P) & antisym (order P) &
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                       trans (order P)}"
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definition
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  CompleteLattice :: "('a potype) set" where
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  "CompleteLattice = {cl. cl: PartialOrder &
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                        (\<forall>S. S \<subseteq> pset cl --> (\<exists>L. isLub S cl L)) &
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                        (\<forall>S. S \<subseteq> pset cl --> (\<exists>G. isGlb S cl G))}"
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definition
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  CLF_set :: "('a potype * ('a => 'a)) set" where
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  "CLF_set = (SIGMA cl: CompleteLattice.
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            {f. f: pset cl \<rightarrow> pset cl & monotone f (pset cl) (order cl)})"
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definition
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  induced :: "['a set, ('a * 'a) set] => ('a *'a)set" where
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  "induced A r = {(a,b). a : A & b: A & (a,b): r}"
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definition
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  sublattice :: "('a potype * 'a set)set" where
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  "sublattice =
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      (SIGMA cl: CompleteLattice.
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          {S. S \<subseteq> pset cl &
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           (| pset = S, order = induced S (order cl) |): CompleteLattice})"
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abbreviation
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  sublat :: "['a set, 'a potype] => bool"  ("_ <<= _" [51,50]50) where
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  "S <<= cl == S : sublattice `` {cl}"
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definition
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  dual :: "'a potype => 'a potype" where
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  "dual po = (| pset = pset po, order = converse (order po) |)"
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locale S =
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  fixes cl :: "'a potype"
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    and A  :: "'a set"
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    and r  :: "('a * 'a) set"
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  defines A_def: "A == pset cl"
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     and  r_def: "r == order cl"
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locale PO = S +
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  assumes cl_po:  "cl : PartialOrder"
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locale CL = S +
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  assumes cl_co:  "cl : CompleteLattice"
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sublocale CL < po?: PO
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apply (simp_all add: A_def r_def)
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apply unfold_locales
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using cl_co unfolding CompleteLattice_def by auto
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locale CLF = S +
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  fixes f :: "'a => 'a"
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    and P :: "'a set"
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  assumes f_cl:  "(cl,f) : CLF_set" (*was the equivalent "f : CLF_set``{cl}"*)
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  defines P_def: "P == fix f A"
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sublocale CLF < cl?: CL
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apply (simp_all add: A_def r_def)
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apply unfold_locales
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using f_cl unfolding CLF_set_def by auto
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locale Tarski = CLF +
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  fixes Y     :: "'a set"
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    and intY1 :: "'a set"
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    and v     :: "'a"
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  assumes
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    Y_ss: "Y \<subseteq> P"
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  defines
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    intY1_def: "intY1 == interval r (lub Y cl) (Top cl)"
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    and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r &
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                             x: intY1}
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                      (| pset=intY1, order=induced intY1 r|)"
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subsection \<open>Partial Order\<close>
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lemma (in PO) dual:
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  "PO (dual cl)"
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apply unfold_locales
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using cl_po
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unfolding PartialOrder_def dual_def
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by auto
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lemma (in PO) PO_imp_refl_on [simp]: "refl_on A r"
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apply (insert cl_po)
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apply (simp add: PartialOrder_def A_def r_def)
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done
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lemma (in PO) PO_imp_sym [simp]: "antisym r"
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apply (insert cl_po)
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apply (simp add: PartialOrder_def r_def)
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done
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lemma (in PO) PO_imp_trans [simp]: "trans r"
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apply (insert cl_po)
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apply (simp add: PartialOrder_def r_def)
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done
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lemma (in PO) reflE: "x \<in> A ==> (x, x) \<in> r"
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apply (insert cl_po)
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apply (simp add: PartialOrder_def refl_on_def A_def r_def)
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done
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lemma (in PO) antisymE: "[| (a, b) \<in> r; (b, a) \<in> r |] ==> a = b"
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apply (insert cl_po)
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apply (simp add: PartialOrder_def antisym_def r_def)
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done
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lemma (in PO) transE: "[| (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r"
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apply (insert cl_po)
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apply (simp add: PartialOrder_def r_def)
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apply (unfold trans_def, fast)
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done
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lemma (in PO) monotoneE:
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     "[| monotone f A r;  x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r"
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by (simp add: monotone_def)
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lemma (in PO) po_subset_po:
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     "S \<subseteq> A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
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apply (simp (no_asm) add: PartialOrder_def)
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apply auto
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\<comment> \<open>refl\<close>
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apply (simp add: refl_on_def induced_def)
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apply (blast intro: reflE)
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\<comment> \<open>antisym\<close>
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apply (simp add: antisym_def induced_def)
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apply (blast intro: antisymE)
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\<comment> \<open>trans\<close>
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apply (simp add: trans_def induced_def)
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apply (blast intro: transE)
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done
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lemma (in PO) indE: "[| (x, y) \<in> induced S r; S \<subseteq> A |] ==> (x, y) \<in> r"
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by (simp add: add: induced_def)
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lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r"
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by (simp add: add: induced_def)
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lemma (in CL) CL_imp_ex_isLub: "S \<subseteq> A ==> \<exists>L. isLub S cl L"
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apply (insert cl_co)
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apply (simp add: CompleteLattice_def A_def)
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done
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declare (in CL) cl_co [simp]
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lemma isLub_lub: "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)"
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by (simp add: lub_def least_def isLub_def some_eq_ex [symmetric])
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lemma isGlb_glb: "(\<exists>G. isGlb S cl G) = isGlb S cl (glb S cl)"
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by (simp add: glb_def greatest_def isGlb_def some_eq_ex [symmetric])
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lemma isGlb_dual_isLub: "isGlb S cl = isLub S (dual cl)"
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by (simp add: isLub_def isGlb_def dual_def converse_unfold)
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lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)"
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by (simp add: isLub_def isGlb_def dual_def converse_unfold)
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lemma (in PO) dualPO: "dual cl \<in> PartialOrder"
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apply (insert cl_po)
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apply (simp add: PartialOrder_def dual_def refl_on_converse
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                 trans_converse antisym_converse)
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done
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lemma Rdual:
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     "\<forall>S. (S \<subseteq> A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
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      ==> \<forall>S. (S \<subseteq> A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"
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apply safe
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apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)}
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                      (|pset = A, order = r|) " in exI)
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apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> r) }" in spec)
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apply (drule mp, fast)
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apply (simp add: isLub_lub isGlb_def)
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apply (simp add: isLub_def, blast)
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done
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lemma lub_dual_glb: "lub S cl = glb S (dual cl)"
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by (simp add: lub_def glb_def least_def greatest_def dual_def converse_unfold)
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lemma glb_dual_lub: "glb S cl = lub S (dual cl)"
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by (simp add: lub_def glb_def least_def greatest_def dual_def converse_unfold)
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lemma CL_subset_PO: "CompleteLattice \<subseteq> PartialOrder"
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by (simp add: PartialOrder_def CompleteLattice_def, fast)
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lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
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(*declare CL_imp_PO [THEN PO.PO_imp_refl, simp]
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declare CL_imp_PO [THEN PO.PO_imp_sym, simp]
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declare CL_imp_PO [THEN PO.PO_imp_trans, simp]*)
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lemma (in CL) CO_refl_on: "refl_on A r"
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by (rule PO_imp_refl_on)
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lemma (in CL) CO_antisym: "antisym r"
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by (rule PO_imp_sym)
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lemma (in CL) CO_trans: "trans r"
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by (rule PO_imp_trans)
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lemma CompleteLatticeI:
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     "[| po \<in> PartialOrder; (\<forall>S. S \<subseteq> pset po --> (\<exists>L. isLub S po L));
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         (\<forall>S. S \<subseteq> pset po --> (\<exists>G. isGlb S po G))|]
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      ==> po \<in> CompleteLattice"
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apply (unfold CompleteLattice_def, blast)
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done
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lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
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apply (insert cl_co)
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apply (simp add: CompleteLattice_def dual_def)
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apply (fold dual_def)
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apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric]
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                 dualPO)
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done
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lemma (in PO) dualA_iff: "pset (dual cl) = pset cl"
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by (simp add: dual_def)
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lemma (in PO) dualr_iff: "((x, y) \<in> (order(dual cl))) = ((y, x) \<in> order cl)"
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by (simp add: dual_def)
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lemma (in PO) monotone_dual:
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     "monotone f (pset cl) (order cl) 
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     ==> monotone f (pset (dual cl)) (order(dual cl))"
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by (simp add: monotone_def dualA_iff dualr_iff)
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lemma (in PO) interval_dual:
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     "[| x \<in> A; y \<in> A|] ==> interval r x y = interval (order(dual cl)) y x"
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apply (simp add: interval_def dualr_iff)
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apply (fold r_def, fast)
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done
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lemma (in PO) trans:
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  "(x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r"
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using cl_po apply (auto simp add: PartialOrder_def r_def)
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unfolding trans_def by blast 
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lemma (in PO) interval_not_empty:
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  "interval r a b \<noteq> {} ==> (a, b) \<in> r"
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apply (simp add: interval_def)
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using trans by blast
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lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r"
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by (simp add: interval_def)
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lemma (in PO) left_in_interval:
paulson@13115
   323
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b"
paulson@13115
   324
apply (simp (no_asm_simp) add: interval_def)
paulson@13115
   325
apply (simp add: PO_imp_trans interval_not_empty)
paulson@18705
   326
apply (simp add: reflE)
paulson@13115
   327
done
paulson@13115
   328
paulson@13115
   329
lemma (in PO) right_in_interval:
paulson@13115
   330
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b"
paulson@13115
   331
apply (simp (no_asm_simp) add: interval_def)
paulson@13115
   332
apply (simp add: PO_imp_trans interval_not_empty)
paulson@18705
   333
apply (simp add: reflE)
paulson@13115
   334
done
paulson@13115
   335
wenzelm@13383
   336
wenzelm@61343
   337
subsection \<open>sublattice\<close>
wenzelm@13383
   338
paulson@13115
   339
lemma (in PO) sublattice_imp_CL:
paulson@18750
   340
     "S <<= cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
paulson@19316
   341
by (simp add: sublattice_def CompleteLattice_def r_def)
paulson@13115
   342
paulson@13115
   343
lemma (in CL) sublatticeI:
paulson@17841
   344
     "[| S \<subseteq> A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
paulson@18750
   345
      ==> S <<= cl"
paulson@13115
   346
by (simp add: sublattice_def A_def r_def)
paulson@13115
   347
haftmann@27681
   348
lemma (in CL) dual:
haftmann@27681
   349
  "CL (dual cl)"
haftmann@27681
   350
apply unfold_locales
haftmann@27681
   351
using cl_co unfolding CompleteLattice_def
haftmann@27681
   352
apply (simp add: dualPO isGlb_dual_isLub [symmetric] isLub_dual_isGlb [symmetric] dualA_iff)
haftmann@27681
   353
done
haftmann@27681
   354
wenzelm@13383
   355
wenzelm@61343
   356
subsection \<open>lub\<close>
wenzelm@13383
   357
paulson@17841
   358
lemma (in CL) lub_unique: "[| S \<subseteq> A; isLub S cl x; isLub S cl L|] ==> x = L"
paulson@13115
   359
apply (rule antisymE)
paulson@13115
   360
apply (auto simp add: isLub_def r_def)
paulson@13115
   361
done
paulson@13115
   362
paulson@17841
   363
lemma (in CL) lub_upper: "[|S \<subseteq> A; x \<in> S|] ==> (x, lub S cl) \<in> r"
paulson@13115
   364
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
paulson@13115
   365
apply (unfold lub_def least_def)
paulson@13115
   366
apply (rule some_equality [THEN ssubst])
paulson@13115
   367
  apply (simp add: isLub_def)
wenzelm@13383
   368
 apply (simp add: lub_unique A_def isLub_def)
paulson@13115
   369
apply (simp add: isLub_def r_def)
paulson@13115
   370
done
paulson@13115
   371
paulson@13115
   372
lemma (in CL) lub_least:
paulson@17841
   373
     "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
paulson@13115
   374
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
paulson@13115
   375
apply (unfold lub_def least_def)
paulson@13115
   376
apply (rule_tac s=x in some_equality [THEN ssubst])
paulson@13115
   377
  apply (simp add: isLub_def)
wenzelm@13383
   378
 apply (simp add: lub_unique A_def isLub_def)
paulson@13115
   379
apply (simp add: isLub_def r_def A_def)
paulson@13115
   380
done
paulson@13115
   381
paulson@17841
   382
lemma (in CL) lub_in_lattice: "S \<subseteq> A ==> lub S cl \<in> A"
paulson@13115
   383
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
paulson@13115
   384
apply (unfold lub_def least_def)
paulson@13115
   385
apply (subst some_equality)
paulson@13115
   386
apply (simp add: isLub_def)
paulson@13115
   387
prefer 2 apply (simp add: isLub_def A_def)
wenzelm@13383
   388
apply (simp add: lub_unique A_def isLub_def)
paulson@13115
   389
done
paulson@13115
   390
paulson@13115
   391
lemma (in CL) lubI:
paulson@17841
   392
     "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
paulson@13115
   393
         \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
paulson@13115
   394
apply (rule lub_unique, assumption)
paulson@13115
   395
apply (simp add: isLub_def A_def r_def)
paulson@13115
   396
apply (unfold isLub_def)
paulson@13115
   397
apply (rule conjI)
paulson@13115
   398
apply (fold A_def r_def)
paulson@13115
   399
apply (rule lub_in_lattice, assumption)
paulson@13115
   400
apply (simp add: lub_upper lub_least)
paulson@13115
   401
done
paulson@13115
   402
paulson@17841
   403
lemma (in CL) lubIa: "[| S \<subseteq> A; isLub S cl L |] ==> L = lub S cl"
paulson@13115
   404
by (simp add: lubI isLub_def A_def r_def)
paulson@13115
   405
paulson@13115
   406
lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
paulson@13115
   407
by (simp add: isLub_def  A_def)
paulson@13115
   408
paulson@13115
   409
lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
paulson@13115
   410
by (simp add: isLub_def r_def)
paulson@13115
   411
paulson@13115
   412
lemma (in CL) isLub_least:
paulson@13115
   413
     "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
paulson@13115
   414
by (simp add: isLub_def A_def r_def)
paulson@13115
   415
paulson@13115
   416
lemma (in CL) isLubI:
wenzelm@13383
   417
     "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
paulson@13115
   418
         (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
paulson@13115
   419
by (simp add: isLub_def A_def r_def)
paulson@13115
   420
wenzelm@13383
   421
wenzelm@61343
   422
subsection \<open>glb\<close>
wenzelm@13383
   423
paulson@17841
   424
lemma (in CL) glb_in_lattice: "S \<subseteq> A ==> glb S cl \<in> A"
paulson@13115
   425
apply (subst glb_dual_lub)
paulson@13115
   426
apply (simp add: A_def)
paulson@13115
   427
apply (rule dualA_iff [THEN subst])
wenzelm@21232
   428
apply (rule CL.lub_in_lattice)
haftmann@27681
   429
apply (rule dual)
paulson@13115
   430
apply (simp add: dualA_iff)
paulson@13115
   431
done
paulson@13115
   432
paulson@17841
   433
lemma (in CL) glb_lower: "[|S \<subseteq> A; x \<in> S|] ==> (glb S cl, x) \<in> r"
paulson@13115
   434
apply (subst glb_dual_lub)
paulson@13115
   435
apply (simp add: r_def)
paulson@13115
   436
apply (rule dualr_iff [THEN subst])
wenzelm@21232
   437
apply (rule CL.lub_upper)
haftmann@27681
   438
apply (rule dual)
paulson@13115
   439
apply (simp add: dualA_iff A_def, assumption)
paulson@13115
   440
done
paulson@13115
   441
wenzelm@61343
   442
text \<open>
wenzelm@13383
   443
  Reduce the sublattice property by using substructural properties;
wenzelm@61933
   444
  abandoned see \<open>Tarski_4.ML\<close>.
wenzelm@61343
   445
\<close>
paulson@13115
   446
paulson@13115
   447
lemma (in CLF) [simp]:
wenzelm@61384
   448
    "f: pset cl \<rightarrow> pset cl & monotone f (pset cl) (order cl)"
wenzelm@13383
   449
apply (insert f_cl)
haftmann@27681
   450
apply (simp add: CLF_set_def)
paulson@13115
   451
done
paulson@13115
   452
paulson@13115
   453
declare (in CLF) f_cl [simp]
paulson@13115
   454
paulson@13115
   455
wenzelm@61384
   456
lemma (in CLF) f_in_funcset: "f \<in> A \<rightarrow> A"
paulson@13115
   457
by (simp add: A_def)
paulson@13115
   458
paulson@13115
   459
lemma (in CLF) monotone_f: "monotone f A r"
paulson@13115
   460
by (simp add: A_def r_def)
paulson@13115
   461
haftmann@27681
   462
lemma (in CLF) CLF_dual: "(dual cl, f) \<in> CLF_set"
haftmann@27681
   463
apply (simp add: CLF_set_def  CL_dualCL monotone_dual)
paulson@13115
   464
apply (simp add: dualA_iff)
paulson@13115
   465
done
paulson@13115
   466
haftmann@27681
   467
lemma (in CLF) dual:
haftmann@27681
   468
  "CLF (dual cl) f"
haftmann@27681
   469
apply (rule CLF.intro)
haftmann@27681
   470
apply (rule CLF_dual)
haftmann@27681
   471
done
haftmann@27681
   472
wenzelm@13383
   473
wenzelm@61343
   474
subsection \<open>fixed points\<close>
wenzelm@13383
   475
paulson@17841
   476
lemma fix_subset: "fix f A \<subseteq> A"
paulson@13115
   477
by (simp add: fix_def, fast)
paulson@13115
   478
paulson@13115
   479
lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
paulson@13115
   480
by (simp add: fix_def)
paulson@13115
   481
paulson@13115
   482
lemma fixf_subset:
paulson@17841
   483
     "[| A \<subseteq> B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
paulson@17841
   484
by (simp add: fix_def, auto)
paulson@13115
   485
wenzelm@13383
   486
wenzelm@61343
   487
subsection \<open>lemmas for Tarski, lub\<close>
paulson@13115
   488
lemma (in CLF) lubH_le_flubH:
paulson@13115
   489
     "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
paulson@13115
   490
apply (rule lub_least, fast)
paulson@13115
   491
apply (rule f_in_funcset [THEN funcset_mem])
paulson@13115
   492
apply (rule lub_in_lattice, fast)
wenzelm@61933
   493
\<comment> \<open>\<open>\<forall>x:H. (x, f (lub H r)) \<in> r\<close>\<close>
paulson@13115
   494
apply (rule ballI)
paulson@13115
   495
apply (rule transE)
wenzelm@61933
   496
\<comment> \<open>instantiates \<open>(x, ???z) \<in> order cl to (x, f x)\<close>,\<close>
wenzelm@61933
   497
\<comment> \<open>because of the def of \<open>H\<close>\<close>
paulson@13115
   498
apply fast
wenzelm@61933
   499
\<comment> \<open>so it remains to show \<open>(f x, f (lub H cl)) \<in> r\<close>\<close>
paulson@13115
   500
apply (rule_tac f = "f" in monotoneE)
paulson@13115
   501
apply (rule monotone_f, fast)
paulson@13115
   502
apply (rule lub_in_lattice, fast)
paulson@13115
   503
apply (rule lub_upper, fast)
paulson@13115
   504
apply assumption
paulson@13115
   505
done
paulson@13115
   506
paulson@13115
   507
lemma (in CLF) flubH_le_lubH:
paulson@13115
   508
     "[|  H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
paulson@13115
   509
apply (rule lub_upper, fast)
paulson@13115
   510
apply (rule_tac t = "H" in ssubst, assumption)
paulson@13115
   511
apply (rule CollectI)
paulson@13115
   512
apply (rule conjI)
paulson@13115
   513
apply (rule_tac [2] f_in_funcset [THEN funcset_mem])
paulson@13115
   514
apply (rule_tac [2] lub_in_lattice)
paulson@13115
   515
prefer 2 apply fast
paulson@13115
   516
apply (rule_tac f = "f" in monotoneE)
paulson@13115
   517
apply (rule monotone_f)
wenzelm@13383
   518
  apply (blast intro: lub_in_lattice)
wenzelm@13383
   519
 apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])
paulson@13115
   520
apply (simp add: lubH_le_flubH)
paulson@13115
   521
done
paulson@13115
   522
paulson@13115
   523
lemma (in CLF) lubH_is_fixp:
paulson@13115
   524
     "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
paulson@13115
   525
apply (simp add: fix_def)
paulson@13115
   526
apply (rule conjI)
paulson@13115
   527
apply (rule lub_in_lattice, fast)
paulson@13115
   528
apply (rule antisymE)
paulson@13115
   529
apply (simp add: flubH_le_lubH)
paulson@13115
   530
apply (simp add: lubH_le_flubH)
paulson@13115
   531
done
paulson@13115
   532
paulson@13115
   533
lemma (in CLF) fix_in_H:
paulson@13115
   534
     "[| H = {x. (x, f x) \<in> r & x \<in> A};  x \<in> P |] ==> x \<in> H"
nipkow@30198
   535
by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl_on
wenzelm@13383
   536
                    fix_subset [of f A, THEN subsetD])
paulson@13115
   537
paulson@13115
   538
lemma (in CLF) fixf_le_lubH:
paulson@13115
   539
     "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
paulson@13115
   540
apply (rule ballI)
paulson@13115
   541
apply (rule lub_upper, fast)
paulson@13115
   542
apply (rule fix_in_H)
wenzelm@13383
   543
apply (simp_all add: P_def)
paulson@13115
   544
done
paulson@13115
   545
paulson@13115
   546
lemma (in CLF) lubH_least_fixf:
wenzelm@13383
   547
     "H = {x. (x, f x) \<in> r & x \<in> A}
paulson@13115
   548
      ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
paulson@13115
   549
apply (rule allI)
paulson@13115
   550
apply (rule impI)
paulson@13115
   551
apply (erule bspec)
paulson@13115
   552
apply (rule lubH_is_fixp, assumption)
paulson@13115
   553
done
paulson@13115
   554
wenzelm@61343
   555
subsection \<open>Tarski fixpoint theorem 1, first part\<close>
paulson@13115
   556
lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
paulson@13115
   557
apply (rule sym)
wenzelm@13383
   558
apply (simp add: P_def)
paulson@13115
   559
apply (rule lubI)
paulson@13115
   560
apply (rule fix_subset)
paulson@13115
   561
apply (rule lub_in_lattice, fast)
paulson@13115
   562
apply (simp add: fixf_le_lubH)
paulson@13115
   563
apply (simp add: lubH_least_fixf)
paulson@13115
   564
done
paulson@13115
   565
paulson@13115
   566
lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
wenzelm@61933
   567
  \<comment> \<open>Tarski for glb\<close>
paulson@13115
   568
apply (simp add: glb_dual_lub P_def A_def r_def)
paulson@13115
   569
apply (rule dualA_iff [THEN subst])
wenzelm@21232
   570
apply (rule CLF.lubH_is_fixp)
haftmann@27681
   571
apply (rule dual)
paulson@13115
   572
apply (simp add: dualr_iff dualA_iff)
paulson@13115
   573
done
paulson@13115
   574
paulson@13115
   575
lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
paulson@13115
   576
apply (simp add: glb_dual_lub P_def A_def r_def)
paulson@13115
   577
apply (rule dualA_iff [THEN subst])
haftmann@27681
   578
apply (simp add: CLF.T_thm_1_lub [of _ f, OF dual]
paulson@13115
   579
                 dualPO CL_dualCL CLF_dual dualr_iff)
paulson@13115
   580
done
paulson@13115
   581
wenzelm@61343
   582
subsection \<open>interval\<close>
wenzelm@13383
   583
paulson@13115
   584
lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
nipkow@30198
   585
apply (insert CO_refl_on)
nipkow@30198
   586
apply (simp add: refl_on_def, blast)
paulson@13115
   587
done
paulson@13115
   588
paulson@17841
   589
lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b \<subseteq> A"
paulson@13115
   590
apply (simp add: interval_def)
paulson@13115
   591
apply (blast intro: rel_imp_elem)
paulson@13115
   592
done
paulson@13115
   593
paulson@13115
   594
lemma (in CLF) intervalI:
paulson@13115
   595
     "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
paulson@17841
   596
by (simp add: interval_def)
paulson@13115
   597
paulson@13115
   598
lemma (in CLF) interval_lemma1:
paulson@17841
   599
     "[| S \<subseteq> interval r a b; x \<in> S |] ==> (a, x) \<in> r"
paulson@17841
   600
by (unfold interval_def, fast)
paulson@13115
   601
paulson@13115
   602
lemma (in CLF) interval_lemma2:
paulson@17841
   603
     "[| S \<subseteq> interval r a b; x \<in> S |] ==> (x, b) \<in> r"
paulson@17841
   604
by (unfold interval_def, fast)
paulson@13115
   605
paulson@13115
   606
lemma (in CLF) a_less_lub:
paulson@17841
   607
     "[| S \<subseteq> A; S \<noteq> {};
paulson@13115
   608
         \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
paulson@18705
   609
by (blast intro: transE)
paulson@13115
   610
paulson@13115
   611
lemma (in CLF) glb_less_b:
paulson@17841
   612
     "[| S \<subseteq> A; S \<noteq> {};
paulson@13115
   613
         \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
paulson@18705
   614
by (blast intro: transE)
paulson@13115
   615
paulson@13115
   616
lemma (in CLF) S_intv_cl:
paulson@17841
   617
     "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b |]==> S \<subseteq> A"
paulson@13115
   618
by (simp add: subset_trans [OF _ interval_subset])
paulson@13115
   619
paulson@13115
   620
lemma (in CLF) L_in_interval:
paulson@17841
   621
     "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b;
paulson@13115
   622
         S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
paulson@13115
   623
apply (rule intervalI)
paulson@13115
   624
apply (rule a_less_lub)
paulson@13115
   625
prefer 2 apply assumption
paulson@13115
   626
apply (simp add: S_intv_cl)
paulson@13115
   627
apply (rule ballI)
paulson@13115
   628
apply (simp add: interval_lemma1)
paulson@13115
   629
apply (simp add: isLub_upper)
wenzelm@61933
   630
\<comment> \<open>\<open>(L, b) \<in> r\<close>\<close>
paulson@13115
   631
apply (simp add: isLub_least interval_lemma2)
paulson@13115
   632
done
paulson@13115
   633
paulson@13115
   634
lemma (in CLF) G_in_interval:
paulson@17841
   635
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> interval r a b; isGlb S cl G;
paulson@13115
   636
         S \<noteq> {} |] ==> G \<in> interval r a b"
paulson@13115
   637
apply (simp add: interval_dual)
haftmann@27681
   638
apply (simp add: CLF.L_in_interval [of _ f, OF dual]
haftmann@27681
   639
                 dualA_iff A_def isGlb_dual_isLub)
paulson@13115
   640
done
paulson@13115
   641
paulson@13115
   642
lemma (in CLF) intervalPO:
wenzelm@13383
   643
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
paulson@13115
   644
      ==> (| pset = interval r a b, order = induced (interval r a b) r |)
paulson@13115
   645
          \<in> PartialOrder"
paulson@13115
   646
apply (rule po_subset_po)
paulson@13115
   647
apply (simp add: interval_subset)
paulson@13115
   648
done
paulson@13115
   649
paulson@13115
   650
lemma (in CLF) intv_CL_lub:
wenzelm@13383
   651
 "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
paulson@17841
   652
  ==> \<forall>S. S \<subseteq> interval r a b -->
wenzelm@13383
   653
          (\<exists>L. isLub S (| pset = interval r a b,
paulson@13115
   654
                          order = induced (interval r a b) r |)  L)"
paulson@13115
   655
apply (intro strip)
paulson@13115
   656
apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
paulson@13115
   657
prefer 2 apply assumption
paulson@13115
   658
apply assumption
paulson@13115
   659
apply (erule exE)
wenzelm@61933
   660
\<comment> \<open>define the lub for the interval as\<close>
paulson@13115
   661
apply (rule_tac x = "if S = {} then a else L" in exI)
nipkow@62390
   662
apply (simp (no_asm_simp) add: isLub_def split del: if_split)
wenzelm@13383
   663
apply (intro impI conjI)
wenzelm@61933
   664
\<comment> \<open>\<open>(if S = {} then a else L) \<in> interval r a b\<close>\<close>
paulson@13115
   665
apply (simp add: CL_imp_PO L_in_interval)
paulson@13115
   666
apply (simp add: left_in_interval)
wenzelm@61933
   667
\<comment> \<open>lub prop 1\<close>
paulson@13115
   668
apply (case_tac "S = {}")
wenzelm@61933
   669
\<comment> \<open>\<open>S = {}, y \<in> S = False => everything\<close>\<close>
paulson@13115
   670
apply fast
wenzelm@61933
   671
\<comment> \<open>\<open>S \<noteq> {}\<close>\<close>
paulson@13115
   672
apply simp
wenzelm@61933
   673
\<comment> \<open>\<open>\<forall>y:S. (y, L) \<in> induced (interval r a b) r\<close>\<close>
paulson@13115
   674
apply (rule ballI)
paulson@13115
   675
apply (simp add: induced_def  L_in_interval)
paulson@13115
   676
apply (rule conjI)
paulson@13115
   677
apply (rule subsetD)
paulson@13115
   678
apply (simp add: S_intv_cl, assumption)
paulson@13115
   679
apply (simp add: isLub_upper)
wenzelm@61933
   680
\<comment> \<open>\<open>\<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\<close>\<close>
paulson@13115
   681
apply (rule ballI)
paulson@13115
   682
apply (rule impI)
paulson@13115
   683
apply (case_tac "S = {}")
wenzelm@61933
   684
\<comment> \<open>\<open>S = {}\<close>\<close>
paulson@13115
   685
apply simp
paulson@13115
   686
apply (simp add: induced_def  interval_def)
paulson@13115
   687
apply (rule conjI)
paulson@18705
   688
apply (rule reflE, assumption)
paulson@13115
   689
apply (rule interval_not_empty)
paulson@13115
   690
apply (simp add: interval_def)
wenzelm@61933
   691
\<comment> \<open>\<open>S \<noteq> {}\<close>\<close>
paulson@13115
   692
apply simp
paulson@13115
   693
apply (simp add: induced_def  L_in_interval)
paulson@13115
   694
apply (rule isLub_least, assumption)
paulson@13115
   695
apply (rule subsetD)
paulson@13115
   696
prefer 2 apply assumption
paulson@13115
   697
apply (simp add: S_intv_cl, fast)
paulson@13115
   698
done
paulson@13115
   699
paulson@13115
   700
lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
paulson@13115
   701
paulson@13115
   702
lemma (in CLF) interval_is_sublattice:
wenzelm@13383
   703
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
paulson@18750
   704
        ==> interval r a b <<= cl"
paulson@13115
   705
apply (rule sublatticeI)
paulson@13115
   706
apply (simp add: interval_subset)
paulson@13115
   707
apply (rule CompleteLatticeI)
paulson@13115
   708
apply (simp add: intervalPO)
paulson@13115
   709
 apply (simp add: intv_CL_lub)
paulson@13115
   710
apply (simp add: intv_CL_glb)
paulson@13115
   711
done
paulson@13115
   712
wenzelm@13383
   713
lemmas (in CLF) interv_is_compl_latt =
paulson@13115
   714
    interval_is_sublattice [THEN sublattice_imp_CL]
paulson@13115
   715
wenzelm@13383
   716
wenzelm@61343
   717
subsection \<open>Top and Bottom\<close>
paulson@13115
   718
lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
paulson@13115
   719
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
paulson@13115
   720
paulson@13115
   721
lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
paulson@13115
   722
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
paulson@13115
   723
paulson@13115
   724
lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
paulson@13115
   725
apply (simp add: Bot_def least_def)
paulson@17841
   726
apply (rule_tac a="glb A cl" in someI2)
paulson@17841
   727
apply (simp_all add: glb_in_lattice glb_lower 
paulson@17841
   728
                     r_def [symmetric] A_def [symmetric])
paulson@13115
   729
done
paulson@13115
   730
paulson@13115
   731
lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
paulson@13115
   732
apply (simp add: Top_dual_Bot A_def)
wenzelm@13383
   733
apply (rule dualA_iff [THEN subst])
haftmann@27681
   734
apply (rule CLF.Bot_in_lattice [OF dual])
paulson@13115
   735
done
paulson@13115
   736
paulson@13115
   737
lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
paulson@13115
   738
apply (simp add: Top_def greatest_def)
paulson@17841
   739
apply (rule_tac a="lub A cl" in someI2)
paulson@13115
   740
apply (rule someI2)
paulson@17841
   741
apply (simp_all add: lub_in_lattice lub_upper 
paulson@17841
   742
                     r_def [symmetric] A_def [symmetric])
paulson@13115
   743
done
paulson@13115
   744
paulson@13115
   745
lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
paulson@13115
   746
apply (simp add: Bot_dual_Top r_def)
paulson@13115
   747
apply (rule dualr_iff [THEN subst])
haftmann@27681
   748
apply (rule CLF.Top_prop [OF dual])
haftmann@27681
   749
apply (simp add: dualA_iff A_def)
paulson@13115
   750
done
paulson@13115
   751
paulson@13115
   752
lemma (in CLF) Top_intv_not_empty: "x \<in> A  ==> interval r x (Top cl) \<noteq> {}"
paulson@13115
   753
apply (rule notI)
paulson@13115
   754
apply (drule_tac a = "Top cl" in equals0D)
paulson@13115
   755
apply (simp add: interval_def)
nipkow@30198
   756
apply (simp add: refl_on_def Top_in_lattice Top_prop)
paulson@13115
   757
done
paulson@13115
   758
paulson@13115
   759
lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
paulson@13115
   760
apply (simp add: Bot_dual_Top)
paulson@13115
   761
apply (subst interval_dual)
paulson@13115
   762
prefer 2 apply assumption
paulson@13115
   763
apply (simp add: A_def)
paulson@13115
   764
apply (rule dualA_iff [THEN subst])
haftmann@27681
   765
apply (rule CLF.Top_in_lattice [OF dual])
haftmann@27681
   766
apply (rule CLF.Top_intv_not_empty [OF dual])
haftmann@27681
   767
apply (simp add: dualA_iff A_def)
paulson@13115
   768
done
paulson@13115
   769
wenzelm@61343
   770
subsection \<open>fixed points form a partial order\<close>
wenzelm@13383
   771
paulson@13115
   772
lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
paulson@13115
   773
by (simp add: P_def fix_subset po_subset_po)
paulson@13115
   774
paulson@17841
   775
lemma (in Tarski) Y_subset_A: "Y \<subseteq> A"
paulson@13115
   776
apply (rule subset_trans [OF _ fix_subset])
paulson@13115
   777
apply (rule Y_ss [simplified P_def])
paulson@13115
   778
done
paulson@13115
   779
paulson@13115
   780
lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
paulson@18750
   781
  by (rule Y_subset_A [THEN lub_in_lattice])
paulson@13115
   782
paulson@13115
   783
lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
paulson@13115
   784
apply (rule lub_least)
paulson@13115
   785
apply (rule Y_subset_A)
paulson@13115
   786
apply (rule f_in_funcset [THEN funcset_mem])
paulson@13115
   787
apply (rule lubY_in_A)
wenzelm@61933
   788
\<comment> \<open>\<open>Y \<subseteq> P ==> f x = x\<close>\<close>
paulson@13115
   789
apply (rule ballI)
paulson@13115
   790
apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
paulson@13115
   791
apply (erule Y_ss [simplified P_def, THEN subsetD])
wenzelm@61933
   792
\<comment> \<open>\<open>reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r\<close> by monotonicity\<close>
paulson@13115
   793
apply (rule_tac f = "f" in monotoneE)
paulson@13115
   794
apply (rule monotone_f)
paulson@13115
   795
apply (simp add: Y_subset_A [THEN subsetD])
paulson@13115
   796
apply (rule lubY_in_A)
paulson@13115
   797
apply (simp add: lub_upper Y_subset_A)
paulson@13115
   798
done
paulson@13115
   799
paulson@17841
   800
lemma (in Tarski) intY1_subset: "intY1 \<subseteq> A"
paulson@13115
   801
apply (unfold intY1_def)
paulson@13115
   802
apply (rule interval_subset)
paulson@13115
   803
apply (rule lubY_in_A)
paulson@13115
   804
apply (rule Top_in_lattice)
paulson@13115
   805
done
paulson@13115
   806
paulson@13115
   807
lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
paulson@13115
   808
paulson@13115
   809
lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
paulson@13115
   810
apply (simp add: intY1_def  interval_def)
paulson@13115
   811
apply (rule conjI)
paulson@13115
   812
apply (rule transE)
paulson@13115
   813
apply (rule lubY_le_flubY)
wenzelm@61933
   814
\<comment> \<open>\<open>(f (lub Y cl), f x) \<in> r\<close>\<close>
paulson@13115
   815
apply (rule_tac f=f in monotoneE)
paulson@13115
   816
apply (rule monotone_f)
paulson@13115
   817
apply (rule lubY_in_A)
paulson@13115
   818
apply (simp add: intY1_def interval_def  intY1_elem)
paulson@13115
   819
apply (simp add: intY1_def  interval_def)
wenzelm@61933
   820
\<comment> \<open>\<open>(f x, Top cl) \<in> r\<close>\<close>
paulson@13115
   821
apply (rule Top_prop)
paulson@13115
   822
apply (rule f_in_funcset [THEN funcset_mem])
paulson@13115
   823
apply (simp add: intY1_def interval_def  intY1_elem)
paulson@13115
   824
done
paulson@13115
   825
paulson@13115
   826
lemma (in Tarski) intY1_mono:
paulson@13115
   827
     "monotone (%x: intY1. f x) intY1 (induced intY1 r)"
paulson@13115
   828
apply (auto simp add: monotone_def induced_def intY1_f_closed)
paulson@13115
   829
apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
paulson@13115
   830
done
paulson@13115
   831
wenzelm@13383
   832
lemma (in Tarski) intY1_is_cl:
paulson@13115
   833
    "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
paulson@13115
   834
apply (unfold intY1_def)
paulson@13115
   835
apply (rule interv_is_compl_latt)
paulson@13115
   836
apply (rule lubY_in_A)
paulson@13115
   837
apply (rule Top_in_lattice)
paulson@13115
   838
apply (rule Top_intv_not_empty)
paulson@13115
   839
apply (rule lubY_in_A)
paulson@13115
   840
done
paulson@13115
   841
paulson@13115
   842
lemma (in Tarski) v_in_P: "v \<in> P"
paulson@13115
   843
apply (unfold P_def)
paulson@13115
   844
apply (rule_tac A = "intY1" in fixf_subset)
paulson@13115
   845
apply (rule intY1_subset)
haftmann@27681
   846
unfolding v_def
haftmann@27681
   847
apply (rule CLF.glbH_is_fixp [OF CLF.intro, unfolded CLF_set_def, of "\<lparr>pset = intY1, order = induced intY1 r\<rparr>", simplified])
haftmann@27681
   848
apply auto
haftmann@27681
   849
apply (rule intY1_is_cl)
nipkow@31754
   850
apply (erule intY1_f_closed)
haftmann@27681
   851
apply (rule intY1_mono)
paulson@13115
   852
done
paulson@13115
   853
wenzelm@13383
   854
lemma (in Tarski) z_in_interval:
paulson@13115
   855
     "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
paulson@13115
   856
apply (unfold intY1_def P_def)
paulson@13115
   857
apply (rule intervalI)
wenzelm@13383
   858
prefer 2
paulson@13115
   859
 apply (erule fix_subset [THEN subsetD, THEN Top_prop])
paulson@13115
   860
apply (rule lub_least)
paulson@13115
   861
apply (rule Y_subset_A)
paulson@13115
   862
apply (fast elim!: fix_subset [THEN subsetD])
paulson@13115
   863
apply (simp add: induced_def)
paulson@13115
   864
done
paulson@13115
   865
wenzelm@13383
   866
lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
paulson@13115
   867
      ==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"
paulson@13115
   868
apply (simp add: induced_def  intY1_f_closed z_in_interval P_def)
wenzelm@13383
   869
apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD]
paulson@18705
   870
                 reflE)
paulson@13115
   871
done
paulson@13115
   872
paulson@13115
   873
lemma (in Tarski) tarski_full_lemma:
paulson@13115
   874
     "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
paulson@13115
   875
apply (rule_tac x = "v" in exI)
paulson@13115
   876
apply (simp add: isLub_def)
wenzelm@61933
   877
\<comment> \<open>\<open>v \<in> P\<close>\<close>
paulson@13115
   878
apply (simp add: v_in_P)
paulson@13115
   879
apply (rule conjI)
wenzelm@61933
   880
\<comment> \<open>\<open>v\<close> is lub\<close>
wenzelm@61933
   881
\<comment> \<open>\<open>1. \<forall>y:Y. (y, v) \<in> induced P r\<close>\<close>
paulson@13115
   882
apply (rule ballI)
paulson@13115
   883
apply (simp add: induced_def subsetD v_in_P)
paulson@13115
   884
apply (rule conjI)
paulson@13115
   885
apply (erule Y_ss [THEN subsetD])
paulson@13115
   886
apply (rule_tac b = "lub Y cl" in transE)
paulson@13115
   887
apply (rule lub_upper)
paulson@13115
   888
apply (rule Y_subset_A, assumption)
paulson@13115
   889
apply (rule_tac b = "Top cl" in interval_imp_mem)
paulson@13115
   890
apply (simp add: v_def)
paulson@13115
   891
apply (fold intY1_def)
haftmann@27681
   892
apply (rule CL.glb_in_lattice [OF CL.intro [OF intY1_is_cl], simplified])
haftmann@27681
   893
apply auto
paulson@13115
   894
apply (rule indI)
paulson@13115
   895
  prefer 3 apply assumption
paulson@13115
   896
 prefer 2 apply (simp add: v_in_P)
paulson@13115
   897
apply (unfold v_def)
paulson@13115
   898
apply (rule indE)
paulson@13115
   899
apply (rule_tac [2] intY1_subset)
haftmann@27681
   900
apply (rule CL.glb_lower [OF CL.intro [OF intY1_is_cl], simplified])
wenzelm@13383
   901
  apply (simp add: CL_imp_PO intY1_is_cl)
paulson@13115
   902
 apply force
paulson@13115
   903
apply (simp add: induced_def intY1_f_closed z_in_interval)
paulson@18705
   904
apply (simp add: P_def fix_imp_eq [of _ f A] reflE
paulson@18705
   905
                 fix_subset [of f A, THEN subsetD])
paulson@13115
   906
done
paulson@13115
   907
paulson@13115
   908
lemma CompleteLatticeI_simp:
wenzelm@13383
   909
     "[| (| pset = A, order = r |) \<in> PartialOrder;
paulson@17841
   910
         \<forall>S. S \<subseteq> A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
paulson@13115
   911
    ==> (| pset = A, order = r |) \<in> CompleteLattice"
paulson@13115
   912
by (simp add: CompleteLatticeI Rdual)
paulson@13115
   913
paulson@13115
   914
theorem (in CLF) Tarski_full:
paulson@13115
   915
     "(| pset = P, order = induced P r|) \<in> CompleteLattice"
paulson@13115
   916
apply (rule CompleteLatticeI_simp)
paulson@13115
   917
apply (rule fixf_po, clarify)
wenzelm@13383
   918
apply (simp add: P_def A_def r_def)
haftmann@27681
   919
apply (rule Tarski.tarski_full_lemma [OF Tarski.intro [OF _ Tarski_axioms.intro]])
haftmann@28823
   920
proof - show "CLF cl f" .. qed
wenzelm@7112
   921
wenzelm@7112
   922
end