src/HOL/ex/Tarski.thy
author paulson
Thu Sep 26 10:51:29 2002 +0200 (2002-09-26)
changeset 13585 db4005b40cc6
parent 13383 041d78bf9403
child 14569 78b75a9eec01
permissions -rw-r--r--
Converted Fun to Isar style.
Moved Pi, funcset, restrict from Fun.thy to Library/FuncSet.thy.
Renamed constant "Fun.op o" to "Fun.comp"
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(*  Title:      HOL/ex/Tarski.thy
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    ID:         $Id$
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    Author:     Florian Kammüller, Cambridge University Computer Laboratory
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*)
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header {* The Full Theorem of Tarski *}
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theory Tarski = Main + FuncSet:
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text {*
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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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*}
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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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constdefs
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  monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
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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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  least :: "['a => bool, 'a potype] => 'a"
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  "least P po == @ 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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  greatest :: "['a => bool, 'a potype] => 'a"
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  "greatest P po == @ 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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  lub  :: "['a set, 'a potype] => 'a"
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  "lub S po == least (%x. \<forall>y\<in>S. (y,x): order po) po"
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  glb  :: "['a set, 'a potype] => 'a"
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  "glb S po == greatest (%x. \<forall>y\<in>S. (x,y): order po) po"
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  isLub :: "['a set, 'a potype, 'a] => bool"
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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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  isGlb :: "['a set, 'a potype, 'a] => bool"
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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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  "fix"    :: "[('a => 'a), 'a set] => 'a set"
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  "fix f A  == {x. x: A & f x = x}"
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  interval :: "[('a*'a) set,'a, 'a ] => 'a set"
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  "interval r a b == {x. (a,x): r & (x,b): r}"
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constdefs
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  Bot :: "'a potype => 'a"
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  "Bot po == least (%x. True) po"
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  Top :: "'a potype => 'a"
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  "Top po == greatest (%x. True) po"
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  PartialOrder :: "('a potype) set"
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  "PartialOrder == {P. refl (pset P) (order P) & antisym (order P) &
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                       trans (order P)}"
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  CompleteLattice :: "('a potype) set"
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  "CompleteLattice == {cl. cl: PartialOrder &
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                        (\<forall>S. S <= pset cl --> (\<exists>L. isLub S cl L)) &
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                        (\<forall>S. S <= pset cl --> (\<exists>G. isGlb S cl G))}"
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  CLF :: "('a potype * ('a => 'a)) set"
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  "CLF == SIGMA cl: CompleteLattice.
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            {f. f: pset cl -> pset cl & monotone f (pset cl) (order cl)}"
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  induced :: "['a set, ('a * 'a) set] => ('a *'a)set"
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  "induced A r == {(a,b). a : A & b: A & (a,b): r}"
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constdefs
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  sublattice :: "('a potype * 'a set)set"
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  "sublattice ==
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      SIGMA cl: CompleteLattice.
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          {S. S <= pset cl &
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           (| pset = S, order = induced S (order cl) |): CompleteLattice }"
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syntax
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  "@SL"  :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
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translations
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  "S <<= cl" == "S : sublattice `` {cl}"
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constdefs
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  dual :: "'a potype => 'a potype"
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  "dual po == (| pset = pset po, order = converse (order po) |)"
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locale (open) PO =
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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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  assumes cl_po:  "cl : PartialOrder"
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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 (open) CL = PO +
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  assumes cl_co:  "cl : CompleteLattice"
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locale (open) CLF = CL +
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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" (*was the equivalent "f : CLF``{cl}"*)
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  defines P_def: "P == fix f A"
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locale (open) 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 <= 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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subsubsection {* Partial Order *}
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lemma (in PO) PO_imp_refl: "refl 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: "antisym 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_trans: "trans 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) reflE: "[| refl A r; 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_def)
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done
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lemma (in PO) antisymE: "[| antisym r; (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)
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done
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lemma (in PO) transE: "[| trans r; (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)
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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 <= 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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-- {* refl *}
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apply (simp add: refl_def induced_def)
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apply (blast intro: PO_imp_refl [THEN reflE])
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-- {* antisym *}
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apply (simp add: antisym_def induced_def)
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apply (blast intro: PO_imp_sym [THEN antisymE])
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-- {* trans *}
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apply (simp add: trans_def induced_def)
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apply (blast intro: PO_imp_trans [THEN transE])
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done
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lemma (in PO) indE: "[| (x, y) \<in> induced S r; S <= 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 <= 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_def)
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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_def)
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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_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 <= A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
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      ==> \<forall>S. (S <= 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_def)
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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_def)
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lemma CL_subset_PO: "CompleteLattice <= 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 Tarski.PO_imp_refl, simp]
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declare CL_imp_PO [THEN Tarski.PO_imp_sym, simp]
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declare CL_imp_PO [THEN Tarski.PO_imp_trans, simp]
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lemma (in CL) CO_refl: "refl A r"
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by (rule PO_imp_refl)
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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 <= pset po --> (\<exists>L. isLub S po L));
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         (\<forall>S. S <= 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) interval_not_empty:
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     "[| trans r; interval r a b \<noteq> {} |] ==> (a, b) \<in> r"
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apply (simp add: interval_def)
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apply (unfold trans_def, blast)
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done
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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:
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     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b"
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apply (simp (no_asm_simp) add: interval_def)
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apply (simp add: PO_imp_trans interval_not_empty)
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apply (simp add: PO_imp_refl [THEN reflE])
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done
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lemma (in PO) right_in_interval:
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     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b"
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apply (simp (no_asm_simp) add: interval_def)
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apply (simp add: PO_imp_trans interval_not_empty)
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apply (simp add: PO_imp_refl [THEN reflE])
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done
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subsubsection {* sublattice *}
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lemma (in PO) sublattice_imp_CL:
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     "S <<= cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
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by (simp add: sublattice_def CompleteLattice_def A_def r_def)
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lemma (in CL) sublatticeI:
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     "[| S <= A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
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      ==> S <<= cl"
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by (simp add: sublattice_def A_def r_def)
paulson@13115
   313
wenzelm@13383
   314
wenzelm@13383
   315
subsubsection {* lub *}
wenzelm@13383
   316
paulson@13115
   317
lemma (in CL) lub_unique: "[| S <= A; isLub S cl x; isLub S cl L|] ==> x = L"
paulson@13115
   318
apply (rule antisymE)
paulson@13115
   319
apply (rule CO_antisym)
paulson@13115
   320
apply (auto simp add: isLub_def r_def)
paulson@13115
   321
done
paulson@13115
   322
paulson@13115
   323
lemma (in CL) lub_upper: "[|S <= A; x \<in> S|] ==> (x, lub S cl) \<in> r"
paulson@13115
   324
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
paulson@13115
   325
apply (unfold lub_def least_def)
paulson@13115
   326
apply (rule some_equality [THEN ssubst])
paulson@13115
   327
  apply (simp add: isLub_def)
wenzelm@13383
   328
 apply (simp add: lub_unique A_def isLub_def)
paulson@13115
   329
apply (simp add: isLub_def r_def)
paulson@13115
   330
done
paulson@13115
   331
paulson@13115
   332
lemma (in CL) lub_least:
paulson@13115
   333
     "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
paulson@13115
   334
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
paulson@13115
   335
apply (unfold lub_def least_def)
paulson@13115
   336
apply (rule_tac s=x in some_equality [THEN ssubst])
paulson@13115
   337
  apply (simp add: isLub_def)
wenzelm@13383
   338
 apply (simp add: lub_unique A_def isLub_def)
paulson@13115
   339
apply (simp add: isLub_def r_def A_def)
paulson@13115
   340
done
paulson@13115
   341
paulson@13115
   342
lemma (in CL) lub_in_lattice: "S <= A ==> lub S cl \<in> A"
paulson@13115
   343
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
paulson@13115
   344
apply (unfold lub_def least_def)
paulson@13115
   345
apply (subst some_equality)
paulson@13115
   346
apply (simp add: isLub_def)
paulson@13115
   347
prefer 2 apply (simp add: isLub_def A_def)
wenzelm@13383
   348
apply (simp add: lub_unique A_def isLub_def)
paulson@13115
   349
done
paulson@13115
   350
paulson@13115
   351
lemma (in CL) lubI:
wenzelm@13383
   352
     "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
paulson@13115
   353
         \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
paulson@13115
   354
apply (rule lub_unique, assumption)
paulson@13115
   355
apply (simp add: isLub_def A_def r_def)
paulson@13115
   356
apply (unfold isLub_def)
paulson@13115
   357
apply (rule conjI)
paulson@13115
   358
apply (fold A_def r_def)
paulson@13115
   359
apply (rule lub_in_lattice, assumption)
paulson@13115
   360
apply (simp add: lub_upper lub_least)
paulson@13115
   361
done
paulson@13115
   362
paulson@13115
   363
lemma (in CL) lubIa: "[| S <= A; isLub S cl L |] ==> L = lub S cl"
paulson@13115
   364
by (simp add: lubI isLub_def A_def r_def)
paulson@13115
   365
paulson@13115
   366
lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
paulson@13115
   367
by (simp add: isLub_def  A_def)
paulson@13115
   368
paulson@13115
   369
lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
paulson@13115
   370
by (simp add: isLub_def r_def)
paulson@13115
   371
paulson@13115
   372
lemma (in CL) isLub_least:
paulson@13115
   373
     "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
paulson@13115
   374
by (simp add: isLub_def A_def r_def)
paulson@13115
   375
paulson@13115
   376
lemma (in CL) isLubI:
wenzelm@13383
   377
     "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
paulson@13115
   378
         (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
paulson@13115
   379
by (simp add: isLub_def A_def r_def)
paulson@13115
   380
wenzelm@13383
   381
wenzelm@13383
   382
subsubsection {* glb *}
wenzelm@13383
   383
paulson@13115
   384
lemma (in CL) glb_in_lattice: "S <= A ==> glb S cl \<in> A"
paulson@13115
   385
apply (subst glb_dual_lub)
paulson@13115
   386
apply (simp add: A_def)
paulson@13115
   387
apply (rule dualA_iff [THEN subst])
paulson@13115
   388
apply (rule Tarski.lub_in_lattice)
wenzelm@13383
   389
apply (rule dualPO)
paulson@13115
   390
apply (rule CL_dualCL)
paulson@13115
   391
apply (simp add: dualA_iff)
paulson@13115
   392
done
paulson@13115
   393
paulson@13115
   394
lemma (in CL) glb_lower: "[|S <= A; x \<in> S|] ==> (glb S cl, x) \<in> r"
paulson@13115
   395
apply (subst glb_dual_lub)
paulson@13115
   396
apply (simp add: r_def)
paulson@13115
   397
apply (rule dualr_iff [THEN subst])
paulson@13115
   398
apply (rule Tarski.lub_upper [rule_format])
wenzelm@13383
   399
apply (rule dualPO)
paulson@13115
   400
apply (rule CL_dualCL)
paulson@13115
   401
apply (simp add: dualA_iff A_def, assumption)
paulson@13115
   402
done
paulson@13115
   403
wenzelm@13383
   404
text {*
wenzelm@13383
   405
  Reduce the sublattice property by using substructural properties;
wenzelm@13383
   406
  abandoned see @{text "Tarski_4.ML"}.
wenzelm@13383
   407
*}
paulson@13115
   408
paulson@13115
   409
lemma (in CLF) [simp]:
paulson@13585
   410
    "f: pset cl -> pset cl & monotone f (pset cl) (order cl)"
wenzelm@13383
   411
apply (insert f_cl)
wenzelm@13383
   412
apply (simp add: CLF_def)
paulson@13115
   413
done
paulson@13115
   414
paulson@13115
   415
declare (in CLF) f_cl [simp]
paulson@13115
   416
paulson@13115
   417
paulson@13585
   418
lemma (in CLF) f_in_funcset: "f \<in> A -> A"
paulson@13115
   419
by (simp add: A_def)
paulson@13115
   420
paulson@13115
   421
lemma (in CLF) monotone_f: "monotone f A r"
paulson@13115
   422
by (simp add: A_def r_def)
paulson@13115
   423
paulson@13115
   424
lemma (in CLF) CLF_dual: "(cl,f) \<in> CLF ==> (dual cl, f) \<in> CLF"
paulson@13115
   425
apply (simp add: CLF_def  CL_dualCL monotone_dual)
paulson@13115
   426
apply (simp add: dualA_iff)
paulson@13115
   427
done
paulson@13115
   428
wenzelm@13383
   429
wenzelm@13383
   430
subsubsection {* fixed points *}
wenzelm@13383
   431
paulson@13115
   432
lemma fix_subset: "fix f A <= A"
paulson@13115
   433
by (simp add: fix_def, fast)
paulson@13115
   434
paulson@13115
   435
lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
paulson@13115
   436
by (simp add: fix_def)
paulson@13115
   437
paulson@13115
   438
lemma fixf_subset:
paulson@13115
   439
     "[| A <= B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
wenzelm@13383
   440
apply (simp add: fix_def, auto)
paulson@13115
   441
done
paulson@13115
   442
wenzelm@13383
   443
wenzelm@13383
   444
subsubsection {* lemmas for Tarski, lub *}
paulson@13115
   445
lemma (in CLF) lubH_le_flubH:
paulson@13115
   446
     "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
paulson@13115
   447
apply (rule lub_least, fast)
paulson@13115
   448
apply (rule f_in_funcset [THEN funcset_mem])
paulson@13115
   449
apply (rule lub_in_lattice, fast)
wenzelm@13383
   450
-- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *}
paulson@13115
   451
apply (rule ballI)
paulson@13115
   452
apply (rule transE)
paulson@13115
   453
apply (rule CO_trans)
paulson@13585
   454
-- {* instantiates @{text "(x, ???z) \<in> order cl to (x, f x)"}, *}
wenzelm@13383
   455
-- {* because of the def of @{text H} *}
paulson@13115
   456
apply fast
wenzelm@13383
   457
-- {* so it remains to show @{text "(f x, f (lub H cl)) \<in> r"} *}
paulson@13115
   458
apply (rule_tac f = "f" in monotoneE)
paulson@13115
   459
apply (rule monotone_f, fast)
paulson@13115
   460
apply (rule lub_in_lattice, fast)
paulson@13115
   461
apply (rule lub_upper, fast)
paulson@13115
   462
apply assumption
paulson@13115
   463
done
paulson@13115
   464
paulson@13115
   465
lemma (in CLF) flubH_le_lubH:
paulson@13115
   466
     "[|  H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
paulson@13115
   467
apply (rule lub_upper, fast)
paulson@13115
   468
apply (rule_tac t = "H" in ssubst, assumption)
paulson@13115
   469
apply (rule CollectI)
paulson@13115
   470
apply (rule conjI)
paulson@13115
   471
apply (rule_tac [2] f_in_funcset [THEN funcset_mem])
paulson@13115
   472
apply (rule_tac [2] lub_in_lattice)
paulson@13115
   473
prefer 2 apply fast
paulson@13115
   474
apply (rule_tac f = "f" in monotoneE)
paulson@13115
   475
apply (rule monotone_f)
wenzelm@13383
   476
  apply (blast intro: lub_in_lattice)
wenzelm@13383
   477
 apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])
paulson@13115
   478
apply (simp add: lubH_le_flubH)
paulson@13115
   479
done
paulson@13115
   480
paulson@13115
   481
lemma (in CLF) lubH_is_fixp:
paulson@13115
   482
     "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
paulson@13115
   483
apply (simp add: fix_def)
paulson@13115
   484
apply (rule conjI)
paulson@13115
   485
apply (rule lub_in_lattice, fast)
paulson@13115
   486
apply (rule antisymE)
paulson@13115
   487
apply (rule CO_antisym)
paulson@13115
   488
apply (simp add: flubH_le_lubH)
paulson@13115
   489
apply (simp add: lubH_le_flubH)
paulson@13115
   490
done
paulson@13115
   491
paulson@13115
   492
lemma (in CLF) fix_in_H:
paulson@13115
   493
     "[| H = {x. (x, f x) \<in> r & x \<in> A};  x \<in> P |] ==> x \<in> H"
wenzelm@13383
   494
by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl
wenzelm@13383
   495
                    fix_subset [of f A, THEN subsetD])
paulson@13115
   496
paulson@13115
   497
lemma (in CLF) fixf_le_lubH:
paulson@13115
   498
     "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
paulson@13115
   499
apply (rule ballI)
paulson@13115
   500
apply (rule lub_upper, fast)
paulson@13115
   501
apply (rule fix_in_H)
wenzelm@13383
   502
apply (simp_all add: P_def)
paulson@13115
   503
done
paulson@13115
   504
paulson@13115
   505
lemma (in CLF) lubH_least_fixf:
wenzelm@13383
   506
     "H = {x. (x, f x) \<in> r & x \<in> A}
paulson@13115
   507
      ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
paulson@13115
   508
apply (rule allI)
paulson@13115
   509
apply (rule impI)
paulson@13115
   510
apply (erule bspec)
paulson@13115
   511
apply (rule lubH_is_fixp, assumption)
paulson@13115
   512
done
paulson@13115
   513
wenzelm@13383
   514
subsubsection {* Tarski fixpoint theorem 1, first part *}
paulson@13115
   515
lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
paulson@13115
   516
apply (rule sym)
wenzelm@13383
   517
apply (simp add: P_def)
paulson@13115
   518
apply (rule lubI)
paulson@13115
   519
apply (rule fix_subset)
paulson@13115
   520
apply (rule lub_in_lattice, fast)
paulson@13115
   521
apply (simp add: fixf_le_lubH)
paulson@13115
   522
apply (simp add: lubH_least_fixf)
paulson@13115
   523
done
paulson@13115
   524
paulson@13115
   525
lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
wenzelm@13383
   526
  -- {* Tarski for glb *}
paulson@13115
   527
apply (simp add: glb_dual_lub P_def A_def r_def)
paulson@13115
   528
apply (rule dualA_iff [THEN subst])
paulson@13115
   529
apply (rule Tarski.lubH_is_fixp)
wenzelm@13383
   530
apply (rule dualPO)
paulson@13115
   531
apply (rule CL_dualCL)
paulson@13115
   532
apply (rule f_cl [THEN CLF_dual])
paulson@13115
   533
apply (simp add: dualr_iff dualA_iff)
paulson@13115
   534
done
paulson@13115
   535
paulson@13115
   536
lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
paulson@13115
   537
apply (simp add: glb_dual_lub P_def A_def r_def)
paulson@13115
   538
apply (rule dualA_iff [THEN subst])
wenzelm@13383
   539
apply (simp add: Tarski.T_thm_1_lub [of _ f, OF dualPO CL_dualCL]
paulson@13115
   540
                 dualPO CL_dualCL CLF_dual dualr_iff)
paulson@13115
   541
done
paulson@13115
   542
wenzelm@13383
   543
subsubsection {* interval *}
wenzelm@13383
   544
paulson@13115
   545
lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
wenzelm@13383
   546
apply (insert CO_refl)
wenzelm@13383
   547
apply (simp add: refl_def, blast)
paulson@13115
   548
done
paulson@13115
   549
paulson@13115
   550
lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b <= A"
paulson@13115
   551
apply (simp add: interval_def)
paulson@13115
   552
apply (blast intro: rel_imp_elem)
paulson@13115
   553
done
paulson@13115
   554
paulson@13115
   555
lemma (in CLF) intervalI:
paulson@13115
   556
     "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
paulson@13115
   557
apply (simp add: interval_def)
paulson@13115
   558
done
paulson@13115
   559
paulson@13115
   560
lemma (in CLF) interval_lemma1:
paulson@13115
   561
     "[| S <= interval r a b; x \<in> S |] ==> (a, x) \<in> r"
paulson@13115
   562
apply (unfold interval_def, fast)
paulson@13115
   563
done
paulson@13115
   564
paulson@13115
   565
lemma (in CLF) interval_lemma2:
paulson@13115
   566
     "[| S <= interval r a b; x \<in> S |] ==> (x, b) \<in> r"
paulson@13115
   567
apply (unfold interval_def, fast)
paulson@13115
   568
done
paulson@13115
   569
paulson@13115
   570
lemma (in CLF) a_less_lub:
wenzelm@13383
   571
     "[| S <= A; S \<noteq> {};
paulson@13115
   572
         \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
paulson@13115
   573
by (blast intro: transE PO_imp_trans)
paulson@13115
   574
paulson@13115
   575
lemma (in CLF) glb_less_b:
wenzelm@13383
   576
     "[| S <= A; S \<noteq> {};
paulson@13115
   577
         \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
paulson@13115
   578
by (blast intro: transE PO_imp_trans)
paulson@13115
   579
paulson@13115
   580
lemma (in CLF) S_intv_cl:
paulson@13115
   581
     "[| a \<in> A; b \<in> A; S <= interval r a b |]==> S <= A"
paulson@13115
   582
by (simp add: subset_trans [OF _ interval_subset])
paulson@13115
   583
paulson@13115
   584
lemma (in CLF) L_in_interval:
wenzelm@13383
   585
     "[| a \<in> A; b \<in> A; S <= interval r a b;
paulson@13115
   586
         S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
paulson@13115
   587
apply (rule intervalI)
paulson@13115
   588
apply (rule a_less_lub)
paulson@13115
   589
prefer 2 apply assumption
paulson@13115
   590
apply (simp add: S_intv_cl)
paulson@13115
   591
apply (rule ballI)
paulson@13115
   592
apply (simp add: interval_lemma1)
paulson@13115
   593
apply (simp add: isLub_upper)
wenzelm@13383
   594
-- {* @{text "(L, b) \<in> r"} *}
paulson@13115
   595
apply (simp add: isLub_least interval_lemma2)
paulson@13115
   596
done
paulson@13115
   597
paulson@13115
   598
lemma (in CLF) G_in_interval:
paulson@13115
   599
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S <= interval r a b; isGlb S cl G;
paulson@13115
   600
         S \<noteq> {} |] ==> G \<in> interval r a b"
paulson@13115
   601
apply (simp add: interval_dual)
wenzelm@13383
   602
apply (simp add: Tarski.L_in_interval [of _ f]
paulson@13115
   603
                 dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub)
paulson@13115
   604
done
paulson@13115
   605
paulson@13115
   606
lemma (in CLF) intervalPO:
wenzelm@13383
   607
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
paulson@13115
   608
      ==> (| pset = interval r a b, order = induced (interval r a b) r |)
paulson@13115
   609
          \<in> PartialOrder"
paulson@13115
   610
apply (rule po_subset_po)
paulson@13115
   611
apply (simp add: interval_subset)
paulson@13115
   612
done
paulson@13115
   613
paulson@13115
   614
lemma (in CLF) intv_CL_lub:
wenzelm@13383
   615
 "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
wenzelm@13383
   616
  ==> \<forall>S. S <= interval r a b -->
wenzelm@13383
   617
          (\<exists>L. isLub S (| pset = interval r a b,
paulson@13115
   618
                          order = induced (interval r a b) r |)  L)"
paulson@13115
   619
apply (intro strip)
paulson@13115
   620
apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
paulson@13115
   621
prefer 2 apply assumption
paulson@13115
   622
apply assumption
paulson@13115
   623
apply (erule exE)
wenzelm@13383
   624
-- {* define the lub for the interval as *}
paulson@13115
   625
apply (rule_tac x = "if S = {} then a else L" in exI)
paulson@13115
   626
apply (simp (no_asm_simp) add: isLub_def split del: split_if)
wenzelm@13383
   627
apply (intro impI conjI)
wenzelm@13383
   628
-- {* @{text "(if S = {} then a else L) \<in> interval r a b"} *}
paulson@13115
   629
apply (simp add: CL_imp_PO L_in_interval)
paulson@13115
   630
apply (simp add: left_in_interval)
wenzelm@13383
   631
-- {* lub prop 1 *}
paulson@13115
   632
apply (case_tac "S = {}")
wenzelm@13383
   633
-- {* @{text "S = {}, y \<in> S = False => everything"} *}
paulson@13115
   634
apply fast
wenzelm@13383
   635
-- {* @{text "S \<noteq> {}"} *}
paulson@13115
   636
apply simp
wenzelm@13383
   637
-- {* @{text "\<forall>y:S. (y, L) \<in> induced (interval r a b) r"} *}
paulson@13115
   638
apply (rule ballI)
paulson@13115
   639
apply (simp add: induced_def  L_in_interval)
paulson@13115
   640
apply (rule conjI)
paulson@13115
   641
apply (rule subsetD)
paulson@13115
   642
apply (simp add: S_intv_cl, assumption)
paulson@13115
   643
apply (simp add: isLub_upper)
wenzelm@13383
   644
-- {* @{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"} *}
paulson@13115
   645
apply (rule ballI)
paulson@13115
   646
apply (rule impI)
paulson@13115
   647
apply (case_tac "S = {}")
wenzelm@13383
   648
-- {* @{text "S = {}"} *}
paulson@13115
   649
apply simp
paulson@13115
   650
apply (simp add: induced_def  interval_def)
paulson@13115
   651
apply (rule conjI)
paulson@13115
   652
apply (rule reflE)
paulson@13115
   653
apply (rule CO_refl, assumption)
paulson@13115
   654
apply (rule interval_not_empty)
paulson@13115
   655
apply (rule CO_trans)
paulson@13115
   656
apply (simp add: interval_def)
wenzelm@13383
   657
-- {* @{text "S \<noteq> {}"} *}
paulson@13115
   658
apply simp
paulson@13115
   659
apply (simp add: induced_def  L_in_interval)
paulson@13115
   660
apply (rule isLub_least, assumption)
paulson@13115
   661
apply (rule subsetD)
paulson@13115
   662
prefer 2 apply assumption
paulson@13115
   663
apply (simp add: S_intv_cl, fast)
paulson@13115
   664
done
paulson@13115
   665
paulson@13115
   666
lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
paulson@13115
   667
paulson@13115
   668
lemma (in CLF) interval_is_sublattice:
wenzelm@13383
   669
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
paulson@13115
   670
        ==> interval r a b <<= cl"
paulson@13115
   671
apply (rule sublatticeI)
paulson@13115
   672
apply (simp add: interval_subset)
paulson@13115
   673
apply (rule CompleteLatticeI)
paulson@13115
   674
apply (simp add: intervalPO)
paulson@13115
   675
 apply (simp add: intv_CL_lub)
paulson@13115
   676
apply (simp add: intv_CL_glb)
paulson@13115
   677
done
paulson@13115
   678
wenzelm@13383
   679
lemmas (in CLF) interv_is_compl_latt =
paulson@13115
   680
    interval_is_sublattice [THEN sublattice_imp_CL]
paulson@13115
   681
wenzelm@13383
   682
wenzelm@13383
   683
subsubsection {* Top and Bottom *}
paulson@13115
   684
lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
paulson@13115
   685
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
paulson@13115
   686
paulson@13115
   687
lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
paulson@13115
   688
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
paulson@13115
   689
paulson@13115
   690
lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
paulson@13115
   691
apply (simp add: Bot_def least_def)
paulson@13115
   692
apply (rule someI2)
paulson@13115
   693
apply (fold A_def)
paulson@13115
   694
apply (erule_tac [2] conjunct1)
paulson@13115
   695
apply (rule conjI)
paulson@13115
   696
apply (rule glb_in_lattice)
paulson@13115
   697
apply (rule subset_refl)
paulson@13115
   698
apply (fold r_def)
paulson@13115
   699
apply (simp add: glb_lower)
paulson@13115
   700
done
paulson@13115
   701
paulson@13115
   702
lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
paulson@13115
   703
apply (simp add: Top_dual_Bot A_def)
wenzelm@13383
   704
apply (rule dualA_iff [THEN subst])
wenzelm@13383
   705
apply (blast intro!: Tarski.Bot_in_lattice dualPO CL_dualCL CLF_dual f_cl)
paulson@13115
   706
done
paulson@13115
   707
paulson@13115
   708
lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
paulson@13115
   709
apply (simp add: Top_def greatest_def)
paulson@13115
   710
apply (rule someI2)
paulson@13115
   711
apply (fold r_def  A_def)
paulson@13115
   712
prefer 2 apply fast
paulson@13115
   713
apply (intro conjI ballI)
paulson@13115
   714
apply (rule_tac [2] lub_upper)
paulson@13115
   715
apply (auto simp add: lub_in_lattice)
paulson@13115
   716
done
paulson@13115
   717
paulson@13115
   718
lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
paulson@13115
   719
apply (simp add: Bot_dual_Top r_def)
paulson@13115
   720
apply (rule dualr_iff [THEN subst])
wenzelm@13383
   721
apply (simp add: Tarski.Top_prop [of _ f]
paulson@13115
   722
                 dualA_iff A_def dualPO CL_dualCL CLF_dual)
paulson@13115
   723
done
paulson@13115
   724
paulson@13115
   725
lemma (in CLF) Top_intv_not_empty: "x \<in> A  ==> interval r x (Top cl) \<noteq> {}"
paulson@13115
   726
apply (rule notI)
paulson@13115
   727
apply (drule_tac a = "Top cl" in equals0D)
paulson@13115
   728
apply (simp add: interval_def)
paulson@13115
   729
apply (simp add: refl_def Top_in_lattice Top_prop)
paulson@13115
   730
done
paulson@13115
   731
paulson@13115
   732
lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
paulson@13115
   733
apply (simp add: Bot_dual_Top)
paulson@13115
   734
apply (subst interval_dual)
paulson@13115
   735
prefer 2 apply assumption
paulson@13115
   736
apply (simp add: A_def)
paulson@13115
   737
apply (rule dualA_iff [THEN subst])
paulson@13115
   738
apply (blast intro!: Tarski.Top_in_lattice
paulson@13115
   739
                 f_cl dualPO CL_dualCL CLF_dual)
wenzelm@13383
   740
apply (simp add: Tarski.Top_intv_not_empty [of _ f]
paulson@13115
   741
                 dualA_iff A_def dualPO CL_dualCL CLF_dual)
paulson@13115
   742
done
paulson@13115
   743
wenzelm@13383
   744
subsubsection {* fixed points form a partial order *}
wenzelm@13383
   745
paulson@13115
   746
lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
paulson@13115
   747
by (simp add: P_def fix_subset po_subset_po)
paulson@13115
   748
paulson@13115
   749
lemma (in Tarski) Y_subset_A: "Y <= A"
paulson@13115
   750
apply (rule subset_trans [OF _ fix_subset])
paulson@13115
   751
apply (rule Y_ss [simplified P_def])
paulson@13115
   752
done
paulson@13115
   753
paulson@13115
   754
lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
paulson@13115
   755
by (simp add: Y_subset_A [THEN lub_in_lattice])
paulson@13115
   756
paulson@13115
   757
lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
paulson@13115
   758
apply (rule lub_least)
paulson@13115
   759
apply (rule Y_subset_A)
paulson@13115
   760
apply (rule f_in_funcset [THEN funcset_mem])
paulson@13115
   761
apply (rule lubY_in_A)
wenzelm@13383
   762
-- {* @{text "Y <= P ==> f x = x"} *}
paulson@13115
   763
apply (rule ballI)
paulson@13115
   764
apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
paulson@13115
   765
apply (erule Y_ss [simplified P_def, THEN subsetD])
wenzelm@13383
   766
-- {* @{text "reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r"} by monotonicity *}
paulson@13115
   767
apply (rule_tac f = "f" in monotoneE)
paulson@13115
   768
apply (rule monotone_f)
paulson@13115
   769
apply (simp add: Y_subset_A [THEN subsetD])
paulson@13115
   770
apply (rule lubY_in_A)
paulson@13115
   771
apply (simp add: lub_upper Y_subset_A)
paulson@13115
   772
done
paulson@13115
   773
paulson@13115
   774
lemma (in Tarski) intY1_subset: "intY1 <= A"
paulson@13115
   775
apply (unfold intY1_def)
paulson@13115
   776
apply (rule interval_subset)
paulson@13115
   777
apply (rule lubY_in_A)
paulson@13115
   778
apply (rule Top_in_lattice)
paulson@13115
   779
done
paulson@13115
   780
paulson@13115
   781
lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
paulson@13115
   782
paulson@13115
   783
lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
paulson@13115
   784
apply (simp add: intY1_def  interval_def)
paulson@13115
   785
apply (rule conjI)
paulson@13115
   786
apply (rule transE)
paulson@13115
   787
apply (rule CO_trans)
paulson@13115
   788
apply (rule lubY_le_flubY)
wenzelm@13383
   789
-- {* @{text "(f (lub Y cl), f x) \<in> r"} *}
paulson@13115
   790
apply (rule_tac f=f in monotoneE)
paulson@13115
   791
apply (rule monotone_f)
paulson@13115
   792
apply (rule lubY_in_A)
paulson@13115
   793
apply (simp add: intY1_def interval_def  intY1_elem)
paulson@13115
   794
apply (simp add: intY1_def  interval_def)
wenzelm@13383
   795
-- {* @{text "(f x, Top cl) \<in> r"} *}
paulson@13115
   796
apply (rule Top_prop)
paulson@13115
   797
apply (rule f_in_funcset [THEN funcset_mem])
paulson@13115
   798
apply (simp add: intY1_def interval_def  intY1_elem)
paulson@13115
   799
done
paulson@13115
   800
paulson@13585
   801
lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 -> intY1"
paulson@13115
   802
apply (rule restrictI)
paulson@13115
   803
apply (erule intY1_f_closed)
paulson@13115
   804
done
paulson@13115
   805
paulson@13115
   806
lemma (in Tarski) intY1_mono:
paulson@13115
   807
     "monotone (%x: intY1. f x) intY1 (induced intY1 r)"
paulson@13115
   808
apply (auto simp add: monotone_def induced_def intY1_f_closed)
paulson@13115
   809
apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
paulson@13115
   810
done
paulson@13115
   811
wenzelm@13383
   812
lemma (in Tarski) intY1_is_cl:
paulson@13115
   813
    "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
paulson@13115
   814
apply (unfold intY1_def)
paulson@13115
   815
apply (rule interv_is_compl_latt)
paulson@13115
   816
apply (rule lubY_in_A)
paulson@13115
   817
apply (rule Top_in_lattice)
paulson@13115
   818
apply (rule Top_intv_not_empty)
paulson@13115
   819
apply (rule lubY_in_A)
paulson@13115
   820
done
paulson@13115
   821
paulson@13115
   822
lemma (in Tarski) v_in_P: "v \<in> P"
paulson@13115
   823
apply (unfold P_def)
paulson@13115
   824
apply (rule_tac A = "intY1" in fixf_subset)
paulson@13115
   825
apply (rule intY1_subset)
paulson@13115
   826
apply (simp add: Tarski.glbH_is_fixp [OF _ intY1_is_cl, simplified]
paulson@13115
   827
                 v_def CL_imp_PO intY1_is_cl CLF_def intY1_func intY1_mono)
paulson@13115
   828
done
paulson@13115
   829
wenzelm@13383
   830
lemma (in Tarski) z_in_interval:
paulson@13115
   831
     "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
paulson@13115
   832
apply (unfold intY1_def P_def)
paulson@13115
   833
apply (rule intervalI)
wenzelm@13383
   834
prefer 2
paulson@13115
   835
 apply (erule fix_subset [THEN subsetD, THEN Top_prop])
paulson@13115
   836
apply (rule lub_least)
paulson@13115
   837
apply (rule Y_subset_A)
paulson@13115
   838
apply (fast elim!: fix_subset [THEN subsetD])
paulson@13115
   839
apply (simp add: induced_def)
paulson@13115
   840
done
paulson@13115
   841
wenzelm@13383
   842
lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
paulson@13115
   843
      ==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"
paulson@13115
   844
apply (simp add: induced_def  intY1_f_closed z_in_interval P_def)
wenzelm@13383
   845
apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD]
paulson@13115
   846
                 CO_refl [THEN reflE])
paulson@13115
   847
done
paulson@13115
   848
paulson@13115
   849
lemma (in Tarski) tarski_full_lemma:
paulson@13115
   850
     "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
paulson@13115
   851
apply (rule_tac x = "v" in exI)
paulson@13115
   852
apply (simp add: isLub_def)
wenzelm@13383
   853
-- {* @{text "v \<in> P"} *}
paulson@13115
   854
apply (simp add: v_in_P)
paulson@13115
   855
apply (rule conjI)
wenzelm@13383
   856
-- {* @{text v} is lub *}
wenzelm@13383
   857
-- {* @{text "1. \<forall>y:Y. (y, v) \<in> induced P r"} *}
paulson@13115
   858
apply (rule ballI)
paulson@13115
   859
apply (simp add: induced_def subsetD v_in_P)
paulson@13115
   860
apply (rule conjI)
paulson@13115
   861
apply (erule Y_ss [THEN subsetD])
paulson@13115
   862
apply (rule_tac b = "lub Y cl" in transE)
paulson@13115
   863
apply (rule CO_trans)
paulson@13115
   864
apply (rule lub_upper)
paulson@13115
   865
apply (rule Y_subset_A, assumption)
paulson@13115
   866
apply (rule_tac b = "Top cl" in interval_imp_mem)
paulson@13115
   867
apply (simp add: v_def)
paulson@13115
   868
apply (fold intY1_def)
paulson@13115
   869
apply (rule Tarski.glb_in_lattice [OF _ intY1_is_cl, simplified])
paulson@13115
   870
 apply (simp add: CL_imp_PO intY1_is_cl, force)
wenzelm@13383
   871
-- {* @{text v} is LEAST ub *}
paulson@13115
   872
apply clarify
paulson@13115
   873
apply (rule indI)
paulson@13115
   874
  prefer 3 apply assumption
paulson@13115
   875
 prefer 2 apply (simp add: v_in_P)
paulson@13115
   876
apply (unfold v_def)
paulson@13115
   877
apply (rule indE)
paulson@13115
   878
apply (rule_tac [2] intY1_subset)
paulson@13115
   879
apply (rule Tarski.glb_lower [OF _ intY1_is_cl, simplified])
wenzelm@13383
   880
  apply (simp add: CL_imp_PO intY1_is_cl)
paulson@13115
   881
 apply force
paulson@13115
   882
apply (simp add: induced_def intY1_f_closed z_in_interval)
wenzelm@13383
   883
apply (simp add: P_def fix_imp_eq [of _ f A]
wenzelm@13383
   884
                 fix_subset [of f A, THEN subsetD]
paulson@13115
   885
                 CO_refl [THEN reflE])
paulson@13115
   886
done
paulson@13115
   887
paulson@13115
   888
lemma CompleteLatticeI_simp:
wenzelm@13383
   889
     "[| (| pset = A, order = r |) \<in> PartialOrder;
wenzelm@13383
   890
         \<forall>S. S <= A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
paulson@13115
   891
    ==> (| pset = A, order = r |) \<in> CompleteLattice"
paulson@13115
   892
by (simp add: CompleteLatticeI Rdual)
paulson@13115
   893
paulson@13115
   894
theorem (in CLF) Tarski_full:
paulson@13115
   895
     "(| pset = P, order = induced P r|) \<in> CompleteLattice"
paulson@13115
   896
apply (rule CompleteLatticeI_simp)
paulson@13115
   897
apply (rule fixf_po, clarify)
wenzelm@13383
   898
apply (simp add: P_def A_def r_def)
wenzelm@13383
   899
apply (blast intro!: Tarski.tarski_full_lemma cl_po cl_co f_cl)
paulson@13115
   900
done
wenzelm@7112
   901
wenzelm@7112
   902
end