(*  Title:      HOL/ex/Tarski.thy

    Author:     Florian Kammüller, Cambridge University Computer Laboratory

*)

section {* The Full Theorem of Tarski *}

theory Tarski

imports Main "~~/src/HOL/Library/FuncSet"

begin

text {*

  Minimal version of lattice theory plus the full theorem of Tarski:

  The fixedpoints of a complete lattice themselves form a complete

  lattice.

  Illustrates first-class theories, using the Sigma representation of

  structures.  Tidied and converted to Isar by lcp.

*}

record 'a potype =

  pset  :: "'a set"

  order :: "('a * 'a) set"

definition

  monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool" where

  "monotone f A r = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r)"

definition

  least :: "['a => bool, 'a potype] => 'a" where

  "least P po = (SOME x. x: pset po & P x &

                       (\<forall>y \<in> pset po. P y --> (x,y): order po))"

definition

  greatest :: "['a => bool, 'a potype] => 'a" where

  "greatest P po = (SOME x. x: pset po & P x &

                          (\<forall>y \<in> pset po. P y --> (y,x): order po))"

definition

  lub  :: "['a set, 'a potype] => 'a" where

  "lub S po = least (%x. \<forall>y\<in>S. (y,x): order po) po"

definition

  glb  :: "['a set, 'a potype] => 'a" where

  "glb S po = greatest (%x. \<forall>y\<in>S. (x,y): order po) po"

definition

  isLub :: "['a set, 'a potype, 'a] => bool" where

  "isLub S po = (%L. (L: pset po & (\<forall>y\<in>S. (y,L): order po) &

                   (\<forall>z\<in>pset po. (\<forall>y\<in>S. (y,z): order po) --> (L,z): order po)))"

definition

  isGlb :: "['a set, 'a potype, 'a] => bool" where

  "isGlb S po = (%G. (G: pset po & (\<forall>y\<in>S. (G,y): order po) &

                 (\<forall>z \<in> pset po. (\<forall>y\<in>S. (z,y): order po) --> (z,G): order po)))"

definition

  "fix"    :: "[('a => 'a), 'a set] => 'a set" where

  "fix f A  = {x. x: A & f x = x}"

definition

  interval :: "[('a*'a) set,'a, 'a ] => 'a set" where

  "interval r a b = {x. (a,x): r & (x,b): r}"

definition

  Bot :: "'a potype => 'a" where

  "Bot po = least (%x. True) po"

definition

  Top :: "'a potype => 'a" where

  "Top po = greatest (%x. True) po"

definition

  PartialOrder :: "('a potype) set" where

  "PartialOrder = {P. refl_on (pset P) (order P) & antisym (order P) &

                       trans (order P)}"

definition

  CompleteLattice :: "('a potype) set" where

  "CompleteLattice = {cl. cl: PartialOrder &

                        (\<forall>S. S \<subseteq> pset cl --> (\<exists>L. isLub S cl L)) &

                        (\<forall>S. S \<subseteq> pset cl --> (\<exists>G. isGlb S cl G))}"

definition

  CLF_set :: "('a potype * ('a => 'a)) set" where

  "CLF_set = (SIGMA cl: CompleteLattice.

            {f. f: pset cl -> pset cl & monotone f (pset cl) (order cl)})"

definition

  induced :: "['a set, ('a * 'a) set] => ('a *'a)set" where

  "induced A r = {(a,b). a : A & b: A & (a,b): r}"

definition

  sublattice :: "('a potype * 'a set)set" where

  "sublattice =

      (SIGMA cl: CompleteLattice.

          {S. S \<subseteq> pset cl &

           (| pset = S, order = induced S (order cl) |): CompleteLattice})"

abbreviation

  sublat :: "['a set, 'a potype] => bool"  ("_ <<= _" [51,50]50) where

  "S <<= cl == S : sublattice `` {cl}"

definition

  dual :: "'a potype => 'a potype" where

  "dual po = (| pset = pset po, order = converse (order po) |)"

locale S =

  fixes cl :: "'a potype"

    and A  :: "'a set"

    and r  :: "('a * 'a) set"

  defines A_def: "A == pset cl"

     and  r_def: "r == order cl"

locale PO = S +

  assumes cl_po:  "cl : PartialOrder"

locale CL = S +

  assumes cl_co:  "cl : CompleteLattice"

sublocale CL < po: PO

apply (simp_all add: A_def r_def)

apply unfold_locales

using cl_co unfolding CompleteLattice_def by auto

locale CLF = S +

  fixes f :: "'a => 'a"

    and P :: "'a set"

  assumes f_cl:  "(cl,f) : CLF_set" (*was the equivalent "f : CLF_set``{cl}"*)

  defines P_def: "P == fix f A"

sublocale CLF < cl: CL

apply (simp_all add: A_def r_def)

apply unfold_locales

using f_cl unfolding CLF_set_def by auto

locale Tarski = CLF +

  fixes Y     :: "'a set"

    and intY1 :: "'a set"

    and v     :: "'a"

  assumes

    Y_ss: "Y \<subseteq> P"

  defines

    intY1_def: "intY1 == interval r (lub Y cl) (Top cl)"

    and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r &

                             x: intY1}

                      (| pset=intY1, order=induced intY1 r|)"

subsection {* Partial Order *}

lemma (in PO) dual:

  "PO (dual cl)"

apply unfold_locales

using cl_po

unfolding PartialOrder_def dual_def

by auto

lemma (in PO) PO_imp_refl_on [simp]: "refl_on A r"

apply (insert cl_po)

apply (simp add: PartialOrder_def A_def r_def)

done

lemma (in PO) PO_imp_sym [simp]: "antisym r"

apply (insert cl_po)

apply (simp add: PartialOrder_def r_def)

done

lemma (in PO) PO_imp_trans [simp]: "trans r"

apply (insert cl_po)

apply (simp add: PartialOrder_def r_def)

done

lemma (in PO) reflE: "x \<in> A ==> (x, x) \<in> r"

apply (insert cl_po)

apply (simp add: PartialOrder_def refl_on_def A_def r_def)

done

lemma (in PO) antisymE: "[| (a, b) \<in> r; (b, a) \<in> r |] ==> a = b"

apply (insert cl_po)

apply (simp add: PartialOrder_def antisym_def r_def)

done

lemma (in PO) transE: "[| (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r"

apply (insert cl_po)

apply (simp add: PartialOrder_def r_def)

apply (unfold trans_def, fast)

done

lemma (in PO) monotoneE:

     "[| monotone f A r;  x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r"

by (simp add: monotone_def)

lemma (in PO) po_subset_po:

     "S \<subseteq> A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"

apply (simp (no_asm) add: PartialOrder_def)

apply auto

-- {* refl *}

apply (simp add: refl_on_def induced_def)

apply (blast intro: reflE)

-- {* antisym *}

apply (simp add: antisym_def induced_def)

apply (blast intro: antisymE)

-- {* trans *}

apply (simp add: trans_def induced_def)

apply (blast intro: transE)

done

lemma (in PO) indE: "[| (x, y) \<in> induced S r; S \<subseteq> A |] ==> (x, y) \<in> r"

by (simp add: add: induced_def)

lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r"

by (simp add: add: induced_def)

lemma (in CL) CL_imp_ex_isLub: "S \<subseteq> A ==> \<exists>L. isLub S cl L"

apply (insert cl_co)

apply (simp add: CompleteLattice_def A_def)

done

declare (in CL) cl_co [simp]

lemma isLub_lub: "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)"

by (simp add: lub_def least_def isLub_def some_eq_ex [symmetric])

lemma isGlb_glb: "(\<exists>G. isGlb S cl G) = isGlb S cl (glb S cl)"

by (simp add: glb_def greatest_def isGlb_def some_eq_ex [symmetric])

lemma isGlb_dual_isLub: "isGlb S cl = isLub S (dual cl)"

by (simp add: isLub_def isGlb_def dual_def converse_unfold)

lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)"

by (simp add: isLub_def isGlb_def dual_def converse_unfold)

lemma (in PO) dualPO: "dual cl \<in> PartialOrder"

apply (insert cl_po)

apply (simp add: PartialOrder_def dual_def refl_on_converse

                 trans_converse antisym_converse)

done

lemma Rdual:

     "\<forall>S. (S \<subseteq> A -->( \<exists>L. isLub S (| pset = A, order = r|) L))

      ==> \<forall>S. (S \<subseteq> A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"

apply safe

apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)}

                      (|pset = A, order = r|) " in exI)

apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> r) }" in spec)

apply (drule mp, fast)

apply (simp add: isLub_lub isGlb_def)

apply (simp add: isLub_def, blast)

done

lemma lub_dual_glb: "lub S cl = glb S (dual cl)"

by (simp add: lub_def glb_def least_def greatest_def dual_def converse_unfold)

lemma glb_dual_lub: "glb S cl = lub S (dual cl)"

by (simp add: lub_def glb_def least_def greatest_def dual_def converse_unfold)

lemma CL_subset_PO: "CompleteLattice \<subseteq> PartialOrder"

by (simp add: PartialOrder_def CompleteLattice_def, fast)

lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]

(*declare CL_imp_PO [THEN PO.PO_imp_refl, simp]

declare CL_imp_PO [THEN PO.PO_imp_sym, simp]

declare CL_imp_PO [THEN PO.PO_imp_trans, simp]*)

lemma (in CL) CO_refl_on: "refl_on A r"

by (rule PO_imp_refl_on)

lemma (in CL) CO_antisym: "antisym r"

by (rule PO_imp_sym)

lemma (in CL) CO_trans: "trans r"

by (rule PO_imp_trans)

lemma CompleteLatticeI:

     "[| po \<in> PartialOrder; (\<forall>S. S \<subseteq> pset po --> (\<exists>L. isLub S po L));

         (\<forall>S. S \<subseteq> pset po --> (\<exists>G. isGlb S po G))|]

      ==> po \<in> CompleteLattice"

apply (unfold CompleteLattice_def, blast)

done

lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"

apply (insert cl_co)

apply (simp add: CompleteLattice_def dual_def)

apply (fold dual_def)

apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric]

                 dualPO)

done

lemma (in PO) dualA_iff: "pset (dual cl) = pset cl"

by (simp add: dual_def)

lemma (in PO) dualr_iff: "((x, y) \<in> (order(dual cl))) = ((y, x) \<in> order cl)"

by (simp add: dual_def)

lemma (in PO) monotone_dual:

     "monotone f (pset cl) (order cl) 

     ==> monotone f (pset (dual cl)) (order(dual cl))"

by (simp add: monotone_def dualA_iff dualr_iff)

lemma (in PO) interval_dual:

     "[| x \<in> A; y \<in> A|] ==> interval r x y = interval (order(dual cl)) y x"

apply (simp add: interval_def dualr_iff)

apply (fold r_def, fast)

done

lemma (in PO) trans:

  "(x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r"

using cl_po apply (auto simp add: PartialOrder_def r_def)

unfolding trans_def by blast 

lemma (in PO) interval_not_empty:

  "interval r a b \<noteq> {} ==> (a, b) \<in> r"

apply (simp add: interval_def)

using trans by blast

lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r"

by (simp add: interval_def)

lemma (in PO) left_in_interval:

     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b"

apply (simp (no_asm_simp) add: interval_def)

apply (simp add: PO_imp_trans interval_not_empty)

apply (simp add: reflE)

done

lemma (in PO) right_in_interval:

     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b"

apply (simp (no_asm_simp) add: interval_def)

apply (simp add: PO_imp_trans interval_not_empty)

apply (simp add: reflE)

done

subsection {* sublattice *}

lemma (in PO) sublattice_imp_CL:

     "S <<= cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"

by (simp add: sublattice_def CompleteLattice_def r_def)

lemma (in CL) sublatticeI:

     "[| S \<subseteq> A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]

      ==> S <<= cl"

by (simp add: sublattice_def A_def r_def)

lemma (in CL) dual:

  "CL (dual cl)"

apply unfold_locales

using cl_co unfolding CompleteLattice_def

apply (simp add: dualPO isGlb_dual_isLub [symmetric] isLub_dual_isGlb [symmetric] dualA_iff)

done

subsection {* lub *}

lemma (in CL) lub_unique: "[| S \<subseteq> A; isLub S cl x; isLub S cl L|] ==> x = L"

apply (rule antisymE)

apply (auto simp add: isLub_def r_def)

done

lemma (in CL) lub_upper: "[|S \<subseteq> A; x \<in> S|] ==> (x, lub S cl) \<in> r"

apply (rule CL_imp_ex_isLub [THEN exE], assumption)

apply (unfold lub_def least_def)

apply (rule some_equality [THEN ssubst])

  apply (simp add: isLub_def)

 apply (simp add: lub_unique A_def isLub_def)

apply (simp add: isLub_def r_def)

done

lemma (in CL) lub_least:

     "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"

apply (rule CL_imp_ex_isLub [THEN exE], assumption)

apply (unfold lub_def least_def)

apply (rule_tac s=x in some_equality [THEN ssubst])

  apply (simp add: isLub_def)

 apply (simp add: lub_unique A_def isLub_def)

apply (simp add: isLub_def r_def A_def)

done

lemma (in CL) lub_in_lattice: "S \<subseteq> A ==> lub S cl \<in> A"

apply (rule CL_imp_ex_isLub [THEN exE], assumption)

apply (unfold lub_def least_def)

apply (subst some_equality)

apply (simp add: isLub_def)

prefer 2 apply (simp add: isLub_def A_def)

apply (simp add: lub_unique A_def isLub_def)

done

lemma (in CL) lubI:

     "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;

         \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"

apply (rule lub_unique, assumption)

apply (simp add: isLub_def A_def r_def)

apply (unfold isLub_def)

apply (rule conjI)

apply (fold A_def r_def)

apply (rule lub_in_lattice, assumption)

apply (simp add: lub_upper lub_least)

done

lemma (in CL) lubIa: "[| S \<subseteq> A; isLub S cl L |] ==> L = lub S cl"

by (simp add: lubI isLub_def A_def r_def)

lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"

by (simp add: isLub_def  A_def)

lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"

by (simp add: isLub_def r_def)

lemma (in CL) isLub_least:

     "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"

by (simp add: isLub_def A_def r_def)

lemma (in CL) isLubI:

     "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;

         (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"

by (simp add: isLub_def A_def r_def)

subsection {* glb *}

lemma (in CL) glb_in_lattice: "S \<subseteq> A ==> glb S cl \<in> A"

apply (subst glb_dual_lub)

apply (simp add: A_def)

apply (rule dualA_iff [THEN subst])

apply (rule CL.lub_in_lattice)

apply (rule dual)

apply (simp add: dualA_iff)

done

lemma (in CL) glb_lower: "[|S \<subseteq> A; x \<in> S|] ==> (glb S cl, x) \<in> r"

apply (subst glb_dual_lub)

apply (simp add: r_def)

apply (rule dualr_iff [THEN subst])

apply (rule CL.lub_upper)

apply (rule dual)

apply (simp add: dualA_iff A_def, assumption)

done

text {*

  Reduce the sublattice property by using substructural properties;

  abandoned see @{text "Tarski_4.ML"}.

*}

lemma (in CLF) [simp]:

    "f: pset cl -> pset cl & monotone f (pset cl) (order cl)"

apply (insert f_cl)

apply (simp add: CLF_set_def)

done

declare (in CLF) f_cl [simp]

lemma (in CLF) f_in_funcset: "f \<in> A -> A"

by (simp add: A_def)

lemma (in CLF) monotone_f: "monotone f A r"

by (simp add: A_def r_def)

lemma (in CLF) CLF_dual: "(dual cl, f) \<in> CLF_set"

apply (simp add: CLF_set_def  CL_dualCL monotone_dual)

apply (simp add: dualA_iff)

done

lemma (in CLF) dual:

  "CLF (dual cl) f"

apply (rule CLF.intro)

apply (rule CLF_dual)

done

subsection {* fixed points *}

lemma fix_subset: "fix f A \<subseteq> A"

by (simp add: fix_def, fast)

lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"

by (simp add: fix_def)

lemma fixf_subset:

     "[| A \<subseteq> B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"

by (simp add: fix_def, auto)

subsection {* lemmas for Tarski, lub *}

lemma (in CLF) lubH_le_flubH:

     "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"

apply (rule lub_least, fast)

apply (rule f_in_funcset [THEN funcset_mem])

apply (rule lub_in_lattice, fast)

-- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *}

apply (rule ballI)

apply (rule transE)

-- {* instantiates @{text "(x, ???z) \<in> order cl to (x, f x)"}, *}

-- {* because of the def of @{text H} *}

apply fast

-- {* so it remains to show @{text "(f x, f (lub H cl)) \<in> r"} *}

apply (rule_tac f = "f" in monotoneE)

apply (rule monotone_f, fast)

apply (rule lub_in_lattice, fast)

apply (rule lub_upper, fast)

apply assumption

done

lemma (in CLF) flubH_le_lubH:

     "[|  H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"

apply (rule lub_upper, fast)

apply (rule_tac t = "H" in ssubst, assumption)

apply (rule CollectI)

apply (rule conjI)

apply (rule_tac [2] f_in_funcset [THEN funcset_mem])

apply (rule_tac [2] lub_in_lattice)

prefer 2 apply fast

apply (rule_tac f = "f" in monotoneE)

apply (rule monotone_f)

  apply (blast intro: lub_in_lattice)

 apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])

apply (simp add: lubH_le_flubH)

done

lemma (in CLF) lubH_is_fixp:

     "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"

apply (simp add: fix_def)

apply (rule conjI)

apply (rule lub_in_lattice, fast)

apply (rule antisymE)

apply (simp add: flubH_le_lubH)

apply (simp add: lubH_le_flubH)

done

lemma (in CLF) fix_in_H:

     "[| H = {x. (x, f x) \<in> r & x \<in> A};  x \<in> P |] ==> x \<in> H"

by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl_on

                    fix_subset [of f A, THEN subsetD])

lemma (in CLF) fixf_le_lubH:

     "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"

apply (rule ballI)

apply (rule lub_upper, fast)

apply (rule fix_in_H)

apply (simp_all add: P_def)

done

lemma (in CLF) lubH_least_fixf:

     "H = {x. (x, f x) \<in> r & x \<in> A}

      ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"

apply (rule allI)

apply (rule impI)

apply (erule bspec)

apply (rule lubH_is_fixp, assumption)

done

subsection {* Tarski fixpoint theorem 1, first part *}

lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"

apply (rule sym)

apply (simp add: P_def)

apply (rule lubI)

apply (rule fix_subset)

apply (rule lub_in_lattice, fast)

apply (simp add: fixf_le_lubH)

apply (simp add: lubH_least_fixf)

done

lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"

  -- {* Tarski for glb *}

apply (simp add: glb_dual_lub P_def A_def r_def)

apply (rule dualA_iff [THEN subst])

apply (rule CLF.lubH_is_fixp)

apply (rule dual)

apply (simp add: dualr_iff dualA_iff)

done

lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"

apply (simp add: glb_dual_lub P_def A_def r_def)

apply (rule dualA_iff [THEN subst])

apply (simp add: CLF.T_thm_1_lub [of _ f, OF dual]

                 dualPO CL_dualCL CLF_dual dualr_iff)

done

subsection {* interval *}

lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"

apply (insert CO_refl_on)

apply (simp add: refl_on_def, blast)

done

lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b \<subseteq> A"

apply (simp add: interval_def)

apply (blast intro: rel_imp_elem)

done

lemma (in CLF) intervalI:

     "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"

by (simp add: interval_def)

lemma (in CLF) interval_lemma1:

     "[| S \<subseteq> interval r a b; x \<in> S |] ==> (a, x) \<in> r"

by (unfold interval_def, fast)

lemma (in CLF) interval_lemma2:

     "[| S \<subseteq> interval r a b; x \<in> S |] ==> (x, b) \<in> r"

by (unfold interval_def, fast)

lemma (in CLF) a_less_lub:

     "[| S \<subseteq> A; S \<noteq> {};

         \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"

by (blast intro: transE)

lemma (in CLF) glb_less_b:

     "[| S \<subseteq> A; S \<noteq> {};

         \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"

by (blast intro: transE)

lemma (in CLF) S_intv_cl:

     "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b |]==> S \<subseteq> A"

by (simp add: subset_trans [OF _ interval_subset])

lemma (in CLF) L_in_interval:

     "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b;

         S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"

apply (rule intervalI)

apply (rule a_less_lub)

prefer 2 apply assumption

apply (simp add: S_intv_cl)

apply (rule ballI)

apply (simp add: interval_lemma1)

apply (simp add: isLub_upper)

-- {* @{text "(L, b) \<in> r"} *}

apply (simp add: isLub_least interval_lemma2)

done

lemma (in CLF) G_in_interval:

     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> interval r a b; isGlb S cl G;

         S \<noteq> {} |] ==> G \<in> interval r a b"

apply (simp add: interval_dual)

apply (simp add: CLF.L_in_interval [of _ f, OF dual]

                 dualA_iff A_def isGlb_dual_isLub)

done

lemma (in CLF) intervalPO:

     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]

      ==> (| pset = interval r a b, order = induced (interval r a b) r |)

          \<in> PartialOrder"

apply (rule po_subset_po)

apply (simp add: interval_subset)

done

lemma (in CLF) intv_CL_lub:

 "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]

  ==> \<forall>S. S \<subseteq> interval r a b -->

          (\<exists>L. isLub S (| pset = interval r a b,

                          order = induced (interval r a b) r |)  L)"

apply (intro strip)

apply (frule S_intv_cl [THEN CL_imp_ex_isLub])

prefer 2 apply assumption

apply assumption

apply (erule exE)

-- {* define the lub for the interval as *}

apply (rule_tac x = "if S = {} then a else L" in exI)

apply (simp (no_asm_simp) add: isLub_def split del: split_if)

apply (intro impI conjI)

-- {* @{text "(if S = {} then a else L) \<in> interval r a b"} *}

apply (simp add: CL_imp_PO L_in_interval)

apply (simp add: left_in_interval)

-- {* lub prop 1 *}

apply (case_tac "S = {}")

-- {* @{text "S = {}, y \<in> S = False => everything"} *}

apply fast

-- {* @{text "S \<noteq> {}"} *}

apply simp

-- {* @{text "\<forall>y:S. (y, L) \<in> induced (interval r a b) r"} *}

apply (rule ballI)

apply (simp add: induced_def  L_in_interval)

apply (rule conjI)

apply (rule subsetD)

apply (simp add: S_intv_cl, assumption)

apply (simp add: isLub_upper)

-- {* @{text "\<forall>z:interval r a b. (\<forall>y:S. (y, z) \<in> induced (interval r a b) r \<longrightarrow> (if S = {} then a else L, z) \<in> induced (interval r a b) r"} *}

apply (rule ballI)

apply (rule impI)

apply (case_tac "S = {}")

-- {* @{text "S = {}"} *}

apply simp

apply (simp add: induced_def  interval_def)

apply (rule conjI)

apply (rule reflE, assumption)

apply (rule interval_not_empty)

apply (simp add: interval_def)

-- {* @{text "S \<noteq> {}"} *}

apply simp

apply (simp add: induced_def  L_in_interval)

apply (rule isLub_least, assumption)

apply (rule subsetD)

prefer 2 apply assumption

apply (simp add: S_intv_cl, fast)

done

lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]

lemma (in CLF) interval_is_sublattice:

     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]

        ==> interval r a b <<= cl"

apply (rule sublatticeI)

apply (simp add: interval_subset)

apply (rule CompleteLatticeI)

apply (simp add: intervalPO)

 apply (simp add: intv_CL_lub)

apply (simp add: intv_CL_glb)

done

lemmas (in CLF) interv_is_compl_latt =

    interval_is_sublattice [THEN sublattice_imp_CL]

subsection {* Top and Bottom *}

lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"

by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)

lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"

by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)

lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"

apply (simp add: Bot_def least_def)

apply (rule_tac a="glb A cl" in someI2)

apply (simp_all add: glb_in_lattice glb_lower 

                     r_def [symmetric] A_def [symmetric])

done

lemma (in CLF) Top_in_lattice: "Top cl \<in> A"

apply (simp add: Top_dual_Bot A_def)

apply (rule dualA_iff [THEN subst])

apply (rule CLF.Bot_in_lattice [OF dual])

done

lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"

apply (simp add: Top_def greatest_def)

apply (rule_tac a="lub A cl" in someI2)

apply (rule someI2)

apply (simp_all add: lub_in_lattice lub_upper 

                     r_def [symmetric] A_def [symmetric])

done

lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"

apply (simp add: Bot_dual_Top r_def)

apply (rule dualr_iff [THEN subst])

apply (rule CLF.Top_prop [OF dual])

apply (simp add: dualA_iff A_def)

done

lemma (in CLF) Top_intv_not_empty: "x \<in> A  ==> interval r x (Top cl) \<noteq> {}"

apply (rule notI)

apply (drule_tac a = "Top cl" in equals0D)

apply (simp add: interval_def)

apply (simp add: refl_on_def Top_in_lattice Top_prop)

done

lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"

apply (simp add: Bot_dual_Top)

apply (subst interval_dual)

prefer 2 apply assumption

apply (simp add: A_def)

apply (rule dualA_iff [THEN subst])

apply (rule CLF.Top_in_lattice [OF dual])

apply (rule CLF.Top_intv_not_empty [OF dual])

apply (simp add: dualA_iff A_def)

done

subsection {* fixed points form a partial order *}

lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"

by (simp add: P_def fix_subset po_subset_po)

lemma (in Tarski) Y_subset_A: "Y \<subseteq> A"

apply (rule subset_trans [OF _ fix_subset])

apply (rule Y_ss [simplified P_def])

done

lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"

  by (rule Y_subset_A [THEN lub_in_lattice])

lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"

apply (rule lub_least)

apply (rule Y_subset_A)

apply (rule f_in_funcset [THEN funcset_mem])

apply (rule lubY_in_A)

-- {* @{text "Y \<subseteq> P ==> f x = x"} *}

apply (rule ballI)

apply (rule_tac t = "x" in fix_imp_eq [THEN subst])

apply (erule Y_ss [simplified P_def, THEN subsetD])

-- {* @{text "reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r"} by monotonicity *}

apply (rule_tac f = "f" in monotoneE)

apply (rule monotone_f)

apply (simp add: Y_subset_A [THEN subsetD])

apply (rule lubY_in_A)

apply (simp add: lub_upper Y_subset_A)

done

lemma (in Tarski) intY1_subset: "intY1 \<subseteq> A"

apply (unfold intY1_def)

apply (rule interval_subset)

apply (rule lubY_in_A)

apply (rule Top_in_lattice)

done

lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]

lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"

apply (simp add: intY1_def  interval_def)

apply (rule conjI)

apply (rule transE)

apply (rule lubY_le_flubY)

-- {* @{text "(f (lub Y cl), f x) \<in> r"} *}

apply (rule_tac f=f in monotoneE)

apply (rule monotone_f)

apply (rule lubY_in_A)

apply (simp add: intY1_def interval_def  intY1_elem)

apply (simp add: intY1_def  interval_def)

-- {* @{text "(f x, Top cl) \<in> r"} *}

apply (rule Top_prop)

apply (rule f_in_funcset [THEN funcset_mem])

apply (simp add: intY1_def interval_def  intY1_elem)

done

lemma (in Tarski) intY1_mono:

     "monotone (%x: intY1. f x) intY1 (induced intY1 r)"

apply (auto simp add: monotone_def induced_def intY1_f_closed)

apply (blast intro: intY1_elem monotone_f [THEN monotoneE])

done

lemma (in Tarski) intY1_is_cl:

    "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"

apply (unfold intY1_def)

apply (rule interv_is_compl_latt)

apply (rule lubY_in_A)

apply (rule Top_in_lattice)

apply (rule Top_intv_not_empty)

apply (rule lubY_in_A)

done

lemma (in Tarski) v_in_P: "v \<in> P"

apply (unfold P_def)

apply (rule_tac A = "intY1" in fixf_subset)

apply (rule intY1_subset)

unfolding v_def

apply (rule CLF.glbH_is_fixp [OF CLF.intro, unfolded CLF_set_def, of "\<lparr>pset = intY1, order = induced intY1 r\<rparr>", simplified])

apply auto

apply (rule intY1_is_cl)

apply (erule intY1_f_closed)

apply (rule intY1_mono)

done

lemma (in Tarski) z_in_interval:

     "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"

apply (unfold intY1_def P_def)

apply (rule intervalI)

prefer 2

 apply (erule fix_subset [THEN subsetD, THEN Top_prop])

apply (rule lub_least)

apply (rule Y_subset_A)

apply (fast elim!: fix_subset [THEN subsetD])

apply (simp add: induced_def)

done

lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]

      ==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"

apply (simp add: induced_def  intY1_f_closed z_in_interval P_def)

apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD]

                 reflE)

done

lemma (in Tarski) tarski_full_lemma:

     "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"

apply (rule_tac x = "v" in exI)

apply (simp add: isLub_def)

-- {* @{text "v \<in> P"} *}

apply (simp add: v_in_P)

apply (rule conjI)

-- {* @{text v} is lub *}

-- {* @{text "1. \<forall>y:Y. (y, v) \<in> induced P r"} *}

apply (rule ballI)

apply (simp add: induced_def subsetD v_in_P)

apply (rule conjI)

apply (erule Y_ss [THEN subsetD])

apply (rule_tac b = "lub Y cl" in transE)

apply (rule lub_upper)

apply (rule Y_subset_A, assumption)

apply (rule_tac b = "Top cl" in interval_imp_mem)

apply (simp add: v_def)

apply (fold intY1_def)

apply (rule CL.glb_in_lattice [OF CL.intro [OF intY1_is_cl], simplified])

apply auto

apply (rule indI)

  prefer 3 apply assumption

 prefer 2 apply (simp add: v_in_P)

apply (unfold v_def)

apply (rule indE)

apply (rule_tac [2] intY1_subset)

apply (rule CL.glb_lower [OF CL.intro [OF intY1_is_cl], simplified])

  apply (simp add: CL_imp_PO intY1_is_cl)

 apply force

apply (simp add: induced_def intY1_f_closed z_in_interval)

apply (simp add: P_def fix_imp_eq [of _ f A] reflE

                 fix_subset [of f A, THEN subsetD])

done

lemma CompleteLatticeI_simp:

     "[| (| pset = A, order = r |) \<in> PartialOrder;

         \<forall>S. S \<subseteq> A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]

    ==> (| pset = A, order = r |) \<in> CompleteLattice"

by (simp add: CompleteLatticeI Rdual)

theorem (in CLF) Tarski_full:

     "(| pset = P, order = induced P r|) \<in> CompleteLattice"

apply (rule CompleteLatticeI_simp)

apply (rule fixf_po, clarify)

apply (simp add: P_def A_def r_def)

apply (rule Tarski.tarski_full_lemma [OF Tarski.intro [OF _ Tarski_axioms.intro]])

proof - show "CLF cl f" .. qed

end