Theory Tarski

theory Tarski
imports FuncSet
(*  Title:      HOL/ex/Tarski.thy
Author: Florian Kammüller, Cambridge University Computer Laboratory
*)


header {* 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 = (∀x∈A. ∀y∈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 &
(∀y ∈ 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 &
(∀y ∈ pset po. P y --> (y,x): order po))"


definition
lub :: "['a set, 'a potype] => 'a" where
"lub S po = least (%x. ∀y∈S. (y,x): order po) po"

definition
glb :: "['a set, 'a potype] => 'a" where
"glb S po = greatest (%x. ∀y∈S. (x,y): order po) po"

definition
isLub :: "['a set, 'a potype, 'a] => bool" where
"isLub S po = (%L. (L: pset po & (∀y∈S. (y,L): order po) &
(∀z∈pset po. (∀y∈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 & (∀y∈S. (G,y): order po) &
(∀z ∈ pset po. (∀y∈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 &
(∀S. S ⊆ pset cl --> (∃L. isLub S cl L)) &
(∀S. S ⊆ pset cl --> (∃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 ⊆ 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 ⊆ 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 ∈ A ==> (x, x) ∈ r"
apply (insert cl_po)
apply (simp add: PartialOrder_def refl_on_def A_def r_def)
done

lemma (in PO) antisymE: "[| (a, b) ∈ r; (b, a) ∈ r |] ==> a = b"
apply (insert cl_po)
apply (simp add: PartialOrder_def antisym_def r_def)
done

lemma (in PO) transE: "[| (a, b) ∈ r; (b, c) ∈ r|] ==> (a,c) ∈ 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 ∈ A; y ∈ A; (x, y) ∈ r |] ==> (f x, f y) ∈ r"
by (simp add: monotone_def)

lemma (in PO) po_subset_po:
"S ⊆ A ==> (| pset = S, order = induced S r |) ∈ 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) ∈ induced S r; S ⊆ A |] ==> (x, y) ∈ r"
by (simp add: add: induced_def)

lemma (in PO) indI: "[| (x, y) ∈ r; x ∈ S; y ∈ S |] ==> (x, y) ∈ induced S r"
by (simp add: add: induced_def)

lemma (in CL) CL_imp_ex_isLub: "S ⊆ A ==> ∃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: "(∃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: "(∃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 ∈ PartialOrder"
apply (insert cl_po)
apply (simp add: PartialOrder_def dual_def refl_on_converse
trans_converse antisym_converse)
done

lemma Rdual:
"∀S. (S ⊆ A -->( ∃L. isLub S (| pset = A, order = r|) L))
==> ∀S. (S ⊆ A --> (∃G. isGlb S (| pset = A, order = r|) G))"

apply safe
apply (rule_tac x = "lub {y. y ∈ A & (∀k ∈ S. (y, k) ∈ r)}
(|pset = A, order = r|) "
in exI)
apply (drule_tac x = "{y. y ∈ A & (∀k ∈ S. (y,k) ∈ 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 ⊆ 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 ∈ PartialOrder; (∀S. S ⊆ pset po --> (∃L. isLub S po L));
(∀S. S ⊆ pset po --> (∃G. isGlb S po G))|]
==> po ∈ CompleteLattice"

apply (unfold CompleteLattice_def, blast)
done

lemma (in CL) CL_dualCL: "dual cl ∈ 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) ∈ (order(dual cl))) = ((y, x) ∈ 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 ∈ A; y ∈ 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) ∈ r ==> (y, z) ∈ r ==> (x, z) ∈ 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 ≠ {} ==> (a, b) ∈ r"
apply (simp add: interval_def)
using trans by blast

lemma (in PO) interval_imp_mem: "x ∈ interval r a b ==> (a, x) ∈ r"
by (simp add: interval_def)

lemma (in PO) left_in_interval:
"[| a ∈ A; b ∈ A; interval r a b ≠ {} |] ==> a ∈ 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 ∈ A; b ∈ A; interval r a b ≠ {} |] ==> b ∈ 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 |) ∈ CompleteLattice"
by (simp add: sublattice_def CompleteLattice_def r_def)

lemma (in CL) sublatticeI:
"[| S ⊆ A; (| pset = S, order = induced S r |) ∈ 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 ⊆ 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 ⊆ A; x ∈ S|] ==> (x, lub S cl) ∈ r"
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
apply (unfold lub_def least_def)
apply (rule some_equality [THEN ssubst])
apply (simp add: isLub_def)
apply (simp add: lub_unique A_def isLub_def)
apply (simp add: isLub_def r_def)
done

lemma (in CL) lub_least:
"[| S ⊆ A; L ∈ A; ∀x ∈ S. (x,L) ∈ r |] ==> (lub S cl, L) ∈ r"
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
apply (unfold lub_def least_def)
apply (rule_tac s=x in some_equality [THEN ssubst])
apply (simp add: isLub_def)
apply (simp add: lub_unique A_def isLub_def)
apply (simp add: isLub_def r_def A_def)
done

lemma (in CL) lub_in_lattice: "S ⊆ A ==> lub S cl ∈ A"
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
apply (unfold lub_def least_def)
apply (subst some_equality)
apply (simp add: isLub_def)
prefer 2 apply (simp add: isLub_def A_def)
apply (simp add: lub_unique A_def isLub_def)
done

lemma (in CL) lubI:
"[| S ⊆ A; L ∈ A; ∀x ∈ S. (x,L) ∈ r;
∀z ∈ A. (∀y ∈ S. (y,z) ∈ r) --> (L,z) ∈ r |] ==> L = lub S cl"

apply (rule lub_unique, assumption)
apply (simp add: isLub_def A_def r_def)
apply (unfold isLub_def)
apply (rule conjI)
apply (fold A_def r_def)
apply (rule lub_in_lattice, assumption)
apply (simp add: lub_upper lub_least)
done

lemma (in CL) lubIa: "[| S ⊆ A; isLub S cl L |] ==> L = lub S cl"
by (simp add: lubI isLub_def A_def r_def)

lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L ∈ A"
by (simp add: isLub_def A_def)

lemma (in CL) isLub_upper: "[|isLub S cl L; y ∈ S|] ==> (y, L) ∈ r"
by (simp add: isLub_def r_def)

lemma (in CL) isLub_least:
"[| isLub S cl L; z ∈ A; ∀y ∈ S. (y, z) ∈ r|] ==> (L, z) ∈ r"
by (simp add: isLub_def A_def r_def)

lemma (in CL) isLubI:
"[| L ∈ A; ∀y ∈ S. (y, L) ∈ r;
(∀z ∈ A. (∀y ∈ S. (y, z):r) --> (L, z) ∈ r)|] ==> isLub S cl L"

by (simp add: isLub_def A_def r_def)


subsection {* glb *}

lemma (in CL) glb_in_lattice: "S ⊆ A ==> glb S cl ∈ 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 ⊆ A; x ∈ S|] ==> (glb S cl, x) ∈ 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 ∈ 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) ∈ 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 ⊆ A"
by (simp add: fix_def, fast)

lemma fix_imp_eq: "x ∈ fix f A ==> f x = x"
by (simp add: fix_def)

lemma fixf_subset:
"[| A ⊆ B; x ∈ fix (%y: A. f y) A |] ==> x ∈ 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) ∈ r & x ∈ A} ==> (lub H cl, f (lub H cl)) ∈ r"
apply (rule lub_least, fast)
apply (rule f_in_funcset [THEN funcset_mem])
apply (rule lub_in_lattice, fast)
-- {* @{text "∀x:H. (x, f (lub H r)) ∈ r"} *}
apply (rule ballI)
apply (rule transE)
-- {* instantiates @{text "(x, ???z) ∈ 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)) ∈ 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) ∈ r & x ∈ A} |] ==> (f (lub H cl), lub H cl) ∈ 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) ∈ r & x ∈ A} ==> lub H cl ∈ 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) ∈ r & x ∈ A}; x ∈ P |] ==> x ∈ 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) ∈ r & x ∈ A} ==> ∀x ∈ fix f A. (x, lub H cl) ∈ 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) ∈ r & x ∈ A}
==> ∀L. (∀y ∈ fix f A. (y,L) ∈ r) --> (lub H cl, L) ∈ 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) ∈ r & x ∈ 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) ∈ r & x ∈ A} ==> glb H cl ∈ 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) ∈ r & x ∈ 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) ∈ r ==> x ∈ A"
apply (insert CO_refl_on)
apply (simp add: refl_on_def, blast)
done

lemma (in CLF) interval_subset: "[| a ∈ A; b ∈ A |] ==> interval r a b ⊆ A"
apply (simp add: interval_def)
apply (blast intro: rel_imp_elem)
done

lemma (in CLF) intervalI:
"[| (a, x) ∈ r; (x, b) ∈ r |] ==> x ∈ interval r a b"
by (simp add: interval_def)

lemma (in CLF) interval_lemma1:
"[| S ⊆ interval r a b; x ∈ S |] ==> (a, x) ∈ r"
by (unfold interval_def, fast)

lemma (in CLF) interval_lemma2:
"[| S ⊆ interval r a b; x ∈ S |] ==> (x, b) ∈ r"
by (unfold interval_def, fast)

lemma (in CLF) a_less_lub:
"[| S ⊆ A; S ≠ {};
∀x ∈ S. (a,x) ∈ r; ∀y ∈ S. (y, L) ∈ r |] ==> (a,L) ∈ r"

by (blast intro: transE)

lemma (in CLF) glb_less_b:
"[| S ⊆ A; S ≠ {};
∀x ∈ S. (x,b) ∈ r; ∀y ∈ S. (G, y) ∈ r |] ==> (G,b) ∈ r"

by (blast intro: transE)

lemma (in CLF) S_intv_cl:
"[| a ∈ A; b ∈ A; S ⊆ interval r a b |]==> S ⊆ A"
by (simp add: subset_trans [OF _ interval_subset])

lemma (in CLF) L_in_interval:
"[| a ∈ A; b ∈ A; S ⊆ interval r a b;
S ≠ {}; isLub S cl L; interval r a b ≠ {} |] ==> L ∈ 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) ∈ r"} *}
apply (simp add: isLub_least interval_lemma2)
done

lemma (in CLF) G_in_interval:
"[| a ∈ A; b ∈ A; interval r a b ≠ {}; S ⊆ interval r a b; isGlb S cl G;
S ≠ {} |] ==> G ∈ 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 ∈ A; b ∈ A; interval r a b ≠ {} |]
==> (| pset = interval r a b, order = induced (interval r a b) r |)
∈ PartialOrder"

apply (rule po_subset_po)
apply (simp add: interval_subset)
done

lemma (in CLF) intv_CL_lub:
"[| a ∈ A; b ∈ A; interval r a b ≠ {} |]
==> ∀S. S ⊆ interval r a b -->
(∃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) ∈ 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 ∈ S = False => everything"} *}
apply fast
-- {* @{text "S ≠ {}"} *}
apply simp
-- {* @{text "∀y:S. (y, L) ∈ 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 "∀z:interval r a b. (∀y:S. (y, z) ∈ induced (interval r a b) r --> (if S = {} then a else L, z) ∈ 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 ≠ {}"} *}
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 ∈ A; b ∈ A; interval r a b ≠ {} |]
==> 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 ∈ 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 ∈ 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 ∈ A ==> (x, Top cl) ∈ 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 ∈ A ==> (Bot cl, x) ∈ 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 ∈ A ==> interval r x (Top cl) ≠ {}"
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 ∈ A ==> interval r (Bot cl) x ≠ {}"
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|) ∈ PartialOrder"
by (simp add: P_def fix_subset po_subset_po)

lemma (in Tarski) Y_subset_A: "Y ⊆ A"
apply (rule subset_trans [OF _ fix_subset])
apply (rule Y_ss [simplified P_def])
done

lemma (in Tarski) lubY_in_A: "lub Y cl ∈ A"
by (rule Y_subset_A [THEN lub_in_lattice])

lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) ∈ r"
apply (rule lub_least)
apply (rule Y_subset_A)
apply (rule f_in_funcset [THEN funcset_mem])
apply (rule lubY_in_A)
-- {* @{text "Y ⊆ P ==> f x = x"} *}
apply (rule ballI)
apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
apply (erule Y_ss [simplified P_def, THEN subsetD])
-- {* @{text "reduce (f x, f (lub Y cl)) ∈ r to (x, lub Y cl) ∈ r"} by monotonicity *}
apply (rule_tac f = "f" in monotoneE)
apply (rule monotone_f)
apply (simp add: Y_subset_A [THEN subsetD])
apply (rule lubY_in_A)
apply (simp add: lub_upper Y_subset_A)
done

lemma (in Tarski) intY1_subset: "intY1 ⊆ A"
apply (unfold intY1_def)
apply (rule interval_subset)
apply (rule lubY_in_A)
apply (rule Top_in_lattice)
done

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

lemma (in Tarski) intY1_f_closed: "x ∈ intY1 ==> f x ∈ 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) ∈ 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) ∈ 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 |) ∈ 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 ∈ 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 "(|pset = intY1, order = induced intY1 r|)),", 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 ∈ P; ∀y∈Y. (y, z) ∈ induced P r |] ==> z ∈ 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 ∈ P; ∀y∈Y. (y, z) ∈ induced P r |]
==> ((%x: intY1. f x) z, z) ∈ 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:
"∃L. isLub Y (| pset = P, order = induced P r |) L"
apply (rule_tac x = "v" in exI)
apply (simp add: isLub_def)
-- {* @{text "v ∈ P"} *}
apply (simp add: v_in_P)
apply (rule conjI)
-- {* @{text v} is lub *}
-- {* @{text "1. ∀y:Y. (y, v) ∈ 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 |) ∈ PartialOrder;
∀S. S ⊆ A --> (∃L. isLub S (| pset = A, order = r |) L) |]
==> (| pset = A, order = r |) ∈ CompleteLattice"

by (simp add: CompleteLatticeI Rdual)

theorem (in CLF) Tarski_full:
"(| pset = P, order = induced P r|) ∈ 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