# Theory Tarski

theory Tarski
imports FuncSet
```(*  Title:      HOL/ex/Tarski.thy
Author:     Florian KammÃ¼ller, Cambridge University Computer Laboratory
*)

section ‹The Full Theorem of Tarski›

theory Tarski
imports Main "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 (λx. ∀y∈S. (y, x) ∈ order po) po"

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

definition isLub :: "['a set, 'a potype, 'a] ⇒ bool"
where "isLub S po =
(λL. 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 =
(λG. (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 (λx. True) po"

definition Top :: "'a potype ⇒ 'a"
where "Top po = greatest (λx. 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 unfold_locales
using cl_co unfolding CompleteLattice_def
apply auto
done

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 unfold_locales
using f_cl unfolding CLF_set_def
apply auto
done

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. ((λx ∈ intY1. f x) x, x) ∈ induced intY1 r ∧ x ∈ intY1}
⦇pset = intY1, order = induced intY1 r⦈"

subsection ‹Partial Order›

context PO
begin

lemma dual: "PO (dual cl)"
apply unfold_locales
using cl_po
unfolding PartialOrder_def dual_def
apply auto
done

lemma PO_imp_refl_on [simp]: "refl_on A r"
using cl_po by (simp add: PartialOrder_def A_def r_def)

lemma PO_imp_sym [simp]: "antisym r"
using cl_po by (simp add: PartialOrder_def r_def)

lemma PO_imp_trans [simp]: "trans r"
using cl_po by (simp add: PartialOrder_def r_def)

lemma reflE: "x ∈ A ⟹ (x, x) ∈ r"
using cl_po by (simp add: PartialOrder_def refl_on_def A_def r_def)

lemma antisymE: "⟦(a, b) ∈ r; (b, a) ∈ r⟧ ⟹ a = b"
using cl_po by (simp add: PartialOrder_def antisym_def r_def)

lemma transE: "⟦(a, b) ∈ r; (b, c) ∈ r⟧ ⟹ (a, c) ∈ r"
using cl_po by (simp add: PartialOrder_def r_def) (unfold trans_def, fast)

lemma monotoneE: "⟦monotone f A r; x ∈ A; y ∈ A; (x, y) ∈ r⟧ ⟹ (f x, f y) ∈ r"

lemma po_subset_po: "S ⊆ A ⟹ ⦇pset = S, order = induced S r⦈ ∈ PartialOrder"
apply auto
― ‹refl›
apply (blast intro: reflE)
― ‹antisym›
apply (blast intro: antisymE)
― ‹trans›
apply (blast intro: transE)
done

lemma indE: "⟦(x, y) ∈ induced S r; S ⊆ A⟧ ⟹ (x, y) ∈ r"

lemma indI: "⟦(x, y) ∈ r; x ∈ S; y ∈ S⟧ ⟹ (x, y) ∈ induced S r"

end

lemma (in CL) CL_imp_ex_isLub: "S ⊆ A ⟹ ∃L. isLub S cl L"
using cl_co by (simp add: CompleteLattice_def A_def)

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"
using cl_po by (simp add: PartialOrder_def dual_def)

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)
apply fast
apply 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 (auto simp: PartialOrder_def CompleteLattice_def)

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]*)

context CL
begin

lemma CO_refl_on: "refl_on A r"
by (rule PO_imp_refl_on)

lemma CO_antisym: "antisym r"
by (rule PO_imp_sym)

lemma CO_trans: "trans r"
by (rule PO_imp_trans)

end

lemma CompleteLatticeI:
"⟦po ∈ PartialOrder; ∀S. S ⊆ pset po ⟶ (∃L. isLub S po L);
∀S. S ⊆ pset po ⟶ (∃G. isGlb S po G)⟧
⟹ po ∈ CompleteLattice"
unfolding CompleteLattice_def by blast

lemma (in CL) CL_dualCL: "dual cl ∈ CompleteLattice"
using cl_co
apply (fold dual_def)
apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric] dualPO)
done

context PO
begin

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

lemma dualr_iff: "(x, y) ∈ (order (dual cl)) ⟷ (y, x) ∈ order cl"

lemma 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 interval_dual: "⟦x ∈ A; y ∈ A⟧ ⟹ interval r x y = interval (order(dual cl)) y x"
apply (fold r_def)
apply fast
done

lemma trans: "(x, y) ∈ r ⟹ (y, z) ∈ r ⟹ (x, z) ∈ r"
using cl_po
apply (auto simp add: PartialOrder_def r_def)
unfolding trans_def
apply blast
done

lemma interval_not_empty: "interval r a b ≠ {} ⟹ (a, b) ∈ r"
by (simp add: interval_def) (use trans in blast)

lemma interval_imp_mem: "x ∈ interval r a b ⟹ (a, x) ∈ r"

lemma left_in_interval: "⟦a ∈ A; b ∈ A; interval r a b ≠ {}⟧ ⟹ a ∈ interval r a b"
done

lemma right_in_interval: "⟦a ∈ A; b ∈ A; interval r a b ≠ {}⟧ ⟹ b ∈ interval r a b"
done

end

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›

context CL
begin

lemma lub_unique: "⟦S ⊆ A; isLub S cl x; isLub S cl L⟧ ⟹ x = L"
by (rule antisymE) (auto simp add: isLub_def r_def)

lemma 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: lub_unique A_def isLub_def)
done

lemma 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: lub_unique A_def isLub_def)
apply (simp add: isLub_def r_def A_def)
done

lemma 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)
prefer 2 apply (simp add: isLub_def A_def)
apply (simp add: lub_unique A_def isLub_def)
done

lemma 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)
done

lemma lubIa: "⟦S ⊆ A; isLub S cl L⟧ ⟹ L = lub S cl"
by (simp add: lubI isLub_def A_def r_def)

lemma isLub_in_lattice: "isLub S cl L ⟹ L ∈ A"

lemma isLub_upper: "⟦isLub S cl L; y ∈ S⟧ ⟹ (y, L) ∈ r"

lemma 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 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)

end

subsection ‹glb›

context CL
begin

lemma glb_in_lattice: "S ⊆ A ⟹ glb S cl ∈ A"
apply (subst glb_dual_lub)
apply (rule dualA_iff [THEN subst])
apply (rule CL.lub_in_lattice)
apply (rule dual)
done

lemma glb_lower: "⟦S ⊆ A; x ∈ S⟧ ⟹ (glb S cl, x) ∈ r"
apply (subst glb_dual_lub)
apply (rule dualr_iff [THEN subst])
apply (rule CL.lub_upper)
apply (rule dual)
apply (simp add: dualA_iff A_def, assumption)
done

end

text ‹
Reduce the sublattice property by using substructural properties;
abandoned see ‹Tarski_4.ML›.
›

context CLF
begin

lemma [simp]: "f ∈ pset cl → pset cl ∧ monotone f (pset cl) (order cl)"
using f_cl by (simp add: CLF_set_def)

declare f_cl [simp]

lemma f_in_funcset: "f ∈ A → A"

lemma monotone_f: "monotone f A r"

lemma CLF_dual: "(dual cl, f) ∈ CLF_set"

lemma dual: "CLF (dual cl) f"
by (rule CLF.intro) (rule CLF_dual)

end

subsection ‹fixed points›

lemma fix_subset: "fix f A ⊆ A"
by (auto simp: fix_def)

lemma fix_imp_eq: "x ∈ fix f A ⟹ f x = x"

lemma fixf_subset: "⟦A ⊆ B; x ∈ fix (λy ∈ A. f y) A⟧ ⟹ x ∈ fix f B"
by (auto simp: fix_def)

subsection ‹lemmas for Tarski, lub›

context CLF
begin

lemma 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)
― ‹‹∀x:H. (x, f (lub H r)) ∈ r››
apply (rule ballI)
apply (rule transE)
― ‹instantiates ‹(x, ???z) ∈ order cl to (x, f x)›,›
― ‹because of the def of ‹H››
apply fast
― ‹so it remains to show ‹(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 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])
done

lemma lubH_is_fixp: "H = {x. (x, f x) ∈ r ∧ x ∈ A} ⟹ lub H cl ∈ fix f A"
apply (rule conjI)
apply (rule lub_in_lattice, fast)
apply (rule antisymE)
done

lemma 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 fix_subset [of f A, THEN subsetD])

lemma 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)
apply fast
apply (rule fix_in_H)
done

lemma 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 T_thm_1_lub: "lub P cl = lub {x. (x, f x) ∈ r ∧ x ∈ A} cl"
apply (rule sym)
apply (rule lubI)
apply (rule fix_subset)
apply (rule lub_in_lattice, fast)
done

lemma 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)
done

lemma 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 rel_imp_elem: "(x, y) ∈ r ⟹ x ∈ A"
using CO_refl_on by (auto simp: refl_on_def)

lemma interval_subset: "⟦a ∈ A; b ∈ A⟧ ⟹ interval r a b ⊆ A"
by (simp add: interval_def) (blast intro: rel_imp_elem)

lemma intervalI: "⟦(a, x) ∈ r; (x, b) ∈ r⟧ ⟹ x ∈ interval r a b"

lemma interval_lemma1: "⟦S ⊆ interval r a b; x ∈ S⟧ ⟹ (a, x) ∈ r"
unfolding interval_def by fast

lemma interval_lemma2: "⟦S ⊆ interval r a b; x ∈ S⟧ ⟹ (x, b) ∈ r"
unfolding interval_def by fast

lemma a_less_lub: "⟦S ⊆ A; S ≠ {}; ∀x ∈ S. (a,x) ∈ r; ∀y ∈ S. (y, L) ∈ r⟧ ⟹ (a, L) ∈ r"
by (blast intro: transE)

lemma glb_less_b: "⟦S ⊆ A; S ≠ {}; ∀x ∈ S. (x,b) ∈ r; ∀y ∈ S. (G, y) ∈ r⟧ ⟹ (G, b) ∈ r"
by (blast intro: transE)

lemma S_intv_cl: "⟦a ∈ A; b ∈ A; S ⊆ interval r a b⟧ ⟹ S ⊆ A"
by (simp add: subset_trans [OF _ interval_subset])

lemma 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 (rule ballI)
― ‹‹(L, b) ∈ r››
done

lemma 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"
(simp add: CLF.L_in_interval [of _ f, OF dual] dualA_iff A_def isGlb_dual_isLub)

lemma intervalPO:
"⟦a ∈ A; b ∈ A; interval r a b ≠ {}⟧
⟹ ⦇pset = interval r a b, order = induced (interval r a b) r⦈ ∈ PartialOrder"
by (rule po_subset_po) (simp add: interval_subset)

lemma 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: if_split)
apply (intro impI conjI)
― ‹‹(if S = {} then a else L) ∈ interval r a b››
― ‹lub prop 1›
apply (case_tac "S = {}")
― ‹‹S = {}, y ∈ S = False ⟹ everything››
apply fast
― ‹‹S ≠ {}››
apply simp
― ‹‹∀y∈S. (y, L) ∈ induced (interval r a b) r››
apply (rule ballI)
apply (rule conjI)
apply (rule subsetD)
― ‹‹∀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 = {}")
― ‹‹S = {}››
apply simp
apply (rule conjI)
apply (rule reflE, assumption)
apply (rule interval_not_empty)
― ‹‹S ≠ {}››
apply simp
apply (rule isLub_least, assumption)
apply (rule subsetD)
prefer 2 apply assumption
done

lemmas intv_CL_glb = intv_CL_lub [THEN Rdual]

lemma interval_is_sublattice: "⟦a ∈ A; b ∈ A; interval r a b ≠ {}⟧ ⟹ interval r a b <<= cl"
apply (rule sublatticeI)
apply (rule CompleteLatticeI)
done

lemmas interv_is_compl_latt = interval_is_sublattice [THEN sublattice_imp_CL]

subsection ‹Top and Bottom›

lemma Top_dual_Bot: "Top cl = Bot (dual cl)"
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)

lemma Bot_dual_Top: "Bot cl = Top (dual cl)"
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)

lemma Bot_in_lattice: "Bot cl ∈ A"
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 Top_in_lattice: "Top cl ∈ A"
apply (rule dualA_iff [THEN subst])
apply (rule CLF.Bot_in_lattice [OF dual])
done

lemma Top_prop: "x ∈ A ⟹ (x, Top cl) ∈ r"
apply (rule_tac a = "lub A cl" in someI2)
apply (rule someI2)
r_def [symmetric] A_def [symmetric])
done

lemma Bot_prop: "x ∈ A ⟹ (Bot cl, x) ∈ r"
apply (rule dualr_iff [THEN subst])
apply (rule CLF.Top_prop [OF dual])
done

lemma 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: refl_on_def Top_in_lattice Top_prop)
done

lemma Bot_intv_not_empty: "x ∈ A ⟹ interval r (Bot cl) x ≠ {}"
apply (subst interval_dual)
prefer 2 apply assumption
apply (rule dualA_iff [THEN subst])
apply (rule CLF.Top_in_lattice [OF dual])
apply (rule CLF.Top_intv_not_empty [OF dual])
done

subsection ‹fixed points form a partial order›

lemma fixf_po: "⦇pset = P, order = induced P r⦈ ∈ PartialOrder"
by (simp add: P_def fix_subset po_subset_po)

end

context Tarski
begin

lemma Y_subset_A: "Y ⊆ A"
by (rule subset_trans [OF _ fix_subset]) (rule Y_ss [simplified P_def])

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

lemma 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)
― ‹‹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])
― ‹‹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)
done

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

lemmas intY1_elem = intY1_subset [THEN subsetD]

lemma intY1_f_closed: "x ∈ intY1 ⟹ f x ∈ intY1"
apply (rule conjI)
apply (rule transE)
apply (rule lubY_le_flubY)
― ‹‹(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)
― ‹‹(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 intY1_mono: "monotone (λ 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 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 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 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])
done

lemma f'z_in_int_rel: "⟦z ∈ P; ∀y∈Y. (y, z) ∈ induced P r⟧
⟹ ((λx ∈ intY1. f x) z, z) ∈ induced intY1 r"
by (simp add: induced_def  intY1_f_closed z_in_interval P_def)
(simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD] reflE)

lemma tarski_full_lemma: "∃L. isLub Y ⦇pset = P, order = induced P r⦈ L"
apply (rule_tac x = "v" in exI)
― ‹‹v ∈ P››
apply (rule conjI)
― ‹‹v› is lub›
― ‹‹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 (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 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

end

lemma CompleteLatticeI_simp:
"⟦⦇pset = A, order = r⦈ ∈ PartialOrder;
∀S. S ⊆ A ⟶ (∃L. isLub S ⦇pset = A, order = r⦈  L)⟧
⟹ ⦇pset = A, order = r⦈ ∈ CompleteLattice"