# Theory Tarski

theory Tarski
imports FuncSet
```(*  Title:      HOL/Metis_Examples/Tarski.thy
Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
Author:     Jasmin Blanchette, TU Muenchen

Metis example featuring the full theorem of Tarski.
*)

section ‹Metis Example Featuring the Full Theorem of Tarski›

theory Tarski
imports Main "HOL-Library.FuncSet"
begin

declare [[metis_new_skolem]]

(*Many of these higher-order problems appear to be impossible using the
current linkup. They often seem to need either higher-order unification
or explicit reasoning about connectives such as conjunction. The numerous
set comprehensions are to blame.*)

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 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
sublattice_syntax :: "['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 PO =
fixes cl :: "'a potype"
and A  :: "'a set"
and r  :: "('a * 'a) set"
assumes cl_po:  "cl ∈ PartialOrder"
defines A_def: "A == pset cl"
and  r_def: "r == order cl"

locale CL = PO +
assumes cl_co:  "cl ∈ CompleteLattice"

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)})"

locale CLF = CL +
fixes f :: "'a => 'a"
and P :: "'a set"
assumes f_cl:  "(cl,f) ∈ CLF_set" (*was the equivalent "f : CLF``{cl}"*)
defines P_def: "P == fix f A"

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›

lemma (in PO) PO_imp_refl_on: "refl_on A r"
apply (insert cl_po)
apply (simp add: PartialOrder_def A_def r_def)
done

lemma (in PO) PO_imp_sym: "antisym r"
apply (insert cl_po)
apply (simp add: PartialOrder_def r_def)
done

lemma (in PO) PO_imp_trans: "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"

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

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)
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 PO.PO_imp_refl_on  [OF PO.intro [OF CL_imp_PO], simp]
declare PO.PO_imp_sym   [OF PO.intro [OF CL_imp_PO], simp]
declare PO.PO_imp_trans [OF PO.intro [OF CL_imp_PO], 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) interval_not_empty:
"[| trans r; interval r a b ≠ {} |] ==> (a, b) ∈ r"
apply (simp add: interval_def)
apply (unfold trans_def, blast)
done

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 A_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)

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 CL.intro)
apply (rule PO.intro)
apply (rule dualPO)
apply (rule CL_axioms.intro)
apply (rule CL_dualCL)
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 CL.intro)
apply (rule PO.intro)
apply (rule dualPO)
apply (rule CL_axioms.intro)
apply (rule CL_dualCL)
apply (simp add: dualA_iff A_def, assumption)
done

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

declare (in CLF) f_cl [simp]

lemma (in CLF) [simp]:
"f ∈ pset cl → pset cl ∧ monotone f (pset cl) (order cl)"
proof -
have "∀u v. (v, u) ∈ CLF_set ⟶ u ∈ {R ∈ pset v → pset v. monotone R (pset v) (order v)}"
unfolding CLF_set_def using SigmaE2 by blast
hence F1: "∀u v. (v, u) ∈ CLF_set ⟶ u ∈ pset v → pset v ∧ monotone u (pset v) (order v)"
using CollectE by blast
hence "Tarski.monotone f (pset cl) (order cl)" by (metis f_cl)
hence "(cl, f) ∈ CLF_set ∧ Tarski.monotone f (pset cl) (order cl)"
by (metis f_cl)
thus "f ∈ pset cl → pset cl ∧ Tarski.monotone f (pset cl) (order cl)"
using F1 by metis
qed

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)

(*never proved, 2007-01-22*)

declare (in CLF) CLF_set_def [simp] CL_dualCL [simp] monotone_dual [simp] dualA_iff [simp]

lemma (in CLF) CLF_dual: "(dual cl, f) ∈ CLF_set"
apply (simp del: dualA_iff)
apply (simp)
done

declare (in CLF) CLF_set_def[simp del] CL_dualCL[simp del] monotone_dual[simp del]
dualA_iff[simp del]

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 (λy ∈ A. f y) A |] ==> x ∈ fix f B"
by (simp add: fix_def, auto)

subsection ‹lemmas for Tarski, lub›

(*never proved, 2007-01-22*)

declare CL.lub_least[intro] CLF.f_in_funcset[intro] funcset_mem[intro] CL.lub_in_lattice[intro] PO.transE[intro] PO.monotoneE[intro] CLF.monotone_f[intro] CL.lub_upper[intro]

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)
― ‹‹∀x:H. (x, f (lub H r)) ∈ r››
apply (rule ballI)
(*never proved, 2007-01-22*)
apply (rule transE)
― ‹instantiates ‹(x, ?z) ∈ order cl to (x, f x)›,›
― ‹because of the definition 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

declare CL.lub_least[rule del] CLF.f_in_funcset[rule del]
funcset_mem[rule del] CL.lub_in_lattice[rule del]
PO.transE[rule del] PO.monotoneE[rule del]
CLF.monotone_f[rule del] CL.lub_upper[rule del]

(*never proved, 2007-01-22*)

declare CLF.f_in_funcset[intro] funcset_mem[intro] CL.lub_in_lattice[intro]
PO.monotoneE[intro] CLF.monotone_f[intro] CL.lub_upper[intro]
CLF.lubH_le_flubH[simp]

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)
by (metis (lifting) CO_refl_on lubH_le_flubH monotone_def monotone_f refl_onD1 refl_onD2)

declare CLF.f_in_funcset[rule del] funcset_mem[rule del]
CL.lub_in_lattice[rule del] PO.monotoneE[rule del]
CLF.monotone_f[rule del] CL.lub_upper[rule del]
CLF.lubH_le_flubH[simp del]

(*never proved, 2007-01-22*)

(* Single-step version fails. The conjecture clauses refer to local abstraction
functions (Frees). *)
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)
proof -
assume A1: "H = {x. (x, f x) ∈ r ∧ x ∈ A}"
have F1: "∀u v. v ∩ u ⊆ u" by (metis Int_commute Int_lower1)
have "{R. (R, f R) ∈ r} ∩ {R. R ∈ A} = H" using A1 by (metis Collect_conj_eq)
hence "H ⊆ {R. R ∈ A}" using F1 by metis
hence "H ⊆ A" by (metis Collect_mem_eq)
hence "lub H cl ∈ A" by (metis lub_in_lattice)
thus "lub {x. (x, f x) ∈ r ∧ x ∈ A} cl ∈ A" using A1 by metis
next
assume A1: "H = {x. (x, f x) ∈ r ∧ x ∈ A}"
have F1: "∀v. {R. R ∈ v} = v" by (metis Collect_mem_eq)
have F2: "∀w u. {R. R ∈ u ∧ R ∈ w} = u ∩ w"
by (metis Collect_conj_eq Collect_mem_eq)
have F3: "∀x v. {R. v R ∈ x} = v -` x" by (metis vimage_def)
hence F4: "A ∩ (λR. (R, f R)) -` r = H" using A1 by auto
hence F5: "(f (lub H cl), lub H cl) ∈ r"
by (metis A1 flubH_le_lubH)
have F6: "(lub H cl, f (lub H cl)) ∈ r"
by (metis A1 lubH_le_flubH)
have "(lub H cl, f (lub H cl)) ∈ r ⟶ f (lub H cl) = lub H cl"
using F5 by (metis antisymE)
hence "f (lub H cl) = lub H cl" using F6 by metis
thus "H = {x. (x, f x) ∈ r ∧ x ∈ A}
⟹ f (lub {x. (x, f x) ∈ r ∧ x ∈ A} cl) =
lub {x. (x, f x) ∈ r ∧ x ∈ A} cl"
by metis
qed

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 (metis CO_refl_on lubH_le_flubH refl_onD1)
apply (metis antisymE flubH_le_lubH 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 (metis P_def lubH_is_fixp)
done

subsection ‹Tarski fixpoint theorem 1, first part›

declare CL.lubI[intro] fix_subset[intro] CL.lub_in_lattice[intro]
CLF.fixf_le_lubH[simp] CLF.lubH_least_fixf[simp]

lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) ∈ r & x ∈ A} cl"
(*sledgehammer;*)
apply (rule sym)
apply (simp add: P_def)
apply (rule lubI)
apply (simp add: fix_subset)
using fix_subset lubH_is_fixp apply fastforce
apply (simp add: fixf_le_lubH)
using lubH_is_fixp apply blast
done

declare CL.lubI[rule del] fix_subset[rule del] CL.lub_in_lattice[rule del]
CLF.fixf_le_lubH[simp del] CLF.lubH_least_fixf[simp del]

(*never proved, 2007-01-22*)

declare glb_dual_lub[simp] PO.dualA_iff[intro] CLF.lubH_is_fixp[intro]
PO.dualPO[intro] CL.CL_dualCL[intro] PO.dualr_iff[simp]

lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) ∈ r & x ∈ A} ==> glb H cl ∈ P"
― ‹Tarski for glb›
(*sledgehammer;*)
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 CLF.intro)
apply (rule CL.intro)
apply (rule PO.intro)
apply (rule dualPO)
apply (rule CL_axioms.intro)
apply (rule CL_dualCL)
apply (rule CLF_axioms.intro)
apply (rule CLF_dual)
apply (simp add: dualr_iff dualA_iff)
done

declare glb_dual_lub[simp del] PO.dualA_iff[rule del] CLF.lubH_is_fixp[rule del]
PO.dualPO[rule del] CL.CL_dualCL[rule del] PO.dualr_iff[simp del]

(*never proved, 2007-01-22*)

lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) ∈ r & x ∈ A} cl"
(*sledgehammer;*)
apply (simp add: glb_dual_lub P_def A_def r_def)
apply (rule dualA_iff [THEN subst])
(*never proved, 2007-01-22*)
(*sledgehammer;*)
apply (simp add: CLF.T_thm_1_lub [of _ f, OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro,
OF dualPO CL_dualCL] dualPO CL_dualCL CLF_dual dualr_iff)
done

subsection ‹interval›

declare (in CLF) CO_refl_on[simp] refl_on_def [simp]

lemma (in CLF) rel_imp_elem: "(x, y) ∈ r ==> x ∈ A"
by (metis CO_refl_on refl_onD1)

declare (in CLF) CO_refl_on[simp del]  refl_on_def [simp del]

declare (in CLF) rel_imp_elem[intro]
declare interval_def [simp]

lemma (in CLF) interval_subset: "[| a ∈ A; b ∈ A |] ==> interval r a b ⊆ A"
by (metis CO_refl_on interval_imp_mem refl_onD refl_onD2 rel_imp_elem subset_eq)

declare (in CLF) rel_imp_elem[rule del]
declare interval_def [simp del]

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"
(*WON'T TERMINATE
apply (metis CO_trans intervalI interval_lemma1 interval_lemma2 isLub_least isLub_upper subset_empty subset_iff trans_def)
*)
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)
― ‹‹(L, b) ∈ r››
apply (simp add: isLub_least interval_lemma2)
done

(*never proved, 2007-01-22*)

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 CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro]
dualA_iff A_def dualPO CL_dualCL CLF_dual 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"
proof -
assume A1: "a ∈ A"
assume "b ∈ A"
hence "∀u. u ∈ A ⟶ interval r u b ⊆ A" by (metis interval_subset)
hence "interval r a b ⊆ A" using A1 by metis
hence "interval r a b ⊆ A" by metis
thus ?thesis by (metis po_subset_po)
qed

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: if_split)
apply (intro impI conjI)
― ‹‹(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 = {}")
― ‹‹S = {}, y ∈ S = False => everything››
apply fast
― ‹‹S ≠ {}››
apply simp
― ‹‹∀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)
― ‹‹∀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 (simp add: induced_def  interval_def)
apply (rule conjI)
apply (rule reflE, assumption)
apply (rule interval_not_empty)
apply (rule CO_trans)
apply (simp add: interval_def)
― ‹‹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]

(*never proved, 2007-01-22*)

lemma (in CLF) interval_is_sublattice:
"[| a ∈ A; b ∈ A; interval r a b ≠ {} |]
==> interval r a b <<= cl"
(*sledgehammer *)
apply (rule sublatticeI)
apply (simp add: interval_subset)
(*never proved, 2007-01-22*)
(*sledgehammer *)
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"
(*sledgehammer; *)
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

(*first proved 2007-01-25 after relaxing relevance*)

lemma (in CLF) Top_in_lattice: "Top cl ∈ A"
(*sledgehammer;*)
apply (simp add: Top_dual_Bot A_def)
(*first proved 2007-01-25 after relaxing relevance*)
(*sledgehammer*)
apply (rule dualA_iff [THEN subst])
apply (blast intro!: CLF.Bot_in_lattice [OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro] dualPO CL_dualCL CLF_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

(*never proved, 2007-01-22*)

lemma (in CLF) Bot_prop: "x ∈ A ==> (Bot cl, x) ∈ r"
(*sledgehammer*)
apply (simp add: Bot_dual_Top r_def)
apply (rule dualr_iff [THEN subst])
apply (simp add: CLF.Top_prop [of _ f, OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro]
dualA_iff A_def dualPO CL_dualCL CLF_dual)
done

lemma (in CLF) Top_intv_not_empty: "x ∈ A  ==> interval r x (Top cl) ≠ {}"
apply (metis Top_in_lattice Top_prop empty_iff intervalI reflE)
done

lemma (in CLF) Bot_intv_not_empty: "x ∈ A ==> interval r (Bot cl) x ≠ {}"
apply (metis Bot_prop ex_in_conv intervalI reflE rel_imp_elem)
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)

(*first proved 2007-01-25 after relaxing relevance*)

declare (in Tarski) P_def[simp] Y_ss [simp]
declare fix_subset [intro] subset_trans [intro]

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

declare (in Tarski) P_def[simp del] Y_ss [simp del]
declare fix_subset [rule del] subset_trans [rule del]

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

(*never proved, 2007-01-22*)

lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) ∈ r"
(*sledgehammer*)
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)
(*sledgehammer *)
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›
(*sledgehammer*)
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

(*first proved 2007-01-25 after relaxing relevance*)

lemma (in Tarski) intY1_subset: "intY1 ⊆ A"
(*sledgehammer*)
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]

(*never proved, 2007-01-22*)

lemma (in Tarski) intY1_f_closed: "x ∈ intY1 ⟹ f x ∈ intY1"
(*sledgehammer*)
apply (simp add: intY1_def  interval_def)
apply (rule conjI)
apply (rule transE)
apply (rule lubY_le_flubY)
― ‹‹(f (lub Y cl), f x) ∈ r››
(*sledgehammer [has been proved before now...]*)
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)
― ‹‹(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_func: "(λx ∈ intY1. f x) ∈ intY1 → intY1"
apply (rule restrict_in_funcset)
apply (metis intY1_f_closed restrict_in_funcset)
done

lemma (in Tarski) intY1_mono:
"monotone (λx ∈ intY1. f x) intY1 (induced intY1 r)"
(*sledgehammer *)
apply (auto simp add: monotone_def induced_def intY1_f_closed)
apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
done

(*proof requires relaxing relevance: 2007-01-25*)

lemma (in Tarski) intY1_is_cl:
"(| pset = intY1, order = induced intY1 r |) ∈ CompleteLattice"
(*sledgehammer*)
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

(*never proved, 2007-01-22*)

lemma (in Tarski) v_in_P: "v ∈ P"
(*sledgehammer*)
apply (unfold P_def)
apply (rule_tac A = "intY1" in fixf_subset)
apply (rule intY1_subset)
apply (simp add: CLF.glbH_is_fixp [OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro, OF _ intY1_is_cl, simplified]
v_def CL_imp_PO intY1_is_cl CLF_set_def intY1_func intY1_mono)
done

lemma (in Tarski) z_in_interval:
"[| z ∈ P; ∀y∈Y. (y, z) ∈ induced P r |] ==> z ∈ intY1"
(*sledgehammer *)
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 |]
==> ((λx ∈ intY1. f x) z, z) ∈ induced intY1 r"
using P_def fix_imp_eq indI intY1_elem reflE z_in_interval by fastforce

(*never proved, 2007-01-22*)

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)
― ‹‹v ∈ P››
apply (simp add: v_in_P)
apply (rule conjI)
(*sledgehammer*)
― ‹‹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 (simp add: v_def)
apply (fold intY1_def)
apply (rule CL.glb_in_lattice [OF CL.intro, OF PO.intro CL_axioms.intro, OF _ intY1_is_cl, simplified])
apply (simp add: CL_imp_PO intY1_is_cl, force)
― ‹‹v› is LEAST ub›
apply clarify
apply (rule indI)
prefer 3 apply assumption
prefer 2 apply (simp add: v_in_P)
apply (unfold v_def)
(*never proved, 2007-01-22*)
(*sledgehammer*)
apply (rule indE)
apply (rule_tac [2] intY1_subset)
(*never proved, 2007-01-22*)
(*sledgehammer*)
apply (rule CL.glb_lower [OF CL.intro, OF PO.intro CL_axioms.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)

(*never proved, 2007-01-22*)

declare (in CLF) fixf_po[intro] P_def [simp] A_def [simp] r_def [simp]
Tarski.tarski_full_lemma [intro] cl_po [intro] cl_co [intro]
CompleteLatticeI_simp [intro]

theorem (in CLF) Tarski_full:
"(| pset = P, order = induced P r|) ∈ CompleteLattice"
(*sledgehammer*)
apply (rule CompleteLatticeI_simp)
apply (rule fixf_po, clarify)
(*never proved, 2007-01-22*)
(*sledgehammer*)
apply (simp add: P_def A_def r_def)
apply (blast intro!: Tarski.tarski_full_lemma [OF Tarski.intro, OF CLF.intro Tarski_axioms.intro,
OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro] cl_po cl_co f_cl)
done

declare (in CLF) fixf_po [rule del] P_def [simp del] A_def [simp del] r_def [simp del]
Tarski.tarski_full_lemma [rule del] cl_po [rule del] cl_co [rule del]
CompleteLatticeI_simp [rule del]

end
```