(* Title: HOL/ex/Tarski.thy ID: $Id: Tarski.thy,v 1.20 2007/03/29 09:59:54 paulson Exp $ Author: Florian Kammüller, Cambridge University Computer Laboratory *) header {* The Full Theorem of Tarski *} theory Tarski imports Main 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 (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 :: "('a potype * ('a => 'a)) set" where "CLF = (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 (open) 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 (open) CL = PO + assumes cl_co: "cl : CompleteLattice" locale (open) CLF = CL + fixes f :: "'a => 'a" and P :: "'a set" assumes f_cl: "(cl,f) : CLF" (*was the equivalent "f : CLF``{cl}"*) defines P_def: "P == fix f A" locale (open) 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) PO_imp_refl: "refl 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_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_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_def) lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)" by (simp add: isLub_def isGlb_def dual_def converse_def) lemma (in PO) dualPO: "dual cl ∈ PartialOrder" apply (insert cl_po) apply (simp add: PartialOrder_def dual_def refl_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_def) 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_def) 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: "refl A r" by (rule PO_imp_refl) 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 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 dualPO) 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 dualPO) apply (rule CL_dualCL) 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_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" apply (simp add: CLF_def CL_dualCL monotone_dual) apply (simp add: dualA_iff) 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 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 dualPO) apply (rule CL_dualCL) apply (rule CLF_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 dualPO CL_dualCL] 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) apply (simp add: refl_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] 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" 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 (rule CO_trans) 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 (blast intro!: CLF.Bot_in_lattice 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 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 (simp add: CLF.Top_prop [of _ f] 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 (rule notI) apply (drule_tac a = "Top cl" in equals0D) apply (simp add: interval_def) apply (simp add: refl_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 (blast intro!: CLF.Top_in_lattice dualPO CL_dualCL CLF_dual) apply (simp add: CLF.Top_intv_not_empty [of _ f] dualA_iff A_def dualPO CL_dualCL CLF_dual) 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_func: "(%x: intY1. f x) ∈ intY1 -> intY1" apply (rule restrictI) apply (erule intY1_f_closed) 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) apply (simp add: CLF.glbH_is_fixp [OF _ intY1_is_cl, simplified] v_def CL_imp_PO intY1_is_cl CLF_def intY1_func 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 _ intY1_is_cl, simplified]) apply (simp add: CL_imp_PO intY1_is_cl, force) -- {* @{text 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) apply (rule indE) apply (rule_tac [2] intY1_subset) apply (rule CL.glb_lower [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 (blast intro!: Tarski.tarski_full_lemma cl_po cl_co f_cl) done end
lemma PO_imp_refl:
refl A r
lemma PO_imp_sym:
antisym r
lemma PO_imp_trans:
trans r
lemma reflE:
x ∈ A ==> (x, x) ∈ r
lemma antisymE:
[| (a, b) ∈ r; (b, a) ∈ r |] ==> a = b
lemma transE:
[| (a, b) ∈ r; (b, c) ∈ r |] ==> (a, c) ∈ r
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
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
lemma CL_imp_ex_isLub:
S ⊆ A ==> ∃L. isLub S cl L
lemma isLub_lub:
(∃L. isLub S cl L) = isLub S cl (lub S cl)
lemma isGlb_glb:
(∃G. isGlb S cl G) = isGlb S cl (glb S cl)
lemma isGlb_dual_isLub:
isGlb S cl = isLub S (dual cl)
lemma isLub_dual_isGlb:
isLub S cl = isGlb S (dual cl)
lemma dualPO:
dual cl ∈ PartialOrder
lemma Rdual:
∀S⊆A. ∃L. isLub S (| pset = A, order = r |) L
==> ∀S⊆A. ∃G. isGlb S (| pset = A, order = r |) G
lemma lub_dual_glb:
lub S cl = glb S (dual cl)
lemma glb_dual_lub:
glb S cl = lub S (dual cl)
lemma CL_subset_PO:
CompleteLattice ⊆ PartialOrder
lemma CL_imp_PO:
c ∈ CompleteLattice ==> c ∈ PartialOrder
lemma CO_refl:
refl A r
lemma CO_antisym:
antisym r
lemma CO_trans:
trans r
lemma CompleteLatticeI:
[| po ∈ PartialOrder; ∀S⊆pset po. ∃L. isLub S po L;
∀S⊆pset po. ∃G. isGlb S po G |]
==> po ∈ CompleteLattice
lemma CL_dualCL:
dual cl ∈ CompleteLattice
lemma dualA_iff:
pset (dual cl) = pset cl
lemma dualr_iff:
((x, y) ∈ potype.order (dual cl)) = ((y, x) ∈ potype.order cl)
lemma monotone_dual:
monotone f (pset cl) (potype.order cl)
==> monotone f (pset (dual cl)) (potype.order (dual cl))
lemma interval_dual:
[| x ∈ A; y ∈ A |] ==> interval r x y = interval (potype.order (dual cl)) y x
lemma interval_not_empty:
[| trans r; interval r a b ≠ {} |] ==> (a, b) ∈ r
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
lemma right_in_interval:
[| a ∈ A; b ∈ A; interval r a b ≠ {} |] ==> b ∈ interval r a b
lemma sublattice_imp_CL:
S <<= cl ==> (| pset = S, order = induced S r |) ∈ CompleteLattice
lemma sublatticeI:
[| S ⊆ A; (| pset = S, order = induced S r |) ∈ CompleteLattice |] ==> S <<= cl
lemma lub_unique:
[| S ⊆ A; isLub S cl x; isLub S cl L |] ==> x = L
lemma lub_upper:
[| S ⊆ A; x ∈ S |] ==> (x, lub S cl) ∈ r
lemma lub_least:
[| S ⊆ A; L ∈ A; ∀x∈S. (x, L) ∈ r |] ==> (lub S cl, L) ∈ r
lemma lub_in_lattice:
S ⊆ A ==> lub S cl ∈ A
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
lemma lubIa:
[| S ⊆ A; isLub S cl L |] ==> L = lub S cl
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
lemma isLubI:
[| L ∈ A; ∀y∈S. (y, L) ∈ r; ∀z∈A. (∀y∈S. (y, z) ∈ r) --> (L, z) ∈ r |]
==> isLub S cl L
lemma glb_in_lattice:
S ⊆ A ==> glb S cl ∈ A
lemma glb_lower:
[| S ⊆ A; x ∈ S |] ==> (glb S cl, x) ∈ r
lemma
f ∈ pset cl -> pset cl ∧ monotone f (pset cl) (potype.order cl)
lemma f_in_funcset:
f ∈ A -> A
lemma monotone_f:
monotone f A r
lemma CLF_dual:
(dual cl, f) ∈ CLF
lemma fix_subset:
fix f A ⊆ A
lemma fix_imp_eq:
x ∈ fix f A ==> f x = x
lemma fixf_subset:
[| A ⊆ B; x ∈ fix (restrict f A) A |] ==> x ∈ fix f B
lemma lubH_le_flubH:
H = {x. (x, f x) ∈ r ∧ x ∈ A} ==> (lub H cl, f (lub H cl)) ∈ r
lemma flubH_le_lubH:
H = {x. (x, f x) ∈ r ∧ x ∈ A} ==> (f (lub H cl), lub H cl) ∈ r
lemma lubH_is_fixp:
H = {x. (x, f x) ∈ r ∧ x ∈ A} ==> lub H cl ∈ fix f A
lemma fix_in_H:
[| H = {x. (x, f x) ∈ r ∧ x ∈ A}; x ∈ P |] ==> x ∈ H
lemma fixf_le_lubH:
H = {x. (x, f x) ∈ r ∧ x ∈ A} ==> ∀x∈fix f A. (x, lub H cl) ∈ r
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
lemma T_thm_1_lub:
lub P cl = lub {x. (x, f x) ∈ r ∧ x ∈ A} cl
lemma glbH_is_fixp:
H = {x. (f x, x) ∈ r ∧ x ∈ A} ==> glb H cl ∈ P
lemma T_thm_1_glb:
glb P cl = glb {x. (f x, x) ∈ r ∧ x ∈ A} cl
lemma rel_imp_elem:
(x, y) ∈ r ==> x ∈ A
lemma interval_subset:
[| a ∈ A; b ∈ A |] ==> interval r a b ⊆ A
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
lemma interval_lemma2:
[| S ⊆ interval r a b; x ∈ S |] ==> (x, b) ∈ r
lemma a_less_lub:
[| S ⊆ A; S ≠ {}; ∀x∈S. (a, x) ∈ r; ∀y∈S. (y, L) ∈ r |] ==> (a, L) ∈ r
lemma glb_less_b:
[| S ⊆ A; S ≠ {}; ∀x∈S. (x, b) ∈ r; ∀y∈S. (G, y) ∈ r |] ==> (G, b) ∈ r
lemma S_intv_cl:
[| a ∈ A; b ∈ A; S ⊆ interval r a b |] ==> S ⊆ A
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
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
lemma intervalPO:
[| a ∈ A; b ∈ A; interval r a b ≠ {} |]
==> (| pset = interval r a b, order = induced (interval r a b) r |)
∈ PartialOrder
lemma intv_CL_lub:
[| a ∈ A; b ∈ A; interval r a b ≠ {} |]
==> ∀S⊆interval r a b.
∃L. isLub S
(| pset = interval r a b, order = induced (interval r a b) r |) L
lemma intv_CL_glb:
[| a4 ∈ A; b5 ∈ A; interval r a4 b5 ≠ {} |]
==> ∀S⊆interval r a4 b5.
∃G. isGlb S
(| pset = interval r a4 b5, order = induced (interval r a4 b5) r |)
G
lemma interval_is_sublattice:
[| a ∈ A; b ∈ A; interval r a b ≠ {} |] ==> interval r a b <<= cl
lemma interv_is_compl_latt:
[| a1 ∈ A; b1 ∈ A; interval r a1 b1 ≠ {} |]
==> (| pset = interval r a1 b1, order = induced (interval r a1 b1) r |)
∈ CompleteLattice
lemma Top_dual_Bot:
Top cl = Bot (dual cl)
lemma Bot_dual_Top:
Bot cl = Top (dual cl)
lemma Bot_in_lattice:
Bot cl ∈ A
lemma Top_in_lattice:
Top cl ∈ A
lemma Top_prop:
x ∈ A ==> (x, Top cl) ∈ r
lemma Bot_prop:
x ∈ A ==> (Bot cl, x) ∈ r
lemma Top_intv_not_empty:
x ∈ A ==> interval r x (Top cl) ≠ {}
lemma Bot_intv_not_empty:
x ∈ A ==> interval r (Bot cl) x ≠ {}
lemma fixf_po:
(| pset = P, order = induced P r |) ∈ PartialOrder
lemma Y_subset_A:
Y ⊆ A
lemma lubY_in_A:
lub Y cl ∈ A
lemma lubY_le_flubY:
(lub Y cl, f (lub Y cl)) ∈ r
lemma intY1_subset:
intY1 ⊆ A
lemma intY1_elem:
c ∈ intY1 ==> c ∈ A
lemma intY1_f_closed:
x ∈ intY1 ==> f x ∈ intY1
lemma intY1_func:
restrict f intY1 ∈ intY1 -> intY1
lemma intY1_mono:
monotone (restrict f intY1) intY1 (induced intY1 r)
lemma intY1_is_cl:
(| pset = intY1, order = induced intY1 r |) ∈ CompleteLattice
lemma v_in_P:
v ∈ P
lemma z_in_interval:
[| z ∈ P; ∀y∈Y. (y, z) ∈ induced P r |] ==> z ∈ intY1
lemma f'z_in_int_rel:
[| z ∈ P; ∀y∈Y. (y, z) ∈ induced P r |]
==> (restrict f intY1 z, z) ∈ induced intY1 r
lemma tarski_full_lemma:
∃L. isLub Y (| pset = P, order = induced P r |) L
lemma CompleteLatticeI_simp:
[| (| pset = A, order = r |) ∈ PartialOrder;
∀S⊆A. ∃L. isLub S (| pset = A, order = r |) L |]
==> (| pset = A, order = r |) ∈ CompleteLattice
theorem Tarski_full:
(| pset = P, order = induced P r |) ∈ CompleteLattice