(* Title: HOL/MetisTest/Tarski.thy ID: $Id: Tarski.thy,v 1.4 2007/10/05 07:59:22 paulson Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Testing the metis method *) header {* The Full Theorem of Tarski *} theory Tarski imports FuncSet begin (*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" constdefs monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool" "monotone f A r == ∀x∈A. ∀y∈A. (x, y): r --> ((f x), (f y)) : r" least :: "['a => bool, 'a potype] => 'a" "least P po == @ x. x: pset po & P x & (∀y ∈ pset po. P y --> (x,y): order po)" greatest :: "['a => bool, 'a potype] => 'a" "greatest P po == @ x. x: pset po & P x & (∀y ∈ pset po. P y --> (y,x): order po)" lub :: "['a set, 'a potype] => 'a" "lub S po == least (%x. ∀y∈S. (y,x): order po) po" glb :: "['a set, 'a potype] => 'a" "glb S po == greatest (%x. ∀y∈S. (x,y): order po) po" isLub :: "['a set, 'a potype, 'a] => bool" "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))" isGlb :: "['a set, 'a potype, 'a] => bool" "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))" "fix" :: "[('a => 'a), 'a set] => 'a set" "fix f A == {x. x: A & f x = x}" interval :: "[('a*'a) set,'a, 'a ] => 'a set" "interval r a b == {x. (a,x): r & (x,b): r}" constdefs Bot :: "'a potype => 'a" "Bot po == least (%x. True) po" Top :: "'a potype => 'a" "Top po == greatest (%x. True) po" PartialOrder :: "('a potype) set" "PartialOrder == {P. refl (pset P) (order P) & antisym (order P) & trans (order P)}" CompleteLattice :: "('a potype) set" "CompleteLattice == {cl. cl: PartialOrder & (∀S. S ⊆ pset cl --> (∃L. isLub S cl L)) & (∀S. S ⊆ pset cl --> (∃G. isGlb S cl G))}" CLF :: "('a potype * ('a => 'a)) set" "CLF == SIGMA cl: CompleteLattice. {f. f: pset cl -> pset cl & monotone f (pset cl) (order cl)}" induced :: "['a set, ('a * 'a) set] => ('a *'a)set" "induced A r == {(a,b). a : A & b: A & (a,b): r}" constdefs sublattice :: "('a potype * 'a set)set" "sublattice == SIGMA cl: CompleteLattice. {S. S ⊆ pset cl & (| pset = S, order = induced S (order cl) |): CompleteLattice }" syntax "@SL" :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50) translations "S <<= cl" == "S : sublattice `` {cl}" constdefs dual :: "'a potype => 'a potype" "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 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 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"}. *} declare (in CLF) f_cl [simp] (*never proved, 2007-01-22: Tarski__CLF_unnamed_lemma NOT PROVABLE because of the conjunction used in the definition: we don't allow reasoning with rules like conjE, which is essential here.*) ML{*ResAtp.problem_name:="Tarski__CLF_unnamed_lemma"*} lemma (in CLF) [simp]: "f: pset cl -> pset cl & monotone f (pset cl) (order cl)" apply (insert f_cl) apply (unfold CLF_def) apply (erule SigmaE2) apply (erule CollectE) apply assumption; done 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*) ML{*ResAtp.problem_name:="Tarski__CLF_CLF_dual"*} declare (in CLF) CLF_def[simp] CL_dualCL[simp] monotone_dual[simp] dualA_iff[simp] lemma (in CLF) CLF_dual: "(dual cl, f) ∈ CLF" apply (simp del: dualA_iff) apply (simp) done declare (in CLF) CLF_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 (%y: A. f y) A |] ==> x ∈ fix f B" by (simp add: fix_def, auto) subsection {* lemmas for Tarski, lub *} (*never proved, 2007-01-22*) ML{*ResAtp.problem_name:="Tarski__CLF_lubH_le_flubH"*} 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) -- {* @{text "∀x:H. (x, f (lub H r)) ∈ r"} *} apply (rule ballI) (*never proved, 2007-01-22*) ML{*ResAtp.problem_name:="Tarski__CLF_lubH_le_flubH_simpler"*} 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 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*) ML{*ResAtp.problem_name:="Tarski__CLF_flubH_le_lubH"*} 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) apply (rule conjI) ML{*ResAtp.problem_name:="Tarski__CLF_flubH_le_lubH_simpler"*} (*??no longer terminates, with combinators apply (metis CO_refl lubH_le_flubH monotone_def monotone_f reflD1 reflD2) *) apply (metis CO_refl lubH_le_flubH monotoneE [OF monotone_f] reflD1 reflD2) apply (metis CO_refl lubH_le_flubH reflD2) done 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*) ML{*ResAtp.problem_name:="Tarski__CLF_lubH_is_fixp"*} (*Single-step version fails. The conjecture clauses refer to local abstraction functions (Frees), which prevents expand_defs_tac from removing those "definitions" at the end of the proof. *) 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 (neg_clausify) assume 0: "H = Collect (COMBS (COMBB op ∧ (COMBC (COMBB op ∈ (COMBS Pair f)) r)) (COMBC op ∈ A))" assume 1: "lub (Collect (COMBS (COMBB op ∧ (COMBC (COMBB op ∈ (COMBS Pair f)) r)) (COMBC op ∈ A))) cl ∉ A" have 2: "lub H cl ∉ A" by (metis 1 0) have 3: "(lub H cl, f (lub H cl)) ∈ r" by (metis lubH_le_flubH 0) have 4: "(f (lub H cl), lub H cl) ∈ r" by (metis flubH_le_lubH 0) have 5: "lub H cl = f (lub H cl) ∨ (lub H cl, f (lub H cl)) ∉ r" by (metis antisymE 4) have 6: "lub H cl = f (lub H cl)" by (metis 5 3) have 7: "(lub H cl, lub H cl) ∈ r" by (metis 6 4) have 8: "!!X1. lub H cl ∈ X1 ∨ ¬ refl X1 r" by (metis 7 reflD2) have 9: "¬ refl A r" by (metis 8 2) show "False" by (metis CO_refl 9); next --{*apparently the way to insert a second structured proof*} show "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" proof (neg_clausify) assume 0: "H = Collect (COMBS (COMBB op ∧ (COMBC (COMBB op ∈ (COMBS Pair f)) r)) (COMBC op ∈ A))" assume 1: "f (lub (Collect (COMBS (COMBB op ∧ (COMBC (COMBB op ∈ (COMBS Pair f)) r)) (COMBC op ∈ A))) cl) ≠ lub (Collect (COMBS (COMBB op ∧ (COMBC (COMBB op ∈ (COMBS Pair f)) r)) (COMBC op ∈ A))) cl" have 2: "f (lub H cl) ≠ lub (Collect (COMBS (COMBB op ∧ (COMBC (COMBB op ∈ (COMBS Pair f)) r)) (COMBC op ∈ A))) cl" by (metis 1 0) have 3: "f (lub H cl) ≠ lub H cl" by (metis 2 0) have 4: "(lub H cl, f (lub H cl)) ∈ r" by (metis lubH_le_flubH 0) have 5: "(f (lub H cl), lub H cl) ∈ r" by (metis flubH_le_lubH 0) have 6: "lub H cl = f (lub H cl) ∨ (lub H cl, f (lub H cl)) ∉ r" by (metis antisymE 5) have 7: "lub H cl = f (lub H cl)" by (metis 6 4) show "False" by (metis 3 7) qed 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) ML{*ResAtp.problem_name:="Tarski__CLF_lubH_is_fixp_simpler"*} apply (metis CO_refl lubH_le_flubH reflD1) 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 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 ML{*ResAtp.problem_name:="Tarski__CLF_lubH_least_fixf"*} 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 *} ML{*ResAtp.problem_name:="Tarski__CLF_T_thm_1_lub"*} 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) ML{*ResAtp.problem_name:="Tarski__CLF_T_thm_1_lub_simpler"*} apply (metis P_def fix_subset) apply (metis Collect_conj_eq Collect_mem_eq Int_commute Int_lower1 lub_in_lattice vimage_def) (*??no longer terminates, with combinators apply (metis P_def fix_def fixf_le_lubH) apply (metis P_def fix_def lubH_least_fixf) *) apply (simp add: fixf_le_lubH) apply (simp add: lubH_least_fixf) 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*) ML{*ResAtp.problem_name:="Tarski__CLF_glbH_is_fixp"*} 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 dualPO) apply (rule CL_dualCL) 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*) ML{*ResAtp.problem_name:="Tarski__T_thm_1_glb"*} (*ALL THEOREMS*) 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*) ML{*ResAtp.problem_name:="Tarski__T_thm_1_glb_simpler"*} (*ALL THEOREMS*) (*sledgehammer;*) apply (simp add: CLF.T_thm_1_lub [of _ f, OF dualPO CL_dualCL] dualPO CL_dualCL CLF_dual dualr_iff) done subsection {* interval *} ML{*ResAtp.problem_name:="Tarski__rel_imp_elem"*} declare (in CLF) CO_refl[simp] refl_def [simp] lemma (in CLF) rel_imp_elem: "(x, y) ∈ r ==> x ∈ A" by (metis CO_refl reflD1) declare (in CLF) CO_refl[simp del] refl_def [simp del] ML{*ResAtp.problem_name:="Tarski__interval_subset"*} 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 interval_imp_mem reflD reflD2 rel_imp_elem subset_def) 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]) ML{*ResAtp.problem_name:="Tarski__L_in_interval"*} (*ALL THEOREMS*) 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) -- {* @{text "(L, b) ∈ r"} *} apply (simp add: isLub_least interval_lemma2) done (*never proved, 2007-01-22*) ML{*ResAtp.problem_name:="Tarski__G_in_interval"*} (*ALL THEOREMS*) 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 ML{*ResAtp.problem_name:="Tarski__intervalPO"*} (*ALL THEOREMS*) 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 (neg_clausify) assume 0: "a ∈ A" assume 1: "b ∈ A" assume 2: "(|pset = interval r a b, order = induced (interval r a b) r|)), ∉ PartialOrder" have 3: "¬ interval r a b ⊆ A" by (metis 2 po_subset_po) have 4: "b ∉ A ∨ a ∉ A" by (metis 3 interval_subset) have 5: "a ∉ A" by (metis 4 1) show "False" by (metis 5 0) 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: 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] (*never proved, 2007-01-22*) ML{*ResAtp.problem_name:="Tarski__interval_is_sublattice"*} (*ALL THEOREMS*) 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*) ML{*ResAtp.problem_name:="Tarski__interval_is_sublattice_simpler"*} (*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) ML{*ResAtp.problem_name:="Tarski__Bot_in_lattice"*} (*ALL THEOREMS*) 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*) ML{*ResAtp.problem_name:="Tarski__Top_in_lattice"*} (*ALL THEOREMS*) 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*) ML{*ResAtp.problem_name:="Tarski__Top_in_lattice_simpler"*} (*ALL THEOREMS*) (*sledgehammer*) 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 (*never proved, 2007-01-22*) ML{*ResAtp.problem_name:="Tarski__Bot_prop"*} (*ALL THEOREMS*) 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] dualA_iff A_def dualPO CL_dualCL CLF_dual) done ML{*ResAtp.problem_name:="Tarski__Bot_in_lattice"*} (*ALL THEOREMS*) 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 ML{*ResAtp.problem_name:="Tarski__Bot_intv_not_empty"*} (*ALL THEOREMS*) 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*) ML{*ResAtp.problem_name:="Tarski__Y_subset_A"*} 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*) ML{*ResAtp.problem_name:="Tarski__lubY_le_flubY"*} (*ALL THEOREMS*) 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) -- {* @{text "Y ⊆ P ==> f x = x"} *} apply (rule ballI) ML{*ResAtp.problem_name:="Tarski__lubY_le_flubY_simpler"*} (*ALL THEOREMS*) (*sledgehammer *) 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 *} ML{*ResAtp.problem_name:="Tarski__lubY_le_flubY_simplest"*} (*ALL THEOREMS*) (*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*) ML{*ResAtp.problem_name:="Tarski__intY1_subset"*} (*ALL THEOREMS*) 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*) ML{*ResAtp.problem_name:="Tarski__intY1_f_closed"*} (*ALL THEOREMS*) 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) -- {* @{text "(f (lub Y cl), f x) ∈ r"} *} ML{*ResAtp.problem_name:="Tarski__intY1_f_closed_simpler"*} (*ALL THEOREMS*) (*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) -- {* @{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 ML{*ResAtp.problem_name:="Tarski__intY1_func"*} (*ALL THEOREMS*) lemma (in Tarski) intY1_func: "(%x: intY1. f x) ∈ intY1 -> intY1" by (metis intY1_f_closed restrict_in_funcset) ML{*ResAtp.problem_name:="Tarski__intY1_mono"*} (*ALL THEOREMS*) lemma (in Tarski) intY1_mono: "monotone (%x: 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*) ML{*ResAtp.problem_name:="Tarski__intY1_is_cl"*} (*ALL THEOREMS*) 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*) ML{*ResAtp.problem_name:="Tarski__v_in_P"*} (*ALL THEOREMS*) 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 _ intY1_is_cl, simplified] v_def CL_imp_PO intY1_is_cl CLF_def intY1_func intY1_mono) done ML{*ResAtp.problem_name:="Tarski__z_in_interval"*} (*ALL THEOREMS*) 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 ML{*ResAtp.problem_name:="Tarski__fz_in_int_rel"*} (*ALL THEOREMS*) 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 (metis P_def acc_def fix_imp_eq fix_subset indI reflE restrict_apply subset_def z_in_interval) done (*never proved, 2007-01-22*) ML{*ResAtp.problem_name:="Tarski__tarski_full_lemma"*} (*ALL THEOREMS*) 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) (*sledgehammer*) -- {* @{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) (*never proved, 2007-01-22*) ML{*ResAtp.problem_name:="Tarski__tarski_full_lemma_simpler"*} (*sledgehammer*) apply (rule indE) apply (rule_tac [2] intY1_subset) (*never proved, 2007-01-22*) ML{*ResAtp.problem_name:="Tarski__tarski_full_lemma_simplest"*} (*sledgehammer*) 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) (*never proved, 2007-01-22*) ML{*ResAtp.problem_name:="Tarski__Tarski_full"*} 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*) ML{*ResAtp.problem_name:="Tarski__Tarski_full_simpler"*} (*sledgehammer*) apply (simp add: P_def A_def r_def) apply (blast intro!: Tarski.tarski_full_lemma 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
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 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