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author | David Aspinall <da@inf.ed.ac.uk> | 2004-04-14 21:21:41 +0000 |
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committer | David Aspinall <da@inf.ed.ac.uk> | 2004-04-14 21:21:41 +0000 |
commit | 4e409735c837314d28f96afa0a7cad9697181fc6 (patch) | |
tree | 48d454c459a8c19766ae8d0e897abb147a61e598 /isar/Tarski.thy | |
parent | 6de9181263800be877e5bbf8dfa0a60234677253 (diff) |
New files.
Diffstat (limited to 'isar/Tarski.thy')
-rw-r--r-- | isar/Tarski.thy | 904 |
1 files changed, 904 insertions, 0 deletions
diff --git a/isar/Tarski.thy b/isar/Tarski.thy new file mode 100644 index 00000000..021b5228 --- /dev/null +++ b/isar/Tarski.thy @@ -0,0 +1,904 @@ +(********** This file is from the Isabelle distribution **********) + +(* Title: HOL/ex/Tarski.thy + ID: Id: Tarski.thy,v 1.10 2002/09/26 08:51:32 paulson Exp + Author: Florian Kammüller, Cambridge University Computer Laboratory +*) + +header {* The Full Theorem of Tarski *} + +theory Tarski = Main + FuncSet: + +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" + +constdefs + monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool" + "monotone f A r == \<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r" + + least :: "['a => bool, 'a potype] => 'a" + "least P po == @ x. x: pset po & P x & + (\<forall>y \<in> pset po. P y --> (x,y): order po)" + + greatest :: "['a => bool, 'a potype] => 'a" + "greatest P po == @ x. x: pset po & P x & + (\<forall>y \<in> pset po. P y --> (y,x): order po)" + + lub :: "['a set, 'a potype] => 'a" + "lub S po == least (%x. \<forall>y\<in>S. (y,x): order po) po" + + glb :: "['a set, 'a potype] => 'a" + "glb S po == greatest (%x. \<forall>y\<in>S. (x,y): order po) po" + + isLub :: "['a set, 'a potype, 'a] => bool" + "isLub S po == %L. (L: pset po & (\<forall>y\<in>S. (y,L): order po) & + (\<forall>z\<in>pset po. (\<forall>y\<in>S. (y,z): order po) --> (L,z): order po))" + + isGlb :: "['a set, 'a potype, 'a] => bool" + "isGlb S po == %G. (G: pset po & (\<forall>y\<in>S. (G,y): order po) & + (\<forall>z \<in> pset po. (\<forall>y\<in>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 & + (\<forall>S. S <= pset cl --> (\<exists>L. isLub S cl L)) & + (\<forall>S. S <= pset cl --> (\<exists>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|)" + + +subsubsection {* 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 A_def r_def) +done + +lemma (in PO) PO_imp_trans: "trans r" +apply (insert cl_po) +apply (simp add: PartialOrder_def A_def r_def) +done + +lemma (in PO) reflE: "[| refl A r; x \<in> A|] ==> (x, x) \<in> r" +apply (insert cl_po) +apply (simp add: PartialOrder_def refl_def) +done + +lemma (in PO) antisymE: "[| antisym r; (a, b) \<in> r; (b, a) \<in> r |] ==> a = b" +apply (insert cl_po) +apply (simp add: PartialOrder_def antisym_def) +done + +lemma (in PO) transE: "[| trans r; (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r" +apply (insert cl_po) +apply (simp add: PartialOrder_def) +apply (unfold trans_def, fast) +done + +lemma (in PO) monotoneE: + "[| monotone f A r; x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r" +by (simp add: monotone_def) + +lemma (in PO) po_subset_po: + "S <= A ==> (| pset = S, order = induced S r |) \<in> PartialOrder" +apply (simp (no_asm) add: PartialOrder_def) +apply auto +-- {* refl *} +apply (simp add: refl_def induced_def) +apply (blast intro: PO_imp_refl [THEN reflE]) +-- {* antisym *} +apply (simp add: antisym_def induced_def) +apply (blast intro: PO_imp_sym [THEN antisymE]) +-- {* trans *} +apply (simp add: trans_def induced_def) +apply (blast intro: PO_imp_trans [THEN transE]) +done + +lemma (in PO) indE: "[| (x, y) \<in> induced S r; S <= A |] ==> (x, y) \<in> r" +by (simp add: add: induced_def) + +lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r" +by (simp add: add: induced_def) + +lemma (in CL) CL_imp_ex_isLub: "S <= A ==> \<exists>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: "(\<exists>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: "(\<exists>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 \<in> PartialOrder" +apply (insert cl_po) +apply (simp add: PartialOrder_def dual_def refl_converse + trans_converse antisym_converse) +done + +lemma Rdual: + "\<forall>S. (S <= A -->( \<exists>L. isLub S (| pset = A, order = r|) L)) + ==> \<forall>S. (S <= A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))" +apply safe +apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)} + (|pset = A, order = r|) " in exI) +apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> 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 Tarski.PO_imp_refl, simp] +declare CL_imp_PO [THEN Tarski.PO_imp_sym, simp] +declare CL_imp_PO [THEN Tarski.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 \<in> PartialOrder; (\<forall>S. S <= pset po --> (\<exists>L. isLub S po L)); + (\<forall>S. S <= pset po --> (\<exists>G. isGlb S po G))|] + ==> po \<in> CompleteLattice" +apply (unfold CompleteLattice_def, blast) +done + +lemma (in CL) CL_dualCL: "dual cl \<in> 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) \<in> (order(dual cl))) = ((y, x) \<in> 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 \<in> A; y \<in> 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 \<noteq> {} |] ==> (a, b) \<in> r" +apply (simp add: interval_def) +apply (unfold trans_def, blast) +done + +lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r" +by (simp add: interval_def) + +lemma (in PO) left_in_interval: + "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b" +apply (simp (no_asm_simp) add: interval_def) +apply (simp add: PO_imp_trans interval_not_empty) +apply (simp add: PO_imp_refl [THEN reflE]) +done + +lemma (in PO) right_in_interval: + "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b" +apply (simp (no_asm_simp) add: interval_def) +apply (simp add: PO_imp_trans interval_not_empty) +apply (simp add: PO_imp_refl [THEN reflE]) +done + + +subsubsection {* sublattice *} + +lemma (in PO) sublattice_imp_CL: + "S <<= cl ==> (| pset = S, order = induced S r |) \<in> CompleteLattice" +by (simp add: sublattice_def CompleteLattice_def A_def r_def) + +lemma (in CL) sublatticeI: + "[| S <= A; (| pset = S, order = induced S r |) \<in> CompleteLattice |] + ==> S <<= cl" +by (simp add: sublattice_def A_def r_def) + + +subsubsection {* lub *} + +lemma (in CL) lub_unique: "[| S <= A; isLub S cl x; isLub S cl L|] ==> x = L" +apply (rule antisymE) +apply (rule CO_antisym) +apply (auto simp add: isLub_def r_def) +done + +lemma (in CL) lub_upper: "[|S <= A; x \<in> S|] ==> (x, lub S cl) \<in> 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 \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> 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 \<in> 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 \<in> A; \<forall>x \<in> S. (x,L) \<in> r; + \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> 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 \<in> A" +by (simp add: isLub_def A_def) + +lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r" +by (simp add: isLub_def r_def) + +lemma (in CL) isLub_least: + "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r" +by (simp add: isLub_def A_def r_def) + +lemma (in CL) isLubI: + "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r; + (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L" +by (simp add: isLub_def A_def r_def) + + +subsubsection {* glb *} + +lemma (in CL) glb_in_lattice: "S <= A ==> glb S cl \<in> A" +apply (subst glb_dual_lub) +apply (simp add: A_def) +apply (rule dualA_iff [THEN subst]) +apply (rule Tarski.lub_in_lattice) +apply (rule dualPO) +apply (rule CL_dualCL) +apply (simp add: dualA_iff) +done + +lemma (in CL) glb_lower: "[|S <= A; x \<in> S|] ==> (glb S cl, x) \<in> r" +apply (subst glb_dual_lub) +apply (simp add: r_def) +apply (rule dualr_iff [THEN subst]) +apply (rule Tarski.lub_upper [rule_format]) +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 \<in> 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: "(cl,f) \<in> CLF ==> (dual cl, f) \<in> CLF" +apply (simp add: CLF_def CL_dualCL monotone_dual) +apply (simp add: dualA_iff) +done + + +subsubsection {* fixed points *} + +lemma fix_subset: "fix f A <= A" +by (simp add: fix_def, fast) + +lemma fix_imp_eq: "x \<in> fix f A ==> f x = x" +by (simp add: fix_def) + +lemma fixf_subset: + "[| A <= B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B" +apply (simp add: fix_def, auto) +done + + +subsubsection {* lemmas for Tarski, lub *} +lemma (in CLF) lubH_le_flubH: + "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r" +apply (rule lub_least, fast) +apply (rule f_in_funcset [THEN funcset_mem]) +apply (rule lub_in_lattice, fast) +-- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *} +apply (rule ballI) +apply (rule transE) +apply (rule CO_trans) +-- {* instantiates @{text "(x, ???z) \<in> 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)) \<in> 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) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> 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) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A" +apply (simp add: fix_def) +apply (rule conjI) +apply (rule lub_in_lattice, fast) +apply (rule antisymE) +apply (rule CO_antisym) +apply (simp add: flubH_le_lubH) +apply (simp add: lubH_le_flubH) +done + +lemma (in CLF) fix_in_H: + "[| H = {x. (x, f x) \<in> r & x \<in> A}; x \<in> P |] ==> x \<in> 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) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> 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) \<in> r & x \<in> A} + ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r" +apply (rule allI) +apply (rule impI) +apply (erule bspec) +apply (rule lubH_is_fixp, assumption) +done + +subsubsection {* Tarski fixpoint theorem 1, first part *} +lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> 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) \<in> r & x \<in> A} ==> glb H cl \<in> P" + -- {* Tarski for glb *} +apply (simp add: glb_dual_lub P_def A_def r_def) +apply (rule dualA_iff [THEN subst]) +apply (rule Tarski.lubH_is_fixp) +apply (rule dualPO) +apply (rule CL_dualCL) +apply (rule f_cl [THEN 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) \<in> r & x \<in> A} cl" +apply (simp add: glb_dual_lub P_def A_def r_def) +apply (rule dualA_iff [THEN subst]) +apply (simp add: Tarski.T_thm_1_lub [of _ f, OF dualPO CL_dualCL] + dualPO CL_dualCL CLF_dual dualr_iff) +done + +subsubsection {* interval *} + +lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A" +apply (insert CO_refl) +apply (simp add: refl_def, blast) +done + +lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b <= A" +apply (simp add: interval_def) +apply (blast intro: rel_imp_elem) +done + +lemma (in CLF) intervalI: + "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b" +apply (simp add: interval_def) +done + +lemma (in CLF) interval_lemma1: + "[| S <= interval r a b; x \<in> S |] ==> (a, x) \<in> r" +apply (unfold interval_def, fast) +done + +lemma (in CLF) interval_lemma2: + "[| S <= interval r a b; x \<in> S |] ==> (x, b) \<in> r" +apply (unfold interval_def, fast) +done + +lemma (in CLF) a_less_lub: + "[| S <= A; S \<noteq> {}; + \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r" +by (blast intro: transE PO_imp_trans) + +lemma (in CLF) glb_less_b: + "[| S <= A; S \<noteq> {}; + \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r" +by (blast intro: transE PO_imp_trans) + +lemma (in CLF) S_intv_cl: + "[| a \<in> A; b \<in> A; S <= interval r a b |]==> S <= A" +by (simp add: subset_trans [OF _ interval_subset]) + +lemma (in CLF) L_in_interval: + "[| a \<in> A; b \<in> A; S <= interval r a b; + S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> 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) \<in> r"} *} +apply (simp add: isLub_least interval_lemma2) +done + +lemma (in CLF) G_in_interval: + "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S <= interval r a b; isGlb S cl G; + S \<noteq> {} |] ==> G \<in> interval r a b" +apply (simp add: interval_dual) +apply (simp add: Tarski.L_in_interval [of _ f] + dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub) +done + +lemma (in CLF) intervalPO: + "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] + ==> (| pset = interval r a b, order = induced (interval r a b) r |) + \<in> PartialOrder" +apply (rule po_subset_po) +apply (simp add: interval_subset) +done + +lemma (in CLF) intv_CL_lub: + "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] + ==> \<forall>S. S <= interval r a b --> + (\<exists>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) \<in> 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 \<in> S = False => everything"} *} +apply fast +-- {* @{text "S \<noteq> {}"} *} +apply simp +-- {* @{text "\<forall>y:S. (y, L) \<in> 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 "\<forall>z:interval r a b. (\<forall>y:S. (y, z) \<in> induced (interval r a b) r \<longrightarrow> (if S = {} then a else L, z) \<in> 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) +apply (rule CO_refl, assumption) +apply (rule interval_not_empty) +apply (rule CO_trans) +apply (simp add: interval_def) +-- {* @{text "S \<noteq> {}"} *} +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 \<in> A; b \<in> A; interval r a b \<noteq> {} |] + ==> 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] + + +subsubsection {* 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 \<in> A" +apply (simp add: Bot_def least_def) +apply (rule someI2) +apply (fold A_def) +apply (erule_tac [2] conjunct1) +apply (rule conjI) +apply (rule glb_in_lattice) +apply (rule subset_refl) +apply (fold r_def) +apply (simp add: glb_lower) +done + +lemma (in CLF) Top_in_lattice: "Top cl \<in> A" +apply (simp add: Top_dual_Bot A_def) +apply (rule dualA_iff [THEN subst]) +apply (blast intro!: Tarski.Bot_in_lattice dualPO CL_dualCL CLF_dual f_cl) +done + +lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r" +apply (simp add: Top_def greatest_def) +apply (rule someI2) +apply (fold r_def A_def) +prefer 2 apply fast +apply (intro conjI ballI) +apply (rule_tac [2] lub_upper) +apply (auto simp add: lub_in_lattice) +done + +lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r" +apply (simp add: Bot_dual_Top r_def) +apply (rule dualr_iff [THEN subst]) +apply (simp add: Tarski.Top_prop [of _ f] + dualA_iff A_def dualPO CL_dualCL CLF_dual) +done + +lemma (in CLF) Top_intv_not_empty: "x \<in> A ==> interval r x (Top cl) \<noteq> {}" +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 \<in> A ==> interval r (Bot cl) x \<noteq> {}" +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!: Tarski.Top_in_lattice + f_cl dualPO CL_dualCL CLF_dual) +apply (simp add: Tarski.Top_intv_not_empty [of _ f] + dualA_iff A_def dualPO CL_dualCL CLF_dual) +done + +subsubsection {* fixed points form a partial order *} + +lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> 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 \<in> A" +by (simp add: Y_subset_A [THEN lub_in_lattice]) + +lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> 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)) \<in> r to (x, lub Y cl) \<in> 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 \<in> intY1 \<Longrightarrow> f x \<in> intY1" +apply (simp add: intY1_def interval_def) +apply (rule conjI) +apply (rule transE) +apply (rule CO_trans) +apply (rule lubY_le_flubY) +-- {* @{text "(f (lub Y cl), f x) \<in> 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) \<in> 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) \<in> 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 |) \<in> 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 \<in> P" +apply (unfold P_def) +apply (rule_tac A = "intY1" in fixf_subset) +apply (rule intY1_subset) +apply (simp add: Tarski.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 \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> 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 \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] + ==> ((%x: intY1. f x) z, z) \<in> 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] + CO_refl [THEN reflE]) +done + +lemma (in Tarski) tarski_full_lemma: + "\<exists>L. isLub Y (| pset = P, order = induced P r |) L" +apply (rule_tac x = "v" in exI) +apply (simp add: isLub_def) +-- {* @{text "v \<in> P"} *} +apply (simp add: v_in_P) +apply (rule conjI) +-- {* @{text v} is lub *} +-- {* @{text "1. \<forall>y:Y. (y, v) \<in> 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 CO_trans) +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 Tarski.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 Tarski.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] + fix_subset [of f A, THEN subsetD] + CO_refl [THEN reflE]) +done + +lemma CompleteLatticeI_simp: + "[| (| pset = A, order = r |) \<in> PartialOrder; + \<forall>S. S <= A --> (\<exists>L. isLub S (| pset = A, order = r |) L) |] + ==> (| pset = A, order = r |) \<in> CompleteLattice" +by (simp add: CompleteLatticeI Rdual) + +theorem (in CLF) Tarski_full: + "(| pset = P, order = induced P r|) \<in> 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 |