author | haftmann |
Tue, 20 Mar 2007 08:27:15 +0100 | |
changeset 22473 | 753123c89d72 |
parent 21404 | eb85850d3eb7 |
child 22547 | c3290f4382e4 |
permissions | -rw-r--r-- |
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(* Title: HOL/ex/Tarski.thy |
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ID: $Id$ |
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Author: Florian Kammüller, Cambridge University Computer Laboratory |
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*) |
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header {* The Full Theorem of Tarski *} |
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theory Tarski imports Main FuncSet begin |
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text {* |
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Minimal version of lattice theory plus the full theorem of Tarski: |
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The fixedpoints of a complete lattice themselves form a complete |
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lattice. |
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Illustrates first-class theories, using the Sigma representation of |
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structures. Tidied and converted to Isar by lcp. |
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*} |
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record 'a potype = |
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pset :: "'a set" |
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order :: "('a * 'a) set" |
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definition |
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monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool" where |
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"monotone f A r = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r)" |
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definition |
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least :: "['a => bool, 'a potype] => 'a" where |
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"least P po = (SOME x. x: pset po & P x & |
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(\<forall>y \<in> pset po. P y --> (x,y): order po))" |
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definition |
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greatest :: "['a => bool, 'a potype] => 'a" where |
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"greatest P po = (SOME x. x: pset po & P x & |
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(\<forall>y \<in> pset po. P y --> (y,x): order po))" |
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definition |
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lub :: "['a set, 'a potype] => 'a" where |
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"lub S po = least (%x. \<forall>y\<in>S. (y,x): order po) po" |
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definition |
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glb :: "['a set, 'a potype] => 'a" where |
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"glb S po = greatest (%x. \<forall>y\<in>S. (x,y): order po) po" |
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definition |
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isLub :: "['a set, 'a potype, 'a] => bool" where |
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"isLub S po = (%L. (L: pset po & (\<forall>y\<in>S. (y,L): order po) & |
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(\<forall>z\<in>pset po. (\<forall>y\<in>S. (y,z): order po) --> (L,z): order po)))" |
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definition |
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isGlb :: "['a set, 'a potype, 'a] => bool" where |
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"isGlb S po = (%G. (G: pset po & (\<forall>y\<in>S. (G,y): order po) & |
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(\<forall>z \<in> pset po. (\<forall>y\<in>S. (z,y): order po) --> (z,G): order po)))" |
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definition |
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"fix" :: "[('a => 'a), 'a set] => 'a set" where |
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"fix f A = {x. x: A & f x = x}" |
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definition |
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interval :: "[('a*'a) set,'a, 'a ] => 'a set" where |
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"interval r a b = {x. (a,x): r & (x,b): r}" |
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definition |
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Bot :: "'a potype => 'a" where |
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"Bot po = least (%x. True) po" |
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definition |
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Top :: "'a potype => 'a" where |
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"Top po = greatest (%x. True) po" |
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definition |
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PartialOrder :: "('a potype) set" where |
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"PartialOrder = {P. refl (pset P) (order P) & antisym (order P) & |
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trans (order P)}" |
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definition |
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CompleteLattice :: "('a potype) set" where |
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"CompleteLattice = {cl. cl: PartialOrder & |
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(\<forall>S. S \<subseteq> pset cl --> (\<exists>L. isLub S cl L)) & |
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(\<forall>S. S \<subseteq> pset cl --> (\<exists>G. isGlb S cl G))}" |
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definition |
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CLF :: "('a potype * ('a => 'a)) set" where |
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"CLF = (SIGMA cl: CompleteLattice. |
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{f. f: pset cl -> pset cl & monotone f (pset cl) (order cl)})" |
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definition |
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induced :: "['a set, ('a * 'a) set] => ('a *'a)set" where |
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"induced A r = {(a,b). a : A & b: A & (a,b): r}" |
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definition |
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sublattice :: "('a potype * 'a set)set" where |
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"sublattice = |
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(SIGMA cl: CompleteLattice. |
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{S. S \<subseteq> pset cl & |
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(| pset = S, order = induced S (order cl) |): CompleteLattice})" |
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abbreviation |
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sublat :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50) where |
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"S <<= cl == S : sublattice `` {cl}" |
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definition |
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dual :: "'a potype => 'a potype" where |
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"dual po = (| pset = pset po, order = converse (order po) |)" |
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locale (open) PO = |
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fixes cl :: "'a potype" |
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and A :: "'a set" |
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and r :: "('a * 'a) set" |
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assumes cl_po: "cl : PartialOrder" |
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defines A_def: "A == pset cl" |
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and r_def: "r == order cl" |
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locale (open) CL = PO + |
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assumes cl_co: "cl : CompleteLattice" |
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locale (open) CLF = CL + |
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fixes f :: "'a => 'a" |
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and P :: "'a set" |
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assumes f_cl: "(cl,f) : CLF" (*was the equivalent "f : CLF``{cl}"*) |
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defines P_def: "P == fix f A" |
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locale (open) Tarski = CLF + |
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fixes Y :: "'a set" |
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and intY1 :: "'a set" |
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and v :: "'a" |
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assumes |
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Y_ss: "Y \<subseteq> P" |
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defines |
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intY1_def: "intY1 == interval r (lub Y cl) (Top cl)" |
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and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r & |
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x: intY1} |
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(| pset=intY1, order=induced intY1 r|)" |
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subsection {* Partial Order *} |
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lemma (in PO) PO_imp_refl: "refl A r" |
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apply (insert cl_po) |
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apply (simp add: PartialOrder_def A_def r_def) |
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done |
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lemma (in PO) PO_imp_sym: "antisym r" |
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apply (insert cl_po) |
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apply (simp add: PartialOrder_def r_def) |
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done |
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lemma (in PO) PO_imp_trans: "trans r" |
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apply (insert cl_po) |
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apply (simp add: PartialOrder_def r_def) |
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done |
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lemma (in PO) reflE: "x \<in> A ==> (x, x) \<in> r" |
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apply (insert cl_po) |
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apply (simp add: PartialOrder_def refl_def A_def r_def) |
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done |
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lemma (in PO) antisymE: "[| (a, b) \<in> r; (b, a) \<in> r |] ==> a = b" |
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apply (insert cl_po) |
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apply (simp add: PartialOrder_def antisym_def r_def) |
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done |
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lemma (in PO) transE: "[| (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r" |
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apply (insert cl_po) |
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apply (simp add: PartialOrder_def r_def) |
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apply (unfold trans_def, fast) |
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done |
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lemma (in PO) monotoneE: |
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"[| monotone f A r; x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r" |
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by (simp add: monotone_def) |
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lemma (in PO) po_subset_po: |
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"S \<subseteq> A ==> (| pset = S, order = induced S r |) \<in> PartialOrder" |
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apply (simp (no_asm) add: PartialOrder_def) |
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apply auto |
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-- {* refl *} |
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apply (simp add: refl_def induced_def) |
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apply (blast intro: reflE) |
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-- {* antisym *} |
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apply (simp add: antisym_def induced_def) |
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apply (blast intro: antisymE) |
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-- {* trans *} |
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apply (simp add: trans_def induced_def) |
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apply (blast intro: transE) |
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done |
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lemma (in PO) indE: "[| (x, y) \<in> induced S r; S \<subseteq> A |] ==> (x, y) \<in> r" |
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by (simp add: add: induced_def) |
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lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r" |
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by (simp add: add: induced_def) |
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lemma (in CL) CL_imp_ex_isLub: "S \<subseteq> A ==> \<exists>L. isLub S cl L" |
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apply (insert cl_co) |
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apply (simp add: CompleteLattice_def A_def) |
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done |
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declare (in CL) cl_co [simp] |
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lemma isLub_lub: "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)" |
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by (simp add: lub_def least_def isLub_def some_eq_ex [symmetric]) |
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lemma isGlb_glb: "(\<exists>G. isGlb S cl G) = isGlb S cl (glb S cl)" |
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by (simp add: glb_def greatest_def isGlb_def some_eq_ex [symmetric]) |
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lemma isGlb_dual_isLub: "isGlb S cl = isLub S (dual cl)" |
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by (simp add: isLub_def isGlb_def dual_def converse_def) |
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lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)" |
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by (simp add: isLub_def isGlb_def dual_def converse_def) |
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lemma (in PO) dualPO: "dual cl \<in> PartialOrder" |
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apply (insert cl_po) |
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apply (simp add: PartialOrder_def dual_def refl_converse |
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trans_converse antisym_converse) |
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done |
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lemma Rdual: |
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"\<forall>S. (S \<subseteq> A -->( \<exists>L. isLub S (| pset = A, order = r|) L)) |
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==> \<forall>S. (S \<subseteq> A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))" |
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apply safe |
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apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)} |
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(|pset = A, order = r|) " in exI) |
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apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> r) }" in spec) |
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apply (drule mp, fast) |
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apply (simp add: isLub_lub isGlb_def) |
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apply (simp add: isLub_def, blast) |
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done |
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lemma lub_dual_glb: "lub S cl = glb S (dual cl)" |
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by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def) |
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lemma glb_dual_lub: "glb S cl = lub S (dual cl)" |
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by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def) |
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lemma CL_subset_PO: "CompleteLattice \<subseteq> PartialOrder" |
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by (simp add: PartialOrder_def CompleteLattice_def, fast) |
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lemmas CL_imp_PO = CL_subset_PO [THEN subsetD] |
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declare CL_imp_PO [THEN PO.PO_imp_refl, simp] |
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declare CL_imp_PO [THEN PO.PO_imp_sym, simp] |
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declare CL_imp_PO [THEN PO.PO_imp_trans, simp] |
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lemma (in CL) CO_refl: "refl A r" |
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by (rule PO_imp_refl) |
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lemma (in CL) CO_antisym: "antisym r" |
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by (rule PO_imp_sym) |
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lemma (in CL) CO_trans: "trans r" |
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by (rule PO_imp_trans) |
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lemma CompleteLatticeI: |
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"[| po \<in> PartialOrder; (\<forall>S. S \<subseteq> pset po --> (\<exists>L. isLub S po L)); |
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(\<forall>S. S \<subseteq> pset po --> (\<exists>G. isGlb S po G))|] |
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==> po \<in> CompleteLattice" |
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apply (unfold CompleteLattice_def, blast) |
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done |
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lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice" |
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apply (insert cl_co) |
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apply (simp add: CompleteLattice_def dual_def) |
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apply (fold dual_def) |
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apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric] |
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dualPO) |
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done |
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lemma (in PO) dualA_iff: "pset (dual cl) = pset cl" |
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by (simp add: dual_def) |
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lemma (in PO) dualr_iff: "((x, y) \<in> (order(dual cl))) = ((y, x) \<in> order cl)" |
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by (simp add: dual_def) |
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lemma (in PO) monotone_dual: |
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"monotone f (pset cl) (order cl) |
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==> monotone f (pset (dual cl)) (order(dual cl))" |
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by (simp add: monotone_def dualA_iff dualr_iff) |
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lemma (in PO) interval_dual: |
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"[| x \<in> A; y \<in> A|] ==> interval r x y = interval (order(dual cl)) y x" |
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apply (simp add: interval_def dualr_iff) |
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apply (fold r_def, fast) |
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done |
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lemma (in PO) interval_not_empty: |
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"[| trans r; interval r a b \<noteq> {} |] ==> (a, b) \<in> r" |
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apply (simp add: interval_def) |
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apply (unfold trans_def, blast) |
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done |
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lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r" |
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by (simp add: interval_def) |
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lemma (in PO) left_in_interval: |
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"[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b" |
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apply (simp (no_asm_simp) add: interval_def) |
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apply (simp add: PO_imp_trans interval_not_empty) |
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apply (simp add: reflE) |
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done |
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lemma (in PO) right_in_interval: |
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"[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b" |
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apply (simp (no_asm_simp) add: interval_def) |
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apply (simp add: PO_imp_trans interval_not_empty) |
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apply (simp add: reflE) |
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done |
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subsection {* sublattice *} |
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lemma (in PO) sublattice_imp_CL: |
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"S <<= cl ==> (| pset = S, order = induced S r |) \<in> CompleteLattice" |
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by (simp add: sublattice_def CompleteLattice_def r_def) |
13115 | 319 |
|
320 |
lemma (in CL) sublatticeI: |
|
17841 | 321 |
"[| S \<subseteq> A; (| pset = S, order = induced S r |) \<in> CompleteLattice |] |
18750 | 322 |
==> S <<= cl" |
13115 | 323 |
by (simp add: sublattice_def A_def r_def) |
324 |
||
13383 | 325 |
|
14569 | 326 |
subsection {* lub *} |
13383 | 327 |
|
17841 | 328 |
lemma (in CL) lub_unique: "[| S \<subseteq> A; isLub S cl x; isLub S cl L|] ==> x = L" |
13115 | 329 |
apply (rule antisymE) |
330 |
apply (auto simp add: isLub_def r_def) |
|
331 |
done |
|
332 |
||
17841 | 333 |
lemma (in CL) lub_upper: "[|S \<subseteq> A; x \<in> S|] ==> (x, lub S cl) \<in> r" |
13115 | 334 |
apply (rule CL_imp_ex_isLub [THEN exE], assumption) |
335 |
apply (unfold lub_def least_def) |
|
336 |
apply (rule some_equality [THEN ssubst]) |
|
337 |
apply (simp add: isLub_def) |
|
13383 | 338 |
apply (simp add: lub_unique A_def isLub_def) |
13115 | 339 |
apply (simp add: isLub_def r_def) |
340 |
done |
|
341 |
||
342 |
lemma (in CL) lub_least: |
|
17841 | 343 |
"[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r" |
13115 | 344 |
apply (rule CL_imp_ex_isLub [THEN exE], assumption) |
345 |
apply (unfold lub_def least_def) |
|
346 |
apply (rule_tac s=x in some_equality [THEN ssubst]) |
|
347 |
apply (simp add: isLub_def) |
|
13383 | 348 |
apply (simp add: lub_unique A_def isLub_def) |
13115 | 349 |
apply (simp add: isLub_def r_def A_def) |
350 |
done |
|
351 |
||
17841 | 352 |
lemma (in CL) lub_in_lattice: "S \<subseteq> A ==> lub S cl \<in> A" |
13115 | 353 |
apply (rule CL_imp_ex_isLub [THEN exE], assumption) |
354 |
apply (unfold lub_def least_def) |
|
355 |
apply (subst some_equality) |
|
356 |
apply (simp add: isLub_def) |
|
357 |
prefer 2 apply (simp add: isLub_def A_def) |
|
13383 | 358 |
apply (simp add: lub_unique A_def isLub_def) |
13115 | 359 |
done |
360 |
||
361 |
lemma (in CL) lubI: |
|
17841 | 362 |
"[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r; |
13115 | 363 |
\<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl" |
364 |
apply (rule lub_unique, assumption) |
|
365 |
apply (simp add: isLub_def A_def r_def) |
|
366 |
apply (unfold isLub_def) |
|
367 |
apply (rule conjI) |
|
368 |
apply (fold A_def r_def) |
|
369 |
apply (rule lub_in_lattice, assumption) |
|
370 |
apply (simp add: lub_upper lub_least) |
|
371 |
done |
|
372 |
||
17841 | 373 |
lemma (in CL) lubIa: "[| S \<subseteq> A; isLub S cl L |] ==> L = lub S cl" |
13115 | 374 |
by (simp add: lubI isLub_def A_def r_def) |
375 |
||
376 |
lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A" |
|
377 |
by (simp add: isLub_def A_def) |
|
378 |
||
379 |
lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r" |
|
380 |
by (simp add: isLub_def r_def) |
|
381 |
||
382 |
lemma (in CL) isLub_least: |
|
383 |
"[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r" |
|
384 |
by (simp add: isLub_def A_def r_def) |
|
385 |
||
386 |
lemma (in CL) isLubI: |
|
13383 | 387 |
"[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r; |
13115 | 388 |
(\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L" |
389 |
by (simp add: isLub_def A_def r_def) |
|
390 |
||
13383 | 391 |
|
14569 | 392 |
subsection {* glb *} |
13383 | 393 |
|
17841 | 394 |
lemma (in CL) glb_in_lattice: "S \<subseteq> A ==> glb S cl \<in> A" |
13115 | 395 |
apply (subst glb_dual_lub) |
396 |
apply (simp add: A_def) |
|
397 |
apply (rule dualA_iff [THEN subst]) |
|
21232 | 398 |
apply (rule CL.lub_in_lattice) |
13383 | 399 |
apply (rule dualPO) |
13115 | 400 |
apply (rule CL_dualCL) |
401 |
apply (simp add: dualA_iff) |
|
402 |
done |
|
403 |
||
17841 | 404 |
lemma (in CL) glb_lower: "[|S \<subseteq> A; x \<in> S|] ==> (glb S cl, x) \<in> r" |
13115 | 405 |
apply (subst glb_dual_lub) |
406 |
apply (simp add: r_def) |
|
407 |
apply (rule dualr_iff [THEN subst]) |
|
21232 | 408 |
apply (rule CL.lub_upper) |
13383 | 409 |
apply (rule dualPO) |
13115 | 410 |
apply (rule CL_dualCL) |
411 |
apply (simp add: dualA_iff A_def, assumption) |
|
412 |
done |
|
413 |
||
13383 | 414 |
text {* |
415 |
Reduce the sublattice property by using substructural properties; |
|
416 |
abandoned see @{text "Tarski_4.ML"}. |
|
417 |
*} |
|
13115 | 418 |
|
419 |
lemma (in CLF) [simp]: |
|
13585 | 420 |
"f: pset cl -> pset cl & monotone f (pset cl) (order cl)" |
13383 | 421 |
apply (insert f_cl) |
422 |
apply (simp add: CLF_def) |
|
13115 | 423 |
done |
424 |
||
425 |
declare (in CLF) f_cl [simp] |
|
426 |
||
427 |
||
13585 | 428 |
lemma (in CLF) f_in_funcset: "f \<in> A -> A" |
13115 | 429 |
by (simp add: A_def) |
430 |
||
431 |
lemma (in CLF) monotone_f: "monotone f A r" |
|
432 |
by (simp add: A_def r_def) |
|
433 |
||
434 |
lemma (in CLF) CLF_dual: "(cl,f) \<in> CLF ==> (dual cl, f) \<in> CLF" |
|
435 |
apply (simp add: CLF_def CL_dualCL monotone_dual) |
|
436 |
apply (simp add: dualA_iff) |
|
437 |
done |
|
438 |
||
13383 | 439 |
|
14569 | 440 |
subsection {* fixed points *} |
13383 | 441 |
|
17841 | 442 |
lemma fix_subset: "fix f A \<subseteq> A" |
13115 | 443 |
by (simp add: fix_def, fast) |
444 |
||
445 |
lemma fix_imp_eq: "x \<in> fix f A ==> f x = x" |
|
446 |
by (simp add: fix_def) |
|
447 |
||
448 |
lemma fixf_subset: |
|
17841 | 449 |
"[| A \<subseteq> B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B" |
450 |
by (simp add: fix_def, auto) |
|
13115 | 451 |
|
13383 | 452 |
|
14569 | 453 |
subsection {* lemmas for Tarski, lub *} |
13115 | 454 |
lemma (in CLF) lubH_le_flubH: |
455 |
"H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r" |
|
456 |
apply (rule lub_least, fast) |
|
457 |
apply (rule f_in_funcset [THEN funcset_mem]) |
|
458 |
apply (rule lub_in_lattice, fast) |
|
13383 | 459 |
-- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *} |
13115 | 460 |
apply (rule ballI) |
461 |
apply (rule transE) |
|
13585 | 462 |
-- {* instantiates @{text "(x, ???z) \<in> order cl to (x, f x)"}, *} |
13383 | 463 |
-- {* because of the def of @{text H} *} |
13115 | 464 |
apply fast |
13383 | 465 |
-- {* so it remains to show @{text "(f x, f (lub H cl)) \<in> r"} *} |
13115 | 466 |
apply (rule_tac f = "f" in monotoneE) |
467 |
apply (rule monotone_f, fast) |
|
468 |
apply (rule lub_in_lattice, fast) |
|
469 |
apply (rule lub_upper, fast) |
|
470 |
apply assumption |
|
471 |
done |
|
472 |
||
473 |
lemma (in CLF) flubH_le_lubH: |
|
474 |
"[| H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r" |
|
475 |
apply (rule lub_upper, fast) |
|
476 |
apply (rule_tac t = "H" in ssubst, assumption) |
|
477 |
apply (rule CollectI) |
|
478 |
apply (rule conjI) |
|
479 |
apply (rule_tac [2] f_in_funcset [THEN funcset_mem]) |
|
480 |
apply (rule_tac [2] lub_in_lattice) |
|
481 |
prefer 2 apply fast |
|
482 |
apply (rule_tac f = "f" in monotoneE) |
|
483 |
apply (rule monotone_f) |
|
13383 | 484 |
apply (blast intro: lub_in_lattice) |
485 |
apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem]) |
|
13115 | 486 |
apply (simp add: lubH_le_flubH) |
487 |
done |
|
488 |
||
489 |
lemma (in CLF) lubH_is_fixp: |
|
490 |
"H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A" |
|
491 |
apply (simp add: fix_def) |
|
492 |
apply (rule conjI) |
|
493 |
apply (rule lub_in_lattice, fast) |
|
494 |
apply (rule antisymE) |
|
495 |
apply (simp add: flubH_le_lubH) |
|
496 |
apply (simp add: lubH_le_flubH) |
|
497 |
done |
|
498 |
||
499 |
lemma (in CLF) fix_in_H: |
|
500 |
"[| H = {x. (x, f x) \<in> r & x \<in> A}; x \<in> P |] ==> x \<in> H" |
|
13383 | 501 |
by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl |
502 |
fix_subset [of f A, THEN subsetD]) |
|
13115 | 503 |
|
504 |
lemma (in CLF) fixf_le_lubH: |
|
505 |
"H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r" |
|
506 |
apply (rule ballI) |
|
507 |
apply (rule lub_upper, fast) |
|
508 |
apply (rule fix_in_H) |
|
13383 | 509 |
apply (simp_all add: P_def) |
13115 | 510 |
done |
511 |
||
512 |
lemma (in CLF) lubH_least_fixf: |
|
13383 | 513 |
"H = {x. (x, f x) \<in> r & x \<in> A} |
13115 | 514 |
==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r" |
515 |
apply (rule allI) |
|
516 |
apply (rule impI) |
|
517 |
apply (erule bspec) |
|
518 |
apply (rule lubH_is_fixp, assumption) |
|
519 |
done |
|
520 |
||
14569 | 521 |
subsection {* Tarski fixpoint theorem 1, first part *} |
13115 | 522 |
lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl" |
523 |
apply (rule sym) |
|
13383 | 524 |
apply (simp add: P_def) |
13115 | 525 |
apply (rule lubI) |
526 |
apply (rule fix_subset) |
|
527 |
apply (rule lub_in_lattice, fast) |
|
528 |
apply (simp add: fixf_le_lubH) |
|
529 |
apply (simp add: lubH_least_fixf) |
|
530 |
done |
|
531 |
||
532 |
lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P" |
|
13383 | 533 |
-- {* Tarski for glb *} |
13115 | 534 |
apply (simp add: glb_dual_lub P_def A_def r_def) |
535 |
apply (rule dualA_iff [THEN subst]) |
|
21232 | 536 |
apply (rule CLF.lubH_is_fixp) |
13383 | 537 |
apply (rule dualPO) |
13115 | 538 |
apply (rule CL_dualCL) |
539 |
apply (rule f_cl [THEN CLF_dual]) |
|
540 |
apply (simp add: dualr_iff dualA_iff) |
|
541 |
done |
|
542 |
||
543 |
lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl" |
|
544 |
apply (simp add: glb_dual_lub P_def A_def r_def) |
|
545 |
apply (rule dualA_iff [THEN subst]) |
|
21232 | 546 |
apply (simp add: CLF.T_thm_1_lub [of _ f, OF dualPO CL_dualCL] |
13115 | 547 |
dualPO CL_dualCL CLF_dual dualr_iff) |
548 |
done |
|
549 |
||
14569 | 550 |
subsection {* interval *} |
13383 | 551 |
|
13115 | 552 |
lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A" |
13383 | 553 |
apply (insert CO_refl) |
554 |
apply (simp add: refl_def, blast) |
|
13115 | 555 |
done |
556 |
||
17841 | 557 |
lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b \<subseteq> A" |
13115 | 558 |
apply (simp add: interval_def) |
559 |
apply (blast intro: rel_imp_elem) |
|
560 |
done |
|
561 |
||
562 |
lemma (in CLF) intervalI: |
|
563 |
"[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b" |
|
17841 | 564 |
by (simp add: interval_def) |
13115 | 565 |
|
566 |
lemma (in CLF) interval_lemma1: |
|
17841 | 567 |
"[| S \<subseteq> interval r a b; x \<in> S |] ==> (a, x) \<in> r" |
568 |
by (unfold interval_def, fast) |
|
13115 | 569 |
|
570 |
lemma (in CLF) interval_lemma2: |
|
17841 | 571 |
"[| S \<subseteq> interval r a b; x \<in> S |] ==> (x, b) \<in> r" |
572 |
by (unfold interval_def, fast) |
|
13115 | 573 |
|
574 |
lemma (in CLF) a_less_lub: |
|
17841 | 575 |
"[| S \<subseteq> A; S \<noteq> {}; |
13115 | 576 |
\<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r" |
18705 | 577 |
by (blast intro: transE) |
13115 | 578 |
|
579 |
lemma (in CLF) glb_less_b: |
|
17841 | 580 |
"[| S \<subseteq> A; S \<noteq> {}; |
13115 | 581 |
\<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r" |
18705 | 582 |
by (blast intro: transE) |
13115 | 583 |
|
584 |
lemma (in CLF) S_intv_cl: |
|
17841 | 585 |
"[| a \<in> A; b \<in> A; S \<subseteq> interval r a b |]==> S \<subseteq> A" |
13115 | 586 |
by (simp add: subset_trans [OF _ interval_subset]) |
587 |
||
588 |
lemma (in CLF) L_in_interval: |
|
17841 | 589 |
"[| a \<in> A; b \<in> A; S \<subseteq> interval r a b; |
13115 | 590 |
S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b" |
591 |
apply (rule intervalI) |
|
592 |
apply (rule a_less_lub) |
|
593 |
prefer 2 apply assumption |
|
594 |
apply (simp add: S_intv_cl) |
|
595 |
apply (rule ballI) |
|
596 |
apply (simp add: interval_lemma1) |
|
597 |
apply (simp add: isLub_upper) |
|
13383 | 598 |
-- {* @{text "(L, b) \<in> r"} *} |
13115 | 599 |
apply (simp add: isLub_least interval_lemma2) |
600 |
done |
|
601 |
||
602 |
lemma (in CLF) G_in_interval: |
|
17841 | 603 |
"[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> interval r a b; isGlb S cl G; |
13115 | 604 |
S \<noteq> {} |] ==> G \<in> interval r a b" |
605 |
apply (simp add: interval_dual) |
|
21232 | 606 |
apply (simp add: CLF.L_in_interval [of _ f] |
13115 | 607 |
dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub) |
608 |
done |
|
609 |
||
610 |
lemma (in CLF) intervalPO: |
|
13383 | 611 |
"[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] |
13115 | 612 |
==> (| pset = interval r a b, order = induced (interval r a b) r |) |
613 |
\<in> PartialOrder" |
|
614 |
apply (rule po_subset_po) |
|
615 |
apply (simp add: interval_subset) |
|
616 |
done |
|
617 |
||
618 |
lemma (in CLF) intv_CL_lub: |
|
13383 | 619 |
"[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] |
17841 | 620 |
==> \<forall>S. S \<subseteq> interval r a b --> |
13383 | 621 |
(\<exists>L. isLub S (| pset = interval r a b, |
13115 | 622 |
order = induced (interval r a b) r |) L)" |
623 |
apply (intro strip) |
|
624 |
apply (frule S_intv_cl [THEN CL_imp_ex_isLub]) |
|
625 |
prefer 2 apply assumption |
|
626 |
apply assumption |
|
627 |
apply (erule exE) |
|
13383 | 628 |
-- {* define the lub for the interval as *} |
13115 | 629 |
apply (rule_tac x = "if S = {} then a else L" in exI) |
630 |
apply (simp (no_asm_simp) add: isLub_def split del: split_if) |
|
13383 | 631 |
apply (intro impI conjI) |
632 |
-- {* @{text "(if S = {} then a else L) \<in> interval r a b"} *} |
|
13115 | 633 |
apply (simp add: CL_imp_PO L_in_interval) |
634 |
apply (simp add: left_in_interval) |
|
13383 | 635 |
-- {* lub prop 1 *} |
13115 | 636 |
apply (case_tac "S = {}") |
13383 | 637 |
-- {* @{text "S = {}, y \<in> S = False => everything"} *} |
13115 | 638 |
apply fast |
13383 | 639 |
-- {* @{text "S \<noteq> {}"} *} |
13115 | 640 |
apply simp |
13383 | 641 |
-- {* @{text "\<forall>y:S. (y, L) \<in> induced (interval r a b) r"} *} |
13115 | 642 |
apply (rule ballI) |
643 |
apply (simp add: induced_def L_in_interval) |
|
644 |
apply (rule conjI) |
|
645 |
apply (rule subsetD) |
|
646 |
apply (simp add: S_intv_cl, assumption) |
|
647 |
apply (simp add: isLub_upper) |
|
13383 | 648 |
-- {* @{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"} *} |
13115 | 649 |
apply (rule ballI) |
650 |
apply (rule impI) |
|
651 |
apply (case_tac "S = {}") |
|
13383 | 652 |
-- {* @{text "S = {}"} *} |
13115 | 653 |
apply simp |
654 |
apply (simp add: induced_def interval_def) |
|
655 |
apply (rule conjI) |
|
18705 | 656 |
apply (rule reflE, assumption) |
13115 | 657 |
apply (rule interval_not_empty) |
658 |
apply (rule CO_trans) |
|
659 |
apply (simp add: interval_def) |
|
13383 | 660 |
-- {* @{text "S \<noteq> {}"} *} |
13115 | 661 |
apply simp |
662 |
apply (simp add: induced_def L_in_interval) |
|
663 |
apply (rule isLub_least, assumption) |
|
664 |
apply (rule subsetD) |
|
665 |
prefer 2 apply assumption |
|
666 |
apply (simp add: S_intv_cl, fast) |
|
667 |
done |
|
668 |
||
669 |
lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual] |
|
670 |
||
671 |
lemma (in CLF) interval_is_sublattice: |
|
13383 | 672 |
"[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] |
18750 | 673 |
==> interval r a b <<= cl" |
13115 | 674 |
apply (rule sublatticeI) |
675 |
apply (simp add: interval_subset) |
|
676 |
apply (rule CompleteLatticeI) |
|
677 |
apply (simp add: intervalPO) |
|
678 |
apply (simp add: intv_CL_lub) |
|
679 |
apply (simp add: intv_CL_glb) |
|
680 |
done |
|
681 |
||
13383 | 682 |
lemmas (in CLF) interv_is_compl_latt = |
13115 | 683 |
interval_is_sublattice [THEN sublattice_imp_CL] |
684 |
||
13383 | 685 |
|
14569 | 686 |
subsection {* Top and Bottom *} |
13115 | 687 |
lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)" |
688 |
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff) |
|
689 |
||
690 |
lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)" |
|
691 |
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff) |
|
692 |
||
693 |
lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A" |
|
694 |
apply (simp add: Bot_def least_def) |
|
17841 | 695 |
apply (rule_tac a="glb A cl" in someI2) |
696 |
apply (simp_all add: glb_in_lattice glb_lower |
|
697 |
r_def [symmetric] A_def [symmetric]) |
|
13115 | 698 |
done |
699 |
||
700 |
lemma (in CLF) Top_in_lattice: "Top cl \<in> A" |
|
701 |
apply (simp add: Top_dual_Bot A_def) |
|
13383 | 702 |
apply (rule dualA_iff [THEN subst]) |
21232 | 703 |
apply (blast intro!: CLF.Bot_in_lattice dualPO CL_dualCL CLF_dual f_cl) |
13115 | 704 |
done |
705 |
||
706 |
lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r" |
|
707 |
apply (simp add: Top_def greatest_def) |
|
17841 | 708 |
apply (rule_tac a="lub A cl" in someI2) |
13115 | 709 |
apply (rule someI2) |
17841 | 710 |
apply (simp_all add: lub_in_lattice lub_upper |
711 |
r_def [symmetric] A_def [symmetric]) |
|
13115 | 712 |
done |
713 |
||
714 |
lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r" |
|
715 |
apply (simp add: Bot_dual_Top r_def) |
|
716 |
apply (rule dualr_iff [THEN subst]) |
|
21232 | 717 |
apply (simp add: CLF.Top_prop [of _ f] |
13115 | 718 |
dualA_iff A_def dualPO CL_dualCL CLF_dual) |
719 |
done |
|
720 |
||
721 |
lemma (in CLF) Top_intv_not_empty: "x \<in> A ==> interval r x (Top cl) \<noteq> {}" |
|
722 |
apply (rule notI) |
|
723 |
apply (drule_tac a = "Top cl" in equals0D) |
|
724 |
apply (simp add: interval_def) |
|
725 |
apply (simp add: refl_def Top_in_lattice Top_prop) |
|
726 |
done |
|
727 |
||
728 |
lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}" |
|
729 |
apply (simp add: Bot_dual_Top) |
|
730 |
apply (subst interval_dual) |
|
731 |
prefer 2 apply assumption |
|
732 |
apply (simp add: A_def) |
|
733 |
apply (rule dualA_iff [THEN subst]) |
|
21232 | 734 |
apply (blast intro!: CLF.Top_in_lattice |
13115 | 735 |
f_cl dualPO CL_dualCL CLF_dual) |
21232 | 736 |
apply (simp add: CLF.Top_intv_not_empty [of _ f] |
13115 | 737 |
dualA_iff A_def dualPO CL_dualCL CLF_dual) |
738 |
done |
|
739 |
||
14569 | 740 |
subsection {* fixed points form a partial order *} |
13383 | 741 |
|
13115 | 742 |
lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder" |
743 |
by (simp add: P_def fix_subset po_subset_po) |
|
744 |
||
17841 | 745 |
lemma (in Tarski) Y_subset_A: "Y \<subseteq> A" |
13115 | 746 |
apply (rule subset_trans [OF _ fix_subset]) |
747 |
apply (rule Y_ss [simplified P_def]) |
|
748 |
done |
|
749 |
||
750 |
lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A" |
|
18750 | 751 |
by (rule Y_subset_A [THEN lub_in_lattice]) |
13115 | 752 |
|
753 |
lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r" |
|
754 |
apply (rule lub_least) |
|
755 |
apply (rule Y_subset_A) |
|
756 |
apply (rule f_in_funcset [THEN funcset_mem]) |
|
757 |
apply (rule lubY_in_A) |
|
17841 | 758 |
-- {* @{text "Y \<subseteq> P ==> f x = x"} *} |
13115 | 759 |
apply (rule ballI) |
760 |
apply (rule_tac t = "x" in fix_imp_eq [THEN subst]) |
|
761 |
apply (erule Y_ss [simplified P_def, THEN subsetD]) |
|
13383 | 762 |
-- {* @{text "reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r"} by monotonicity *} |
13115 | 763 |
apply (rule_tac f = "f" in monotoneE) |
764 |
apply (rule monotone_f) |
|
765 |
apply (simp add: Y_subset_A [THEN subsetD]) |
|
766 |
apply (rule lubY_in_A) |
|
767 |
apply (simp add: lub_upper Y_subset_A) |
|
768 |
done |
|
769 |
||
17841 | 770 |
lemma (in Tarski) intY1_subset: "intY1 \<subseteq> A" |
13115 | 771 |
apply (unfold intY1_def) |
772 |
apply (rule interval_subset) |
|
773 |
apply (rule lubY_in_A) |
|
774 |
apply (rule Top_in_lattice) |
|
775 |
done |
|
776 |
||
777 |
lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD] |
|
778 |
||
779 |
lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1" |
|
780 |
apply (simp add: intY1_def interval_def) |
|
781 |
apply (rule conjI) |
|
782 |
apply (rule transE) |
|
783 |
apply (rule lubY_le_flubY) |
|
13383 | 784 |
-- {* @{text "(f (lub Y cl), f x) \<in> r"} *} |
13115 | 785 |
apply (rule_tac f=f in monotoneE) |
786 |
apply (rule monotone_f) |
|
787 |
apply (rule lubY_in_A) |
|
788 |
apply (simp add: intY1_def interval_def intY1_elem) |
|
789 |
apply (simp add: intY1_def interval_def) |
|
13383 | 790 |
-- {* @{text "(f x, Top cl) \<in> r"} *} |
13115 | 791 |
apply (rule Top_prop) |
792 |
apply (rule f_in_funcset [THEN funcset_mem]) |
|
793 |
apply (simp add: intY1_def interval_def intY1_elem) |
|
794 |
done |
|
795 |
||
13585 | 796 |
lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 -> intY1" |
13115 | 797 |
apply (rule restrictI) |
798 |
apply (erule intY1_f_closed) |
|
799 |
done |
|
800 |
||
801 |
lemma (in Tarski) intY1_mono: |
|
802 |
"monotone (%x: intY1. f x) intY1 (induced intY1 r)" |
|
803 |
apply (auto simp add: monotone_def induced_def intY1_f_closed) |
|
804 |
apply (blast intro: intY1_elem monotone_f [THEN monotoneE]) |
|
805 |
done |
|
806 |
||
13383 | 807 |
lemma (in Tarski) intY1_is_cl: |
13115 | 808 |
"(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice" |
809 |
apply (unfold intY1_def) |
|
810 |
apply (rule interv_is_compl_latt) |
|
811 |
apply (rule lubY_in_A) |
|
812 |
apply (rule Top_in_lattice) |
|
813 |
apply (rule Top_intv_not_empty) |
|
814 |
apply (rule lubY_in_A) |
|
815 |
done |
|
816 |
||
817 |
lemma (in Tarski) v_in_P: "v \<in> P" |
|
818 |
apply (unfold P_def) |
|
819 |
apply (rule_tac A = "intY1" in fixf_subset) |
|
820 |
apply (rule intY1_subset) |
|
21232 | 821 |
apply (simp add: CLF.glbH_is_fixp [OF _ intY1_is_cl, simplified] |
13115 | 822 |
v_def CL_imp_PO intY1_is_cl CLF_def intY1_func intY1_mono) |
823 |
done |
|
824 |
||
13383 | 825 |
lemma (in Tarski) z_in_interval: |
13115 | 826 |
"[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1" |
827 |
apply (unfold intY1_def P_def) |
|
828 |
apply (rule intervalI) |
|
13383 | 829 |
prefer 2 |
13115 | 830 |
apply (erule fix_subset [THEN subsetD, THEN Top_prop]) |
831 |
apply (rule lub_least) |
|
832 |
apply (rule Y_subset_A) |
|
833 |
apply (fast elim!: fix_subset [THEN subsetD]) |
|
834 |
apply (simp add: induced_def) |
|
835 |
done |
|
836 |
||
13383 | 837 |
lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] |
13115 | 838 |
==> ((%x: intY1. f x) z, z) \<in> induced intY1 r" |
839 |
apply (simp add: induced_def intY1_f_closed z_in_interval P_def) |
|
13383 | 840 |
apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD] |
18705 | 841 |
reflE) |
13115 | 842 |
done |
843 |
||
844 |
lemma (in Tarski) tarski_full_lemma: |
|
845 |
"\<exists>L. isLub Y (| pset = P, order = induced P r |) L" |
|
846 |
apply (rule_tac x = "v" in exI) |
|
847 |
apply (simp add: isLub_def) |
|
13383 | 848 |
-- {* @{text "v \<in> P"} *} |
13115 | 849 |
apply (simp add: v_in_P) |
850 |
apply (rule conjI) |
|
13383 | 851 |
-- {* @{text v} is lub *} |
852 |
-- {* @{text "1. \<forall>y:Y. (y, v) \<in> induced P r"} *} |
|
13115 | 853 |
apply (rule ballI) |
854 |
apply (simp add: induced_def subsetD v_in_P) |
|
855 |
apply (rule conjI) |
|
856 |
apply (erule Y_ss [THEN subsetD]) |
|
857 |
apply (rule_tac b = "lub Y cl" in transE) |
|
858 |
apply (rule lub_upper) |
|
859 |
apply (rule Y_subset_A, assumption) |
|
860 |
apply (rule_tac b = "Top cl" in interval_imp_mem) |
|
861 |
apply (simp add: v_def) |
|
862 |
apply (fold intY1_def) |
|
21232 | 863 |
apply (rule CL.glb_in_lattice [OF _ intY1_is_cl, simplified]) |
13115 | 864 |
apply (simp add: CL_imp_PO intY1_is_cl, force) |
13383 | 865 |
-- {* @{text v} is LEAST ub *} |
13115 | 866 |
apply clarify |
867 |
apply (rule indI) |
|
868 |
prefer 3 apply assumption |
|
869 |
prefer 2 apply (simp add: v_in_P) |
|
870 |
apply (unfold v_def) |
|
871 |
apply (rule indE) |
|
872 |
apply (rule_tac [2] intY1_subset) |
|
21232 | 873 |
apply (rule CL.glb_lower [OF _ intY1_is_cl, simplified]) |
13383 | 874 |
apply (simp add: CL_imp_PO intY1_is_cl) |
13115 | 875 |
apply force |
876 |
apply (simp add: induced_def intY1_f_closed z_in_interval) |
|
18705 | 877 |
apply (simp add: P_def fix_imp_eq [of _ f A] reflE |
878 |
fix_subset [of f A, THEN subsetD]) |
|
13115 | 879 |
done |
880 |
||
881 |
lemma CompleteLatticeI_simp: |
|
13383 | 882 |
"[| (| pset = A, order = r |) \<in> PartialOrder; |
17841 | 883 |
\<forall>S. S \<subseteq> A --> (\<exists>L. isLub S (| pset = A, order = r |) L) |] |
13115 | 884 |
==> (| pset = A, order = r |) \<in> CompleteLattice" |
885 |
by (simp add: CompleteLatticeI Rdual) |
|
886 |
||
887 |
theorem (in CLF) Tarski_full: |
|
888 |
"(| pset = P, order = induced P r|) \<in> CompleteLattice" |
|
889 |
apply (rule CompleteLatticeI_simp) |
|
890 |
apply (rule fixf_po, clarify) |
|
13383 | 891 |
apply (simp add: P_def A_def r_def) |
892 |
apply (blast intro!: Tarski.tarski_full_lemma cl_po cl_co f_cl) |
|
13115 | 893 |
done |
7112 | 894 |
|
895 |
end |