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(* Title: HOL/ex/Tarski
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ID: $Id$
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Author: Florian Kammueller, Cambridge University Computer Laboratory
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Copyright 1999 University of Cambridge
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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 lattice.
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Illustrates first-class theories, using the Sigma representation of structures
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Tidied and converted to Isar by lcp
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*)
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theory Tarski = Main:
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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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syntax
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"@pset" :: "'a potype => 'a set" ("_ .<A>" [90] 90)
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"@order" :: "'a potype => ('a *'a)set" ("_ .<r>" [90] 90)
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translations
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"po.<A>" == "pset po"
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"po.<r>" == "order po"
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constdefs
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monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
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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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least :: "['a => bool, 'a potype] => 'a"
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"least P po == @ x. x: po.<A> & P x &
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(\<forall>y \<in> po.<A>. P y --> (x,y): po.<r>)"
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greatest :: "['a => bool, 'a potype] => 'a"
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"greatest P po == @ x. x: po.<A> & P x &
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(\<forall>y \<in> po.<A>. P y --> (y,x): po.<r>)"
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lub :: "['a set, 'a potype] => 'a"
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"lub S po == least (%x. \<forall>y\<in>S. (y,x): po.<r>) po"
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glb :: "['a set, 'a potype] => 'a"
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"glb S po == greatest (%x. \<forall>y\<in>S. (x,y): po.<r>) po"
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isLub :: "['a set, 'a potype, 'a] => bool"
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"isLub S po == %L. (L: po.<A> & (\<forall>y\<in>S. (y,L): po.<r>) &
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(\<forall>z\<in>po.<A>. (\<forall>y\<in>S. (y,z): po.<r>) --> (L,z): po.<r>))"
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isGlb :: "['a set, 'a potype, 'a] => bool"
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"isGlb S po == %G. (G: po.<A> & (\<forall>y\<in>S. (G,y): po.<r>) &
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(\<forall>z \<in> po.<A>. (\<forall>y\<in>S. (z,y): po.<r>) --> (z,G): po.<r>))"
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"fix" :: "[('a => 'a), 'a set] => 'a set"
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"fix f A == {x. x: A & f x = x}"
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interval :: "[('a*'a) set,'a, 'a ] => 'a set"
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"interval r a b == {x. (a,x): r & (x,b): r}"
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constdefs
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Bot :: "'a potype => 'a"
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"Bot po == least (%x. True) po"
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Top :: "'a potype => 'a"
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"Top po == greatest (%x. True) po"
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PartialOrder :: "('a potype) set"
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"PartialOrder == {P. refl (P.<A>) (P.<r>) & antisym (P.<r>) &
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trans (P.<r>)}"
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CompleteLattice :: "('a potype) set"
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"CompleteLattice == {cl. cl: PartialOrder &
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(\<forall>S. S <= cl.<A> --> (\<exists>L. isLub S cl L)) &
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(\<forall>S. S <= cl.<A> --> (\<exists>G. isGlb S cl G))}"
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CLF :: "('a potype * ('a => 'a)) set"
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"CLF == SIGMA cl: CompleteLattice.
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{f. f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)}"
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induced :: "['a set, ('a * 'a) set] => ('a *'a)set"
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"induced A r == {(a,b). a : A & b: A & (a,b): r}"
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constdefs
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sublattice :: "('a potype * 'a set)set"
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"sublattice ==
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SIGMA cl: CompleteLattice.
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{S. S <= cl.<A> &
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(| pset = S, order = induced S (cl.<r>) |): CompleteLattice }"
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syntax
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"@SL" :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
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translations
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"S <<= cl" == "S : sublattice `` {cl}"
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constdefs
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dual :: "'a potype => 'a potype"
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"dual po == (| pset = po.<A>, order = converse (po.<r>) |)"
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locale 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 == cl.<A>"
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and r_def: "r == cl.<r>"
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locale CL = PO +
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assumes cl_co: "cl : CompleteLattice"
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locale 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 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 <= 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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(* 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 A_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 A_def r_def)
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done
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lemma (in PO) reflE: "[| refl A r; 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)
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done
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lemma (in PO) antisymE: "[| antisym r; (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)
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done
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lemma (in PO) transE: "[| trans r; (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)
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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 <= 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: PO_imp_refl [THEN reflE])
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(* antisym *)
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apply (simp add: antisym_def induced_def)
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apply (blast intro: PO_imp_sym [THEN antisymE])
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(* trans *)
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apply (simp add: trans_def induced_def)
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apply (blast intro: PO_imp_trans [THEN transE])
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done
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lemma (in PO) indE: "[| (x, y) \<in> induced S r; S <= 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 <= 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 <= A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
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==> \<forall>S. (S <= 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 <= 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 Tarski.PO_imp_refl, simp]
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declare CL_imp_PO [THEN Tarski.PO_imp_sym, simp]
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declare CL_imp_PO [THEN Tarski.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 <= po.<A> --> (\<exists>L. isLub S po L));
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(\<forall>S. S <= po.<A> --> (\<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: "(dual cl.<A>) = cl.<A>"
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by (simp add: dual_def)
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lemma (in PO) dualr_iff: "((x, y) \<in> (dual cl.<r>)) = ((y, x) \<in> cl.<r>)"
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by (simp add: dual_def)
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lemma (in PO) monotone_dual:
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"monotone f (cl.<A>) (cl.<r>) ==> monotone f (dual cl.<A>) (dual cl.<r>)"
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apply (simp add: monotone_def dualA_iff dualr_iff)
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done
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lemma (in PO) interval_dual:
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"[| x \<in> A; y \<in> A|] ==> interval r x y = interval (dual cl.<r>) 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: PO_imp_refl [THEN 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: PO_imp_refl [THEN reflE])
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done
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(* 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 A_def r_def)
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lemma (in CL) sublatticeI:
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"[| S <= A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
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==> S <<= cl"
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by (simp add: sublattice_def A_def r_def)
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(* lub *)
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lemma (in CL) lub_unique: "[| S <= A; isLub S cl x; isLub S cl L|] ==> x = L"
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apply (rule antisymE)
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apply (rule CO_antisym)
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apply (auto simp add: isLub_def r_def)
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done
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lemma (in CL) lub_upper: "[|S <= A; x \<in> S|] ==> (x, lub S cl) \<in> r"
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apply (rule CL_imp_ex_isLub [THEN exE], assumption)
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apply (unfold lub_def least_def)
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apply (rule some_equality [THEN ssubst])
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apply (simp add: isLub_def)
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apply (simp add: lub_unique A_def isLub_def)
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apply (simp add: isLub_def r_def)
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done
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lemma (in CL) lub_least:
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"[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
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apply (rule CL_imp_ex_isLub [THEN exE], assumption)
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apply (unfold lub_def least_def)
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apply (rule_tac s=x in some_equality [THEN ssubst])
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apply (simp add: isLub_def)
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apply (simp add: lub_unique A_def isLub_def)
|
|
343 |
apply (simp add: isLub_def r_def A_def)
|
|
344 |
done
|
|
345 |
|
|
346 |
lemma (in CL) lub_in_lattice: "S <= A ==> lub S cl \<in> A"
|
|
347 |
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
|
|
348 |
apply (unfold lub_def least_def)
|
|
349 |
apply (subst some_equality)
|
|
350 |
apply (simp add: isLub_def)
|
|
351 |
prefer 2 apply (simp add: isLub_def A_def)
|
|
352 |
apply (simp add: lub_unique A_def isLub_def)
|
|
353 |
done
|
|
354 |
|
|
355 |
lemma (in CL) lubI:
|
|
356 |
"[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
|
|
357 |
\<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
|
|
358 |
apply (rule lub_unique, assumption)
|
|
359 |
apply (simp add: isLub_def A_def r_def)
|
|
360 |
apply (unfold isLub_def)
|
|
361 |
apply (rule conjI)
|
|
362 |
apply (fold A_def r_def)
|
|
363 |
apply (rule lub_in_lattice, assumption)
|
|
364 |
apply (simp add: lub_upper lub_least)
|
|
365 |
done
|
|
366 |
|
|
367 |
lemma (in CL) lubIa: "[| S <= A; isLub S cl L |] ==> L = lub S cl"
|
|
368 |
by (simp add: lubI isLub_def A_def r_def)
|
|
369 |
|
|
370 |
lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
|
|
371 |
by (simp add: isLub_def A_def)
|
|
372 |
|
|
373 |
lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
|
|
374 |
by (simp add: isLub_def r_def)
|
|
375 |
|
|
376 |
lemma (in CL) isLub_least:
|
|
377 |
"[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
|
|
378 |
by (simp add: isLub_def A_def r_def)
|
|
379 |
|
|
380 |
lemma (in CL) isLubI:
|
|
381 |
"[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
|
|
382 |
(\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
|
|
383 |
by (simp add: isLub_def A_def r_def)
|
|
384 |
|
|
385 |
(* glb *)
|
|
386 |
lemma (in CL) glb_in_lattice: "S <= A ==> glb S cl \<in> A"
|
|
387 |
apply (subst glb_dual_lub)
|
|
388 |
apply (simp add: A_def)
|
|
389 |
apply (rule dualA_iff [THEN subst])
|
|
390 |
apply (rule Tarski.lub_in_lattice)
|
|
391 |
apply (rule dualPO)
|
|
392 |
apply (rule CL_dualCL)
|
|
393 |
apply (simp add: dualA_iff)
|
|
394 |
done
|
|
395 |
|
|
396 |
lemma (in CL) glb_lower: "[|S <= A; x \<in> S|] ==> (glb S cl, x) \<in> r"
|
|
397 |
apply (subst glb_dual_lub)
|
|
398 |
apply (simp add: r_def)
|
|
399 |
apply (rule dualr_iff [THEN subst])
|
|
400 |
apply (rule Tarski.lub_upper [rule_format])
|
|
401 |
apply (rule dualPO)
|
|
402 |
apply (rule CL_dualCL)
|
|
403 |
apply (simp add: dualA_iff A_def, assumption)
|
|
404 |
done
|
|
405 |
|
|
406 |
(* Reduce the sublattice property by using substructural properties*)
|
|
407 |
(* abandoned see Tarski_4.ML *)
|
|
408 |
|
|
409 |
lemma (in CLF) [simp]:
|
|
410 |
"f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)"
|
|
411 |
apply (insert f_cl)
|
|
412 |
apply (simp add: CLF_def)
|
|
413 |
done
|
|
414 |
|
|
415 |
declare (in CLF) f_cl [simp]
|
|
416 |
|
|
417 |
|
|
418 |
lemma (in CLF) f_in_funcset: "f \<in> A funcset A"
|
|
419 |
by (simp add: A_def)
|
|
420 |
|
|
421 |
lemma (in CLF) monotone_f: "monotone f A r"
|
|
422 |
by (simp add: A_def r_def)
|
|
423 |
|
|
424 |
lemma (in CLF) CLF_dual: "(cl,f) \<in> CLF ==> (dual cl, f) \<in> CLF"
|
|
425 |
apply (simp add: CLF_def CL_dualCL monotone_dual)
|
|
426 |
apply (simp add: dualA_iff)
|
|
427 |
done
|
|
428 |
|
|
429 |
(* fixed points *)
|
|
430 |
lemma fix_subset: "fix f A <= A"
|
|
431 |
by (simp add: fix_def, fast)
|
|
432 |
|
|
433 |
lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
|
|
434 |
by (simp add: fix_def)
|
|
435 |
|
|
436 |
lemma fixf_subset:
|
|
437 |
"[| A <= B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
|
|
438 |
apply (simp add: fix_def, auto)
|
|
439 |
done
|
|
440 |
|
|
441 |
(* lemmas for Tarski, lub *)
|
|
442 |
lemma (in CLF) lubH_le_flubH:
|
|
443 |
"H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
|
|
444 |
apply (rule lub_least, fast)
|
|
445 |
apply (rule f_in_funcset [THEN funcset_mem])
|
|
446 |
apply (rule lub_in_lattice, fast)
|
|
447 |
(* \<forall>x:H. (x, f (lub H r)) \<in> r *)
|
|
448 |
apply (rule ballI)
|
|
449 |
apply (rule transE)
|
|
450 |
apply (rule CO_trans)
|
|
451 |
(* instantiates (x, ???z) \<in> cl.<r> to (x, f x), because of the def of H *)
|
|
452 |
apply fast
|
|
453 |
(* so it remains to show (f x, f (lub H cl)) \<in> r *)
|
|
454 |
apply (rule_tac f = "f" in monotoneE)
|
|
455 |
apply (rule monotone_f, fast)
|
|
456 |
apply (rule lub_in_lattice, fast)
|
|
457 |
apply (rule lub_upper, fast)
|
|
458 |
apply assumption
|
|
459 |
done
|
|
460 |
|
|
461 |
lemma (in CLF) flubH_le_lubH:
|
|
462 |
"[| H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
|
|
463 |
apply (rule lub_upper, fast)
|
|
464 |
apply (rule_tac t = "H" in ssubst, assumption)
|
|
465 |
apply (rule CollectI)
|
|
466 |
apply (rule conjI)
|
|
467 |
apply (rule_tac [2] f_in_funcset [THEN funcset_mem])
|
|
468 |
apply (rule_tac [2] lub_in_lattice)
|
|
469 |
prefer 2 apply fast
|
|
470 |
apply (rule_tac f = "f" in monotoneE)
|
|
471 |
apply (rule monotone_f)
|
|
472 |
apply (blast intro: lub_in_lattice)
|
|
473 |
apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])
|
|
474 |
apply (simp add: lubH_le_flubH)
|
|
475 |
done
|
|
476 |
|
|
477 |
lemma (in CLF) lubH_is_fixp:
|
|
478 |
"H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
|
|
479 |
apply (simp add: fix_def)
|
|
480 |
apply (rule conjI)
|
|
481 |
apply (rule lub_in_lattice, fast)
|
|
482 |
apply (rule antisymE)
|
|
483 |
apply (rule CO_antisym)
|
|
484 |
apply (simp add: flubH_le_lubH)
|
|
485 |
apply (simp add: lubH_le_flubH)
|
|
486 |
done
|
|
487 |
|
|
488 |
lemma (in CLF) fix_in_H:
|
|
489 |
"[| H = {x. (x, f x) \<in> r & x \<in> A}; x \<in> P |] ==> x \<in> H"
|
|
490 |
by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl
|
|
491 |
fix_subset [of f A, THEN subsetD])
|
|
492 |
|
|
493 |
lemma (in CLF) fixf_le_lubH:
|
|
494 |
"H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
|
|
495 |
apply (rule ballI)
|
|
496 |
apply (rule lub_upper, fast)
|
|
497 |
apply (rule fix_in_H)
|
|
498 |
apply (simp_all add: P_def)
|
|
499 |
done
|
|
500 |
|
|
501 |
lemma (in CLF) lubH_least_fixf:
|
|
502 |
"H = {x. (x, f x) \<in> r & x \<in> A}
|
|
503 |
==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
|
|
504 |
apply (rule allI)
|
|
505 |
apply (rule impI)
|
|
506 |
apply (erule bspec)
|
|
507 |
apply (rule lubH_is_fixp, assumption)
|
|
508 |
done
|
|
509 |
|
|
510 |
(* Tarski fixpoint theorem 1, first part *)
|
|
511 |
lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
|
|
512 |
apply (rule sym)
|
|
513 |
apply (simp add: P_def)
|
|
514 |
apply (rule lubI)
|
|
515 |
apply (rule fix_subset)
|
|
516 |
apply (rule lub_in_lattice, fast)
|
|
517 |
apply (simp add: fixf_le_lubH)
|
|
518 |
apply (simp add: lubH_least_fixf)
|
|
519 |
done
|
|
520 |
|
|
521 |
(* Tarski for glb *)
|
|
522 |
lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
|
|
523 |
apply (simp add: glb_dual_lub P_def A_def r_def)
|
|
524 |
apply (rule dualA_iff [THEN subst])
|
|
525 |
apply (rule Tarski.lubH_is_fixp)
|
|
526 |
apply (rule dualPO)
|
|
527 |
apply (rule CL_dualCL)
|
|
528 |
apply (rule f_cl [THEN CLF_dual])
|
|
529 |
apply (simp add: dualr_iff dualA_iff)
|
|
530 |
done
|
|
531 |
|
|
532 |
lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
|
|
533 |
apply (simp add: glb_dual_lub P_def A_def r_def)
|
|
534 |
apply (rule dualA_iff [THEN subst])
|
|
535 |
apply (simp add: Tarski.T_thm_1_lub [of _ f, OF dualPO CL_dualCL]
|
|
536 |
dualPO CL_dualCL CLF_dual dualr_iff)
|
|
537 |
done
|
|
538 |
|
|
539 |
(* interval *)
|
|
540 |
lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
|
|
541 |
apply (insert CO_refl)
|
|
542 |
apply (simp add: refl_def, blast)
|
|
543 |
done
|
|
544 |
|
|
545 |
lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b <= A"
|
|
546 |
apply (simp add: interval_def)
|
|
547 |
apply (blast intro: rel_imp_elem)
|
|
548 |
done
|
|
549 |
|
|
550 |
lemma (in CLF) intervalI:
|
|
551 |
"[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
|
|
552 |
apply (simp add: interval_def)
|
|
553 |
done
|
|
554 |
|
|
555 |
lemma (in CLF) interval_lemma1:
|
|
556 |
"[| S <= interval r a b; x \<in> S |] ==> (a, x) \<in> r"
|
|
557 |
apply (unfold interval_def, fast)
|
|
558 |
done
|
|
559 |
|
|
560 |
lemma (in CLF) interval_lemma2:
|
|
561 |
"[| S <= interval r a b; x \<in> S |] ==> (x, b) \<in> r"
|
|
562 |
apply (unfold interval_def, fast)
|
|
563 |
done
|
|
564 |
|
|
565 |
lemma (in CLF) a_less_lub:
|
|
566 |
"[| S <= A; S \<noteq> {};
|
|
567 |
\<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
|
|
568 |
by (blast intro: transE PO_imp_trans)
|
|
569 |
|
|
570 |
lemma (in CLF) glb_less_b:
|
|
571 |
"[| S <= A; S \<noteq> {};
|
|
572 |
\<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
|
|
573 |
by (blast intro: transE PO_imp_trans)
|
|
574 |
|
|
575 |
lemma (in CLF) S_intv_cl:
|
|
576 |
"[| a \<in> A; b \<in> A; S <= interval r a b |]==> S <= A"
|
|
577 |
by (simp add: subset_trans [OF _ interval_subset])
|
|
578 |
|
|
579 |
lemma (in CLF) L_in_interval:
|
|
580 |
"[| a \<in> A; b \<in> A; S <= interval r a b;
|
|
581 |
S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
|
|
582 |
apply (rule intervalI)
|
|
583 |
apply (rule a_less_lub)
|
|
584 |
prefer 2 apply assumption
|
|
585 |
apply (simp add: S_intv_cl)
|
|
586 |
apply (rule ballI)
|
|
587 |
apply (simp add: interval_lemma1)
|
|
588 |
apply (simp add: isLub_upper)
|
|
589 |
(* (L, b) \<in> r *)
|
|
590 |
apply (simp add: isLub_least interval_lemma2)
|
|
591 |
done
|
|
592 |
|
|
593 |
lemma (in CLF) G_in_interval:
|
|
594 |
"[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S <= interval r a b; isGlb S cl G;
|
|
595 |
S \<noteq> {} |] ==> G \<in> interval r a b"
|
|
596 |
apply (simp add: interval_dual)
|
|
597 |
apply (simp add: Tarski.L_in_interval [of _ f]
|
|
598 |
dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub)
|
|
599 |
done
|
|
600 |
|
|
601 |
lemma (in CLF) intervalPO:
|
|
602 |
"[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
|
|
603 |
==> (| pset = interval r a b, order = induced (interval r a b) r |)
|
|
604 |
\<in> PartialOrder"
|
|
605 |
apply (rule po_subset_po)
|
|
606 |
apply (simp add: interval_subset)
|
|
607 |
done
|
|
608 |
|
|
609 |
lemma (in CLF) intv_CL_lub:
|
|
610 |
"[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
|
|
611 |
==> \<forall>S. S <= interval r a b -->
|
|
612 |
(\<exists>L. isLub S (| pset = interval r a b,
|
|
613 |
order = induced (interval r a b) r |) L)"
|
|
614 |
apply (intro strip)
|
|
615 |
apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
|
|
616 |
prefer 2 apply assumption
|
|
617 |
apply assumption
|
|
618 |
apply (erule exE)
|
|
619 |
(* define the lub for the interval as *)
|
|
620 |
apply (rule_tac x = "if S = {} then a else L" in exI)
|
|
621 |
apply (simp (no_asm_simp) add: isLub_def split del: split_if)
|
|
622 |
apply (intro impI conjI)
|
|
623 |
(* (if S = {} then a else L) \<in> interval r a b *)
|
|
624 |
apply (simp add: CL_imp_PO L_in_interval)
|
|
625 |
apply (simp add: left_in_interval)
|
|
626 |
(* lub prop 1 *)
|
|
627 |
apply (case_tac "S = {}")
|
|
628 |
(* S = {}, y \<in> S = False => everything *)
|
|
629 |
apply fast
|
|
630 |
(* S \<noteq> {} *)
|
|
631 |
apply simp
|
|
632 |
(* \<forall>y:S. (y, L) \<in> induced (interval r a b) r *)
|
|
633 |
apply (rule ballI)
|
|
634 |
apply (simp add: induced_def L_in_interval)
|
|
635 |
apply (rule conjI)
|
|
636 |
apply (rule subsetD)
|
|
637 |
apply (simp add: S_intv_cl, assumption)
|
|
638 |
apply (simp add: isLub_upper)
|
|
639 |
(* \<forall>z:interval r a b. (\<forall>y:S. (y, z) \<in> induced (interval r a b) r -->
|
|
640 |
(if S = {} then a else L, z) \<in> induced (interval r a b) r *)
|
|
641 |
apply (rule ballI)
|
|
642 |
apply (rule impI)
|
|
643 |
apply (case_tac "S = {}")
|
|
644 |
(* S = {} *)
|
|
645 |
apply simp
|
|
646 |
apply (simp add: induced_def interval_def)
|
|
647 |
apply (rule conjI)
|
|
648 |
apply (rule reflE)
|
|
649 |
apply (rule CO_refl, assumption)
|
|
650 |
apply (rule interval_not_empty)
|
|
651 |
apply (rule CO_trans)
|
|
652 |
apply (simp add: interval_def)
|
|
653 |
(* S \<noteq> {} *)
|
|
654 |
apply simp
|
|
655 |
apply (simp add: induced_def L_in_interval)
|
|
656 |
apply (rule isLub_least, assumption)
|
|
657 |
apply (rule subsetD)
|
|
658 |
prefer 2 apply assumption
|
|
659 |
apply (simp add: S_intv_cl, fast)
|
|
660 |
done
|
|
661 |
|
|
662 |
lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
|
|
663 |
|
|
664 |
lemma (in CLF) interval_is_sublattice:
|
|
665 |
"[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
|
|
666 |
==> interval r a b <<= cl"
|
|
667 |
apply (rule sublatticeI)
|
|
668 |
apply (simp add: interval_subset)
|
|
669 |
apply (rule CompleteLatticeI)
|
|
670 |
apply (simp add: intervalPO)
|
|
671 |
apply (simp add: intv_CL_lub)
|
|
672 |
apply (simp add: intv_CL_glb)
|
|
673 |
done
|
|
674 |
|
|
675 |
lemmas (in CLF) interv_is_compl_latt =
|
|
676 |
interval_is_sublattice [THEN sublattice_imp_CL]
|
|
677 |
|
|
678 |
(* Top and Bottom *)
|
|
679 |
lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
|
|
680 |
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
|
|
681 |
|
|
682 |
lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
|
|
683 |
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
|
|
684 |
|
|
685 |
lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
|
|
686 |
apply (simp add: Bot_def least_def)
|
|
687 |
apply (rule someI2)
|
|
688 |
apply (fold A_def)
|
|
689 |
apply (erule_tac [2] conjunct1)
|
|
690 |
apply (rule conjI)
|
|
691 |
apply (rule glb_in_lattice)
|
|
692 |
apply (rule subset_refl)
|
|
693 |
apply (fold r_def)
|
|
694 |
apply (simp add: glb_lower)
|
|
695 |
done
|
|
696 |
|
|
697 |
lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
|
|
698 |
apply (simp add: Top_dual_Bot A_def)
|
|
699 |
apply (rule dualA_iff [THEN subst])
|
|
700 |
apply (blast intro!: Tarski.Bot_in_lattice dualPO CL_dualCL CLF_dual f_cl)
|
|
701 |
done
|
|
702 |
|
|
703 |
lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
|
|
704 |
apply (simp add: Top_def greatest_def)
|
|
705 |
apply (rule someI2)
|
|
706 |
apply (fold r_def A_def)
|
|
707 |
prefer 2 apply fast
|
|
708 |
apply (intro conjI ballI)
|
|
709 |
apply (rule_tac [2] lub_upper)
|
|
710 |
apply (auto simp add: lub_in_lattice)
|
|
711 |
done
|
|
712 |
|
|
713 |
lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
|
|
714 |
apply (simp add: Bot_dual_Top r_def)
|
|
715 |
apply (rule dualr_iff [THEN subst])
|
|
716 |
apply (simp add: Tarski.Top_prop [of _ f]
|
|
717 |
dualA_iff A_def dualPO CL_dualCL CLF_dual)
|
|
718 |
done
|
|
719 |
|
|
720 |
lemma (in CLF) Top_intv_not_empty: "x \<in> A ==> interval r x (Top cl) \<noteq> {}"
|
|
721 |
apply (rule notI)
|
|
722 |
apply (drule_tac a = "Top cl" in equals0D)
|
|
723 |
apply (simp add: interval_def)
|
|
724 |
apply (simp add: refl_def Top_in_lattice Top_prop)
|
|
725 |
done
|
|
726 |
|
|
727 |
lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
|
|
728 |
apply (simp add: Bot_dual_Top)
|
|
729 |
apply (subst interval_dual)
|
|
730 |
prefer 2 apply assumption
|
|
731 |
apply (simp add: A_def)
|
|
732 |
apply (rule dualA_iff [THEN subst])
|
|
733 |
apply (blast intro!: Tarski.Top_in_lattice
|
|
734 |
f_cl dualPO CL_dualCL CLF_dual)
|
|
735 |
apply (simp add: Tarski.Top_intv_not_empty [of _ f]
|
|
736 |
dualA_iff A_def dualPO CL_dualCL CLF_dual)
|
|
737 |
done
|
|
738 |
|
|
739 |
(* fixed points form a partial order *)
|
|
740 |
lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
|
|
741 |
by (simp add: P_def fix_subset po_subset_po)
|
|
742 |
|
|
743 |
lemma (in Tarski) Y_subset_A: "Y <= A"
|
|
744 |
apply (rule subset_trans [OF _ fix_subset])
|
|
745 |
apply (rule Y_ss [simplified P_def])
|
|
746 |
done
|
|
747 |
|
|
748 |
lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
|
|
749 |
by (simp add: Y_subset_A [THEN lub_in_lattice])
|
|
750 |
|
|
751 |
lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
|
|
752 |
apply (rule lub_least)
|
|
753 |
apply (rule Y_subset_A)
|
|
754 |
apply (rule f_in_funcset [THEN funcset_mem])
|
|
755 |
apply (rule lubY_in_A)
|
|
756 |
(* Y <= P ==> f x = x *)
|
|
757 |
apply (rule ballI)
|
|
758 |
apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
|
|
759 |
apply (erule Y_ss [simplified P_def, THEN subsetD])
|
|
760 |
(* reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r by monotonicity *)
|
|
761 |
apply (rule_tac f = "f" in monotoneE)
|
|
762 |
apply (rule monotone_f)
|
|
763 |
apply (simp add: Y_subset_A [THEN subsetD])
|
|
764 |
apply (rule lubY_in_A)
|
|
765 |
apply (simp add: lub_upper Y_subset_A)
|
|
766 |
done
|
|
767 |
|
|
768 |
lemma (in Tarski) intY1_subset: "intY1 <= A"
|
|
769 |
apply (unfold intY1_def)
|
|
770 |
apply (rule interval_subset)
|
|
771 |
apply (rule lubY_in_A)
|
|
772 |
apply (rule Top_in_lattice)
|
|
773 |
done
|
|
774 |
|
|
775 |
lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
|
|
776 |
|
|
777 |
lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
|
|
778 |
apply (simp add: intY1_def interval_def)
|
|
779 |
apply (rule conjI)
|
|
780 |
apply (rule transE)
|
|
781 |
apply (rule CO_trans)
|
|
782 |
apply (rule lubY_le_flubY)
|
|
783 |
(* (f (lub Y cl), f x) \<in> r *)
|
|
784 |
apply (rule_tac f=f in monotoneE)
|
|
785 |
apply (rule monotone_f)
|
|
786 |
apply (rule lubY_in_A)
|
|
787 |
apply (simp add: intY1_def interval_def intY1_elem)
|
|
788 |
apply (simp add: intY1_def interval_def)
|
|
789 |
(* (f x, Top cl) \<in> r *)
|
|
790 |
apply (rule Top_prop)
|
|
791 |
apply (rule f_in_funcset [THEN funcset_mem])
|
|
792 |
apply (simp add: intY1_def interval_def intY1_elem)
|
|
793 |
done
|
|
794 |
|
|
795 |
lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 funcset intY1"
|
|
796 |
apply (rule restrictI)
|
|
797 |
apply (erule intY1_f_closed)
|
|
798 |
done
|
|
799 |
|
|
800 |
lemma (in Tarski) intY1_mono:
|
|
801 |
"monotone (%x: intY1. f x) intY1 (induced intY1 r)"
|
|
802 |
apply (auto simp add: monotone_def induced_def intY1_f_closed)
|
|
803 |
apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
|
|
804 |
done
|
|
805 |
|
|
806 |
lemma (in Tarski) intY1_is_cl:
|
|
807 |
"(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
|
|
808 |
apply (unfold intY1_def)
|
|
809 |
apply (rule interv_is_compl_latt)
|
|
810 |
apply (rule lubY_in_A)
|
|
811 |
apply (rule Top_in_lattice)
|
|
812 |
apply (rule Top_intv_not_empty)
|
|
813 |
apply (rule lubY_in_A)
|
|
814 |
done
|
|
815 |
|
|
816 |
lemma (in Tarski) v_in_P: "v \<in> P"
|
|
817 |
apply (unfold P_def)
|
|
818 |
apply (rule_tac A = "intY1" in fixf_subset)
|
|
819 |
apply (rule intY1_subset)
|
|
820 |
apply (simp add: Tarski.glbH_is_fixp [OF _ intY1_is_cl, simplified]
|
|
821 |
v_def CL_imp_PO intY1_is_cl CLF_def intY1_func intY1_mono)
|
|
822 |
done
|
|
823 |
|
|
824 |
lemma (in Tarski) z_in_interval:
|
|
825 |
"[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
|
|
826 |
apply (unfold intY1_def P_def)
|
|
827 |
apply (rule intervalI)
|
|
828 |
prefer 2
|
|
829 |
apply (erule fix_subset [THEN subsetD, THEN Top_prop])
|
|
830 |
apply (rule lub_least)
|
|
831 |
apply (rule Y_subset_A)
|
|
832 |
apply (fast elim!: fix_subset [THEN subsetD])
|
|
833 |
apply (simp add: induced_def)
|
|
834 |
done
|
|
835 |
|
|
836 |
lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
|
|
837 |
==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"
|
|
838 |
apply (simp add: induced_def intY1_f_closed z_in_interval P_def)
|
|
839 |
apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD]
|
|
840 |
CO_refl [THEN reflE])
|
|
841 |
done
|
|
842 |
|
|
843 |
lemma (in Tarski) tarski_full_lemma:
|
|
844 |
"\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
|
|
845 |
apply (rule_tac x = "v" in exI)
|
|
846 |
apply (simp add: isLub_def)
|
|
847 |
(* v \<in> P *)
|
|
848 |
apply (simp add: v_in_P)
|
|
849 |
apply (rule conjI)
|
|
850 |
(* v is lub *)
|
|
851 |
(* 1. \<forall>y:Y. (y, v) \<in> induced P r *)
|
|
852 |
apply (rule ballI)
|
|
853 |
apply (simp add: induced_def subsetD v_in_P)
|
|
854 |
apply (rule conjI)
|
|
855 |
apply (erule Y_ss [THEN subsetD])
|
|
856 |
apply (rule_tac b = "lub Y cl" in transE)
|
|
857 |
apply (rule CO_trans)
|
|
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)
|
|
863 |
apply (rule Tarski.glb_in_lattice [OF _ intY1_is_cl, simplified])
|
|
864 |
apply (simp add: CL_imp_PO intY1_is_cl, force)
|
|
865 |
(* v is LEAST ub *)
|
|
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)
|
|
873 |
apply (rule Tarski.glb_lower [OF _ intY1_is_cl, simplified])
|
|
874 |
apply (simp add: CL_imp_PO intY1_is_cl)
|
|
875 |
apply force
|
|
876 |
apply (simp add: induced_def intY1_f_closed z_in_interval)
|
|
877 |
apply (simp add: P_def fix_imp_eq [of _ f A]
|
|
878 |
fix_subset [of f A, THEN subsetD]
|
|
879 |
CO_refl [THEN reflE])
|
|
880 |
done
|
|
881 |
|
|
882 |
|
|
883 |
lemma CompleteLatticeI_simp:
|
|
884 |
"[| (| pset = A, order = r |) \<in> PartialOrder;
|
|
885 |
\<forall>S. S <= A --> (\<exists>L. isLub S (| pset = A, order = r |) L) |]
|
|
886 |
==> (| pset = A, order = r |) \<in> CompleteLattice"
|
|
887 |
by (simp add: CompleteLatticeI Rdual)
|
|
888 |
|
|
889 |
theorem (in CLF) Tarski_full:
|
|
890 |
"(| pset = P, order = induced P r|) \<in> CompleteLattice"
|
|
891 |
apply (rule CompleteLatticeI_simp)
|
|
892 |
apply (rule fixf_po, clarify)
|
|
893 |
apply (simp add: P_def A_def r_def)
|
|
894 |
apply (blast intro!: Tarski.tarski_full_lemma cl_po cl_co f_cl)
|
|
895 |
done
|
7112
|
896 |
|
|
897 |
end
|