src/HOL/Metis_Examples/Tarski.thy
author haftmann
Fri Oct 10 19:55:32 2014 +0200 (2014-10-10)
changeset 58646 cd63a4b12a33
parent 50705 0e943b33d907
child 58889 5b7a9633cfa8
permissions -rw-r--r--
specialized specification: avoid trivial instances
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(*  Title:      HOL/Metis_Examples/Tarski.thy
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    Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
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    Author:     Jasmin Blanchette, TU Muenchen
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Metis example featuring the full theorem of Tarski.
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*)
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header {* Metis Example Featuring the Full Theorem of Tarski *}
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theory Tarski
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imports Main "~~/src/HOL/Library/FuncSet"
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begin
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declare [[metis_new_skolem]]
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(*Many of these higher-order problems appear to be impossible using the
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current linkup. They often seem to need either higher-order unification
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or explicit reasoning about connectives such as conjunction. The numerous
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set comprehensions are to blame.*)
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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 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 least :: "['a => bool, 'a potype] => 'a" where
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  "least P po == @ 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 greatest :: "['a => bool, 'a potype] => 'a" where
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  "greatest P po == @ 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 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 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 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 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 "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 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 Bot :: "'a potype => 'a" where
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  "Bot po == least (%x. True) po"
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definition Top :: "'a potype => 'a" where
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  "Top po == greatest (%x. True) po"
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definition PartialOrder :: "('a potype) set" where
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  "PartialOrder == {P. refl_on (pset P) (order P) & antisym (order P) &
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                       trans (order P)}"
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definition 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 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 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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  sublattice_syntax :: "['a set, 'a potype] => bool" ("_ <<= _" [51, 50] 50)
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  where "S <<= cl \<equiv> S : sublattice `` {cl}"
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definition dual :: "'a potype => 'a potype" where
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  "dual po == (| pset = pset po, order = converse (order po) |)"
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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 == pset cl"
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     and  r_def: "r == order cl"
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locale CL = PO +
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  assumes cl_co:  "cl : CompleteLattice"
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definition CLF_set :: "('a potype * ('a => 'a)) set" where
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  "CLF_set = (SIGMA cl: CompleteLattice.
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            {f. f: pset cl -> pset cl & monotone f (pset cl) (order cl)})"
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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_set" (*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 \<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_on: "refl_on 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_on_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_on_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_unfold)
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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_unfold)
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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)
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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_unfold)
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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_unfold)
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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 PO.PO_imp_refl_on  [OF PO.intro [OF CL_imp_PO], simp]
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declare PO.PO_imp_sym   [OF PO.intro [OF CL_imp_PO], simp]
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declare PO.PO_imp_trans [OF PO.intro [OF CL_imp_PO], simp]
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lemma (in CL) CO_refl_on: "refl_on A r"
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by (rule PO_imp_refl_on)
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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 A_def r_def)
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lemma (in CL) sublatticeI:
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     "[| S \<subseteq> 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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subsection {* lub *}
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lemma (in CL) lub_unique: "[| S \<subseteq> A; isLub S cl x; isLub S cl L|] ==> x = L"
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apply (rule antisymE)
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apply (auto simp add: isLub_def r_def)
paulson@23449
   309
done
paulson@23449
   310
paulson@23449
   311
lemma (in CL) lub_upper: "[|S \<subseteq> A; x \<in> S|] ==> (x, lub S cl) \<in> r"
paulson@23449
   312
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
paulson@23449
   313
apply (unfold lub_def least_def)
paulson@23449
   314
apply (rule some_equality [THEN ssubst])
paulson@23449
   315
  apply (simp add: isLub_def)
paulson@23449
   316
 apply (simp add: lub_unique A_def isLub_def)
paulson@23449
   317
apply (simp add: isLub_def r_def)
paulson@23449
   318
done
paulson@23449
   319
paulson@23449
   320
lemma (in CL) lub_least:
paulson@23449
   321
     "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
paulson@23449
   322
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
paulson@23449
   323
apply (unfold lub_def least_def)
paulson@23449
   324
apply (rule_tac s=x in some_equality [THEN ssubst])
paulson@23449
   325
  apply (simp add: isLub_def)
paulson@23449
   326
 apply (simp add: lub_unique A_def isLub_def)
paulson@23449
   327
apply (simp add: isLub_def r_def A_def)
paulson@23449
   328
done
paulson@23449
   329
paulson@23449
   330
lemma (in CL) lub_in_lattice: "S \<subseteq> A ==> lub S cl \<in> A"
paulson@23449
   331
apply (rule CL_imp_ex_isLub [THEN exE], assumption)
paulson@23449
   332
apply (unfold lub_def least_def)
paulson@23449
   333
apply (subst some_equality)
paulson@23449
   334
apply (simp add: isLub_def)
paulson@23449
   335
prefer 2 apply (simp add: isLub_def A_def)
paulson@23449
   336
apply (simp add: lub_unique A_def isLub_def)
paulson@23449
   337
done
paulson@23449
   338
paulson@23449
   339
lemma (in CL) lubI:
paulson@23449
   340
     "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
paulson@23449
   341
         \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
paulson@23449
   342
apply (rule lub_unique, assumption)
paulson@23449
   343
apply (simp add: isLub_def A_def r_def)
paulson@23449
   344
apply (unfold isLub_def)
paulson@23449
   345
apply (rule conjI)
paulson@23449
   346
apply (fold A_def r_def)
paulson@23449
   347
apply (rule lub_in_lattice, assumption)
paulson@23449
   348
apply (simp add: lub_upper lub_least)
paulson@23449
   349
done
paulson@23449
   350
paulson@23449
   351
lemma (in CL) lubIa: "[| S \<subseteq> A; isLub S cl L |] ==> L = lub S cl"
paulson@23449
   352
by (simp add: lubI isLub_def A_def r_def)
paulson@23449
   353
paulson@23449
   354
lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
paulson@23449
   355
by (simp add: isLub_def  A_def)
paulson@23449
   356
paulson@23449
   357
lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
paulson@23449
   358
by (simp add: isLub_def r_def)
paulson@23449
   359
paulson@23449
   360
lemma (in CL) isLub_least:
paulson@23449
   361
     "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
paulson@23449
   362
by (simp add: isLub_def A_def r_def)
paulson@23449
   363
paulson@23449
   364
lemma (in CL) isLubI:
paulson@23449
   365
     "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
paulson@23449
   366
         (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
paulson@23449
   367
by (simp add: isLub_def A_def r_def)
paulson@23449
   368
paulson@23449
   369
subsection {* glb *}
paulson@23449
   370
paulson@23449
   371
lemma (in CL) glb_in_lattice: "S \<subseteq> A ==> glb S cl \<in> A"
paulson@23449
   372
apply (subst glb_dual_lub)
paulson@23449
   373
apply (simp add: A_def)
paulson@23449
   374
apply (rule dualA_iff [THEN subst])
paulson@23449
   375
apply (rule CL.lub_in_lattice)
haftmann@27681
   376
apply (rule CL.intro)
haftmann@27681
   377
apply (rule PO.intro)
paulson@23449
   378
apply (rule dualPO)
haftmann@27681
   379
apply (rule CL_axioms.intro)
paulson@23449
   380
apply (rule CL_dualCL)
paulson@23449
   381
apply (simp add: dualA_iff)
paulson@23449
   382
done
paulson@23449
   383
paulson@23449
   384
lemma (in CL) glb_lower: "[|S \<subseteq> A; x \<in> S|] ==> (glb S cl, x) \<in> r"
paulson@23449
   385
apply (subst glb_dual_lub)
paulson@23449
   386
apply (simp add: r_def)
paulson@23449
   387
apply (rule dualr_iff [THEN subst])
paulson@23449
   388
apply (rule CL.lub_upper)
haftmann@27681
   389
apply (rule CL.intro)
haftmann@27681
   390
apply (rule PO.intro)
paulson@23449
   391
apply (rule dualPO)
haftmann@27681
   392
apply (rule CL_axioms.intro)
paulson@23449
   393
apply (rule CL_dualCL)
paulson@23449
   394
apply (simp add: dualA_iff A_def, assumption)
paulson@23449
   395
done
paulson@23449
   396
paulson@23449
   397
text {*
paulson@23449
   398
  Reduce the sublattice property by using substructural properties;
paulson@23449
   399
  abandoned see @{text "Tarski_4.ML"}.
paulson@23449
   400
*}
paulson@23449
   401
paulson@23449
   402
declare (in CLF) f_cl [simp]
paulson@23449
   403
paulson@23449
   404
lemma (in CLF) [simp]:
blanchet@42762
   405
    "f: pset cl -> pset cl & monotone f (pset cl) (order cl)"
blanchet@42762
   406
proof -
blanchet@42762
   407
  have "\<forall>u v. (v, u) \<in> CLF_set \<longrightarrow> u \<in> {R \<in> pset v \<rightarrow> pset v. monotone R (pset v) (order v)}"
blanchet@42762
   408
    unfolding CLF_set_def using SigmaE2 by blast
blanchet@42762
   409
  hence F1: "\<forall>u v. (v, u) \<in> CLF_set \<longrightarrow> u \<in> pset v \<rightarrow> pset v \<and> monotone u (pset v) (order v)"
blanchet@42762
   410
    using CollectE by blast
blanchet@42762
   411
  hence "Tarski.monotone f (pset cl) (order cl)" by (metis f_cl)
blanchet@42762
   412
  hence "(cl, f) \<in> CLF_set \<and> Tarski.monotone f (pset cl) (order cl)"
blanchet@42762
   413
    by (metis f_cl)
blanchet@42762
   414
  thus "f \<in> pset cl \<rightarrow> pset cl \<and> Tarski.monotone f (pset cl) (order cl)"
blanchet@42762
   415
    using F1 by metis
blanchet@42762
   416
qed
paulson@23449
   417
paulson@23449
   418
lemma (in CLF) f_in_funcset: "f \<in> A -> A"
paulson@23449
   419
by (simp add: A_def)
paulson@23449
   420
paulson@23449
   421
lemma (in CLF) monotone_f: "monotone f A r"
paulson@23449
   422
by (simp add: A_def r_def)
paulson@23449
   423
paulson@23449
   424
(*never proved, 2007-01-22*)
blanchet@45705
   425
haftmann@27681
   426
declare (in CLF) CLF_set_def [simp] CL_dualCL [simp] monotone_dual [simp] dualA_iff [simp]
haftmann@27681
   427
blanchet@42762
   428
lemma (in CLF) CLF_dual: "(dual cl, f) \<in> CLF_set"
paulson@23449
   429
apply (simp del: dualA_iff)
paulson@23449
   430
apply (simp)
blanchet@43197
   431
done
haftmann@27681
   432
haftmann@27681
   433
declare (in CLF) CLF_set_def[simp del] CL_dualCL[simp del] monotone_dual[simp del]
paulson@23449
   434
          dualA_iff[simp del]
paulson@23449
   435
paulson@23449
   436
subsection {* fixed points *}
paulson@23449
   437
paulson@23449
   438
lemma fix_subset: "fix f A \<subseteq> A"
paulson@23449
   439
by (simp add: fix_def, fast)
paulson@23449
   440
paulson@23449
   441
lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
paulson@23449
   442
by (simp add: fix_def)
paulson@23449
   443
paulson@23449
   444
lemma fixf_subset:
paulson@23449
   445
     "[| A \<subseteq> B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
paulson@23449
   446
by (simp add: fix_def, auto)
paulson@23449
   447
paulson@23449
   448
subsection {* lemmas for Tarski, lub *}
paulson@23449
   449
paulson@23449
   450
(*never proved, 2007-01-22*)
blanchet@45705
   451
blanchet@45705
   452
declare CL.lub_least[intro] CLF.f_in_funcset[intro] funcset_mem[intro] CL.lub_in_lattice[intro] PO.transE[intro] PO.monotoneE[intro] CLF.monotone_f[intro] CL.lub_upper[intro]
blanchet@45705
   453
paulson@23449
   454
lemma (in CLF) lubH_le_flubH:
paulson@23449
   455
     "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
paulson@23449
   456
apply (rule lub_least, fast)
paulson@23449
   457
apply (rule f_in_funcset [THEN funcset_mem])
paulson@23449
   458
apply (rule lub_in_lattice, fast)
paulson@23449
   459
-- {* @{text "\<forall>x:H. (x, f (lub H r)) \<in> r"} *}
paulson@23449
   460
apply (rule ballI)
paulson@23449
   461
(*never proved, 2007-01-22*)
paulson@23449
   462
apply (rule transE)
paulson@23449
   463
-- {* instantiates @{text "(x, ?z) \<in> order cl to (x, f x)"}, *}
paulson@23449
   464
-- {* because of the def of @{text H} *}
paulson@23449
   465
apply fast
paulson@23449
   466
-- {* so it remains to show @{text "(f x, f (lub H cl)) \<in> r"} *}
paulson@23449
   467
apply (rule_tac f = "f" in monotoneE)
paulson@23449
   468
apply (rule monotone_f, fast)
paulson@23449
   469
apply (rule lub_in_lattice, fast)
paulson@23449
   470
apply (rule lub_upper, fast)
paulson@23449
   471
apply assumption
paulson@23449
   472
done
blanchet@45705
   473
blanchet@45705
   474
declare CL.lub_least[rule del] CLF.f_in_funcset[rule del]
blanchet@45705
   475
        funcset_mem[rule del] CL.lub_in_lattice[rule del]
blanchet@45705
   476
        PO.transE[rule del] PO.monotoneE[rule del]
blanchet@45705
   477
        CLF.monotone_f[rule del] CL.lub_upper[rule del]
paulson@23449
   478
paulson@23449
   479
(*never proved, 2007-01-22*)
blanchet@45705
   480
blanchet@45705
   481
declare CLF.f_in_funcset[intro] funcset_mem[intro] CL.lub_in_lattice[intro]
blanchet@45705
   482
     PO.monotoneE[intro] CLF.monotone_f[intro] CL.lub_upper[intro]
blanchet@45705
   483
     CLF.lubH_le_flubH[simp]
blanchet@45705
   484
paulson@23449
   485
lemma (in CLF) flubH_le_lubH:
paulson@23449
   486
     "[|  H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
paulson@23449
   487
apply (rule lub_upper, fast)
paulson@23449
   488
apply (rule_tac t = "H" in ssubst, assumption)
paulson@23449
   489
apply (rule CollectI)
blanchet@47040
   490
by (metis (lifting) CO_refl_on lubH_le_flubH monotone_def monotone_f refl_onD1 refl_onD2)
paulson@23449
   491
blanchet@45705
   492
declare CLF.f_in_funcset[rule del] funcset_mem[rule del]
blanchet@45705
   493
        CL.lub_in_lattice[rule del] PO.monotoneE[rule del]
blanchet@45705
   494
        CLF.monotone_f[rule del] CL.lub_upper[rule del]
blanchet@45705
   495
        CLF.lubH_le_flubH[simp del]
paulson@23449
   496
paulson@23449
   497
(*never proved, 2007-01-22*)
blanchet@45705
   498
blanchet@37622
   499
(* Single-step version fails. The conjecture clauses refer to local abstraction
blanchet@37622
   500
functions (Frees). *)
paulson@23449
   501
lemma (in CLF) lubH_is_fixp:
paulson@23449
   502
     "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
paulson@23449
   503
apply (simp add: fix_def)
paulson@23449
   504
apply (rule conjI)
blanchet@36554
   505
proof -
blanchet@36554
   506
  assume A1: "H = {x. (x, f x) \<in> r \<and> x \<in> A}"
blanchet@42762
   507
  have F1: "\<forall>u v. v \<inter> u \<subseteq> u" by (metis Int_commute Int_lower1)
blanchet@42762
   508
  have "{R. (R, f R) \<in> r} \<inter> {R. R \<in> A} = H" using A1 by (metis Collect_conj_eq)
blanchet@42762
   509
  hence "H \<subseteq> {R. R \<in> A}" using F1 by metis
blanchet@42762
   510
  hence "H \<subseteq> A" by (metis Collect_mem_eq)
blanchet@42762
   511
  hence "lub H cl \<in> A" by (metis lub_in_lattice)
blanchet@42762
   512
  thus "lub {x. (x, f x) \<in> r \<and> x \<in> A} cl \<in> A" using A1 by metis
blanchet@36554
   513
next
blanchet@36554
   514
  assume A1: "H = {x. (x, f x) \<in> r \<and> x \<in> A}"
haftmann@45970
   515
  have F1: "\<forall>v. {R. R \<in> v} = v" by (metis Collect_mem_eq)
haftmann@45970
   516
  have F2: "\<forall>w u. {R. R \<in> u \<and> R \<in> w} = u \<inter> w"
haftmann@45970
   517
    by (metis Collect_conj_eq Collect_mem_eq)
haftmann@45970
   518
  have F3: "\<forall>x v. {R. v R \<in> x} = v -` x" by (metis vimage_def)
blanchet@36554
   519
  hence F4: "A \<inter> (\<lambda>R. (R, f R)) -` r = H" using A1 by auto
haftmann@45970
   520
  hence F5: "(f (lub H cl), lub H cl) \<in> r" 
haftmann@45970
   521
    by (metis A1 flubH_le_lubH)
blanchet@36554
   522
  have F6: "(lub H cl, f (lub H cl)) \<in> r"
haftmann@45970
   523
    by (metis A1 lubH_le_flubH)
blanchet@36554
   524
  have "(lub H cl, f (lub H cl)) \<in> r \<longrightarrow> f (lub H cl) = lub H cl"
blanchet@36554
   525
    using F5 by (metis antisymE)
blanchet@36554
   526
  hence "f (lub H cl) = lub H cl" using F6 by metis
blanchet@36554
   527
  thus "H = {x. (x, f x) \<in> r \<and> x \<in> A}
blanchet@36554
   528
        \<Longrightarrow> f (lub {x. (x, f x) \<in> r \<and> x \<in> A} cl) =
blanchet@36554
   529
           lub {x. (x, f x) \<in> r \<and> x \<in> A} cl"
haftmann@45970
   530
    by metis
paulson@24827
   531
qed
paulson@23449
   532
paulson@25710
   533
lemma (in CLF) (*lubH_is_fixp:*)
paulson@23449
   534
     "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
paulson@23449
   535
apply (simp add: fix_def)
paulson@23449
   536
apply (rule conjI)
nipkow@30198
   537
apply (metis CO_refl_on lubH_le_flubH refl_onD1)
paulson@23449
   538
apply (metis antisymE flubH_le_lubH lubH_le_flubH)
paulson@23449
   539
done
paulson@23449
   540
paulson@23449
   541
lemma (in CLF) fix_in_H:
paulson@23449
   542
     "[| H = {x. (x, f x) \<in> r & x \<in> A};  x \<in> P |] ==> x \<in> H"
nipkow@30198
   543
by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl_on
paulson@23449
   544
                    fix_subset [of f A, THEN subsetD])
paulson@23449
   545
paulson@23449
   546
lemma (in CLF) fixf_le_lubH:
paulson@23449
   547
     "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
paulson@23449
   548
apply (rule ballI)
paulson@23449
   549
apply (rule lub_upper, fast)
paulson@23449
   550
apply (rule fix_in_H)
paulson@23449
   551
apply (simp_all add: P_def)
paulson@23449
   552
done
paulson@23449
   553
paulson@23449
   554
lemma (in CLF) lubH_least_fixf:
paulson@23449
   555
     "H = {x. (x, f x) \<in> r & x \<in> A}
paulson@23449
   556
      ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
paulson@23449
   557
apply (metis P_def lubH_is_fixp)
paulson@23449
   558
done
paulson@23449
   559
paulson@23449
   560
subsection {* Tarski fixpoint theorem 1, first part *}
blanchet@45705
   561
blanchet@45705
   562
declare CL.lubI[intro] fix_subset[intro] CL.lub_in_lattice[intro]
blanchet@45705
   563
        CLF.fixf_le_lubH[simp] CLF.lubH_least_fixf[simp]
blanchet@45705
   564
paulson@23449
   565
lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
paulson@23449
   566
(*sledgehammer;*)
paulson@23449
   567
apply (rule sym)
paulson@23449
   568
apply (simp add: P_def)
paulson@23449
   569
apply (rule lubI)
blanchet@43197
   570
apply (metis P_def fix_subset)
paulson@24827
   571
apply (metis Collect_conj_eq Collect_mem_eq Int_commute Int_lower1 lub_in_lattice vimage_def)
blanchet@47040
   572
apply (metis P_def fixf_le_lubH)
blanchet@47040
   573
by (metis P_def lubH_least_fixf)
paulson@23449
   574
blanchet@45705
   575
declare CL.lubI[rule del] fix_subset[rule del] CL.lub_in_lattice[rule del]
blanchet@45705
   576
        CLF.fixf_le_lubH[simp del] CLF.lubH_least_fixf[simp del]
paulson@23449
   577
paulson@23449
   578
(*never proved, 2007-01-22*)
blanchet@45705
   579
blanchet@45705
   580
declare glb_dual_lub[simp] PO.dualA_iff[intro] CLF.lubH_is_fixp[intro]
blanchet@45705
   581
        PO.dualPO[intro] CL.CL_dualCL[intro] PO.dualr_iff[simp]
blanchet@45705
   582
paulson@23449
   583
lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
paulson@23449
   584
  -- {* Tarski for glb *}
paulson@23449
   585
(*sledgehammer;*)
paulson@23449
   586
apply (simp add: glb_dual_lub P_def A_def r_def)
paulson@23449
   587
apply (rule dualA_iff [THEN subst])
paulson@23449
   588
apply (rule CLF.lubH_is_fixp)
haftmann@27681
   589
apply (rule CLF.intro)
haftmann@27681
   590
apply (rule CL.intro)
haftmann@27681
   591
apply (rule PO.intro)
paulson@23449
   592
apply (rule dualPO)
haftmann@27681
   593
apply (rule CL_axioms.intro)
paulson@23449
   594
apply (rule CL_dualCL)
haftmann@27681
   595
apply (rule CLF_axioms.intro)
paulson@23449
   596
apply (rule CLF_dual)
paulson@23449
   597
apply (simp add: dualr_iff dualA_iff)
paulson@23449
   598
done
paulson@23449
   599
blanchet@45705
   600
declare glb_dual_lub[simp del] PO.dualA_iff[rule del] CLF.lubH_is_fixp[rule del]
blanchet@45705
   601
        PO.dualPO[rule del] CL.CL_dualCL[rule del] PO.dualr_iff[simp del]
paulson@23449
   602
paulson@23449
   603
(*never proved, 2007-01-22*)
blanchet@45705
   604
paulson@23449
   605
lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
paulson@23449
   606
(*sledgehammer;*)
paulson@23449
   607
apply (simp add: glb_dual_lub P_def A_def r_def)
paulson@23449
   608
apply (rule dualA_iff [THEN subst])
paulson@23449
   609
(*never proved, 2007-01-22*)
paulson@23449
   610
(*sledgehammer;*)
haftmann@27681
   611
apply (simp add: CLF.T_thm_1_lub [of _ f, OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro,
haftmann@27681
   612
  OF dualPO CL_dualCL] dualPO CL_dualCL CLF_dual dualr_iff)
paulson@23449
   613
done
paulson@23449
   614
paulson@23449
   615
subsection {* interval *}
paulson@23449
   616
blanchet@45705
   617
declare (in CLF) CO_refl_on[simp] refl_on_def [simp]
paulson@23449
   618
paulson@23449
   619
lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
nipkow@30198
   620
by (metis CO_refl_on refl_onD1)
blanchet@45705
   621
blanchet@45705
   622
declare (in CLF) CO_refl_on[simp del]  refl_on_def [simp del]
paulson@23449
   623
blanchet@45705
   624
declare (in CLF) rel_imp_elem[intro]
blanchet@45705
   625
declare interval_def [simp]
blanchet@45705
   626
paulson@23449
   627
lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b \<subseteq> A"
nipkow@30198
   628
by (metis CO_refl_on interval_imp_mem refl_onD refl_onD2 rel_imp_elem subset_eq)
paulson@23449
   629
blanchet@45705
   630
declare (in CLF) rel_imp_elem[rule del]
blanchet@45705
   631
declare interval_def [simp del]
paulson@23449
   632
paulson@23449
   633
lemma (in CLF) intervalI:
paulson@23449
   634
     "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
paulson@23449
   635
by (simp add: interval_def)
paulson@23449
   636
paulson@23449
   637
lemma (in CLF) interval_lemma1:
paulson@23449
   638
     "[| S \<subseteq> interval r a b; x \<in> S |] ==> (a, x) \<in> r"
paulson@23449
   639
by (unfold interval_def, fast)
paulson@23449
   640
paulson@23449
   641
lemma (in CLF) interval_lemma2:
paulson@23449
   642
     "[| S \<subseteq> interval r a b; x \<in> S |] ==> (x, b) \<in> r"
paulson@23449
   643
by (unfold interval_def, fast)
paulson@23449
   644
paulson@23449
   645
lemma (in CLF) a_less_lub:
paulson@23449
   646
     "[| S \<subseteq> A; S \<noteq> {};
paulson@23449
   647
         \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
paulson@23449
   648
by (blast intro: transE)
paulson@23449
   649
paulson@23449
   650
lemma (in CLF) glb_less_b:
paulson@23449
   651
     "[| S \<subseteq> A; S \<noteq> {};
paulson@23449
   652
         \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
paulson@23449
   653
by (blast intro: transE)
paulson@23449
   654
paulson@23449
   655
lemma (in CLF) S_intv_cl:
paulson@23449
   656
     "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b |]==> S \<subseteq> A"
paulson@23449
   657
by (simp add: subset_trans [OF _ interval_subset])
paulson@23449
   658
blanchet@45705
   659
paulson@23449
   660
lemma (in CLF) L_in_interval:
paulson@23449
   661
     "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b;
blanchet@43197
   662
         S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
paulson@23449
   663
(*WON'T TERMINATE
paulson@23449
   664
apply (metis CO_trans intervalI interval_lemma1 interval_lemma2 isLub_least isLub_upper subset_empty subset_iff trans_def)
paulson@23449
   665
*)
paulson@23449
   666
apply (rule intervalI)
paulson@23449
   667
apply (rule a_less_lub)
paulson@23449
   668
prefer 2 apply assumption
paulson@23449
   669
apply (simp add: S_intv_cl)
paulson@23449
   670
apply (rule ballI)
paulson@23449
   671
apply (simp add: interval_lemma1)
paulson@23449
   672
apply (simp add: isLub_upper)
paulson@23449
   673
-- {* @{text "(L, b) \<in> r"} *}
paulson@23449
   674
apply (simp add: isLub_least interval_lemma2)
paulson@23449
   675
done
paulson@23449
   676
paulson@23449
   677
(*never proved, 2007-01-22*)
blanchet@45705
   678
paulson@23449
   679
lemma (in CLF) G_in_interval:
paulson@23449
   680
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> interval r a b; isGlb S cl G;
paulson@23449
   681
         S \<noteq> {} |] ==> G \<in> interval r a b"
paulson@23449
   682
apply (simp add: interval_dual)
haftmann@27681
   683
apply (simp add: CLF.L_in_interval [of _ f, OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro]
paulson@23449
   684
                 dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub)
paulson@23449
   685
done
paulson@23449
   686
blanchet@45705
   687
paulson@23449
   688
lemma (in CLF) intervalPO:
paulson@23449
   689
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
paulson@23449
   690
      ==> (| pset = interval r a b, order = induced (interval r a b) r |)
paulson@23449
   691
          \<in> PartialOrder"
blanchet@36554
   692
proof -
blanchet@36554
   693
  assume A1: "a \<in> A"
blanchet@36554
   694
  assume "b \<in> A"
blanchet@36554
   695
  hence "\<forall>u. u \<in> A \<longrightarrow> interval r u b \<subseteq> A" by (metis interval_subset)
blanchet@36554
   696
  hence "interval r a b \<subseteq> A" using A1 by metis
blanchet@36554
   697
  hence "interval r a b \<subseteq> A" by metis
blanchet@36554
   698
  thus ?thesis by (metis po_subset_po)
paulson@23449
   699
qed
paulson@23449
   700
paulson@23449
   701
lemma (in CLF) intv_CL_lub:
paulson@23449
   702
 "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
paulson@23449
   703
  ==> \<forall>S. S \<subseteq> interval r a b -->
paulson@23449
   704
          (\<exists>L. isLub S (| pset = interval r a b,
paulson@23449
   705
                          order = induced (interval r a b) r |)  L)"
paulson@23449
   706
apply (intro strip)
paulson@23449
   707
apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
paulson@23449
   708
prefer 2 apply assumption
paulson@23449
   709
apply assumption
paulson@23449
   710
apply (erule exE)
paulson@23449
   711
-- {* define the lub for the interval as *}
paulson@23449
   712
apply (rule_tac x = "if S = {} then a else L" in exI)
paulson@23449
   713
apply (simp (no_asm_simp) add: isLub_def split del: split_if)
paulson@23449
   714
apply (intro impI conjI)
paulson@23449
   715
-- {* @{text "(if S = {} then a else L) \<in> interval r a b"} *}
paulson@23449
   716
apply (simp add: CL_imp_PO L_in_interval)
paulson@23449
   717
apply (simp add: left_in_interval)
paulson@23449
   718
-- {* lub prop 1 *}
paulson@23449
   719
apply (case_tac "S = {}")
paulson@23449
   720
-- {* @{text "S = {}, y \<in> S = False => everything"} *}
paulson@23449
   721
apply fast
paulson@23449
   722
-- {* @{text "S \<noteq> {}"} *}
paulson@23449
   723
apply simp
paulson@23449
   724
-- {* @{text "\<forall>y:S. (y, L) \<in> induced (interval r a b) r"} *}
paulson@23449
   725
apply (rule ballI)
paulson@23449
   726
apply (simp add: induced_def  L_in_interval)
paulson@23449
   727
apply (rule conjI)
paulson@23449
   728
apply (rule subsetD)
paulson@23449
   729
apply (simp add: S_intv_cl, assumption)
paulson@23449
   730
apply (simp add: isLub_upper)
paulson@23449
   731
-- {* @{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"} *}
paulson@23449
   732
apply (rule ballI)
paulson@23449
   733
apply (rule impI)
paulson@23449
   734
apply (case_tac "S = {}")
paulson@23449
   735
-- {* @{text "S = {}"} *}
paulson@23449
   736
apply simp
paulson@23449
   737
apply (simp add: induced_def  interval_def)
paulson@23449
   738
apply (rule conjI)
paulson@23449
   739
apply (rule reflE, assumption)
paulson@23449
   740
apply (rule interval_not_empty)
paulson@23449
   741
apply (rule CO_trans)
paulson@23449
   742
apply (simp add: interval_def)
paulson@23449
   743
-- {* @{text "S \<noteq> {}"} *}
paulson@23449
   744
apply simp
paulson@23449
   745
apply (simp add: induced_def  L_in_interval)
paulson@23449
   746
apply (rule isLub_least, assumption)
paulson@23449
   747
apply (rule subsetD)
paulson@23449
   748
prefer 2 apply assumption
paulson@23449
   749
apply (simp add: S_intv_cl, fast)
paulson@23449
   750
done
paulson@23449
   751
paulson@23449
   752
lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
paulson@23449
   753
paulson@23449
   754
(*never proved, 2007-01-22*)
blanchet@45705
   755
paulson@23449
   756
lemma (in CLF) interval_is_sublattice:
paulson@23449
   757
     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
paulson@23449
   758
        ==> interval r a b <<= cl"
paulson@23449
   759
(*sledgehammer *)
paulson@23449
   760
apply (rule sublatticeI)
paulson@23449
   761
apply (simp add: interval_subset)
paulson@23449
   762
(*never proved, 2007-01-22*)
paulson@23449
   763
(*sledgehammer *)
paulson@23449
   764
apply (rule CompleteLatticeI)
paulson@23449
   765
apply (simp add: intervalPO)
paulson@23449
   766
 apply (simp add: intv_CL_lub)
paulson@23449
   767
apply (simp add: intv_CL_glb)
paulson@23449
   768
done
paulson@23449
   769
paulson@23449
   770
lemmas (in CLF) interv_is_compl_latt =
paulson@23449
   771
    interval_is_sublattice [THEN sublattice_imp_CL]
paulson@23449
   772
paulson@23449
   773
subsection {* Top and Bottom *}
paulson@23449
   774
lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
paulson@23449
   775
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
paulson@23449
   776
paulson@23449
   777
lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
paulson@23449
   778
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
paulson@23449
   779
blanchet@45705
   780
paulson@23449
   781
lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
paulson@23449
   782
(*sledgehammer; *)
paulson@23449
   783
apply (simp add: Bot_def least_def)
paulson@23449
   784
apply (rule_tac a="glb A cl" in someI2)
blanchet@43197
   785
apply (simp_all add: glb_in_lattice glb_lower
paulson@23449
   786
                     r_def [symmetric] A_def [symmetric])
paulson@23449
   787
done
paulson@23449
   788
paulson@23449
   789
(*first proved 2007-01-25 after relaxing relevance*)
blanchet@45705
   790
paulson@23449
   791
lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
paulson@23449
   792
(*sledgehammer;*)
paulson@23449
   793
apply (simp add: Top_dual_Bot A_def)
paulson@23449
   794
(*first proved 2007-01-25 after relaxing relevance*)
paulson@23449
   795
(*sledgehammer*)
paulson@23449
   796
apply (rule dualA_iff [THEN subst])
haftmann@27681
   797
apply (blast intro!: CLF.Bot_in_lattice [OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro] dualPO CL_dualCL CLF_dual)
paulson@23449
   798
done
paulson@23449
   799
paulson@23449
   800
lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
paulson@23449
   801
apply (simp add: Top_def greatest_def)
paulson@23449
   802
apply (rule_tac a="lub A cl" in someI2)
paulson@23449
   803
apply (rule someI2)
blanchet@43197
   804
apply (simp_all add: lub_in_lattice lub_upper
paulson@23449
   805
                     r_def [symmetric] A_def [symmetric])
paulson@23449
   806
done
paulson@23449
   807
paulson@23449
   808
(*never proved, 2007-01-22*)
blanchet@45705
   809
paulson@23449
   810
lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
blanchet@43197
   811
(*sledgehammer*)
paulson@23449
   812
apply (simp add: Bot_dual_Top r_def)
paulson@23449
   813
apply (rule dualr_iff [THEN subst])
haftmann@27681
   814
apply (simp add: CLF.Top_prop [of _ f, OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro]
paulson@23449
   815
                 dualA_iff A_def dualPO CL_dualCL CLF_dual)
paulson@23449
   816
done
paulson@23449
   817
blanchet@45705
   818
blanchet@43197
   819
lemma (in CLF) Top_intv_not_empty: "x \<in> A  ==> interval r x (Top cl) \<noteq> {}"
paulson@23449
   820
apply (metis Top_in_lattice Top_prop empty_iff intervalI reflE)
paulson@23449
   821
done
paulson@23449
   822
blanchet@45705
   823
blanchet@43197
   824
lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
paulson@23449
   825
apply (metis Bot_prop ex_in_conv intervalI reflE rel_imp_elem)
paulson@23449
   826
done
paulson@23449
   827
paulson@23449
   828
subsection {* fixed points form a partial order *}
paulson@23449
   829
paulson@23449
   830
lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
paulson@23449
   831
by (simp add: P_def fix_subset po_subset_po)
paulson@23449
   832
paulson@23449
   833
(*first proved 2007-01-25 after relaxing relevance*)
blanchet@45705
   834
blanchet@45705
   835
declare (in Tarski) P_def[simp] Y_ss [simp]
blanchet@45705
   836
declare fix_subset [intro] subset_trans [intro]
blanchet@45705
   837
paulson@23449
   838
lemma (in Tarski) Y_subset_A: "Y \<subseteq> A"
blanchet@43197
   839
(*sledgehammer*)
paulson@23449
   840
apply (rule subset_trans [OF _ fix_subset])
paulson@23449
   841
apply (rule Y_ss [simplified P_def])
paulson@23449
   842
done
paulson@23449
   843
blanchet@45705
   844
declare (in Tarski) P_def[simp del] Y_ss [simp del]
blanchet@45705
   845
declare fix_subset [rule del] subset_trans [rule del]
paulson@23449
   846
paulson@23449
   847
lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
paulson@23449
   848
  by (rule Y_subset_A [THEN lub_in_lattice])
paulson@23449
   849
paulson@23449
   850
(*never proved, 2007-01-22*)
blanchet@45705
   851
paulson@23449
   852
lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
blanchet@43197
   853
(*sledgehammer*)
paulson@23449
   854
apply (rule lub_least)
paulson@23449
   855
apply (rule Y_subset_A)
paulson@23449
   856
apply (rule f_in_funcset [THEN funcset_mem])
paulson@23449
   857
apply (rule lubY_in_A)
paulson@23449
   858
-- {* @{text "Y \<subseteq> P ==> f x = x"} *}
paulson@23449
   859
apply (rule ballI)
paulson@23449
   860
(*sledgehammer *)
paulson@23449
   861
apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
paulson@23449
   862
apply (erule Y_ss [simplified P_def, THEN subsetD])
paulson@23449
   863
-- {* @{text "reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r"} by monotonicity *}
paulson@23449
   864
(*sledgehammer*)
paulson@23449
   865
apply (rule_tac f = "f" in monotoneE)
paulson@23449
   866
apply (rule monotone_f)
paulson@23449
   867
apply (simp add: Y_subset_A [THEN subsetD])
paulson@23449
   868
apply (rule lubY_in_A)
paulson@23449
   869
apply (simp add: lub_upper Y_subset_A)
paulson@23449
   870
done
paulson@23449
   871
paulson@23449
   872
(*first proved 2007-01-25 after relaxing relevance*)
blanchet@45705
   873
paulson@23449
   874
lemma (in Tarski) intY1_subset: "intY1 \<subseteq> A"
blanchet@43197
   875
(*sledgehammer*)
paulson@23449
   876
apply (unfold intY1_def)
paulson@23449
   877
apply (rule interval_subset)
paulson@23449
   878
apply (rule lubY_in_A)
paulson@23449
   879
apply (rule Top_in_lattice)
paulson@23449
   880
done
paulson@23449
   881
paulson@23449
   882
lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
paulson@23449
   883
paulson@23449
   884
(*never proved, 2007-01-22*)
blanchet@45705
   885
paulson@23449
   886
lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
blanchet@43197
   887
(*sledgehammer*)
paulson@23449
   888
apply (simp add: intY1_def  interval_def)
paulson@23449
   889
apply (rule conjI)
paulson@23449
   890
apply (rule transE)
paulson@23449
   891
apply (rule lubY_le_flubY)
paulson@23449
   892
-- {* @{text "(f (lub Y cl), f x) \<in> r"} *}
paulson@23449
   893
(*sledgehammer [has been proved before now...]*)
paulson@23449
   894
apply (rule_tac f=f in monotoneE)
paulson@23449
   895
apply (rule monotone_f)
paulson@23449
   896
apply (rule lubY_in_A)
paulson@23449
   897
apply (simp add: intY1_def interval_def  intY1_elem)
paulson@23449
   898
apply (simp add: intY1_def  interval_def)
blanchet@43197
   899
-- {* @{text "(f x, Top cl) \<in> r"} *}
paulson@23449
   900
apply (rule Top_prop)
paulson@23449
   901
apply (rule f_in_funcset [THEN funcset_mem])
paulson@23449
   902
apply (simp add: intY1_def interval_def  intY1_elem)
paulson@23449
   903
done
paulson@23449
   904
blanchet@45705
   905
haftmann@27368
   906
lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 -> intY1"
haftmann@27368
   907
apply (rule restrict_in_funcset)
haftmann@27368
   908
apply (metis intY1_f_closed restrict_in_funcset)
haftmann@27368
   909
done
paulson@23449
   910
blanchet@45705
   911
paulson@24855
   912
lemma (in Tarski) intY1_mono:
paulson@23449
   913
     "monotone (%x: intY1. f x) intY1 (induced intY1 r)"
paulson@23449
   914
(*sledgehammer *)
paulson@23449
   915
apply (auto simp add: monotone_def induced_def intY1_f_closed)
paulson@23449
   916
apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
paulson@23449
   917
done
paulson@23449
   918
paulson@23449
   919
(*proof requires relaxing relevance: 2007-01-25*)
blanchet@45705
   920
paulson@23449
   921
lemma (in Tarski) intY1_is_cl:
paulson@23449
   922
    "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
blanchet@43197
   923
(*sledgehammer*)
paulson@23449
   924
apply (unfold intY1_def)
paulson@23449
   925
apply (rule interv_is_compl_latt)
paulson@23449
   926
apply (rule lubY_in_A)
paulson@23449
   927
apply (rule Top_in_lattice)
paulson@23449
   928
apply (rule Top_intv_not_empty)
paulson@23449
   929
apply (rule lubY_in_A)
paulson@23449
   930
done
paulson@23449
   931
paulson@23449
   932
(*never proved, 2007-01-22*)
blanchet@45705
   933
paulson@23449
   934
lemma (in Tarski) v_in_P: "v \<in> P"
blanchet@43197
   935
(*sledgehammer*)
paulson@23449
   936
apply (unfold P_def)
paulson@23449
   937
apply (rule_tac A = "intY1" in fixf_subset)
paulson@23449
   938
apply (rule intY1_subset)
haftmann@27681
   939
apply (simp add: CLF.glbH_is_fixp [OF CLF.intro, OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro, OF _ intY1_is_cl, simplified]
haftmann@27681
   940
                 v_def CL_imp_PO intY1_is_cl CLF_set_def intY1_func intY1_mono)
paulson@23449
   941
done
paulson@23449
   942
blanchet@45705
   943
paulson@23449
   944
lemma (in Tarski) z_in_interval:
paulson@23449
   945
     "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
paulson@23449
   946
(*sledgehammer *)
paulson@23449
   947
apply (unfold intY1_def P_def)
paulson@23449
   948
apply (rule intervalI)
paulson@23449
   949
prefer 2
paulson@23449
   950
 apply (erule fix_subset [THEN subsetD, THEN Top_prop])
paulson@23449
   951
apply (rule lub_least)
paulson@23449
   952
apply (rule Y_subset_A)
paulson@23449
   953
apply (fast elim!: fix_subset [THEN subsetD])
paulson@23449
   954
apply (simp add: induced_def)
paulson@23449
   955
done
paulson@23449
   956
blanchet@45705
   957
paulson@23449
   958
lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
blanchet@43197
   959
      ==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"
berghofe@26806
   960
apply (metis P_def acc_def fix_imp_eq fix_subset indI reflE restrict_apply subset_eq z_in_interval)
paulson@23449
   961
done
paulson@23449
   962
paulson@23449
   963
(*never proved, 2007-01-22*)
blanchet@45705
   964
paulson@23449
   965
lemma (in Tarski) tarski_full_lemma:
paulson@23449
   966
     "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
paulson@23449
   967
apply (rule_tac x = "v" in exI)
paulson@23449
   968
apply (simp add: isLub_def)
paulson@23449
   969
-- {* @{text "v \<in> P"} *}
paulson@23449
   970
apply (simp add: v_in_P)
paulson@23449
   971
apply (rule conjI)
blanchet@43197
   972
(*sledgehammer*)
paulson@23449
   973
-- {* @{text v} is lub *}
paulson@23449
   974
-- {* @{text "1. \<forall>y:Y. (y, v) \<in> induced P r"} *}
paulson@23449
   975
apply (rule ballI)
paulson@23449
   976
apply (simp add: induced_def subsetD v_in_P)
paulson@23449
   977
apply (rule conjI)
paulson@23449
   978
apply (erule Y_ss [THEN subsetD])
paulson@23449
   979
apply (rule_tac b = "lub Y cl" in transE)
paulson@23449
   980
apply (rule lub_upper)
paulson@23449
   981
apply (rule Y_subset_A, assumption)
paulson@23449
   982
apply (rule_tac b = "Top cl" in interval_imp_mem)
paulson@23449
   983
apply (simp add: v_def)
paulson@23449
   984
apply (fold intY1_def)
haftmann@27681
   985
apply (rule CL.glb_in_lattice [OF CL.intro, OF PO.intro CL_axioms.intro, OF _ intY1_is_cl, simplified])
paulson@23449
   986
 apply (simp add: CL_imp_PO intY1_is_cl, force)
paulson@23449
   987
-- {* @{text v} is LEAST ub *}
paulson@23449
   988
apply clarify
paulson@23449
   989
apply (rule indI)
paulson@23449
   990
  prefer 3 apply assumption
paulson@23449
   991
 prefer 2 apply (simp add: v_in_P)
paulson@23449
   992
apply (unfold v_def)
paulson@23449
   993
(*never proved, 2007-01-22*)
blanchet@43197
   994
(*sledgehammer*)
paulson@23449
   995
apply (rule indE)
paulson@23449
   996
apply (rule_tac [2] intY1_subset)
paulson@23449
   997
(*never proved, 2007-01-22*)
blanchet@43197
   998
(*sledgehammer*)
haftmann@27681
   999
apply (rule CL.glb_lower [OF CL.intro, OF PO.intro CL_axioms.intro, OF _ intY1_is_cl, simplified])
paulson@23449
  1000
  apply (simp add: CL_imp_PO intY1_is_cl)
paulson@23449
  1001
 apply force
paulson@23449
  1002
apply (simp add: induced_def intY1_f_closed z_in_interval)
paulson@23449
  1003
apply (simp add: P_def fix_imp_eq [of _ f A] reflE
paulson@23449
  1004
                 fix_subset [of f A, THEN subsetD])
paulson@23449
  1005
done
paulson@23449
  1006
paulson@23449
  1007
lemma CompleteLatticeI_simp:
paulson@23449
  1008
     "[| (| pset = A, order = r |) \<in> PartialOrder;
paulson@23449
  1009
         \<forall>S. S \<subseteq> A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
paulson@23449
  1010
    ==> (| pset = A, order = r |) \<in> CompleteLattice"
paulson@23449
  1011
by (simp add: CompleteLatticeI Rdual)
paulson@23449
  1012
blanchet@45705
  1013
(*never proved, 2007-01-22*)
paulson@23449
  1014
blanchet@45705
  1015
declare (in CLF) fixf_po[intro] P_def [simp] A_def [simp] r_def [simp]
blanchet@45705
  1016
             Tarski.tarski_full_lemma [intro] cl_po [intro] cl_co [intro]
blanchet@45705
  1017
             CompleteLatticeI_simp [intro]
blanchet@45705
  1018
paulson@23449
  1019
theorem (in CLF) Tarski_full:
paulson@23449
  1020
     "(| pset = P, order = induced P r|) \<in> CompleteLattice"
blanchet@43197
  1021
(*sledgehammer*)
paulson@23449
  1022
apply (rule CompleteLatticeI_simp)
paulson@23449
  1023
apply (rule fixf_po, clarify)
paulson@23449
  1024
(*never proved, 2007-01-22*)
blanchet@43197
  1025
(*sledgehammer*)
paulson@23449
  1026
apply (simp add: P_def A_def r_def)
haftmann@27681
  1027
apply (blast intro!: Tarski.tarski_full_lemma [OF Tarski.intro, OF CLF.intro Tarski_axioms.intro,
haftmann@27681
  1028
  OF CL.intro CLF_axioms.intro, OF PO.intro CL_axioms.intro] cl_po cl_co f_cl)
paulson@23449
  1029
done
blanchet@36554
  1030
blanchet@36554
  1031
declare (in CLF) fixf_po [rule del] P_def [simp del] A_def [simp del] r_def [simp del]
paulson@23449
  1032
         Tarski.tarski_full_lemma [rule del] cl_po [rule del] cl_co [rule del]
paulson@23449
  1033
         CompleteLatticeI_simp [rule del]
paulson@23449
  1034
paulson@23449
  1035
end