--- a/src/HOL/ex/Tarski.thy Tue May 07 19:54:29 2002 +0200
+++ b/src/HOL/ex/Tarski.thy Wed May 08 09:08:16 2002 +0200
@@ -7,17 +7,18 @@
The fixedpoints of a complete lattice themselves form a complete lattice.
Illustrates first-class theories, using the Sigma representation of structures
+
+Tidied and converted to Isar by lcp
*)
-Tarski = Main +
-
+theory Tarski = Main:
record 'a potype =
pset :: "'a set"
order :: "('a * 'a) set"
syntax
- "@pset" :: "'a potype => 'a set" ("_ .<A>" [90] 90)
+ "@pset" :: "'a potype => 'a set" ("_ .<A>" [90] 90)
"@order" :: "'a potype => ('a *'a)set" ("_ .<r>" [90] 90)
translations
@@ -26,31 +27,31 @@
constdefs
monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
- "monotone f A r == ! x: A. ! y: A. (x, y): r --> ((f x), (f y)) : r"
+ "monotone f A r == \<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r"
least :: "['a => bool, 'a potype] => 'a"
"least P po == @ x. x: po.<A> & P x &
- (! y: po.<A>. P y --> (x,y): po.<r>)"
+ (\<forall>y \<in> po.<A>. P y --> (x,y): po.<r>)"
greatest :: "['a => bool, 'a potype] => 'a"
"greatest P po == @ x. x: po.<A> & P x &
- (! y: po.<A>. P y --> (y,x): po.<r>)"
+ (\<forall>y \<in> po.<A>. P y --> (y,x): po.<r>)"
lub :: "['a set, 'a potype] => 'a"
- "lub S po == least (%x. ! y: S. (y,x): po.<r>) po"
+ "lub S po == least (%x. \<forall>y\<in>S. (y,x): po.<r>) po"
glb :: "['a set, 'a potype] => 'a"
- "glb S po == greatest (%x. ! y: S. (x,y): po.<r>) po"
+ "glb S po == greatest (%x. \<forall>y\<in>S. (x,y): po.<r>) po"
- islub :: "['a set, 'a potype, 'a] => bool"
- "islub S po == %L. (L: po.<A> & (! y: S. (y,L): po.<r>) &
- (! z:po.<A>. (! y: S. (y,z): po.<r>) --> (L,z): po.<r>))"
+ isLub :: "['a set, 'a potype, 'a] => bool"
+ "isLub S po == %L. (L: po.<A> & (\<forall>y\<in>S. (y,L): po.<r>) &
+ (\<forall>z\<in>po.<A>. (\<forall>y\<in>S. (y,z): po.<r>) --> (L,z): po.<r>))"
- isglb :: "['a set, 'a potype, 'a] => bool"
- "isglb S po == %G. (G: po.<A> & (! y: S. (G,y): po.<r>) &
- (! z: po.<A>. (! y: S. (z,y): po.<r>) --> (z,G): po.<r>))"
+ isGlb :: "['a set, 'a potype, 'a] => bool"
+ "isGlb S po == %G. (G: po.<A> & (\<forall>y\<in>S. (G,y): po.<r>) &
+ (\<forall>z \<in> po.<A>. (\<forall>y\<in>S. (z,y): po.<r>) --> (z,G): po.<r>))"
- fix :: "[('a => 'a), 'a set] => 'a set"
+ "fix" :: "[('a => 'a), 'a set] => 'a set"
"fix f A == {x. x: A & f x = x}"
interval :: "[('a*'a) set,'a, 'a ] => 'a set"
@@ -70,8 +71,8 @@
CompleteLattice :: "('a potype) set"
"CompleteLattice == {cl. cl: PartialOrder &
- (! S. S <= cl.<A> --> (? L. islub S cl L)) &
- (! S. S <= cl.<A> --> (? G. isglb S cl G))}"
+ (\<forall>S. S <= cl.<A> --> (\<exists>L. isLub S cl L)) &
+ (\<forall>S. S <= cl.<A> --> (\<exists>G. isGlb S cl G))}"
CLF :: "('a potype * ('a => 'a)) set"
"CLF == SIGMA cl: CompleteLattice.
@@ -101,41 +102,796 @@
"dual po == (| pset = po.<A>, order = converse (po.<r>) |)"
locale PO =
-fixes
- cl :: "'a potype"
- A :: "'a set"
- r :: "('a * 'a) set"
-assumes
- cl_po "cl : PartialOrder"
-defines
- A_def "A == cl.<A>"
- r_def "r == cl.<r>"
+ fixes cl :: "'a potype"
+ and A :: "'a set"
+ and r :: "('a * 'a) set"
+ assumes cl_po: "cl : PartialOrder"
+ defines A_def: "A == cl.<A>"
+ and r_def: "r == cl.<r>"
locale CL = PO +
-fixes
-assumes
- cl_co "cl : CompleteLattice"
+ assumes cl_co: "cl : CompleteLattice"
locale CLF = CL +
-fixes
- f :: "'a => 'a"
- P :: "'a set"
-assumes
- f_cl "f : CLF``{cl}"
-defines
- P_def "P == fix f A"
+ fixes f :: "'a => 'a"
+ and P :: "'a set"
+ assumes f_cl: "(cl,f) : CLF" (*was the equivalent "f : CLF``{cl}"*)
+ defines P_def: "P == fix f A"
locale Tarski = CLF +
-fixes
- Y :: "'a set"
- intY1 :: "'a set"
- v :: "'a"
-assumes
- Y_ss "Y <= P"
-defines
- intY1_def "intY1 == interval r (lub Y cl) (Top cl)"
- v_def "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r & x: intY1}
- (| pset=intY1, order=induced intY1 r|)"
+ fixes Y :: "'a set"
+ and intY1 :: "'a set"
+ and v :: "'a"
+ assumes
+ Y_ss: "Y <= P"
+ defines
+ intY1_def: "intY1 == interval r (lub Y cl) (Top cl)"
+ and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r &
+ x: intY1}
+ (| pset=intY1, order=induced intY1 r|)"
+
+
+
+(* Partial Order *)
+
+lemma (in PO) PO_imp_refl: "refl A r"
+apply (insert cl_po)
+apply (simp add: PartialOrder_def A_def r_def)
+done
+
+lemma (in PO) PO_imp_sym: "antisym r"
+apply (insert cl_po)
+apply (simp add: PartialOrder_def A_def r_def)
+done
+
+lemma (in PO) PO_imp_trans: "trans r"
+apply (insert cl_po)
+apply (simp add: PartialOrder_def A_def r_def)
+done
+
+lemma (in PO) reflE: "[| refl A r; x \<in> A|] ==> (x, x) \<in> r"
+apply (insert cl_po)
+apply (simp add: PartialOrder_def refl_def)
+done
+
+lemma (in PO) antisymE: "[| antisym r; (a, b) \<in> r; (b, a) \<in> r |] ==> a = b"
+apply (insert cl_po)
+apply (simp add: PartialOrder_def antisym_def)
+done
+
+lemma (in PO) transE: "[| trans r; (a, b) \<in> r; (b, c) \<in> r|] ==> (a,c) \<in> r"
+apply (insert cl_po)
+apply (simp add: PartialOrder_def)
+apply (unfold trans_def, fast)
+done
+
+lemma (in PO) monotoneE:
+ "[| monotone f A r; x \<in> A; y \<in> A; (x, y) \<in> r |] ==> (f x, f y) \<in> r"
+by (simp add: monotone_def)
+
+lemma (in PO) po_subset_po:
+ "S <= A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
+apply (simp (no_asm) add: PartialOrder_def)
+apply auto
+(* refl *)
+apply (simp add: refl_def induced_def)
+apply (blast intro: PO_imp_refl [THEN reflE])
+(* antisym *)
+apply (simp add: antisym_def induced_def)
+apply (blast intro: PO_imp_sym [THEN antisymE])
+(* trans *)
+apply (simp add: trans_def induced_def)
+apply (blast intro: PO_imp_trans [THEN transE])
+done
+
+lemma (in PO) indE: "[| (x, y) \<in> induced S r; S <= A |] ==> (x, y) \<in> r"
+by (simp add: add: induced_def)
+
+lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r"
+by (simp add: add: induced_def)
+
+lemma (in CL) CL_imp_ex_isLub: "S <= A ==> \<exists>L. isLub S cl L"
+apply (insert cl_co)
+apply (simp add: CompleteLattice_def A_def)
+done
+
+declare (in CL) cl_co [simp]
+
+lemma isLub_lub: "(\<exists>L. isLub S cl L) = isLub S cl (lub S cl)"
+by (simp add: lub_def least_def isLub_def some_eq_ex [symmetric])
+
+lemma isGlb_glb: "(\<exists>G. isGlb S cl G) = isGlb S cl (glb S cl)"
+by (simp add: glb_def greatest_def isGlb_def some_eq_ex [symmetric])
+
+lemma isGlb_dual_isLub: "isGlb S cl = isLub S (dual cl)"
+by (simp add: isLub_def isGlb_def dual_def converse_def)
+
+lemma isLub_dual_isGlb: "isLub S cl = isGlb S (dual cl)"
+by (simp add: isLub_def isGlb_def dual_def converse_def)
+
+lemma (in PO) dualPO: "dual cl \<in> PartialOrder"
+apply (insert cl_po)
+apply (simp add: PartialOrder_def dual_def refl_converse
+ trans_converse antisym_converse)
+done
+
+lemma Rdual:
+ "\<forall>S. (S <= A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
+ ==> \<forall>S. (S <= A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"
+apply safe
+apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)}
+ (|pset = A, order = r|) " in exI)
+apply (drule_tac x = "{y. y \<in> A & (\<forall>k \<in> S. (y,k) \<in> r) }" in spec)
+apply (drule mp, fast)
+apply (simp add: isLub_lub isGlb_def)
+apply (simp add: isLub_def, blast)
+done
+
+lemma lub_dual_glb: "lub S cl = glb S (dual cl)"
+by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
+
+lemma glb_dual_lub: "glb S cl = lub S (dual cl)"
+by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
+
+lemma CL_subset_PO: "CompleteLattice <= PartialOrder"
+by (simp add: PartialOrder_def CompleteLattice_def, fast)
+
+lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
+
+declare CL_imp_PO [THEN Tarski.PO_imp_refl, simp]
+declare CL_imp_PO [THEN Tarski.PO_imp_sym, simp]
+declare CL_imp_PO [THEN Tarski.PO_imp_trans, simp]
+
+lemma (in CL) CO_refl: "refl A r"
+by (rule PO_imp_refl)
+
+lemma (in CL) CO_antisym: "antisym r"
+by (rule PO_imp_sym)
+
+lemma (in CL) CO_trans: "trans r"
+by (rule PO_imp_trans)
+
+lemma CompleteLatticeI:
+ "[| po \<in> PartialOrder; (\<forall>S. S <= po.<A> --> (\<exists>L. isLub S po L));
+ (\<forall>S. S <= po.<A> --> (\<exists>G. isGlb S po G))|]
+ ==> po \<in> CompleteLattice"
+apply (unfold CompleteLattice_def, blast)
+done
+
+lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
+apply (insert cl_co)
+apply (simp add: CompleteLattice_def dual_def)
+apply (fold dual_def)
+apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric]
+ dualPO)
+done
+
+lemma (in PO) dualA_iff: "(dual cl.<A>) = cl.<A>"
+by (simp add: dual_def)
+
+lemma (in PO) dualr_iff: "((x, y) \<in> (dual cl.<r>)) = ((y, x) \<in> cl.<r>)"
+by (simp add: dual_def)
+
+lemma (in PO) monotone_dual:
+ "monotone f (cl.<A>) (cl.<r>) ==> monotone f (dual cl.<A>) (dual cl.<r>)"
+apply (simp add: monotone_def dualA_iff dualr_iff)
+done
+
+lemma (in PO) interval_dual:
+ "[| x \<in> A; y \<in> A|] ==> interval r x y = interval (dual cl.<r>) y x"
+apply (simp add: interval_def dualr_iff)
+apply (fold r_def, fast)
+done
+
+lemma (in PO) interval_not_empty:
+ "[| trans r; interval r a b \<noteq> {} |] ==> (a, b) \<in> r"
+apply (simp add: interval_def)
+apply (unfold trans_def, blast)
+done
+
+lemma (in PO) interval_imp_mem: "x \<in> interval r a b ==> (a, x) \<in> r"
+by (simp add: interval_def)
+
+lemma (in PO) left_in_interval:
+ "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> a \<in> interval r a b"
+apply (simp (no_asm_simp) add: interval_def)
+apply (simp add: PO_imp_trans interval_not_empty)
+apply (simp add: PO_imp_refl [THEN reflE])
+done
+
+lemma (in PO) right_in_interval:
+ "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |] ==> b \<in> interval r a b"
+apply (simp (no_asm_simp) add: interval_def)
+apply (simp add: PO_imp_trans interval_not_empty)
+apply (simp add: PO_imp_refl [THEN reflE])
+done
+
+(* sublattice *)
+lemma (in PO) sublattice_imp_CL:
+ "S <<= cl ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
+by (simp add: sublattice_def CompleteLattice_def A_def r_def)
+
+lemma (in CL) sublatticeI:
+ "[| S <= A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
+ ==> S <<= cl"
+by (simp add: sublattice_def A_def r_def)
+
+(* lub *)
+lemma (in CL) lub_unique: "[| S <= A; isLub S cl x; isLub S cl L|] ==> x = L"
+apply (rule antisymE)
+apply (rule CO_antisym)
+apply (auto simp add: isLub_def r_def)
+done
+
+lemma (in CL) lub_upper: "[|S <= A; x \<in> S|] ==> (x, lub S cl) \<in> r"
+apply (rule CL_imp_ex_isLub [THEN exE], assumption)
+apply (unfold lub_def least_def)
+apply (rule some_equality [THEN ssubst])
+ apply (simp add: isLub_def)
+ apply (simp add: lub_unique A_def isLub_def)
+apply (simp add: isLub_def r_def)
+done
+
+lemma (in CL) lub_least:
+ "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
+apply (rule CL_imp_ex_isLub [THEN exE], assumption)
+apply (unfold lub_def least_def)
+apply (rule_tac s=x in some_equality [THEN ssubst])
+ apply (simp add: isLub_def)
+ apply (simp add: lub_unique A_def isLub_def)
+apply (simp add: isLub_def r_def A_def)
+done
+
+lemma (in CL) lub_in_lattice: "S <= A ==> lub S cl \<in> A"
+apply (rule CL_imp_ex_isLub [THEN exE], assumption)
+apply (unfold lub_def least_def)
+apply (subst some_equality)
+apply (simp add: isLub_def)
+prefer 2 apply (simp add: isLub_def A_def)
+apply (simp add: lub_unique A_def isLub_def)
+done
+
+lemma (in CL) lubI:
+ "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
+ \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
+apply (rule lub_unique, assumption)
+apply (simp add: isLub_def A_def r_def)
+apply (unfold isLub_def)
+apply (rule conjI)
+apply (fold A_def r_def)
+apply (rule lub_in_lattice, assumption)
+apply (simp add: lub_upper lub_least)
+done
+
+lemma (in CL) lubIa: "[| S <= A; isLub S cl L |] ==> L = lub S cl"
+by (simp add: lubI isLub_def A_def r_def)
+
+lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
+by (simp add: isLub_def A_def)
+
+lemma (in CL) isLub_upper: "[|isLub S cl L; y \<in> S|] ==> (y, L) \<in> r"
+by (simp add: isLub_def r_def)
+
+lemma (in CL) isLub_least:
+ "[| isLub S cl L; z \<in> A; \<forall>y \<in> S. (y, z) \<in> r|] ==> (L, z) \<in> r"
+by (simp add: isLub_def A_def r_def)
+
+lemma (in CL) isLubI:
+ "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
+ (\<forall>z \<in> A. (\<forall>y \<in> S. (y, z):r) --> (L, z) \<in> r)|] ==> isLub S cl L"
+by (simp add: isLub_def A_def r_def)
+
+(* glb *)
+lemma (in CL) glb_in_lattice: "S <= A ==> glb S cl \<in> A"
+apply (subst glb_dual_lub)
+apply (simp add: A_def)
+apply (rule dualA_iff [THEN subst])
+apply (rule Tarski.lub_in_lattice)
+apply (rule dualPO)
+apply (rule CL_dualCL)
+apply (simp add: dualA_iff)
+done
+
+lemma (in CL) glb_lower: "[|S <= A; x \<in> S|] ==> (glb S cl, x) \<in> r"
+apply (subst glb_dual_lub)
+apply (simp add: r_def)
+apply (rule dualr_iff [THEN subst])
+apply (rule Tarski.lub_upper [rule_format])
+apply (rule dualPO)
+apply (rule CL_dualCL)
+apply (simp add: dualA_iff A_def, assumption)
+done
+
+(* Reduce the sublattice property by using substructural properties*)
+(* abandoned see Tarski_4.ML *)
+
+lemma (in CLF) [simp]:
+ "f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)"
+apply (insert f_cl)
+apply (simp add: CLF_def)
+done
+
+declare (in CLF) f_cl [simp]
+
+
+lemma (in CLF) f_in_funcset: "f \<in> A funcset A"
+by (simp add: A_def)
+
+lemma (in CLF) monotone_f: "monotone f A r"
+by (simp add: A_def r_def)
+
+lemma (in CLF) CLF_dual: "(cl,f) \<in> CLF ==> (dual cl, f) \<in> CLF"
+apply (simp add: CLF_def CL_dualCL monotone_dual)
+apply (simp add: dualA_iff)
+done
+
+(* fixed points *)
+lemma fix_subset: "fix f A <= A"
+by (simp add: fix_def, fast)
+
+lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
+by (simp add: fix_def)
+
+lemma fixf_subset:
+ "[| A <= B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
+apply (simp add: fix_def, auto)
+done
+
+(* lemmas for Tarski, lub *)
+lemma (in CLF) lubH_le_flubH:
+ "H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
+apply (rule lub_least, fast)
+apply (rule f_in_funcset [THEN funcset_mem])
+apply (rule lub_in_lattice, fast)
+(* \<forall>x:H. (x, f (lub H r)) \<in> r *)
+apply (rule ballI)
+apply (rule transE)
+apply (rule CO_trans)
+(* instantiates (x, ???z) \<in> cl.<r> to (x, f x), because of the def of H *)
+apply fast
+(* so it remains to show (f x, f (lub H cl)) \<in> r *)
+apply (rule_tac f = "f" in monotoneE)
+apply (rule monotone_f, fast)
+apply (rule lub_in_lattice, fast)
+apply (rule lub_upper, fast)
+apply assumption
+done
+
+lemma (in CLF) flubH_le_lubH:
+ "[| H = {x. (x, f x) \<in> r & x \<in> A} |] ==> (f (lub H cl), lub H cl) \<in> r"
+apply (rule lub_upper, fast)
+apply (rule_tac t = "H" in ssubst, assumption)
+apply (rule CollectI)
+apply (rule conjI)
+apply (rule_tac [2] f_in_funcset [THEN funcset_mem])
+apply (rule_tac [2] lub_in_lattice)
+prefer 2 apply fast
+apply (rule_tac f = "f" in monotoneE)
+apply (rule monotone_f)
+ apply (blast intro: lub_in_lattice)
+ apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])
+apply (simp add: lubH_le_flubH)
+done
+
+lemma (in CLF) lubH_is_fixp:
+ "H = {x. (x, f x) \<in> r & x \<in> A} ==> lub H cl \<in> fix f A"
+apply (simp add: fix_def)
+apply (rule conjI)
+apply (rule lub_in_lattice, fast)
+apply (rule antisymE)
+apply (rule CO_antisym)
+apply (simp add: flubH_le_lubH)
+apply (simp add: lubH_le_flubH)
+done
+
+lemma (in CLF) fix_in_H:
+ "[| H = {x. (x, f x) \<in> r & x \<in> A}; x \<in> P |] ==> x \<in> H"
+by (simp add: P_def fix_imp_eq [of _ f A] reflE CO_refl
+ fix_subset [of f A, THEN subsetD])
+
+lemma (in CLF) fixf_le_lubH:
+ "H = {x. (x, f x) \<in> r & x \<in> A} ==> \<forall>x \<in> fix f A. (x, lub H cl) \<in> r"
+apply (rule ballI)
+apply (rule lub_upper, fast)
+apply (rule fix_in_H)
+apply (simp_all add: P_def)
+done
+
+lemma (in CLF) lubH_least_fixf:
+ "H = {x. (x, f x) \<in> r & x \<in> A}
+ ==> \<forall>L. (\<forall>y \<in> fix f A. (y,L) \<in> r) --> (lub H cl, L) \<in> r"
+apply (rule allI)
+apply (rule impI)
+apply (erule bspec)
+apply (rule lubH_is_fixp, assumption)
+done
+
+(* Tarski fixpoint theorem 1, first part *)
+lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
+apply (rule sym)
+apply (simp add: P_def)
+apply (rule lubI)
+apply (rule fix_subset)
+apply (rule lub_in_lattice, fast)
+apply (simp add: fixf_le_lubH)
+apply (simp add: lubH_least_fixf)
+done
+
+(* Tarski for glb *)
+lemma (in CLF) glbH_is_fixp: "H = {x. (f x, x) \<in> r & x \<in> A} ==> glb H cl \<in> P"
+apply (simp add: glb_dual_lub P_def A_def r_def)
+apply (rule dualA_iff [THEN subst])
+apply (rule Tarski.lubH_is_fixp)
+apply (rule dualPO)
+apply (rule CL_dualCL)
+apply (rule f_cl [THEN CLF_dual])
+apply (simp add: dualr_iff dualA_iff)
+done
+
+lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl"
+apply (simp add: glb_dual_lub P_def A_def r_def)
+apply (rule dualA_iff [THEN subst])
+apply (simp add: Tarski.T_thm_1_lub [of _ f, OF dualPO CL_dualCL]
+ dualPO CL_dualCL CLF_dual dualr_iff)
+done
+
+(* interval *)
+lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
+apply (insert CO_refl)
+apply (simp add: refl_def, blast)
+done
+
+lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b <= A"
+apply (simp add: interval_def)
+apply (blast intro: rel_imp_elem)
+done
+
+lemma (in CLF) intervalI:
+ "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
+apply (simp add: interval_def)
+done
+
+lemma (in CLF) interval_lemma1:
+ "[| S <= interval r a b; x \<in> S |] ==> (a, x) \<in> r"
+apply (unfold interval_def, fast)
+done
+
+lemma (in CLF) interval_lemma2:
+ "[| S <= interval r a b; x \<in> S |] ==> (x, b) \<in> r"
+apply (unfold interval_def, fast)
+done
+
+lemma (in CLF) a_less_lub:
+ "[| S <= A; S \<noteq> {};
+ \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
+by (blast intro: transE PO_imp_trans)
+
+lemma (in CLF) glb_less_b:
+ "[| S <= A; S \<noteq> {};
+ \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
+by (blast intro: transE PO_imp_trans)
+
+lemma (in CLF) S_intv_cl:
+ "[| a \<in> A; b \<in> A; S <= interval r a b |]==> S <= A"
+by (simp add: subset_trans [OF _ interval_subset])
+
+lemma (in CLF) L_in_interval:
+ "[| a \<in> A; b \<in> A; S <= interval r a b;
+ S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
+apply (rule intervalI)
+apply (rule a_less_lub)
+prefer 2 apply assumption
+apply (simp add: S_intv_cl)
+apply (rule ballI)
+apply (simp add: interval_lemma1)
+apply (simp add: isLub_upper)
+(* (L, b) \<in> r *)
+apply (simp add: isLub_least interval_lemma2)
+done
+
+lemma (in CLF) G_in_interval:
+ "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S <= interval r a b; isGlb S cl G;
+ S \<noteq> {} |] ==> G \<in> interval r a b"
+apply (simp add: interval_dual)
+apply (simp add: Tarski.L_in_interval [of _ f]
+ dualA_iff A_def dualPO CL_dualCL CLF_dual isGlb_dual_isLub)
+done
+
+lemma (in CLF) intervalPO:
+ "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
+ ==> (| pset = interval r a b, order = induced (interval r a b) r |)
+ \<in> PartialOrder"
+apply (rule po_subset_po)
+apply (simp add: interval_subset)
+done
+
+lemma (in CLF) intv_CL_lub:
+ "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
+ ==> \<forall>S. S <= interval r a b -->
+ (\<exists>L. isLub S (| pset = interval r a b,
+ order = induced (interval r a b) r |) L)"
+apply (intro strip)
+apply (frule S_intv_cl [THEN CL_imp_ex_isLub])
+prefer 2 apply assumption
+apply assumption
+apply (erule exE)
+(* define the lub for the interval as *)
+apply (rule_tac x = "if S = {} then a else L" in exI)
+apply (simp (no_asm_simp) add: isLub_def split del: split_if)
+apply (intro impI conjI)
+(* (if S = {} then a else L) \<in> interval r a b *)
+apply (simp add: CL_imp_PO L_in_interval)
+apply (simp add: left_in_interval)
+(* lub prop 1 *)
+apply (case_tac "S = {}")
+(* S = {}, y \<in> S = False => everything *)
+apply fast
+(* S \<noteq> {} *)
+apply simp
+(* \<forall>y:S. (y, L) \<in> induced (interval r a b) r *)
+apply (rule ballI)
+apply (simp add: induced_def L_in_interval)
+apply (rule conjI)
+apply (rule subsetD)
+apply (simp add: S_intv_cl, assumption)
+apply (simp add: isLub_upper)
+(* \<forall>z:interval r a b. (\<forall>y:S. (y, z) \<in> induced (interval r a b) r -->
+ (if S = {} then a else L, z) \<in> induced (interval r a b) r *)
+apply (rule ballI)
+apply (rule impI)
+apply (case_tac "S = {}")
+(* S = {} *)
+apply simp
+apply (simp add: induced_def interval_def)
+apply (rule conjI)
+apply (rule reflE)
+apply (rule CO_refl, assumption)
+apply (rule interval_not_empty)
+apply (rule CO_trans)
+apply (simp add: interval_def)
+(* S \<noteq> {} *)
+apply simp
+apply (simp add: induced_def L_in_interval)
+apply (rule isLub_least, assumption)
+apply (rule subsetD)
+prefer 2 apply assumption
+apply (simp add: S_intv_cl, fast)
+done
+
+lemmas (in CLF) intv_CL_glb = intv_CL_lub [THEN Rdual]
+
+lemma (in CLF) interval_is_sublattice:
+ "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
+ ==> interval r a b <<= cl"
+apply (rule sublatticeI)
+apply (simp add: interval_subset)
+apply (rule CompleteLatticeI)
+apply (simp add: intervalPO)
+ apply (simp add: intv_CL_lub)
+apply (simp add: intv_CL_glb)
+done
+
+lemmas (in CLF) interv_is_compl_latt =
+ interval_is_sublattice [THEN sublattice_imp_CL]
+
+(* Top and Bottom *)
+lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
+by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
+
+lemma (in CLF) Bot_dual_Top: "Bot cl = Top (dual cl)"
+by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
+
+lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
+apply (simp add: Bot_def least_def)
+apply (rule someI2)
+apply (fold A_def)
+apply (erule_tac [2] conjunct1)
+apply (rule conjI)
+apply (rule glb_in_lattice)
+apply (rule subset_refl)
+apply (fold r_def)
+apply (simp add: glb_lower)
+done
+
+lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
+apply (simp add: Top_dual_Bot A_def)
+apply (rule dualA_iff [THEN subst])
+apply (blast intro!: Tarski.Bot_in_lattice dualPO CL_dualCL CLF_dual f_cl)
+done
+
+lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
+apply (simp add: Top_def greatest_def)
+apply (rule someI2)
+apply (fold r_def A_def)
+prefer 2 apply fast
+apply (intro conjI ballI)
+apply (rule_tac [2] lub_upper)
+apply (auto simp add: lub_in_lattice)
+done
+
+lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
+apply (simp add: Bot_dual_Top r_def)
+apply (rule dualr_iff [THEN subst])
+apply (simp add: Tarski.Top_prop [of _ f]
+ dualA_iff A_def dualPO CL_dualCL CLF_dual)
+done
+
+lemma (in CLF) Top_intv_not_empty: "x \<in> A ==> interval r x (Top cl) \<noteq> {}"
+apply (rule notI)
+apply (drule_tac a = "Top cl" in equals0D)
+apply (simp add: interval_def)
+apply (simp add: refl_def Top_in_lattice Top_prop)
+done
+
+lemma (in CLF) Bot_intv_not_empty: "x \<in> A ==> interval r (Bot cl) x \<noteq> {}"
+apply (simp add: Bot_dual_Top)
+apply (subst interval_dual)
+prefer 2 apply assumption
+apply (simp add: A_def)
+apply (rule dualA_iff [THEN subst])
+apply (blast intro!: Tarski.Top_in_lattice
+ f_cl dualPO CL_dualCL CLF_dual)
+apply (simp add: Tarski.Top_intv_not_empty [of _ f]
+ dualA_iff A_def dualPO CL_dualCL CLF_dual)
+done
+
+(* fixed points form a partial order *)
+lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
+by (simp add: P_def fix_subset po_subset_po)
+
+lemma (in Tarski) Y_subset_A: "Y <= A"
+apply (rule subset_trans [OF _ fix_subset])
+apply (rule Y_ss [simplified P_def])
+done
+
+lemma (in Tarski) lubY_in_A: "lub Y cl \<in> A"
+by (simp add: Y_subset_A [THEN lub_in_lattice])
+
+lemma (in Tarski) lubY_le_flubY: "(lub Y cl, f (lub Y cl)) \<in> r"
+apply (rule lub_least)
+apply (rule Y_subset_A)
+apply (rule f_in_funcset [THEN funcset_mem])
+apply (rule lubY_in_A)
+(* Y <= P ==> f x = x *)
+apply (rule ballI)
+apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
+apply (erule Y_ss [simplified P_def, THEN subsetD])
+(* reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r by monotonicity *)
+apply (rule_tac f = "f" in monotoneE)
+apply (rule monotone_f)
+apply (simp add: Y_subset_A [THEN subsetD])
+apply (rule lubY_in_A)
+apply (simp add: lub_upper Y_subset_A)
+done
+
+lemma (in Tarski) intY1_subset: "intY1 <= A"
+apply (unfold intY1_def)
+apply (rule interval_subset)
+apply (rule lubY_in_A)
+apply (rule Top_in_lattice)
+done
+
+lemmas (in Tarski) intY1_elem = intY1_subset [THEN subsetD]
+
+lemma (in Tarski) intY1_f_closed: "x \<in> intY1 \<Longrightarrow> f x \<in> intY1"
+apply (simp add: intY1_def interval_def)
+apply (rule conjI)
+apply (rule transE)
+apply (rule CO_trans)
+apply (rule lubY_le_flubY)
+(* (f (lub Y cl), f x) \<in> r *)
+apply (rule_tac f=f in monotoneE)
+apply (rule monotone_f)
+apply (rule lubY_in_A)
+apply (simp add: intY1_def interval_def intY1_elem)
+apply (simp add: intY1_def interval_def)
+(* (f x, Top cl) \<in> r *)
+apply (rule Top_prop)
+apply (rule f_in_funcset [THEN funcset_mem])
+apply (simp add: intY1_def interval_def intY1_elem)
+done
+
+lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 funcset intY1"
+apply (rule restrictI)
+apply (erule intY1_f_closed)
+done
+
+lemma (in Tarski) intY1_mono:
+ "monotone (%x: intY1. f x) intY1 (induced intY1 r)"
+apply (auto simp add: monotone_def induced_def intY1_f_closed)
+apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
+done
+
+lemma (in Tarski) intY1_is_cl:
+ "(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
+apply (unfold intY1_def)
+apply (rule interv_is_compl_latt)
+apply (rule lubY_in_A)
+apply (rule Top_in_lattice)
+apply (rule Top_intv_not_empty)
+apply (rule lubY_in_A)
+done
+
+lemma (in Tarski) v_in_P: "v \<in> P"
+apply (unfold P_def)
+apply (rule_tac A = "intY1" in fixf_subset)
+apply (rule intY1_subset)
+apply (simp add: Tarski.glbH_is_fixp [OF _ intY1_is_cl, simplified]
+ v_def CL_imp_PO intY1_is_cl CLF_def intY1_func intY1_mono)
+done
+
+lemma (in Tarski) z_in_interval:
+ "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |] ==> z \<in> intY1"
+apply (unfold intY1_def P_def)
+apply (rule intervalI)
+prefer 2
+ apply (erule fix_subset [THEN subsetD, THEN Top_prop])
+apply (rule lub_least)
+apply (rule Y_subset_A)
+apply (fast elim!: fix_subset [THEN subsetD])
+apply (simp add: induced_def)
+done
+
+lemma (in Tarski) f'z_in_int_rel: "[| z \<in> P; \<forall>y\<in>Y. (y, z) \<in> induced P r |]
+ ==> ((%x: intY1. f x) z, z) \<in> induced intY1 r"
+apply (simp add: induced_def intY1_f_closed z_in_interval P_def)
+apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD]
+ CO_refl [THEN reflE])
+done
+
+lemma (in Tarski) tarski_full_lemma:
+ "\<exists>L. isLub Y (| pset = P, order = induced P r |) L"
+apply (rule_tac x = "v" in exI)
+apply (simp add: isLub_def)
+(* v \<in> P *)
+apply (simp add: v_in_P)
+apply (rule conjI)
+(* v is lub *)
+(* 1. \<forall>y:Y. (y, v) \<in> induced P r *)
+apply (rule ballI)
+apply (simp add: induced_def subsetD v_in_P)
+apply (rule conjI)
+apply (erule Y_ss [THEN subsetD])
+apply (rule_tac b = "lub Y cl" in transE)
+apply (rule CO_trans)
+apply (rule lub_upper)
+apply (rule Y_subset_A, assumption)
+apply (rule_tac b = "Top cl" in interval_imp_mem)
+apply (simp add: v_def)
+apply (fold intY1_def)
+apply (rule Tarski.glb_in_lattice [OF _ intY1_is_cl, simplified])
+ apply (simp add: CL_imp_PO intY1_is_cl, force)
+(* v is LEAST ub *)
+apply clarify
+apply (rule indI)
+ prefer 3 apply assumption
+ prefer 2 apply (simp add: v_in_P)
+apply (unfold v_def)
+apply (rule indE)
+apply (rule_tac [2] intY1_subset)
+apply (rule Tarski.glb_lower [OF _ intY1_is_cl, simplified])
+ apply (simp add: CL_imp_PO intY1_is_cl)
+ apply force
+apply (simp add: induced_def intY1_f_closed z_in_interval)
+apply (simp add: P_def fix_imp_eq [of _ f A]
+ fix_subset [of f A, THEN subsetD]
+ CO_refl [THEN reflE])
+done
+
+
+lemma CompleteLatticeI_simp:
+ "[| (| pset = A, order = r |) \<in> PartialOrder;
+ \<forall>S. S <= A --> (\<exists>L. isLub S (| pset = A, order = r |) L) |]
+ ==> (| pset = A, order = r |) \<in> CompleteLattice"
+by (simp add: CompleteLatticeI Rdual)
+
+theorem (in CLF) Tarski_full:
+ "(| pset = P, order = induced P r|) \<in> CompleteLattice"
+apply (rule CompleteLatticeI_simp)
+apply (rule fixf_po, clarify)
+apply (simp add: P_def A_def r_def)
+apply (blast intro!: Tarski.tarski_full_lemma cl_po cl_co f_cl)
+done
end