--- a/src/HOL/ex/Tarski.thy Tue Jul 16 18:46:59 2002 +0200
+++ b/src/HOL/ex/Tarski.thy Tue Jul 16 18:52:26 2002 +0200
@@ -1,25 +1,29 @@
-(* Title: HOL/ex/Tarski
+(* Title: HOL/ex/Tarski.thy
ID: $Id$
- Author: Florian Kammueller, Cambridge University Computer Laboratory
+ Author: Florian Kammüller, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
+*)
-Minimal version of lattice theory plus the full theorem of Tarski:
- The fixedpoints of a complete lattice themselves form a complete lattice.
-
-Illustrates first-class theories, using the Sigma representation of structures
-
-Tidied and converted to Isar by lcp
-*)
+header {* The full theorem of Tarski *}
theory Tarski = Main:
-record 'a potype =
+text {*
+ Minimal version of lattice theory plus the full theorem of Tarski:
+ The fixedpoints of a complete lattice themselves form a complete
+ lattice.
+
+ Illustrates first-class theories, using the Sigma representation of
+ structures. Tidied and converted to Isar by lcp.
+*}
+
+record 'a potype =
pset :: "'a set"
order :: "('a * 'a) set"
syntax
"@pset" :: "'a potype => 'a set" ("_ .<A>" [90] 90)
- "@order" :: "'a potype => ('a *'a)set" ("_ .<r>" [90] 90)
+ "@order" :: "'a potype => ('a *'a)set" ("_ .<r>" [90] 90)
translations
"po.<A>" == "pset po"
@@ -27,69 +31,67 @@
constdefs
monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
- "monotone f A r == \<forall>x\<in>A. \<forall>y\<in>A. (x, y): r --> ((f x), (f y)) : r"
+ "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 &
+ "least P po == @ x. x: po.<A> & P x &
(\<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 &
+ "greatest P po == @ x. x: po.<A> & P x &
(\<forall>y \<in> po.<A>. P y --> (y,x): po.<r>)"
lub :: "['a set, 'a potype] => 'a"
- "lub S po == least (%x. \<forall>y\<in>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. \<forall>y\<in>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> & (\<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>))"
+ "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> & (\<forall>y\<in>S. (G,y): po.<r>) &
+ "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 f A == {x. x: A & f x = x}"
+ "fix f A == {x. x: A & f x = x}"
interval :: "[('a*'a) set,'a, 'a ] => 'a set"
- "interval r a b == {x. (a,x): r & (x,b): r}"
+ "interval r a b == {x. (a,x): r & (x,b): r}"
constdefs
Bot :: "'a potype => 'a"
- "Bot po == least (%x. True) po"
+ "Bot po == least (%x. True) po"
Top :: "'a potype => 'a"
- "Top po == greatest (%x. True) po"
+ "Top po == greatest (%x. True) po"
PartialOrder :: "('a potype) set"
- "PartialOrder == {P. refl (P.<A>) (P.<r>) & antisym (P.<r>) &
- trans (P.<r>)}"
+ "PartialOrder == {P. refl (P.<A>) (P.<r>) & antisym (P.<r>) &
+ trans (P.<r>)}"
CompleteLattice :: "('a potype) set"
- "CompleteLattice == {cl. cl: PartialOrder &
- (\<forall>S. S <= cl.<A> --> (\<exists>L. isLub S cl L)) &
- (\<forall>S. S <= cl.<A> --> (\<exists>G. isGlb S cl G))}"
+ "CompleteLattice == {cl. cl: PartialOrder &
+ (\<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.
- {f. f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)}"
-
+ "CLF == SIGMA cl: CompleteLattice.
+ {f. f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)}"
+
induced :: "['a set, ('a * 'a) set] => ('a *'a)set"
- "induced A r == {(a,b). a : A & b: A & (a,b): r}"
-
-
+ "induced A r == {(a,b). a : A & b: A & (a,b): r}"
constdefs
sublattice :: "('a potype * 'a set)set"
- "sublattice ==
+ "sublattice ==
SIGMA cl: CompleteLattice.
{S. S <= cl.<A> &
- (| pset = S, order = induced S (cl.<r>) |): CompleteLattice }"
+ (| pset = S, order = induced S (cl.<r>) |): CompleteLattice }"
syntax
"@SL" :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
@@ -99,9 +101,9 @@
constdefs
dual :: "'a potype => 'a potype"
- "dual po == (| pset = po.<A>, order = converse (po.<r>) |)"
+ "dual po == (| pset = po.<A>, order = converse (po.<r>) |)"
-locale PO =
+locale (open) PO =
fixes cl :: "'a potype"
and A :: "'a set"
and r :: "('a * 'a) set"
@@ -109,17 +111,17 @@
defines A_def: "A == cl.<A>"
and r_def: "r == cl.<r>"
-locale CL = PO +
+locale (open) CL = PO +
assumes cl_co: "cl : CompleteLattice"
-locale CLF = CL +
+locale (open) CLF = CL +
fixes f :: "'a => 'a"
and P :: "'a set"
assumes f_cl: "(cl,f) : CLF" (*was the equivalent "f : CLF``{cl}"*)
defines P_def: "P == fix f A"
-locale Tarski = CLF +
+locale (open) Tarski = CLF +
fixes Y :: "'a set"
and intY1 :: "'a set"
and v :: "'a"
@@ -127,41 +129,40 @@
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 &
+ and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r &
x: intY1}
- (| pset=intY1, order=induced intY1 r|)"
+ (| pset=intY1, order=induced intY1 r|)"
-
-(* Partial Order *)
+subsubsection {* Partial Order *}
lemma (in PO) PO_imp_refl: "refl A r"
-apply (insert cl_po)
+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 (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 (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 (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 (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 (insert cl_po)
apply (simp add: PartialOrder_def)
apply (unfold trans_def, fast)
done
@@ -174,13 +175,13 @@
"S <= A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
apply (simp (no_asm) add: PartialOrder_def)
apply auto
-(* refl *)
+-- {* refl *}
apply (simp add: refl_def induced_def)
apply (blast intro: PO_imp_refl [THEN reflE])
-(* antisym *)
+-- {* antisym *}
apply (simp add: antisym_def induced_def)
apply (blast intro: PO_imp_sym [THEN antisymE])
-(* trans *)
+-- {* trans *}
apply (simp add: trans_def induced_def)
apply (blast intro: PO_imp_trans [THEN transE])
done
@@ -191,8 +192,8 @@
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)
+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
@@ -211,13 +212,13 @@
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
+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>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)}
@@ -253,17 +254,17 @@
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> 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)
+apply (unfold CompleteLattice_def, blast)
done
lemma (in CL) CL_dualCL: "dual cl \<in> CompleteLattice"
-apply (insert cl_co)
+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]
+apply (fold dual_def)
+apply (simp add: isLub_dual_isGlb [symmetric] isGlb_dual_isLub [symmetric]
dualPO)
done
@@ -307,17 +308,21 @@
apply (simp add: PO_imp_refl [THEN reflE])
done
-(* sublattice *)
+
+subsubsection {* sublattice *}
+
lemma (in PO) sublattice_imp_CL:
"S <<= cl ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
by (simp add: sublattice_def CompleteLattice_def A_def r_def)
lemma (in CL) sublatticeI:
- "[| S <= A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
+ "[| S <= A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
==> S <<= cl"
by (simp add: sublattice_def A_def r_def)
-(* lub *)
+
+subsubsection {* lub *}
+
lemma (in CL) lub_unique: "[| S <= A; isLub S cl x; isLub S cl L|] ==> x = L"
apply (rule antisymE)
apply (rule CO_antisym)
@@ -329,7 +334,7 @@
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: lub_unique A_def isLub_def)
apply (simp add: isLub_def r_def)
done
@@ -339,7 +344,7 @@
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: lub_unique A_def isLub_def)
apply (simp add: isLub_def r_def A_def)
done
@@ -349,11 +354,11 @@
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)
+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;
+ "[| 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)
@@ -378,17 +383,19 @@
by (simp add: isLub_def A_def r_def)
lemma (in CL) isLubI:
- "[| L \<in> A; \<forall>y \<in> S. (y, L) \<in> r;
+ "[| 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 *)
+
+subsubsection {* glb *}
+
lemma (in CL) glb_in_lattice: "S <= A ==> glb S cl \<in> A"
apply (subst glb_dual_lub)
apply (simp add: A_def)
apply (rule dualA_iff [THEN subst])
apply (rule Tarski.lub_in_lattice)
-apply (rule dualPO)
+apply (rule dualPO)
apply (rule CL_dualCL)
apply (simp add: dualA_iff)
done
@@ -398,18 +405,20 @@
apply (simp add: r_def)
apply (rule dualr_iff [THEN subst])
apply (rule Tarski.lub_upper [rule_format])
-apply (rule dualPO)
+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 *)
+text {*
+ Reduce the sublattice property by using substructural properties;
+ abandoned see @{text "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)
+apply (insert f_cl)
+apply (simp add: CLF_def)
done
declare (in CLF) f_cl [simp]
@@ -426,7 +435,9 @@
apply (simp add: dualA_iff)
done
-(* fixed points *)
+
+subsubsection {* fixed points *}
+
lemma fix_subset: "fix f A <= A"
by (simp add: fix_def, fast)
@@ -435,22 +446,24 @@
lemma fixf_subset:
"[| A <= B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
-apply (simp add: fix_def, auto)
+apply (simp add: fix_def, auto)
done
-(* lemmas for Tarski, lub *)
+
+subsubsection {* lemmas for Tarski, lub *}
lemma (in CLF) lubH_le_flubH:
"H = {x. (x, f x) \<in> r & x \<in> A} ==> (lub H cl, f (lub H cl)) \<in> r"
apply (rule lub_least, fast)
apply (rule f_in_funcset [THEN funcset_mem])
apply (rule lub_in_lattice, fast)
-(* \<forall>x:H. (x, f (lub H r)) \<in> r *)
+-- {* @{text "\<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 *)
+-- {* instantiates @{text "(x, ???z) \<in> cl.<r> to (x, f x)"}, *}
+-- {* because of the def of @{text H} *}
apply fast
-(* so it remains to show (f x, f (lub H cl)) \<in> r *)
+-- {* so it remains to show @{text "(f x, f (lub H cl)) \<in> r"} *}
apply (rule_tac f = "f" in monotoneE)
apply (rule monotone_f, fast)
apply (rule lub_in_lattice, fast)
@@ -469,8 +482,8 @@
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 (blast intro: lub_in_lattice)
+ apply (blast intro: lub_in_lattice f_in_funcset [THEN funcset_mem])
apply (simp add: lubH_le_flubH)
done
@@ -487,19 +500,19 @@
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])
+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)
+apply (simp_all add: P_def)
done
lemma (in CLF) lubH_least_fixf:
- "H = {x. (x, f x) \<in> r & x \<in> A}
+ "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)
@@ -507,10 +520,10 @@
apply (rule lubH_is_fixp, assumption)
done
-(* Tarski fixpoint theorem 1, first part *)
+subsubsection {* Tarski fixpoint theorem 1, first part *}
lemma (in CLF) T_thm_1_lub: "lub P cl = lub {x. (x, f x) \<in> r & x \<in> A} cl"
apply (rule sym)
-apply (simp add: P_def)
+apply (simp add: P_def)
apply (rule lubI)
apply (rule fix_subset)
apply (rule lub_in_lattice, fast)
@@ -518,12 +531,12 @@
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"
+ -- {* Tarski for glb *}
apply (simp add: glb_dual_lub P_def A_def r_def)
apply (rule dualA_iff [THEN subst])
apply (rule Tarski.lubH_is_fixp)
-apply (rule dualPO)
+apply (rule dualPO)
apply (rule CL_dualCL)
apply (rule f_cl [THEN CLF_dual])
apply (simp add: dualr_iff dualA_iff)
@@ -532,14 +545,15 @@
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]
+apply (simp add: Tarski.T_thm_1_lub [of _ f, OF dualPO CL_dualCL]
dualPO CL_dualCL CLF_dual dualr_iff)
done
-(* interval *)
+subsubsection {* interval *}
+
lemma (in CLF) rel_imp_elem: "(x, y) \<in> r ==> x \<in> A"
-apply (insert CO_refl)
-apply (simp add: refl_def, blast)
+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"
@@ -563,12 +577,12 @@
done
lemma (in CLF) a_less_lub:
- "[| S <= A; S \<noteq> {};
+ "[| 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> {};
+ "[| 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)
@@ -577,7 +591,7 @@
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;
+ "[| 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)
@@ -586,7 +600,7 @@
apply (rule ballI)
apply (simp add: interval_lemma1)
apply (simp add: isLub_upper)
-(* (L, b) \<in> r *)
+-- {* @{text "(L, b) \<in> r"} *}
apply (simp add: isLub_least interval_lemma2)
done
@@ -594,12 +608,12 @@
"[| 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]
+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> {} |]
+ "[| 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)
@@ -607,41 +621,40 @@
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,
+ "[| 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 *)
+-- {* 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 (intro impI conjI)
+-- {* @{text "(if S = {} then a else L) \<in> interval r a b"} *}
apply (simp add: CL_imp_PO L_in_interval)
apply (simp add: left_in_interval)
-(* lub prop 1 *)
+-- {* lub prop 1 *}
apply (case_tac "S = {}")
-(* S = {}, y \<in> S = False => everything *)
+-- {* @{text "S = {}, y \<in> S = False => everything"} *}
apply fast
-(* S \<noteq> {} *)
+-- {* @{text "S \<noteq> {}"} *}
apply simp
-(* \<forall>y:S. (y, L) \<in> induced (interval r a b) r *)
+-- {* @{text "\<forall>y:S. (y, L) \<in> induced (interval r a b) r"} *}
apply (rule ballI)
apply (simp add: induced_def L_in_interval)
apply (rule conjI)
apply (rule subsetD)
apply (simp add: S_intv_cl, assumption)
apply (simp add: isLub_upper)
-(* \<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 *)
+-- {* @{text "\<forall>z:interval r a b. (\<forall>y:S. (y, z) \<in> induced (interval r a b) r \<longrightarrow> (if S = {} then a else L, z) \<in> induced (interval r a b) r"} *}
apply (rule ballI)
apply (rule impI)
apply (case_tac "S = {}")
-(* S = {} *)
+-- {* @{text "S = {}"} *}
apply simp
apply (simp add: induced_def interval_def)
apply (rule conjI)
@@ -650,7 +663,7 @@
apply (rule interval_not_empty)
apply (rule CO_trans)
apply (simp add: interval_def)
-(* S \<noteq> {} *)
+-- {* @{text "S \<noteq> {}"} *}
apply simp
apply (simp add: induced_def L_in_interval)
apply (rule isLub_least, assumption)
@@ -662,7 +675,7 @@
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> {} |]
+ "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
==> interval r a b <<= cl"
apply (rule sublatticeI)
apply (simp add: interval_subset)
@@ -672,10 +685,11 @@
apply (simp add: intv_CL_glb)
done
-lemmas (in CLF) interv_is_compl_latt =
+lemmas (in CLF) interv_is_compl_latt =
interval_is_sublattice [THEN sublattice_imp_CL]
-(* Top and Bottom *)
+
+subsubsection {* Top and Bottom *}
lemma (in CLF) Top_dual_Bot: "Top cl = Bot (dual cl)"
by (simp add: Top_def Bot_def least_def greatest_def dualA_iff dualr_iff)
@@ -696,8 +710,8 @@
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)
+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"
@@ -713,7 +727,7 @@
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]
+apply (simp add: Tarski.Top_prop [of _ f]
dualA_iff A_def dualPO CL_dualCL CLF_dual)
done
@@ -732,11 +746,12 @@
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]
+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 *)
+subsubsection {* fixed points form a partial order *}
+
lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
by (simp add: P_def fix_subset po_subset_po)
@@ -753,11 +768,11 @@
apply (rule Y_subset_A)
apply (rule f_in_funcset [THEN funcset_mem])
apply (rule lubY_in_A)
-(* Y <= P ==> f x = x *)
+-- {* @{text "Y <= P ==> f x = x"} *}
apply (rule ballI)
apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
apply (erule Y_ss [simplified P_def, THEN subsetD])
-(* reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r by monotonicity *)
+-- {* @{text "reduce (f x, f (lub Y cl)) \<in> r to (x, lub Y cl) \<in> r"} by monotonicity *}
apply (rule_tac f = "f" in monotoneE)
apply (rule monotone_f)
apply (simp add: Y_subset_A [THEN subsetD])
@@ -780,13 +795,13 @@
apply (rule transE)
apply (rule CO_trans)
apply (rule lubY_le_flubY)
-(* (f (lub Y cl), f x) \<in> r *)
+-- {* @{text "(f (lub Y cl), f x) \<in> r"} *}
apply (rule_tac f=f in monotoneE)
apply (rule monotone_f)
apply (rule lubY_in_A)
apply (simp add: intY1_def interval_def intY1_elem)
apply (simp add: intY1_def interval_def)
-(* (f x, Top cl) \<in> r *)
+-- {* @{text "(f x, Top cl) \<in> r"} *}
apply (rule Top_prop)
apply (rule f_in_funcset [THEN funcset_mem])
apply (simp add: intY1_def interval_def intY1_elem)
@@ -803,7 +818,7 @@
apply (blast intro: intY1_elem monotone_f [THEN monotoneE])
done
-lemma (in Tarski) intY1_is_cl:
+lemma (in Tarski) intY1_is_cl:
"(| pset = intY1, order = induced intY1 r |) \<in> CompleteLattice"
apply (unfold intY1_def)
apply (rule interv_is_compl_latt)
@@ -821,11 +836,11 @@
v_def CL_imp_PO intY1_is_cl CLF_def intY1_func intY1_mono)
done
-lemma (in Tarski) z_in_interval:
+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
+prefer 2
apply (erule fix_subset [THEN subsetD, THEN Top_prop])
apply (rule lub_least)
apply (rule Y_subset_A)
@@ -833,10 +848,10 @@
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 |]
+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]
+apply (simp add: fix_imp_eq [of _ f A] fix_subset [of f A, THEN subsetD]
CO_refl [THEN reflE])
done
@@ -844,11 +859,11 @@
"\<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 *)
+-- {* @{text "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 *)
+-- {* @{text v} is lub *}
+-- {* @{text "1. \<forall>y:Y. (y, v) \<in> induced P r"} *}
apply (rule ballI)
apply (simp add: induced_def subsetD v_in_P)
apply (rule conjI)
@@ -862,7 +877,7 @@
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 *)
+-- {* @{text v} is LEAST ub *}
apply clarify
apply (rule indI)
prefer 3 apply assumption
@@ -871,18 +886,17 @@
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 (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]
+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> 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)
@@ -890,8 +904,8 @@
"(| 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)
+apply (simp add: P_def A_def r_def)
+apply (blast intro!: Tarski.tarski_full_lemma cl_po cl_co f_cl)
done
end