--- a/src/HOL/ex/Tarski.thy Wed Oct 12 03:02:18 2005 +0200
+++ b/src/HOL/ex/Tarski.thy Wed Oct 12 10:49:07 2005 +0200
@@ -66,8 +66,8 @@
CompleteLattice :: "('a potype) set"
"CompleteLattice == {cl. cl: PartialOrder &
- (\<forall>S. S <= pset cl --> (\<exists>L. isLub S cl L)) &
- (\<forall>S. S <= pset cl --> (\<exists>G. isGlb S cl G))}"
+ (\<forall>S. S \<subseteq> pset cl --> (\<exists>L. isLub S cl L)) &
+ (\<forall>S. S \<subseteq> pset cl --> (\<exists>G. isGlb S cl G))}"
CLF :: "('a potype * ('a => 'a)) set"
"CLF == SIGMA cl: CompleteLattice.
@@ -81,14 +81,14 @@
sublattice :: "('a potype * 'a set)set"
"sublattice ==
SIGMA cl: CompleteLattice.
- {S. S <= pset cl &
+ {S. S \<subseteq> pset cl &
(| pset = S, order = induced S (order cl) |): CompleteLattice }"
syntax
- "@SL" :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
+ "@SL" :: "['a set, 'a potype] => bool" ("_ <\<subseteq> _" [51,50]50)
translations
- "S <<= cl" == "S : sublattice `` {cl}"
+ "S <\<subseteq> cl" == "S : sublattice `` {cl}"
constdefs
dual :: "'a potype => 'a potype"
@@ -117,7 +117,7 @@
and intY1 :: "'a set"
and v :: "'a"
assumes
- Y_ss: "Y <= P"
+ Y_ss: "Y \<subseteq> 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 &
@@ -163,7 +163,7 @@
by (simp add: monotone_def)
lemma (in PO) po_subset_po:
- "S <= A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
+ "S \<subseteq> A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
apply (simp (no_asm) add: PartialOrder_def)
apply auto
-- {* refl *}
@@ -177,13 +177,13 @@
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"
+lemma (in PO) indE: "[| (x, y) \<in> induced S r; S \<subseteq> 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"
+lemma (in CL) CL_imp_ex_isLub: "S \<subseteq> A ==> \<exists>L. isLub S cl L"
apply (insert cl_co)
apply (simp add: CompleteLattice_def A_def)
done
@@ -209,8 +209,8 @@
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))"
+ "\<forall>S. (S \<subseteq> A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
+ ==> \<forall>S. (S \<subseteq> 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)
@@ -226,7 +226,7 @@
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"
+lemma CL_subset_PO: "CompleteLattice \<subseteq> PartialOrder"
by (simp add: PartialOrder_def CompleteLattice_def, fast)
lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
@@ -245,8 +245,8 @@
by (rule PO_imp_trans)
lemma CompleteLatticeI:
- "[| po \<in> PartialOrder; (\<forall>S. S <= pset po --> (\<exists>L. isLub S po L));
- (\<forall>S. S <= pset po --> (\<exists>G. isGlb S po G))|]
+ "[| po \<in> PartialOrder; (\<forall>S. S \<subseteq> pset po --> (\<exists>L. isLub S po L));
+ (\<forall>S. S \<subseteq> pset po --> (\<exists>G. isGlb S po G))|]
==> po \<in> CompleteLattice"
apply (unfold CompleteLattice_def, blast)
done
@@ -303,24 +303,24 @@
subsection {* sublattice *}
lemma (in PO) sublattice_imp_CL:
- "S <<= cl ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
+ "S <\<subseteq> 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"
+ "[| S \<subseteq> A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
+ ==> S <\<subseteq> cl"
by (simp add: sublattice_def A_def r_def)
subsection {* lub *}
-lemma (in CL) lub_unique: "[| S <= A; isLub S cl x; isLub S cl L|] ==> x = L"
+lemma (in CL) lub_unique: "[| S \<subseteq> 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"
+lemma (in CL) lub_upper: "[|S \<subseteq> 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])
@@ -330,7 +330,7 @@
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"
+ "[| S \<subseteq> 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])
@@ -339,7 +339,7 @@
apply (simp add: isLub_def r_def A_def)
done
-lemma (in CL) lub_in_lattice: "S <= A ==> lub S cl \<in> A"
+lemma (in CL) lub_in_lattice: "S \<subseteq> 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)
@@ -349,7 +349,7 @@
done
lemma (in CL) lubI:
- "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
+ "[| S \<subseteq> 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)
@@ -360,7 +360,7 @@
apply (simp add: lub_upper lub_least)
done
-lemma (in CL) lubIa: "[| S <= A; isLub S cl L |] ==> L = lub S cl"
+lemma (in CL) lubIa: "[| S \<subseteq> 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"
@@ -381,7 +381,7 @@
subsection {* glb *}
-lemma (in CL) glb_in_lattice: "S <= A ==> glb S cl \<in> A"
+lemma (in CL) glb_in_lattice: "S \<subseteq> A ==> glb S cl \<in> A"
apply (subst glb_dual_lub)
apply (simp add: A_def)
apply (rule dualA_iff [THEN subst])
@@ -391,11 +391,11 @@
apply (simp add: dualA_iff)
done
-lemma (in CL) glb_lower: "[|S <= A; x \<in> S|] ==> (glb S cl, x) \<in> r"
+lemma (in CL) glb_lower: "[|S \<subseteq> 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 Tarski.lub_upper)
apply (rule dualPO)
apply (rule CL_dualCL)
apply (simp add: dualA_iff A_def, assumption)
@@ -429,16 +429,15 @@
subsection {* fixed points *}
-lemma fix_subset: "fix f A <= A"
+lemma fix_subset: "fix f A \<subseteq> 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
+ "[| A \<subseteq> B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
+by (simp add: fix_def, auto)
subsection {* lemmas for Tarski, lub *}
@@ -547,42 +546,39 @@
apply (simp add: refl_def, blast)
done
-lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b <= A"
+lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b \<subseteq> 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
+by (simp add: interval_def)
lemma (in CLF) interval_lemma1:
- "[| S <= interval r a b; x \<in> S |] ==> (a, x) \<in> r"
-apply (unfold interval_def, fast)
-done
+ "[| S \<subseteq> interval r a b; x \<in> S |] ==> (a, x) \<in> r"
+by (unfold interval_def, fast)
lemma (in CLF) interval_lemma2:
- "[| S <= interval r a b; x \<in> S |] ==> (x, b) \<in> r"
-apply (unfold interval_def, fast)
-done
+ "[| S \<subseteq> interval r a b; x \<in> S |] ==> (x, b) \<in> r"
+by (unfold interval_def, fast)
lemma (in CLF) a_less_lub:
- "[| S <= A; S \<noteq> {};
+ "[| S \<subseteq> 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 \<subseteq> 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"
+ "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b |]==> S \<subseteq> 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;
+ "[| a \<in> A; b \<in> A; S \<subseteq> 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)
@@ -596,7 +592,7 @@
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;
+ "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> 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]
@@ -613,7 +609,7 @@
lemma (in CLF) intv_CL_lub:
"[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
- ==> \<forall>S. S <= interval r a b -->
+ ==> \<forall>S. S \<subseteq> interval r a b -->
(\<exists>L. isLub S (| pset = interval r a b,
order = induced (interval r a b) r |) L)"
apply (intro strip)
@@ -667,7 +663,7 @@
lemma (in CLF) interval_is_sublattice:
"[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
- ==> interval r a b <<= cl"
+ ==> interval r a b <\<subseteq> cl"
apply (rule sublatticeI)
apply (simp add: interval_subset)
apply (rule CompleteLatticeI)
@@ -689,14 +685,9 @@
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)
+apply (rule_tac a="glb A cl" in someI2)
+apply (simp_all add: glb_in_lattice glb_lower
+ r_def [symmetric] A_def [symmetric])
done
lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
@@ -707,12 +698,10 @@
lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
apply (simp add: Top_def greatest_def)
+apply (rule_tac a="lub A cl" in someI2)
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)
+apply (simp_all add: lub_in_lattice lub_upper
+ r_def [symmetric] A_def [symmetric])
done
lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
@@ -746,7 +735,7 @@
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"
+lemma (in Tarski) Y_subset_A: "Y \<subseteq> A"
apply (rule subset_trans [OF _ fix_subset])
apply (rule Y_ss [simplified P_def])
done
@@ -759,7 +748,7 @@
apply (rule Y_subset_A)
apply (rule f_in_funcset [THEN funcset_mem])
apply (rule lubY_in_A)
--- {* @{text "Y <= P ==> f x = x"} *}
+-- {* @{text "Y \<subseteq> 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])
@@ -771,7 +760,7 @@
apply (simp add: lub_upper Y_subset_A)
done
-lemma (in Tarski) intY1_subset: "intY1 <= A"
+lemma (in Tarski) intY1_subset: "intY1 \<subseteq> A"
apply (unfold intY1_def)
apply (rule interval_subset)
apply (rule lubY_in_A)
@@ -887,7 +876,7 @@
lemma CompleteLatticeI_simp:
"[| (| pset = A, order = r |) \<in> PartialOrder;
- \<forall>S. S <= A --> (\<exists>L. isLub S (| pset = A, order = r |) L) |]
+ \<forall>S. S \<subseteq> A --> (\<exists>L. isLub S (| pset = A, order = r |) L) |]
==> (| pset = A, order = r |) \<in> CompleteLattice"
by (simp add: CompleteLatticeI Rdual)