--- a/src/HOL/HOLCF/CompactBasis.thy Fri Dec 17 23:18:39 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,111 +0,0 @@
-(* Title: HOLCF/CompactBasis.thy
- Author: Brian Huffman
-*)
-
-header {* A compact basis for powerdomains *}
-
-theory CompactBasis
-imports Bifinite
-begin
-
-default_sort "domain"
-
-subsection {* A compact basis for powerdomains *}
-
-typedef 'a pd_basis =
- "{S::'a compact_basis set. finite S \<and> S \<noteq> {}}"
-by (rule_tac x="{arbitrary}" in exI, simp)
-
-lemma finite_Rep_pd_basis [simp]: "finite (Rep_pd_basis u)"
-by (insert Rep_pd_basis [of u, unfolded pd_basis_def]) simp
-
-lemma Rep_pd_basis_nonempty [simp]: "Rep_pd_basis u \<noteq> {}"
-by (insert Rep_pd_basis [of u, unfolded pd_basis_def]) simp
-
-text {* The powerdomain basis type is countable. *}
-
-lemma pd_basis_countable: "\<exists>f::'a pd_basis \<Rightarrow> nat. inj f"
-proof -
- obtain g :: "'a compact_basis \<Rightarrow> nat" where "inj g"
- using compact_basis.countable ..
- hence image_g_eq: "\<And>A B. g ` A = g ` B \<longleftrightarrow> A = B"
- by (rule inj_image_eq_iff)
- have "inj (\<lambda>t. set_encode (g ` Rep_pd_basis t))"
- by (simp add: inj_on_def set_encode_eq image_g_eq Rep_pd_basis_inject)
- thus ?thesis by - (rule exI)
- (* FIXME: why doesn't ".." or "by (rule exI)" work? *)
-qed
-
-subsection {* Unit and plus constructors *}
-
-definition
- PDUnit :: "'a compact_basis \<Rightarrow> 'a pd_basis" where
- "PDUnit = (\<lambda>x. Abs_pd_basis {x})"
-
-definition
- PDPlus :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> 'a pd_basis" where
- "PDPlus t u = Abs_pd_basis (Rep_pd_basis t \<union> Rep_pd_basis u)"
-
-lemma Rep_PDUnit:
- "Rep_pd_basis (PDUnit x) = {x}"
-unfolding PDUnit_def by (rule Abs_pd_basis_inverse) (simp add: pd_basis_def)
-
-lemma Rep_PDPlus:
- "Rep_pd_basis (PDPlus u v) = Rep_pd_basis u \<union> Rep_pd_basis v"
-unfolding PDPlus_def by (rule Abs_pd_basis_inverse) (simp add: pd_basis_def)
-
-lemma PDUnit_inject [simp]: "(PDUnit a = PDUnit b) = (a = b)"
-unfolding Rep_pd_basis_inject [symmetric] Rep_PDUnit by simp
-
-lemma PDPlus_assoc: "PDPlus (PDPlus t u) v = PDPlus t (PDPlus u v)"
-unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_assoc)
-
-lemma PDPlus_commute: "PDPlus t u = PDPlus u t"
-unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_commute)
-
-lemma PDPlus_absorb: "PDPlus t t = t"
-unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_absorb)
-
-lemma pd_basis_induct1:
- assumes PDUnit: "\<And>a. P (PDUnit a)"
- assumes PDPlus: "\<And>a t. P t \<Longrightarrow> P (PDPlus (PDUnit a) t)"
- shows "P x"
-apply (induct x, unfold pd_basis_def, clarify)
-apply (erule (1) finite_ne_induct)
-apply (cut_tac a=x in PDUnit)
-apply (simp add: PDUnit_def)
-apply (drule_tac a=x in PDPlus)
-apply (simp add: PDUnit_def PDPlus_def
- Abs_pd_basis_inverse [unfolded pd_basis_def])
-done
-
-lemma pd_basis_induct:
- assumes PDUnit: "\<And>a. P (PDUnit a)"
- assumes PDPlus: "\<And>t u. \<lbrakk>P t; P u\<rbrakk> \<Longrightarrow> P (PDPlus t u)"
- shows "P x"
-apply (induct x rule: pd_basis_induct1)
-apply (rule PDUnit, erule PDPlus [OF PDUnit])
-done
-
-subsection {* Fold operator *}
-
-definition
- fold_pd ::
- "('a compact_basis \<Rightarrow> 'b::type) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a pd_basis \<Rightarrow> 'b"
- where "fold_pd g f t = fold1 f (g ` Rep_pd_basis t)"
-
-lemma fold_pd_PDUnit:
- assumes "class.ab_semigroup_idem_mult f"
- shows "fold_pd g f (PDUnit x) = g x"
-unfolding fold_pd_def Rep_PDUnit by simp
-
-lemma fold_pd_PDPlus:
- assumes "class.ab_semigroup_idem_mult f"
- shows "fold_pd g f (PDPlus t u) = f (fold_pd g f t) (fold_pd g f u)"
-proof -
- interpret ab_semigroup_idem_mult f by fact
- show ?thesis unfolding fold_pd_def Rep_PDPlus
- by (simp add: image_Un fold1_Un2)
-qed
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/HOLCF/Compact_Basis.thy Fri Dec 17 16:43:45 2010 -0800
@@ -0,0 +1,111 @@
+(* Title: HOLCF/Compact_Basis.thy
+ Author: Brian Huffman
+*)
+
+header {* A compact basis for powerdomains *}
+
+theory Compact_Basis
+imports Bifinite
+begin
+
+default_sort "domain"
+
+subsection {* A compact basis for powerdomains *}
+
+typedef 'a pd_basis =
+ "{S::'a compact_basis set. finite S \<and> S \<noteq> {}}"
+by (rule_tac x="{arbitrary}" in exI, simp)
+
+lemma finite_Rep_pd_basis [simp]: "finite (Rep_pd_basis u)"
+by (insert Rep_pd_basis [of u, unfolded pd_basis_def]) simp
+
+lemma Rep_pd_basis_nonempty [simp]: "Rep_pd_basis u \<noteq> {}"
+by (insert Rep_pd_basis [of u, unfolded pd_basis_def]) simp
+
+text {* The powerdomain basis type is countable. *}
+
+lemma pd_basis_countable: "\<exists>f::'a pd_basis \<Rightarrow> nat. inj f"
+proof -
+ obtain g :: "'a compact_basis \<Rightarrow> nat" where "inj g"
+ using compact_basis.countable ..
+ hence image_g_eq: "\<And>A B. g ` A = g ` B \<longleftrightarrow> A = B"
+ by (rule inj_image_eq_iff)
+ have "inj (\<lambda>t. set_encode (g ` Rep_pd_basis t))"
+ by (simp add: inj_on_def set_encode_eq image_g_eq Rep_pd_basis_inject)
+ thus ?thesis by - (rule exI)
+ (* FIXME: why doesn't ".." or "by (rule exI)" work? *)
+qed
+
+subsection {* Unit and plus constructors *}
+
+definition
+ PDUnit :: "'a compact_basis \<Rightarrow> 'a pd_basis" where
+ "PDUnit = (\<lambda>x. Abs_pd_basis {x})"
+
+definition
+ PDPlus :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> 'a pd_basis" where
+ "PDPlus t u = Abs_pd_basis (Rep_pd_basis t \<union> Rep_pd_basis u)"
+
+lemma Rep_PDUnit:
+ "Rep_pd_basis (PDUnit x) = {x}"
+unfolding PDUnit_def by (rule Abs_pd_basis_inverse) (simp add: pd_basis_def)
+
+lemma Rep_PDPlus:
+ "Rep_pd_basis (PDPlus u v) = Rep_pd_basis u \<union> Rep_pd_basis v"
+unfolding PDPlus_def by (rule Abs_pd_basis_inverse) (simp add: pd_basis_def)
+
+lemma PDUnit_inject [simp]: "(PDUnit a = PDUnit b) = (a = b)"
+unfolding Rep_pd_basis_inject [symmetric] Rep_PDUnit by simp
+
+lemma PDPlus_assoc: "PDPlus (PDPlus t u) v = PDPlus t (PDPlus u v)"
+unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_assoc)
+
+lemma PDPlus_commute: "PDPlus t u = PDPlus u t"
+unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_commute)
+
+lemma PDPlus_absorb: "PDPlus t t = t"
+unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_absorb)
+
+lemma pd_basis_induct1:
+ assumes PDUnit: "\<And>a. P (PDUnit a)"
+ assumes PDPlus: "\<And>a t. P t \<Longrightarrow> P (PDPlus (PDUnit a) t)"
+ shows "P x"
+apply (induct x, unfold pd_basis_def, clarify)
+apply (erule (1) finite_ne_induct)
+apply (cut_tac a=x in PDUnit)
+apply (simp add: PDUnit_def)
+apply (drule_tac a=x in PDPlus)
+apply (simp add: PDUnit_def PDPlus_def
+ Abs_pd_basis_inverse [unfolded pd_basis_def])
+done
+
+lemma pd_basis_induct:
+ assumes PDUnit: "\<And>a. P (PDUnit a)"
+ assumes PDPlus: "\<And>t u. \<lbrakk>P t; P u\<rbrakk> \<Longrightarrow> P (PDPlus t u)"
+ shows "P x"
+apply (induct x rule: pd_basis_induct1)
+apply (rule PDUnit, erule PDPlus [OF PDUnit])
+done
+
+subsection {* Fold operator *}
+
+definition
+ fold_pd ::
+ "('a compact_basis \<Rightarrow> 'b::type) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a pd_basis \<Rightarrow> 'b"
+ where "fold_pd g f t = fold1 f (g ` Rep_pd_basis t)"
+
+lemma fold_pd_PDUnit:
+ assumes "class.ab_semigroup_idem_mult f"
+ shows "fold_pd g f (PDUnit x) = g x"
+unfolding fold_pd_def Rep_PDUnit by simp
+
+lemma fold_pd_PDPlus:
+ assumes "class.ab_semigroup_idem_mult f"
+ shows "fold_pd g f (PDPlus t u) = f (fold_pd g f t) (fold_pd g f u)"
+proof -
+ interpret ab_semigroup_idem_mult f by fact
+ show ?thesis unfolding fold_pd_def Rep_PDPlus
+ by (simp add: image_Un fold1_Un2)
+qed
+
+end
--- a/src/HOL/HOLCF/IsaMakefile Fri Dec 17 23:18:39 2010 +0100
+++ b/src/HOL/HOLCF/IsaMakefile Fri Dec 17 16:43:45 2010 -0800
@@ -39,7 +39,7 @@
Algebraic.thy \
Bifinite.thy \
Cfun.thy \
- CompactBasis.thy \
+ Compact_Basis.thy \
Completion.thy \
Cont.thy \
ConvexPD.thy \
--- a/src/HOL/HOLCF/LowerPD.thy Fri Dec 17 23:18:39 2010 +0100
+++ b/src/HOL/HOLCF/LowerPD.thy Fri Dec 17 16:43:45 2010 -0800
@@ -5,7 +5,7 @@
header {* Lower powerdomain *}
theory LowerPD
-imports CompactBasis
+imports Compact_Basis
begin
subsection {* Basis preorder *}
--- a/src/HOL/HOLCF/UpperPD.thy Fri Dec 17 23:18:39 2010 +0100
+++ b/src/HOL/HOLCF/UpperPD.thy Fri Dec 17 16:43:45 2010 -0800
@@ -5,7 +5,7 @@
header {* Upper powerdomain *}
theory UpperPD
-imports CompactBasis
+imports Compact_Basis
begin
subsection {* Basis preorder *}
--- a/src/HOL/IsaMakefile Fri Dec 17 23:18:39 2010 +0100
+++ b/src/HOL/IsaMakefile Fri Dec 17 16:43:45 2010 -0800
@@ -1406,7 +1406,7 @@
HOLCF/Algebraic.thy \
HOLCF/Bifinite.thy \
HOLCF/Cfun.thy \
- HOLCF/CompactBasis.thy \
+ HOLCF/Compact_Basis.thy \
HOLCF/Completion.thy \
HOLCF/Cont.thy \
HOLCF/ConvexPD.thy \