--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complete_Lattice.thy Wed Jul 22 18:08:45 2009 +0200
@@ -0,0 +1,794 @@
+(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
+
+header {* Complete lattices, with special focus on sets *}
+
+theory Complete_Lattice
+imports Set
+begin
+
+notation
+ less_eq (infix "\<sqsubseteq>" 50) and
+ less (infix "\<sqsubset>" 50) and
+ inf (infixl "\<sqinter>" 70) and
+ sup (infixl "\<squnion>" 65)
+
+
+subsection {* Abstract complete lattices *}
+
+class complete_lattice = lattice + bot + top +
+ fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
+ and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
+ assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
+ and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
+ assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
+ and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
+begin
+
+lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
+ by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
+
+lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
+ by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
+
+lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
+ unfolding Sup_Inf by auto
+
+lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
+ unfolding Inf_Sup by auto
+
+lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
+ by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
+
+lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
+ by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
+
+lemma Inf_singleton [simp]:
+ "\<Sqinter>{a} = a"
+ by (auto intro: antisym Inf_lower Inf_greatest)
+
+lemma Sup_singleton [simp]:
+ "\<Squnion>{a} = a"
+ by (auto intro: antisym Sup_upper Sup_least)
+
+lemma Inf_insert_simp:
+ "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
+ by (cases "A = {}") (simp_all, simp add: Inf_insert)
+
+lemma Sup_insert_simp:
+ "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
+ by (cases "A = {}") (simp_all, simp add: Sup_insert)
+
+lemma Inf_binary:
+ "\<Sqinter>{a, b} = a \<sqinter> b"
+ by (auto simp add: Inf_insert_simp)
+
+lemma Sup_binary:
+ "\<Squnion>{a, b} = a \<squnion> b"
+ by (auto simp add: Sup_insert_simp)
+
+lemma bot_def:
+ "bot = \<Squnion>{}"
+ by (auto intro: antisym Sup_least)
+
+lemma top_def:
+ "top = \<Sqinter>{}"
+ by (auto intro: antisym Inf_greatest)
+
+lemma sup_bot [simp]:
+ "x \<squnion> bot = x"
+ using bot_least [of x] by (simp add: le_iff_sup sup_commute)
+
+lemma inf_top [simp]:
+ "x \<sqinter> top = x"
+ using top_greatest [of x] by (simp add: le_iff_inf inf_commute)
+
+definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
+ "SUPR A f = \<Squnion> (f ` A)"
+
+definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
+ "INFI A f = \<Sqinter> (f ` A)"
+
+end
+
+syntax
+ "_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10)
+ "_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10)
+ "_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10)
+ "_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10)
+
+translations
+ "SUP x y. B" == "SUP x. SUP y. B"
+ "SUP x. B" == "CONST SUPR CONST UNIV (%x. B)"
+ "SUP x. B" == "SUP x:CONST UNIV. B"
+ "SUP x:A. B" == "CONST SUPR A (%x. B)"
+ "INF x y. B" == "INF x. INF y. B"
+ "INF x. B" == "CONST INFI CONST UNIV (%x. B)"
+ "INF x. B" == "INF x:CONST UNIV. B"
+ "INF x:A. B" == "CONST INFI A (%x. B)"
+
+print_translation {* [
+Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} "_SUP",
+Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} "_INF"
+] *} -- {* to avoid eta-contraction of body *}
+
+context complete_lattice
+begin
+
+lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
+ by (auto simp add: SUPR_def intro: Sup_upper)
+
+lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
+ by (auto simp add: SUPR_def intro: Sup_least)
+
+lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
+ by (auto simp add: INFI_def intro: Inf_lower)
+
+lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
+ by (auto simp add: INFI_def intro: Inf_greatest)
+
+lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
+ by (auto intro: antisym SUP_leI le_SUPI)
+
+lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
+ by (auto intro: antisym INF_leI le_INFI)
+
+end
+
+
+subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
+
+instantiation bool :: complete_lattice
+begin
+
+definition
+ Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
+
+definition
+ Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
+
+instance proof
+qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
+
+end
+
+lemma Inf_empty_bool [simp]:
+ "\<Sqinter>{}"
+ unfolding Inf_bool_def by auto
+
+lemma not_Sup_empty_bool [simp]:
+ "\<not> \<Squnion>{}"
+ unfolding Sup_bool_def by auto
+
+lemma INFI_bool_eq:
+ "INFI = Ball"
+proof (rule ext)+
+ fix A :: "'a set"
+ fix P :: "'a \<Rightarrow> bool"
+ show "(INF x:A. P x) \<longleftrightarrow> (\<forall>x \<in> A. P x)"
+ by (auto simp add: Ball_def INFI_def Inf_bool_def)
+qed
+
+lemma SUPR_bool_eq:
+ "SUPR = Bex"
+proof (rule ext)+
+ fix A :: "'a set"
+ fix P :: "'a \<Rightarrow> bool"
+ show "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)"
+ by (auto simp add: Bex_def SUPR_def Sup_bool_def)
+qed
+
+instantiation "fun" :: (type, complete_lattice) complete_lattice
+begin
+
+definition
+ Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
+
+definition
+ Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
+
+instance proof
+qed (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
+ intro: Inf_lower Sup_upper Inf_greatest Sup_least)
+
+end
+
+lemma Inf_empty_fun:
+ "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
+ by (simp add: Inf_fun_def)
+
+lemma Sup_empty_fun:
+ "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
+ by (simp add: Sup_fun_def)
+
+
+subsection {* Union *}
+
+definition Union :: "'a set set \<Rightarrow> 'a set" where
+ Sup_set_eq [symmetric]: "Union S = \<Squnion>S"
+
+notation (xsymbols)
+ Union ("\<Union>_" [90] 90)
+
+lemma Union_eq:
+ "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
+proof (rule set_ext)
+ fix x
+ have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
+ by auto
+ then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
+ by (simp add: Sup_set_eq [symmetric] Sup_fun_def Sup_bool_def) (simp add: mem_def)
+qed
+
+lemma Union_iff [simp, noatp]:
+ "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
+ by (unfold Union_eq) blast
+
+lemma UnionI [intro]:
+ "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
+ -- {* The order of the premises presupposes that @{term C} is rigid;
+ @{term A} may be flexible. *}
+ by auto
+
+lemma UnionE [elim!]:
+ "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"
+ by auto
+
+lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
+ by (iprover intro: subsetI UnionI)
+
+lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
+ by (iprover intro: subsetI elim: UnionE dest: subsetD)
+
+lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
+ by blast
+
+lemma Union_empty [simp]: "Union({}) = {}"
+ by blast
+
+lemma Union_UNIV [simp]: "Union UNIV = UNIV"
+ by blast
+
+lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
+ by blast
+
+lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
+ by blast
+
+lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
+ by blast
+
+lemma Union_empty_conv [simp,noatp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
+ by blast
+
+lemma empty_Union_conv [simp,noatp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
+ by blast
+
+lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
+ by blast
+
+lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
+ by blast
+
+lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
+ by blast
+
+lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
+ by blast
+
+
+subsection {* Unions of families *}
+
+definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
+ SUPR_set_eq [symmetric]: "UNION S f = (SUP x:S. f x)"
+
+syntax
+ "@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10)
+ "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 10] 10)
+
+syntax (xsymbols)
+ "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10)
+ "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
+
+syntax (latex output)
+ "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
+ "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
+
+translations
+ "UN x y. B" == "UN x. UN y. B"
+ "UN x. B" == "CONST UNION CONST UNIV (%x. B)"
+ "UN x. B" == "UN x:CONST UNIV. B"
+ "UN x:A. B" == "CONST UNION A (%x. B)"
+
+text {*
+ Note the difference between ordinary xsymbol syntax of indexed
+ unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
+ and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
+ former does not make the index expression a subscript of the
+ union/intersection symbol because this leads to problems with nested
+ subscripts in Proof General.
+*}
+
+print_translation {* [
+Syntax.preserve_binder_abs2_tr' @{const_syntax UNION} "@UNION"
+] *} -- {* to avoid eta-contraction of body *}
+
+lemma UNION_eq_Union_image:
+ "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
+ by (simp add: SUPR_def SUPR_set_eq [symmetric] Sup_set_eq)
+
+lemma Union_def:
+ "\<Union>S = (\<Union>x\<in>S. x)"
+ by (simp add: UNION_eq_Union_image image_def)
+
+lemma UNION_def [noatp]:
+ "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
+ by (auto simp add: UNION_eq_Union_image Union_eq)
+
+lemma Union_image_eq [simp]:
+ "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
+ by (rule sym) (fact UNION_eq_Union_image)
+
+lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
+ by (unfold UNION_def) blast
+
+lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
+ -- {* The order of the premises presupposes that @{term A} is rigid;
+ @{term b} may be flexible. *}
+ by auto
+
+lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
+ by (unfold UNION_def) blast
+
+lemma UN_cong [cong]:
+ "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
+ by (simp add: UNION_def)
+
+lemma strong_UN_cong:
+ "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
+ by (simp add: UNION_def simp_implies_def)
+
+lemma image_eq_UN: "f`A = (UN x:A. {f x})"
+ by blast
+
+lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
+ by blast
+
+lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
+ by (iprover intro: subsetI elim: UN_E dest: subsetD)
+
+lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
+ by blast
+
+lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
+ by blast
+
+lemma UN_empty [simp,noatp]: "(\<Union>x\<in>{}. B x) = {}"
+ by blast
+
+lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
+ by blast
+
+lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
+ by blast
+
+lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
+ by auto
+
+lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
+ by blast
+
+lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
+ by blast
+
+lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
+ by blast
+
+lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
+ by blast
+
+lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
+ by blast
+
+lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
+ by auto
+
+lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
+ by blast
+
+lemma UNION_empty_conv[simp]:
+ "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
+ "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
+by blast+
+
+lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
+ by blast
+
+lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
+ by blast
+
+lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
+ by blast
+
+lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
+ by (auto simp add: split_if_mem2)
+
+lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
+ by (auto intro: bool_contrapos)
+
+lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
+ by blast
+
+lemma UN_mono:
+ "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
+ (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
+ by (blast dest: subsetD)
+
+lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
+ by blast
+
+lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
+ by blast
+
+lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
+ -- {* NOT suitable for rewriting *}
+ by blast
+
+lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
+by blast
+
+
+subsection {* Inter *}
+
+definition Inter :: "'a set set \<Rightarrow> 'a set" where
+ Inf_set_eq [symmetric]: "Inter S = \<Sqinter>S"
+
+notation (xsymbols)
+ Inter ("\<Inter>_" [90] 90)
+
+lemma Inter_eq [code del]:
+ "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
+proof (rule set_ext)
+ fix x
+ have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
+ by auto
+ then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
+ by (simp add: Inf_fun_def Inf_bool_def Inf_set_eq [symmetric]) (simp add: mem_def)
+qed
+
+lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)"
+ by (unfold Inter_eq) blast
+
+lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
+ by (simp add: Inter_eq)
+
+text {*
+ \medskip A ``destruct'' rule -- every @{term X} in @{term C}
+ contains @{term A} as an element, but @{prop "A:X"} can hold when
+ @{prop "X:C"} does not! This rule is analogous to @{text spec}.
+*}
+
+lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
+ by auto
+
+lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
+ -- {* ``Classical'' elimination rule -- does not require proving
+ @{prop "X:C"}. *}
+ by (unfold Inter_eq) blast
+
+lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
+ by blast
+
+lemma Inter_subset:
+ "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
+ by blast
+
+lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
+ by (iprover intro: InterI subsetI dest: subsetD)
+
+lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
+ by blast
+
+lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
+ by blast
+
+lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
+ by blast
+
+lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
+ by blast
+
+lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
+ by blast
+
+lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
+ by blast
+
+lemma Inter_UNIV_conv [simp,noatp]:
+ "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
+ "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
+ by blast+
+
+lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
+ by blast
+
+
+subsection {* Intersections of families *}
+
+definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
+ INFI_set_eq [symmetric]: "INTER S f = (INF x:S. f x)"
+
+syntax
+ "@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10)
+ "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 10] 10)
+
+syntax (xsymbols)
+ "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10)
+ "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
+
+syntax (latex output)
+ "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
+ "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
+
+translations
+ "INT x y. B" == "INT x. INT y. B"
+ "INT x. B" == "CONST INTER CONST UNIV (%x. B)"
+ "INT x. B" == "INT x:CONST UNIV. B"
+ "INT x:A. B" == "CONST INTER A (%x. B)"
+
+print_translation {* [
+Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} "@INTER"
+] *} -- {* to avoid eta-contraction of body *}
+
+lemma INTER_eq_Inter_image:
+ "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
+ by (simp add: INFI_def INFI_set_eq [symmetric] Inf_set_eq)
+
+lemma Inter_def:
+ "\<Inter>S = (\<Inter>x\<in>S. x)"
+ by (simp add: INTER_eq_Inter_image image_def)
+
+lemma INTER_def:
+ "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
+ by (auto simp add: INTER_eq_Inter_image Inter_eq)
+
+lemma Inter_image_eq [simp]:
+ "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
+ by (rule sym) (fact INTER_eq_Inter_image)
+
+lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
+ by (unfold INTER_def) blast
+
+lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
+ by (unfold INTER_def) blast
+
+lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
+ by auto
+
+lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
+ -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
+ by (unfold INTER_def) blast
+
+lemma INT_cong [cong]:
+ "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
+ by (simp add: INTER_def)
+
+lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
+ by blast
+
+lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
+ by blast
+
+lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
+ by blast
+
+lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
+ by (iprover intro: INT_I subsetI dest: subsetD)
+
+lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
+ by blast
+
+lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
+ by blast
+
+lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
+ by blast
+
+lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
+ by blast
+
+lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
+ by blast
+
+lemma INT_insert_distrib:
+ "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
+ by blast
+
+lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
+ by auto
+
+lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
+ -- {* Look: it has an \emph{existential} quantifier *}
+ by blast
+
+lemma INTER_UNIV_conv[simp]:
+ "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
+ "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
+by blast+
+
+lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
+ by (auto intro: bool_induct)
+
+lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
+ by blast
+
+lemma INT_anti_mono:
+ "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
+ (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
+ -- {* The last inclusion is POSITIVE! *}
+ by (blast dest: subsetD)
+
+lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
+ by blast
+
+
+subsection {* Distributive laws *}
+
+lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
+ by blast
+
+lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
+ by blast
+
+lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
+ -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
+ -- {* Union of a family of unions *}
+ by blast
+
+lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
+ -- {* Equivalent version *}
+ by blast
+
+lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
+ by blast
+
+lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
+ by blast
+
+lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
+ -- {* Equivalent version *}
+ by blast
+
+lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
+ -- {* Halmos, Naive Set Theory, page 35. *}
+ by blast
+
+lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
+ by blast
+
+lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
+ by blast
+
+lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
+ by blast
+
+
+subsection {* Complement *}
+
+lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
+ by blast
+
+lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
+ by blast
+
+
+subsection {* Miniscoping and maxiscoping *}
+
+text {* \medskip Miniscoping: pushing in quantifiers and big Unions
+ and Intersections. *}
+
+lemma UN_simps [simp]:
+ "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
+ "!!A B C. (UN x:C. A x Un B) = ((if C={} then {} else (UN x:C. A x) Un B))"
+ "!!A B C. (UN x:C. A Un B x) = ((if C={} then {} else A Un (UN x:C. B x)))"
+ "!!A B C. (UN x:C. A x Int B) = ((UN x:C. A x) Int B)"
+ "!!A B C. (UN x:C. A Int B x) = (A Int (UN x:C. B x))"
+ "!!A B C. (UN x:C. A x - B) = ((UN x:C. A x) - B)"
+ "!!A B C. (UN x:C. A - B x) = (A - (INT x:C. B x))"
+ "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
+ "!!A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)"
+ "!!A B f. (UN x:f`A. B x) = (UN a:A. B (f a))"
+ by auto
+
+lemma INT_simps [simp]:
+ "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
+ "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
+ "!!A B C. (INT x:C. A x - B) = (if C={} then UNIV else (INT x:C. A x) - B)"
+ "!!A B C. (INT x:C. A - B x) = (if C={} then UNIV else A - (UN x:C. B x))"
+ "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
+ "!!A B C. (INT x:C. A x Un B) = ((INT x:C. A x) Un B)"
+ "!!A B C. (INT x:C. A Un B x) = (A Un (INT x:C. B x))"
+ "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
+ "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
+ "!!A B f. (INT x:f`A. B x) = (INT a:A. B (f a))"
+ by auto
+
+lemma ball_simps [simp,noatp]:
+ "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
+ "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
+ "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
+ "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
+ "!!P. (ALL x:{}. P x) = True"
+ "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
+ "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
+ "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
+ "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
+ "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
+ "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
+ "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
+ by auto
+
+lemma bex_simps [simp,noatp]:
+ "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
+ "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
+ "!!P. (EX x:{}. P x) = False"
+ "!!P. (EX x:UNIV. P x) = (EX x. P x)"
+ "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
+ "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
+ "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
+ "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
+ "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
+ "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
+ by auto
+
+lemma ball_conj_distrib:
+ "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
+ by blast
+
+lemma bex_disj_distrib:
+ "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
+ by blast
+
+
+text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
+
+lemma UN_extend_simps:
+ "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
+ "!!A B C. (UN x:C. A x) Un B = (if C={} then B else (UN x:C. A x Un B))"
+ "!!A B C. A Un (UN x:C. B x) = (if C={} then A else (UN x:C. A Un B x))"
+ "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
+ "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
+ "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
+ "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
+ "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
+ "!!A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
+ "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
+ by auto
+
+lemma INT_extend_simps:
+ "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
+ "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
+ "!!A B C. (INT x:C. A x) - B = (if C={} then UNIV-B else (INT x:C. A x - B))"
+ "!!A B C. A - (UN x:C. B x) = (if C={} then A else (INT x:C. A - B x))"
+ "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
+ "!!A B C. ((INT x:C. A x) Un B) = (INT x:C. A x Un B)"
+ "!!A B C. A Un (INT x:C. B x) = (INT x:C. A Un B x)"
+ "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
+ "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
+ "!!A B f. (INT a:A. B (f a)) = (INT x:f`A. B x)"
+ by auto
+
+
+no_notation
+ less_eq (infix "\<sqsubseteq>" 50) and
+ less (infix "\<sqsubset>" 50) and
+ inf (infixl "\<sqinter>" 70) and
+ sup (infixl "\<squnion>" 65) and
+ Inf ("\<Sqinter>_" [900] 900) and
+ Sup ("\<Squnion>_" [900] 900)
+
+lemmas mem_simps =
+ insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
+ mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
+ -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
+
+end
--- a/src/HOL/Fun.thy Wed Jul 22 15:28:49 2009 +0200
+++ b/src/HOL/Fun.thy Wed Jul 22 18:08:45 2009 +0200
@@ -6,7 +6,7 @@
header {* Notions about functions *}
theory Fun
-imports Set
+imports Complete_Lattice
begin
text{*As a simplification rule, it replaces all function equalities by
--- a/src/HOL/HoareParallel/OG_Hoare.thy Wed Jul 22 15:28:49 2009 +0200
+++ b/src/HOL/HoareParallel/OG_Hoare.thy Wed Jul 22 18:08:45 2009 +0200
@@ -441,7 +441,7 @@
apply clarify
apply(frule Parallel_length_post_PStar)
apply clarify
- apply(drule_tac j=xa in Parallel_Strong_Soundness)
+ apply(drule_tac j=xb in Parallel_Strong_Soundness)
apply clarify
apply simp
apply force
--- a/src/HOL/IsaMakefile Wed Jul 22 15:28:49 2009 +0200
+++ b/src/HOL/IsaMakefile Wed Jul 22 18:08:45 2009 +0200
@@ -117,6 +117,7 @@
@$(ISABELLE_TOOL) usedir -b -f base.ML -d false -g false $(OUT)/Pure HOL-Base
PLAIN_DEPENDENCIES = $(BASE_DEPENDENCIES)\
+ Complete_Lattice.thy \
Datatype.thy \
Divides.thy \
Extraction.thy \
--- a/src/HOL/Library/Executable_Set.thy Wed Jul 22 15:28:49 2009 +0200
+++ b/src/HOL/Library/Executable_Set.thy Wed Jul 22 18:08:45 2009 +0200
@@ -75,8 +75,8 @@
"op \<union>" ("{*Fset.union*}")
"op \<inter>" ("{*Fset.inter*}")
"op - \<Colon> 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" ("{*flip Fset.subtract*}")
- "Set.Union" ("{*Fset.Union*}")
- "Set.Inter" ("{*Fset.Inter*}")
+ "Complete_Lattice.Union" ("{*Fset.Union*}")
+ "Complete_Lattice.Inter" ("{*Fset.Inter*}")
card ("{*Fset.card*}")
fold ("{*foldl o flip*}")
--- a/src/HOL/Library/Fset.thy Wed Jul 22 15:28:49 2009 +0200
+++ b/src/HOL/Library/Fset.thy Wed Jul 22 18:08:45 2009 +0200
@@ -160,7 +160,7 @@
qed
definition Inter :: "'a fset fset \<Rightarrow> 'a fset" where
- [simp]: "Inter A = Fset (Set.Inter (member ` member A))"
+ [simp]: "Inter A = Fset (Complete_Lattice.Inter (member ` member A))"
lemma Inter_inter [code]:
"Inter (Set (A # As)) = foldl inter A As"
@@ -174,7 +174,7 @@
qed
definition Union :: "'a fset fset \<Rightarrow> 'a fset" where
- [simp]: "Union A = Fset (Set.Union (member ` member A))"
+ [simp]: "Union A = Fset (Complete_Lattice.Union (member ` member A))"
lemma Union_union [code]:
"Union (Set As) = foldl union empty As"
--- a/src/HOL/Library/Lattice_Syntax.thy Wed Jul 22 15:28:49 2009 +0200
+++ b/src/HOL/Library/Lattice_Syntax.thy Wed Jul 22 18:08:45 2009 +0200
@@ -4,16 +4,16 @@
(*<*)
theory Lattice_Syntax
-imports Set
+imports Complete_Lattice
begin
notation
+ top ("\<top>") and
+ bot ("\<bottom>") and
inf (infixl "\<sqinter>" 70) and
sup (infixl "\<squnion>" 65) and
Inf ("\<Sqinter>_" [900] 900) and
- Sup ("\<Squnion>_" [900] 900) and
- top ("\<top>") and
- bot ("\<bottom>")
+ Sup ("\<Squnion>_" [900] 900)
end
(*>*)
\ No newline at end of file
--- a/src/HOL/Set.thy Wed Jul 22 15:28:49 2009 +0200
+++ b/src/HOL/Set.thy Wed Jul 22 18:08:45 2009 +0200
@@ -1,6 +1,4 @@
-(* Title: HOL/Set.thy
- Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
-*)
+(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel *)
header {* Set theory for higher-order logic *}
@@ -1733,789 +1731,4 @@
val vimage_Un = @{thm vimage_Un}
*}
-
-subsection {* Complete lattices *}
-
-notation
- less_eq (infix "\<sqsubseteq>" 50) and
- less (infix "\<sqsubset>" 50) and
- inf (infixl "\<sqinter>" 70) and
- sup (infixl "\<squnion>" 65)
-
-class complete_lattice = lattice + bot + top +
- fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
- and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
- assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
- and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
- assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
- and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
-begin
-
-lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
- by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
-
-lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
- by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
-
-lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
- unfolding Sup_Inf by auto
-
-lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
- unfolding Inf_Sup by auto
-
-lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
- by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
-
-lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
- by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
-
-lemma Inf_singleton [simp]:
- "\<Sqinter>{a} = a"
- by (auto intro: antisym Inf_lower Inf_greatest)
-
-lemma Sup_singleton [simp]:
- "\<Squnion>{a} = a"
- by (auto intro: antisym Sup_upper Sup_least)
-
-lemma Inf_insert_simp:
- "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
- by (cases "A = {}") (simp_all, simp add: Inf_insert)
-
-lemma Sup_insert_simp:
- "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
- by (cases "A = {}") (simp_all, simp add: Sup_insert)
-
-lemma Inf_binary:
- "\<Sqinter>{a, b} = a \<sqinter> b"
- by (auto simp add: Inf_insert_simp)
-
-lemma Sup_binary:
- "\<Squnion>{a, b} = a \<squnion> b"
- by (auto simp add: Sup_insert_simp)
-
-lemma bot_def:
- "bot = \<Squnion>{}"
- by (auto intro: antisym Sup_least)
-
-lemma top_def:
- "top = \<Sqinter>{}"
- by (auto intro: antisym Inf_greatest)
-
-lemma sup_bot [simp]:
- "x \<squnion> bot = x"
- using bot_least [of x] by (simp add: le_iff_sup sup_commute)
-
-lemma inf_top [simp]:
- "x \<sqinter> top = x"
- using top_greatest [of x] by (simp add: le_iff_inf inf_commute)
-
-definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
- "SUPR A f = \<Squnion> (f ` A)"
-
-definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
- "INFI A f = \<Sqinter> (f ` A)"
-
end
-
-syntax
- "_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10)
- "_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10)
- "_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10)
- "_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10)
-
-translations
- "SUP x y. B" == "SUP x. SUP y. B"
- "SUP x. B" == "CONST SUPR CONST UNIV (%x. B)"
- "SUP x. B" == "SUP x:CONST UNIV. B"
- "SUP x:A. B" == "CONST SUPR A (%x. B)"
- "INF x y. B" == "INF x. INF y. B"
- "INF x. B" == "CONST INFI CONST UNIV (%x. B)"
- "INF x. B" == "INF x:CONST UNIV. B"
- "INF x:A. B" == "CONST INFI A (%x. B)"
-
-print_translation {* [
-Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} "_SUP",
-Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} "_INF"
-] *} -- {* to avoid eta-contraction of body *}
-
-context complete_lattice
-begin
-
-lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
- by (auto simp add: SUPR_def intro: Sup_upper)
-
-lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
- by (auto simp add: SUPR_def intro: Sup_least)
-
-lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
- by (auto simp add: INFI_def intro: Inf_lower)
-
-lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
- by (auto simp add: INFI_def intro: Inf_greatest)
-
-lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
- by (auto intro: antisym SUP_leI le_SUPI)
-
-lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
- by (auto intro: antisym INF_leI le_INFI)
-
-end
-
-
-subsubsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
-
-instantiation bool :: complete_lattice
-begin
-
-definition
- Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
-
-definition
- Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
-
-instance proof
-qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
-
-end
-
-lemma Inf_empty_bool [simp]:
- "\<Sqinter>{}"
- unfolding Inf_bool_def by auto
-
-lemma not_Sup_empty_bool [simp]:
- "\<not> \<Squnion>{}"
- unfolding Sup_bool_def by auto
-
-lemma INFI_bool_eq:
- "INFI = Ball"
-proof (rule ext)+
- fix A :: "'a set"
- fix P :: "'a \<Rightarrow> bool"
- show "(INF x:A. P x) \<longleftrightarrow> (\<forall>x \<in> A. P x)"
- by (auto simp add: Ball_def INFI_def Inf_bool_def)
-qed
-
-lemma SUPR_bool_eq:
- "SUPR = Bex"
-proof (rule ext)+
- fix A :: "'a set"
- fix P :: "'a \<Rightarrow> bool"
- show "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)"
- by (auto simp add: Bex_def SUPR_def Sup_bool_def)
-qed
-
-instantiation "fun" :: (type, complete_lattice) complete_lattice
-begin
-
-definition
- Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
-
-definition
- Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
-
-instance proof
-qed (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
- intro: Inf_lower Sup_upper Inf_greatest Sup_least)
-
-end
-
-lemma Inf_empty_fun:
- "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
- by (simp add: Inf_fun_def)
-
-lemma Sup_empty_fun:
- "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
- by (simp add: Sup_fun_def)
-
-
-subsubsection {* Union *}
-
-definition Union :: "'a set set \<Rightarrow> 'a set" where
- Sup_set_eq [symmetric]: "Union S = \<Squnion>S"
-
-notation (xsymbols)
- Union ("\<Union>_" [90] 90)
-
-lemma Union_eq:
- "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
-proof (rule set_ext)
- fix x
- have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
- by auto
- then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
- by (simp add: Sup_set_eq [symmetric] Sup_fun_def Sup_bool_def) (simp add: mem_def)
-qed
-
-lemma Union_iff [simp, noatp]:
- "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
- by (unfold Union_eq) blast
-
-lemma UnionI [intro]:
- "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
- -- {* The order of the premises presupposes that @{term C} is rigid;
- @{term A} may be flexible. *}
- by auto
-
-lemma UnionE [elim!]:
- "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"
- by auto
-
-lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
- by (iprover intro: subsetI UnionI)
-
-lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
- by (iprover intro: subsetI elim: UnionE dest: subsetD)
-
-lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
- by blast
-
-lemma Union_empty [simp]: "Union({}) = {}"
- by blast
-
-lemma Union_UNIV [simp]: "Union UNIV = UNIV"
- by blast
-
-lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
- by blast
-
-lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
- by blast
-
-lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
- by blast
-
-lemma Union_empty_conv [simp,noatp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
- by blast
-
-lemma empty_Union_conv [simp,noatp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
- by blast
-
-lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
- by blast
-
-lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
- by blast
-
-lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
- by blast
-
-lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
- by blast
-
-
-subsubsection {* Unions of families *}
-
-definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
- SUPR_set_eq [symmetric]: "UNION S f = (SUP x:S. f x)"
-
-syntax
- "@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10)
- "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 10] 10)
-
-syntax (xsymbols)
- "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10)
- "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
-
-syntax (latex output)
- "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
- "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
-
-translations
- "UN x y. B" == "UN x. UN y. B"
- "UN x. B" == "CONST UNION CONST UNIV (%x. B)"
- "UN x. B" == "UN x:CONST UNIV. B"
- "UN x:A. B" == "CONST UNION A (%x. B)"
-
-text {*
- Note the difference between ordinary xsymbol syntax of indexed
- unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
- and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
- former does not make the index expression a subscript of the
- union/intersection symbol because this leads to problems with nested
- subscripts in Proof General.
-*}
-
-print_translation {* [
-Syntax.preserve_binder_abs2_tr' @{const_syntax UNION} "@UNION"
-] *} -- {* to avoid eta-contraction of body *}
-
-lemma UNION_eq_Union_image:
- "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
- by (simp add: SUPR_def SUPR_set_eq [symmetric] Sup_set_eq)
-
-lemma Union_def:
- "\<Union>S = (\<Union>x\<in>S. x)"
- by (simp add: UNION_eq_Union_image image_def)
-
-lemma UNION_def [noatp]:
- "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
- by (auto simp add: UNION_eq_Union_image Union_eq)
-
-lemma Union_image_eq [simp]:
- "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
- by (rule sym) (fact UNION_eq_Union_image)
-
-lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
- by (unfold UNION_def) blast
-
-lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
- -- {* The order of the premises presupposes that @{term A} is rigid;
- @{term b} may be flexible. *}
- by auto
-
-lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
- by (unfold UNION_def) blast
-
-lemma UN_cong [cong]:
- "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
- by (simp add: UNION_def)
-
-lemma strong_UN_cong:
- "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
- by (simp add: UNION_def simp_implies_def)
-
-lemma image_eq_UN: "f`A = (UN x:A. {f x})"
- by blast
-
-lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
- by blast
-
-lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
- by (iprover intro: subsetI elim: UN_E dest: subsetD)
-
-lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
- by blast
-
-lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
- by blast
-
-lemma UN_empty [simp,noatp]: "(\<Union>x\<in>{}. B x) = {}"
- by blast
-
-lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
- by blast
-
-lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
- by blast
-
-lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
- by auto
-
-lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
- by blast
-
-lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
- by blast
-
-lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
- by blast
-
-lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
- by blast
-
-lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
- by blast
-
-lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
- by auto
-
-lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
- by blast
-
-lemma UNION_empty_conv[simp]:
- "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
- "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
-by blast+
-
-lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
- by blast
-
-lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
- by blast
-
-lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
- by blast
-
-lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
- by (auto simp add: split_if_mem2)
-
-lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
- by (auto intro: bool_contrapos)
-
-lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
- by blast
-
-lemma UN_mono:
- "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
- (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
- by (blast dest: subsetD)
-
-lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
- by blast
-
-lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
- by blast
-
-lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
- -- {* NOT suitable for rewriting *}
- by blast
-
-lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
-by blast
-
-
-subsubsection {* Inter *}
-
-definition Inter :: "'a set set \<Rightarrow> 'a set" where
- Inf_set_eq [symmetric]: "Inter S = \<Sqinter>S"
-
-notation (xsymbols)
- Inter ("\<Inter>_" [90] 90)
-
-lemma Inter_eq [code del]:
- "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
-proof (rule set_ext)
- fix x
- have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
- by auto
- then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
- by (simp add: Inf_fun_def Inf_bool_def Inf_set_eq [symmetric]) (simp add: mem_def)
-qed
-
-lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)"
- by (unfold Inter_eq) blast
-
-lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
- by (simp add: Inter_eq)
-
-text {*
- \medskip A ``destruct'' rule -- every @{term X} in @{term C}
- contains @{term A} as an element, but @{prop "A:X"} can hold when
- @{prop "X:C"} does not! This rule is analogous to @{text spec}.
-*}
-
-lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
- by auto
-
-lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
- -- {* ``Classical'' elimination rule -- does not require proving
- @{prop "X:C"}. *}
- by (unfold Inter_eq) blast
-
-lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
- by blast
-
-lemma Inter_subset:
- "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
- by blast
-
-lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
- by (iprover intro: InterI subsetI dest: subsetD)
-
-lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
- by blast
-
-lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
- by blast
-
-lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
- by blast
-
-lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
- by blast
-
-lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
- by blast
-
-lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
- by blast
-
-lemma Inter_UNIV_conv [simp,noatp]:
- "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
- "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
- by blast+
-
-lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
- by blast
-
-
-subsubsection {* Intersections of families *}
-
-definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
- INFI_set_eq [symmetric]: "INTER S f = (INF x:S. f x)"
-
-syntax
- "@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10)
- "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 10] 10)
-
-syntax (xsymbols)
- "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10)
- "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
-
-syntax (latex output)
- "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
- "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
-
-translations
- "INT x y. B" == "INT x. INT y. B"
- "INT x. B" == "CONST INTER CONST UNIV (%x. B)"
- "INT x. B" == "INT x:CONST UNIV. B"
- "INT x:A. B" == "CONST INTER A (%x. B)"
-
-print_translation {* [
-Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} "@INTER"
-] *} -- {* to avoid eta-contraction of body *}
-
-lemma INTER_eq_Inter_image:
- "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
- by (simp add: INFI_def INFI_set_eq [symmetric] Inf_set_eq)
-
-lemma Inter_def:
- "\<Inter>S = (\<Inter>x\<in>S. x)"
- by (simp add: INTER_eq_Inter_image image_def)
-
-lemma INTER_def:
- "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
- by (auto simp add: INTER_eq_Inter_image Inter_eq)
-
-lemma Inter_image_eq [simp]:
- "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
- by (rule sym) (fact INTER_eq_Inter_image)
-
-lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
- by (unfold INTER_def) blast
-
-lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
- by (unfold INTER_def) blast
-
-lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
- by auto
-
-lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
- -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
- by (unfold INTER_def) blast
-
-lemma INT_cong [cong]:
- "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
- by (simp add: INTER_def)
-
-lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
- by blast
-
-lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
- by blast
-
-lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
- by blast
-
-lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
- by (iprover intro: INT_I subsetI dest: subsetD)
-
-lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
- by blast
-
-lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
- by blast
-
-lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
- by blast
-
-lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
- by blast
-
-lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
- by blast
-
-lemma INT_insert_distrib:
- "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
- by blast
-
-lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
- by auto
-
-lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
- -- {* Look: it has an \emph{existential} quantifier *}
- by blast
-
-lemma INTER_UNIV_conv[simp]:
- "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
- "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
-by blast+
-
-lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
- by (auto intro: bool_induct)
-
-lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
- by blast
-
-lemma INT_anti_mono:
- "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
- (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
- -- {* The last inclusion is POSITIVE! *}
- by (blast dest: subsetD)
-
-lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
- by blast
-
-
-subsubsection {* Distributive laws *}
-
-lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
- by blast
-
-lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
- by blast
-
-lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
- -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
- -- {* Union of a family of unions *}
- by blast
-
-lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
- -- {* Equivalent version *}
- by blast
-
-lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
- by blast
-
-lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
- by blast
-
-lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
- -- {* Equivalent version *}
- by blast
-
-lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
- -- {* Halmos, Naive Set Theory, page 35. *}
- by blast
-
-lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
- by blast
-
-lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
- by blast
-
-lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
- by blast
-
-
-subsubsection {* Complement *}
-
-lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
- by blast
-
-lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
- by blast
-
-
-subsubsection {* Miniscoping and maxiscoping *}
-
-text {* \medskip Miniscoping: pushing in quantifiers and big Unions
- and Intersections. *}
-
-lemma UN_simps [simp]:
- "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
- "!!A B C. (UN x:C. A x Un B) = ((if C={} then {} else (UN x:C. A x) Un B))"
- "!!A B C. (UN x:C. A Un B x) = ((if C={} then {} else A Un (UN x:C. B x)))"
- "!!A B C. (UN x:C. A x Int B) = ((UN x:C. A x) Int B)"
- "!!A B C. (UN x:C. A Int B x) = (A Int (UN x:C. B x))"
- "!!A B C. (UN x:C. A x - B) = ((UN x:C. A x) - B)"
- "!!A B C. (UN x:C. A - B x) = (A - (INT x:C. B x))"
- "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
- "!!A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)"
- "!!A B f. (UN x:f`A. B x) = (UN a:A. B (f a))"
- by auto
-
-lemma INT_simps [simp]:
- "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
- "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
- "!!A B C. (INT x:C. A x - B) = (if C={} then UNIV else (INT x:C. A x) - B)"
- "!!A B C. (INT x:C. A - B x) = (if C={} then UNIV else A - (UN x:C. B x))"
- "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
- "!!A B C. (INT x:C. A x Un B) = ((INT x:C. A x) Un B)"
- "!!A B C. (INT x:C. A Un B x) = (A Un (INT x:C. B x))"
- "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
- "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
- "!!A B f. (INT x:f`A. B x) = (INT a:A. B (f a))"
- by auto
-
-lemma ball_simps [simp,noatp]:
- "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
- "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
- "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
- "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
- "!!P. (ALL x:{}. P x) = True"
- "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
- "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
- "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
- "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
- "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
- "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
- "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
- by auto
-
-lemma bex_simps [simp,noatp]:
- "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
- "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
- "!!P. (EX x:{}. P x) = False"
- "!!P. (EX x:UNIV. P x) = (EX x. P x)"
- "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
- "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
- "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
- "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
- "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
- "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
- by auto
-
-lemma ball_conj_distrib:
- "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
- by blast
-
-lemma bex_disj_distrib:
- "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
- by blast
-
-
-text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
-
-lemma UN_extend_simps:
- "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
- "!!A B C. (UN x:C. A x) Un B = (if C={} then B else (UN x:C. A x Un B))"
- "!!A B C. A Un (UN x:C. B x) = (if C={} then A else (UN x:C. A Un B x))"
- "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
- "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
- "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
- "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
- "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
- "!!A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
- "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
- by auto
-
-lemma INT_extend_simps:
- "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
- "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
- "!!A B C. (INT x:C. A x) - B = (if C={} then UNIV-B else (INT x:C. A x - B))"
- "!!A B C. A - (UN x:C. B x) = (if C={} then A else (INT x:C. A - B x))"
- "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
- "!!A B C. ((INT x:C. A x) Un B) = (INT x:C. A x Un B)"
- "!!A B C. A Un (INT x:C. B x) = (INT x:C. A Un B x)"
- "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
- "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
- "!!A B f. (INT a:A. B (f a)) = (INT x:f`A. B x)"
- by auto
-
-
-no_notation
- less_eq (infix "\<sqsubseteq>" 50) and
- less (infix "\<sqsubset>" 50) and
- inf (infixl "\<sqinter>" 70) and
- sup (infixl "\<squnion>" 65) and
- Inf ("\<Sqinter>_" [900] 900) and
- Sup ("\<Squnion>_" [900] 900)
-
-lemmas mem_simps =
- insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
- mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
- -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
-
-end
--- a/src/HOL/UNITY/Comp/Alloc.thy Wed Jul 22 15:28:49 2009 +0200
+++ b/src/HOL/UNITY/Comp/Alloc.thy Wed Jul 22 18:08:45 2009 +0200
@@ -1021,7 +1021,7 @@
LeadsTo {s. h pfixLe (sub i o allocGiv) s})"
apply (simp only: o_apply sub_def)
apply (insert Alloc_Progress [THEN rename_guarantees_sysOfAlloc_I])
- apply (simp add: o_def del: Set.INT_iff);
+ apply (simp add: o_def del: INT_iff)
apply (erule component_guaranteesD)
apply (auto simp add:
System_Increasing_allocRel [simplified sub_apply o_def]
--- a/src/HOL/UNITY/ProgressSets.thy Wed Jul 22 15:28:49 2009 +0200
+++ b/src/HOL/UNITY/ProgressSets.thy Wed Jul 22 18:08:45 2009 +0200
@@ -44,7 +44,7 @@
lemma UN_in_lattice:
"[|lattice L; !!i. i\<in>I ==> r i \<in> L|] ==> (\<Union>i\<in>I. r i) \<in> L"
-apply (simp add: Set.UN_eq)
+apply (simp add: UN_eq)
apply (blast intro: Union_in_lattice)
done