--- a/src/HOL/Set.thy Mon Jul 20 08:32:07 2009 +0200
+++ b/src/HOL/Set.thy Mon Jul 20 09:52:09 2009 +0200
@@ -10,7 +10,7 @@
text {* A set in HOL is simply a predicate. *}
-subsection {* Basic syntax *}
+subsection {* Basic definitions and syntax *}
global
@@ -19,9 +19,6 @@
consts
Collect :: "('a => bool) => 'a set" -- "comprehension"
"op :" :: "'a => 'a set => bool" -- "membership"
- Ball :: "'a set => ('a => bool) => bool" -- "bounded universal quantifiers"
- Bex :: "'a set => ('a => bool) => bool" -- "bounded existential quantifiers"
- Bex1 :: "'a set => ('a => bool) => bool" -- "bounded unique existential quantifiers"
local
@@ -29,6 +26,10 @@
"op :" ("op :") and
"op :" ("(_/ : _)" [50, 51] 50)
+defs
+ mem_def [code]: "x : S == S x"
+ Collect_def [code]: "Collect P == P"
+
abbreviation
"not_mem x A == ~ (x : A)" -- "non-membership"
@@ -84,6 +85,20 @@
"{x, xs}" == "CONST insert x {xs}"
"{x}" == "CONST insert x {}"
+global
+
+consts
+ Ball :: "'a set => ('a => bool) => bool" -- "bounded universal quantifiers"
+ Bex :: "'a set => ('a => bool) => bool" -- "bounded existential quantifiers"
+ Bex1 :: "'a set => ('a => bool) => bool" -- "bounded unique existential quantifiers"
+
+local
+
+defs
+ Ball_def: "Ball A P == ALL x. x:A --> P(x)"
+ Bex_def: "Bex A P == EX x. x:A & P(x)"
+ Bex1_def: "Bex1 A P == EX! x. x:A & P(x)"
+
syntax
"_Ball" :: "pttrn => 'a set => bool => bool" ("(3ALL _:_./ _)" [0, 0, 10] 10)
"_Bex" :: "pttrn => 'a set => bool => bool" ("(3EX _:_./ _)" [0, 0, 10] 10)
@@ -112,65 +127,18 @@
"EX! x:A. P" == "Bex1 A (%x. P)"
"LEAST x:A. P" => "LEAST x. x:A & P"
-definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
- "INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}"
-
-definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
- "UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}"
-
-definition Inter :: "'a set set \<Rightarrow> 'a set" where
- "Inter S \<equiv> INTER S (\<lambda>x. x)"
-
-definition Union :: "'a set set \<Rightarrow> 'a set" where
- "Union S \<equiv> UNION S (\<lambda>x. x)"
-
-notation (xsymbols)
- Inter ("\<Inter>_" [90] 90) and
- Union ("\<Union>_" [90] 90)
-
subsection {* Additional concrete syntax *}
syntax
"@SetCompr" :: "'a => idts => bool => 'a set" ("(1{_ |/_./ _})")
"@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ :/ _./ _})")
- "@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10)
- "@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10)
- "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 10] 10)
- "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 10] 10)
syntax (xsymbols)
"@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ \<in>/ _./ _})")
- "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10)
- "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10)
- "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
- "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
-
-syntax (latex output)
- "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
- "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
- "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
- "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
translations
"{x:A. P}" => "{x. x:A & P}"
- "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)"
- "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.
-*}
abbreviation
subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
@@ -313,10 +281,7 @@
fun btr' syn [A, Abs abs] =
let val (x, t) = atomic_abs_tr' abs
in Syntax.const syn $ x $ A $ t end
-in
-[(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex"),
- (@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")]
-end
+in [(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex")] end
*}
print_translation {*
@@ -349,30 +314,6 @@
*}
-subsection {* Rules and definitions *}
-
-text {* Isomorphisms between predicates and sets. *}
-
-defs
- mem_def [code]: "x : S == S x"
- Collect_def [code]: "Collect P == P"
-
-defs
- Ball_def: "Ball A P == ALL x. x:A --> P(x)"
- Bex_def: "Bex A P == EX x. x:A & P(x)"
- Bex1_def: "Bex1 A P == EX! x. x:A & P(x)"
-
-definition Pow :: "'a set => 'a set set" where
- Pow_def: "Pow A = {B. B \<le> A}"
-
-definition image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) where
- image_def: "f ` A = {y. EX x:A. y = f(x)}"
-
-abbreviation
- range :: "('a => 'b) => 'b set" where -- "of function"
- "range f == f ` UNIV"
-
-
subsection {* Lemmas and proof tool setup *}
subsubsection {* Relating predicates and sets *}
@@ -671,6 +612,9 @@
subsubsection {* The Powerset operator -- Pow *}
+definition Pow :: "'a set => 'a set set" where
+ Pow_def: "Pow A = {B. B \<le> A}"
+
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
by (simp add: Pow_def)
@@ -846,12 +790,397 @@
by (blast elim: equalityE)
-subsubsection {* Unions of families *}
+subsubsection {* Image of a set under a function *}
+
+text {*
+ Frequently @{term b} does not have the syntactic form of @{term "f x"}.
+*}
+
+definition image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) where
+ image_def [noatp]: "f ` A = {y. EX x:A. y = f(x)}"
+
+abbreviation
+ range :: "('a => 'b) => 'b set" where -- "of function"
+ "range f == f ` UNIV"
+
+lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
+ by (unfold image_def) blast
+
+lemma imageI: "x : A ==> f x : f ` A"
+ by (rule image_eqI) (rule refl)
+
+lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
+ -- {* This version's more effective when we already have the
+ required @{term x}. *}
+ by (unfold image_def) blast
+
+lemma imageE [elim!]:
+ "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
+ -- {* The eta-expansion gives variable-name preservation. *}
+ by (unfold image_def) blast
+
+lemma image_Un: "f`(A Un B) = f`A Un f`B"
+ by blast
+
+lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
+ by blast
+
+lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
+ -- {* This rewrite rule would confuse users if made default. *}
+ by blast
+
+lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
+ apply safe
+ prefer 2 apply fast
+ apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
+ done
+
+lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
+ -- {* Replaces the three steps @{text subsetI}, @{text imageE},
+ @{text hypsubst}, but breaks too many existing proofs. *}
+ by blast
text {*
- @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
+ \medskip Range of a function -- just a translation for image!
+*}
+
+lemma range_eqI: "b = f x ==> b \<in> range f"
+ by simp
+
+lemma rangeI: "f x \<in> range f"
+ by simp
+
+lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
+ by blast
+
+
+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 simp add: UNIV_def)
+
+lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
+ unfolding Inf_Sup by (auto simp add: UNIV_def)
+
+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)"
+
+(* To avoid eta-contraction of body: *)
+print_translation {*
+let
+ fun btr' syn (A :: Abs abs :: ts) =
+ let val (x,t) = atomic_abs_tr' abs
+ in list_comb (Syntax.const syn $ x $ A $ t, ts) end
+ val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
+in
+[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
+end
*}
+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 {* Bool 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
+
+
+subsection {* Fun as complete lattice *}
+
+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 rule (simp add: Inf_fun_def, simp add: empty_def)
+
+lemma Sup_empty_fun:
+ "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
+ by rule (simp add: Sup_fun_def, simp add: empty_def)
+
+
+subsection {* Set as lattice *}
+
+definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
+ "INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}"
+
+definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
+ "UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}"
+
+definition Inter :: "'a set set \<Rightarrow> 'a set" where
+ "Inter S \<equiv> INTER S (\<lambda>x. x)"
+
+definition Union :: "'a set set \<Rightarrow> 'a set" where
+ "Union S \<equiv> UNION S (\<lambda>x. x)"
+
+notation (xsymbols)
+ Inter ("\<Inter>_" [90] 90) and
+ Union ("\<Union>_" [90] 90)
+
+syntax
+ "@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10)
+ "@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10)
+ "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 10] 10)
+ "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 10] 10)
+
+syntax (xsymbols)
+ "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10)
+ "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10)
+ "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)
+ "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
+
+syntax (latex output)
+ "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
+ "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
+ "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
+ "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(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)"
+ "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.
+*}
+
+(* To avoid eta-contraction of body: *)
+print_translation {*
+let
+ fun btr' syn [A, Abs abs] =
+ let val (x, t) = atomic_abs_tr' abs
+ in Syntax.const syn $ x $ A $ t end
+in [(@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")] end
+*}
+
+lemma Inter_image_eq [simp]:
+ "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
+ by (auto simp add: Inter_def INTER_def image_def)
+
+lemma Union_image_eq [simp]:
+ "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
+ by (auto simp add: Union_def UNION_def image_def)
+
+lemma inf_set_eq: "A \<sqinter> B = A \<inter> B"
+ by (simp add: inf_fun_eq inf_bool_eq Int_def Collect_def mem_def)
+
+lemma sup_set_eq: "A \<squnion> B = A \<union> B"
+ by (simp add: sup_fun_eq sup_bool_eq Un_def Collect_def mem_def)
+
+lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
+ apply (fold inf_set_eq sup_set_eq)
+ apply (erule mono_inf)
+ done
+
+lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
+ apply (fold inf_set_eq sup_set_eq)
+ apply (erule mono_sup)
+ done
+
+lemma top_set_eq: "top = UNIV"
+ by (iprover intro!: subset_antisym subset_UNIV top_greatest)
+
+lemma bot_set_eq: "bot = {}"
+ by (iprover intro!: subset_antisym empty_subsetI bot_least)
+
+lemma Inter_eq:
+ "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
+ by (simp add: Inter_def INTER_def)
+
+lemma Union_eq:
+ "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
+ by (simp add: Union_def UNION_def)
+
+lemma Inf_set_eq:
+ "\<Sqinter>S = \<Inter>S"
+proof (rule set_ext)
+ fix x
+ have "(\<forall>Q\<in>{P. \<exists>A\<in>S. P \<longleftrightarrow> x \<in> A}. Q) \<longleftrightarrow> (\<forall>A\<in>S. x \<in> A)"
+ by auto
+ then show "x \<in> \<Sqinter>S \<longleftrightarrow> x \<in> \<Inter>S"
+ by (simp add: Inter_eq Inf_fun_def Inf_bool_def) (simp add: mem_def)
+qed
+
+lemma Sup_set_eq:
+ "\<Squnion>S = \<Union>S"
+proof (rule set_ext)
+ fix x
+ have "(\<exists>Q\<in>{P. \<exists>A\<in>S. P \<longleftrightarrow> x \<in> A}. Q) \<longleftrightarrow> (\<exists>A\<in>S. x \<in> A)"
+ by auto
+ then show "x \<in> \<Squnion>S \<longleftrightarrow> x \<in> \<Union>S"
+ by (simp add: Union_eq Sup_fun_def Sup_bool_def) (simp add: mem_def)
+qed
+
+lemma INFI_set_eq:
+ "(INF x:S. f x) = (\<Inter>x\<in>S. f x)"
+ by (simp add: INFI_def Inf_set_eq)
+
+lemma SUPR_set_eq:
+ "(SUP x:S. f x) = (\<Union>x\<in>S. f x)"
+ by (simp add: SUPR_def Sup_set_eq)
+
+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)
+
+
+subsubsection {* Unions of families *}
+
declare UNION_def [noatp]
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
@@ -873,11 +1202,12 @@
"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
+
subsubsection {* Intersections of families *}
-text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
-
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
by (unfold INTER_def) blast
@@ -932,66 +1262,6 @@
@{prop "X:C"}. *}
by (unfold Inter_def) blast
-text {*
- \medskip Image of a set under a function. Frequently @{term b} does
- not have the syntactic form of @{term "f x"}.
-*}
-
-declare image_def [noatp]
-
-lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
- by (unfold image_def) blast
-
-lemma imageI: "x : A ==> f x : f ` A"
- by (rule image_eqI) (rule refl)
-
-lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
- -- {* This version's more effective when we already have the
- required @{term x}. *}
- by (unfold image_def) blast
-
-lemma imageE [elim!]:
- "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
- -- {* The eta-expansion gives variable-name preservation. *}
- by (unfold image_def) blast
-
-lemma image_Un: "f`(A Un B) = f`A Un f`B"
- by blast
-
-lemma image_eq_UN: "f`A = (UN x:A. {f x})"
- by blast
-
-lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
- by blast
-
-lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
- -- {* This rewrite rule would confuse users if made default. *}
- by blast
-
-lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"
- apply safe
- prefer 2 apply fast
- apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
- done
-
-lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B"
- -- {* Replaces the three steps @{text subsetI}, @{text imageE},
- @{text hypsubst}, but breaks too many existing proofs. *}
- by blast
-
-text {*
- \medskip Range of a function -- just a translation for image!
-*}
-
-lemma range_eqI: "b = f x ==> b \<in> range f"
- by simp
-
-lemma rangeI: "f x \<in> range f"
- by simp
-
-lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
- by blast
-
subsubsection {* Set reasoning tools *}
@@ -1632,15 +1902,9 @@
"u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
by blast
-lemma Union_image_eq [simp]: "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
- by blast
-
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
by blast
-lemma Inter_image_eq [simp]: "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
- by blast
-
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
by auto
@@ -2219,256 +2483,6 @@
unfolding vimage_def Collect_def mem_def ..
-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 (simp add: Inf_insert_simp)
-
-lemma Sup_binary:
- "\<Squnion>{a, b} = a \<squnion> b"
- by (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)"
-
-(* To avoid eta-contraction of body: *)
-print_translation {*
-let
- fun btr' syn (A :: Abs abs :: ts) =
- let val (x,t) = atomic_abs_tr' abs
- in list_comb (Syntax.const syn $ x $ A $ t, ts) end
- val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
-in
-[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
-end
-*}
-
-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 {* Bool 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
- by intro_classes (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
-
-
-subsection {* Fun as complete lattice *}
-
-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
- by intro_classes
- (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 rule (auto simp add: Inf_fun_def)
-
-lemma Sup_empty_fun:
- "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
- by rule (auto simp add: Sup_fun_def)
-
-
-subsection {* Set as lattice *}
-
-lemma inf_set_eq: "A \<sqinter> B = A \<inter> B"
- apply (rule subset_antisym)
- apply (rule Int_greatest)
- apply (rule inf_le1)
- apply (rule inf_le2)
- apply (rule inf_greatest)
- apply (rule Int_lower1)
- apply (rule Int_lower2)
- done
-
-lemma sup_set_eq: "A \<squnion> B = A \<union> B"
- apply (rule subset_antisym)
- apply (rule sup_least)
- apply (rule Un_upper1)
- apply (rule Un_upper2)
- apply (rule Un_least)
- apply (rule sup_ge1)
- apply (rule sup_ge2)
- done
-
-lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
- apply (fold inf_set_eq sup_set_eq)
- apply (erule mono_inf)
- done
-
-lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
- apply (fold inf_set_eq sup_set_eq)
- apply (erule mono_sup)
- done
-
-lemma Inf_set_eq: "\<Sqinter>S = \<Inter>S"
- apply (rule subset_antisym)
- apply (rule Inter_greatest)
- apply (erule Inf_lower)
- apply (rule Inf_greatest)
- apply (erule Inter_lower)
- done
-
-lemma Sup_set_eq: "\<Squnion>S = \<Union>S"
- apply (rule subset_antisym)
- apply (rule Sup_least)
- apply (erule Union_upper)
- apply (rule Union_least)
- apply (erule Sup_upper)
- done
-
-lemma top_set_eq: "top = UNIV"
- by (iprover intro!: subset_antisym subset_UNIV top_greatest)
-
-lemma bot_set_eq: "bot = {}"
- by (iprover intro!: subset_antisym empty_subsetI bot_least)
-
-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)
-
-
subsection {* Misc theorem and ML bindings *}
lemmas equalityI = subset_antisym