--- a/src/HOL/Set.thy Fri Mar 13 19:18:07 2009 +0100
+++ b/src/HOL/Set.thy Sat Mar 14 12:50:29 2009 +0100
@@ -8,28 +8,27 @@
imports Lattices
begin
-subsection {* Basic operations *}
-
-subsubsection {* Comprehension and membership *}
-
text {* A set in HOL is simply a predicate. *}
+
+subsection {* Basic syntax *}
+
global
types 'a set = "'a => bool"
consts
- Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set"
- "op :" :: "'a => 'a set => bool"
+ Collect :: "('a => bool) => 'a set" -- "comprehension"
+ "op :" :: "'a => 'a set => bool" -- "membership"
+ insert :: "'a => 'a set => 'a set"
+ 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"
+ Pow :: "'a set => 'a set set" -- "powerset"
+ image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90)
local
-syntax
- "@Coll" :: "pttrn => bool => 'a set" ("(1{_./ _})")
-
-translations
- "{x. P}" == "Collect (%x. P)"
-
notation
"op :" ("op :") and
"op :" ("(_/ : _)" [50, 51] 50)
@@ -53,51 +52,126 @@
not_mem ("op \<notin>") and
not_mem ("(_/ \<notin> _)" [50, 51] 50)
-defs
- Collect_def [code]: "Collect P \<equiv> P"
- mem_def [code]: "x \<in> S \<equiv> S x"
-
-text {* Relating predicates and sets *}
-
-lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
- by (simp add: Collect_def mem_def)
-
-lemma Collect_mem_eq [simp]: "{x. x:A} = A"
- by (simp add: Collect_def mem_def)
-
-lemma CollectI: "P(a) ==> a : {x. P(x)}"
- by simp
-
-lemma CollectD: "a : {x. P(x)} ==> P(a)"
- by simp
-
-lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
- by simp
-
-lemmas CollectE = CollectD [elim_format]
-
-lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
- apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
- apply (rule Collect_mem_eq)
- apply (rule Collect_mem_eq)
- done
-
-(* Due to Brian Huffman *)
-lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
-by(auto intro:set_ext)
-
-lemma equalityCE [elim]:
- "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
- by blast
-
-lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
- by simp
-
-lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
- by simp
-
-
-subsubsection {* Subset relation, empty and universal set *}
+syntax
+ "@Coll" :: "pttrn => bool => 'a set" ("(1{_./ _})")
+
+translations
+ "{x. P}" == "Collect (%x. P)"
+
+definition empty :: "'a set" ("{}") where
+ "empty \<equiv> {x. False}"
+
+definition UNIV :: "'a set" where
+ "UNIV \<equiv> {x. True}"
+
+syntax
+ "@Finset" :: "args => 'a set" ("{(_)}")
+
+translations
+ "{x, xs}" == "insert x {xs}"
+ "{x}" == "insert x {}"
+
+definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
+ "A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}"
+
+definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
+ "A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}"
+
+notation (xsymbols)
+ "Int" (infixl "\<inter>" 70) and
+ "Un" (infixl "\<union>" 65)
+
+notation (HTML output)
+ "Int" (infixl "\<inter>" 70) and
+ "Un" (infixl "\<union>" 65)
+
+syntax
+ "_Ball" :: "pttrn => 'a set => bool => bool" ("(3ALL _:_./ _)" [0, 0, 10] 10)
+ "_Bex" :: "pttrn => 'a set => bool => bool" ("(3EX _:_./ _)" [0, 0, 10] 10)
+ "_Bex1" :: "pttrn => 'a set => bool => bool" ("(3EX! _:_./ _)" [0, 0, 10] 10)
+ "_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST _:_./ _)" [0, 0, 10] 10)
+
+syntax (HOL)
+ "_Ball" :: "pttrn => 'a set => bool => bool" ("(3! _:_./ _)" [0, 0, 10] 10)
+ "_Bex" :: "pttrn => 'a set => bool => bool" ("(3? _:_./ _)" [0, 0, 10] 10)
+ "_Bex1" :: "pttrn => 'a set => bool => bool" ("(3?! _:_./ _)" [0, 0, 10] 10)
+
+syntax (xsymbols)
+ "_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
+ "_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
+ "_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
+ "_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)
+
+syntax (HTML output)
+ "_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
+ "_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
+ "_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
+
+translations
+ "ALL x:A. P" == "Ball A (%x. P)"
+ "EX x:A. P" == "Bex A (%x. P)"
+ "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
@@ -139,557 +213,12 @@
supset_eq ("op \<supseteq>") and
supset_eq ("(_/ \<supseteq> _)" [50, 51] 50)
-definition empty :: "'a set" ("{}") where
- "empty \<equiv> {x. False}"
-
-definition UNIV :: "'a set" where
- "UNIV \<equiv> {x. True}"
-
-lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
- by (auto simp add: mem_def intro: predicate1I)
-
-text {*
- \medskip Map the type @{text "'a set => anything"} to just @{typ
- 'a}; for overloading constants whose first argument has type @{typ
- "'a set"}.
-*}
-
-lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
- -- {* Rule in Modus Ponens style. *}
- by (unfold mem_def) blast
-
-declare subsetD [intro?] -- FIXME
-
-lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
- -- {* The same, with reversed premises for use with @{text erule} --
- cf @{text rev_mp}. *}
- by (rule subsetD)
-
-declare rev_subsetD [intro?] -- FIXME
-
-text {*
- \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
-*}
-
-ML {*
- fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
-*}
-
-lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
- -- {* Classical elimination rule. *}
- by (unfold mem_def) blast
-
-text {*
- \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
- creates the assumption @{prop "c \<in> B"}.
-*}
-
-ML {*
- fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i
-*}
-
-lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
- by blast
-
-lemma subset_refl [simp,atp]: "A \<subseteq> A"
- by fast
-
-lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
- by blast
-
-lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
- -- {* Anti-symmetry of the subset relation. *}
- by (iprover intro: set_ext subsetD)
-
-text {*
- \medskip Equality rules from ZF set theory -- are they appropriate
- here?
-*}
-
-lemma equalityD1: "A = B ==> A \<subseteq> B"
- by (simp add: subset_refl)
-
-lemma equalityD2: "A = B ==> B \<subseteq> A"
- by (simp add: subset_refl)
-
-text {*
- \medskip Be careful when adding this to the claset as @{text
- subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
- \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
-*}
-
-lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
- by (simp add: subset_refl)
-
-lemma empty_iff [simp]: "(c : {}) = False"
- by (simp add: empty_def)
-
-lemma emptyE [elim!]: "a : {} ==> P"
- by simp
-
-lemma empty_subsetI [iff]: "{} \<subseteq> A"
- -- {* One effect is to delete the ASSUMPTION @{prop "{} \<subseteq> A"} *}
- by blast
-
-lemma bot_set_eq: "bot = {}"
- by (iprover intro!: subset_antisym empty_subsetI bot_least)
-
-lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
- by blast
-
-lemma equals0D: "A = {} ==> a \<notin> A"
- -- {* Use for reasoning about disjointness *}
- by blast
-
-lemma UNIV_I [simp]: "x : UNIV"
- by (simp add: UNIV_def)
-
-declare UNIV_I [intro] -- {* unsafe makes it less likely to cause problems *}
-
-lemma UNIV_witness [intro?]: "EX x. x : UNIV"
- by simp
-
-lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
- by (rule subsetI) (rule UNIV_I)
-
-lemma top_set_eq: "top = UNIV"
- by (iprover intro!: subset_antisym subset_UNIV top_greatest)
-
-lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
- by auto
-
-lemma UNIV_not_empty [iff]: "UNIV ~= {}"
- by blast
-
-lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
- by (unfold less_le) blast
-
-lemma psubsetE [elim!,noatp]:
- "[|A \<subset> B; [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
- by (unfold less_le) blast
-
-lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
- by (simp only: less_le)
-
-lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
- by (simp add: psubset_eq)
-
-lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
-apply (unfold less_le)
-apply (auto dest: subset_antisym)
-done
-
-lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
-apply (unfold less_le)
-apply (auto dest: subsetD)
-done
-
-lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
- by (auto simp add: psubset_eq)
-
-lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
- by (auto simp add: psubset_eq)
-
-lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
-by blast
-
-lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
-by blast
-
-subsubsection {* Intersection and union *}
-
-definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
- "A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}"
-
-definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
- "A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}"
-
-notation (xsymbols)
- "Int" (infixl "\<inter>" 70) and
- "Un" (infixl "\<union>" 65)
-
-notation (HTML output)
- "Int" (infixl "\<inter>" 70) and
- "Un" (infixl "\<union>" 65)
-
-lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
- by (unfold Int_def) blast
-
-lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
- by simp
-
-lemma IntD1: "c : A Int B ==> c:A"
- by simp
-
-lemma IntD2: "c : A Int B ==> c:B"
- by simp
-
-lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
- by simp
-
-lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
- by (unfold Un_def) blast
-
-lemma UnI1 [elim?]: "c:A ==> c : A Un B"
- by simp
-
-lemma UnI2 [elim?]: "c:B ==> c : A Un B"
- by simp
-
-text {*
- \medskip Classical introduction rule: no commitment to @{prop A} vs
- @{prop B}.
-*}
-
-lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
- by auto
-
-lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
- by (unfold Un_def) blast
-
-lemma Int_lower1: "A \<inter> B \<subseteq> A"
- by blast
-
-lemma Int_lower2: "A \<inter> B \<subseteq> B"
- by blast
-
-lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
- by blast
-
-lemma inf_set_eq: "inf A 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 Un_upper1: "A \<subseteq> A \<union> B"
- by blast
-
-lemma Un_upper2: "B \<subseteq> A \<union> B"
- by blast
-
-lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
- by blast
-
-lemma sup_set_eq: "sup A 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 Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
- by blast
-
-lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
- by blast
-
-lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
- by blast
-
-lemma not_psubset_empty [iff]: "\<not> (A \<subset> {})"
- by (unfold less_le) blast
-
-lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
- -- {* supersedes @{text "Collect_False_empty"} *}
- by auto
-
-
-subsubsection {* Complement and set difference *}
-
-instantiation bool :: minus
-begin
-
-definition
- bool_diff_def: "A - B \<longleftrightarrow> A \<and> \<not> B"
-
-instance ..
-
-end
-
-instantiation "fun" :: (type, minus) minus
-begin
-
-definition
- fun_diff_def: "A - B = (\<lambda>x. A x - B x)"
-
-instance ..
-
-end
-
-instantiation bool :: uminus
-begin
-
-definition
- bool_Compl_def: "- A \<longleftrightarrow> \<not> A"
-
-instance ..
-
-end
-
-instantiation "fun" :: (type, uminus) uminus
-begin
-
-definition
- fun_Compl_def: "- A = (\<lambda>x. - A x)"
-
-instance ..
-
-end
-
-lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
- by (simp add: mem_def fun_Compl_def bool_Compl_def)
-
-lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
- by (unfold mem_def fun_Compl_def bool_Compl_def) blast
-
-text {*
- \medskip This form, with negated conclusion, works well with the
- Classical prover. Negated assumptions behave like formulae on the
- right side of the notional turnstile ... *}
-
-lemma ComplD [dest!]: "c : -A ==> c~:A"
- by (simp add: mem_def fun_Compl_def bool_Compl_def)
-
-lemmas ComplE = ComplD [elim_format]
-
-lemma Compl_eq: "- A = {x. ~ x : A}" by blast
-
-lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
- by (simp add: mem_def fun_diff_def bool_diff_def)
-
-lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
- by simp
-
-lemma DiffD1: "c : A - B ==> c : A"
- by simp
-
-lemma DiffD2: "c : A - B ==> c : B ==> P"
- by simp
-
-lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
- by simp
-
-lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
-
-lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
-by blast
-
-lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
- by (unfold less_le) blast
-
-lemma Diff_subset: "A - B \<subseteq> A"
- by blast
-
-lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
-by blast
-
-lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
- by blast
-
-lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
- by blast
-
-
-subsubsection {* Set enumerations *}
-
-global
-
-consts
- insert :: "'a => 'a set => 'a set"
-
-local
-
-defs
- insert_def: "insert a B == {x. x=a} Un B"
-
-syntax
- "@Finset" :: "args => 'a set" ("{(_)}")
-
-translations
- "{x, xs}" == "insert x {xs}"
- "{x}" == "insert x {}"
-
-lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
- by (unfold insert_def) blast
-
-lemma insertI1: "a : insert a B"
- by simp
-
-lemma insertI2: "a : B ==> a : insert b B"
- by simp
-
-lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
- by (unfold insert_def) blast
-
-lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
- -- {* Classical introduction rule. *}
- by auto
-
-lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
- by auto
-
-lemma set_insert:
- assumes "x \<in> A"
- obtains B where "A = insert x B" and "x \<notin> B"
-proof
- from assms show "A = insert x (A - {x})" by blast
-next
- show "x \<notin> A - {x}" by blast
-qed
-
-lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
-by auto
-
-lemma insert_is_Un: "insert a A = {a} Un A"
- -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
- by blast
-
-lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
- by blast
-
-lemmas empty_not_insert = insert_not_empty [symmetric, standard]
-declare empty_not_insert [simp]
-
-lemma insert_absorb: "a \<in> A ==> insert a A = A"
- -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
- -- {* with \emph{quadratic} running time *}
- by blast
-
-lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
- by blast
-
-lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
- by blast
-
-lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
- by blast
-
-lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
- -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
- apply (rule_tac x = "A - {a}" in exI, blast)
- done
-
-lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
- by auto
-
-lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
- by blast
-
-lemma insert_disjoint [simp,noatp]:
- "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
- "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
- by auto
-
-lemma disjoint_insert [simp,noatp]:
- "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
- "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
- by auto
-
-text {* Singletons, using insert *}
-
-lemma singletonI [intro!,noatp]: "a : {a}"
- -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
- by (rule insertI1)
-
-lemma singletonD [dest!,noatp]: "b : {a} ==> b = a"
- by blast
-
-lemmas singletonE = singletonD [elim_format]
-
-lemma singleton_iff: "(b : {a}) = (b = a)"
- by blast
-
-lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
- by blast
-
-lemma singleton_insert_inj_eq [iff,noatp]:
- "({b} = insert a A) = (a = b & A \<subseteq> {b})"
- by blast
-
-lemma singleton_insert_inj_eq' [iff,noatp]:
- "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
- by blast
-
-lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
- by fast
-
-lemma singleton_conv [simp]: "{x. x = a} = {a}"
- by blast
-
-lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
- by blast
-
-lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
- by blast
-
-lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
- by (blast elim: equalityE)
-
-lemma psubset_insert_iff:
- "(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
- by (auto simp add: less_le subset_insert_iff)
-
-lemma subset_insertI: "B \<subseteq> insert a B"
- by (rule subsetI) (erule insertI2)
-
-lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
- by blast
-
-lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
- by blast
-
-
-subsubsection {* Bounded quantifiers and operators *}
-
-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)
- "_Bex1" :: "pttrn => 'a set => bool => bool" ("(3EX! _:_./ _)" [0, 0, 10] 10)
- "_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST _:_./ _)" [0, 0, 10] 10)
-
-syntax (HOL)
- "_Ball" :: "pttrn => 'a set => bool => bool" ("(3! _:_./ _)" [0, 0, 10] 10)
- "_Bex" :: "pttrn => 'a set => bool => bool" ("(3? _:_./ _)" [0, 0, 10] 10)
- "_Bex1" :: "pttrn => 'a set => bool => bool" ("(3?! _:_./ _)" [0, 0, 10] 10)
-
-syntax (xsymbols)
- "_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
- "_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
- "_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
- "_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)
-
-syntax (HTML output)
- "_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
- "_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
- "_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
-
-translations
- "ALL x:A. P" == "Ball A (%x. P)"
- "EX x:A. P" == "Bex A (%x. P)"
- "EX! x:A. P" == "Bex1 A (%x. P)"
- "LEAST x:A. P" => "LEAST x. x:A & P"
+abbreviation
+ range :: "('a => 'b) => 'b set" where -- "of function"
+ "range f == f ` UNIV"
+
+
+subsubsection "Bounded quantifiers"
syntax (output)
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10)
@@ -720,11 +249,11 @@
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
translations
- "\<forall>A\<subset>B. P" => "ALL A. A \<subset> B --> P"
- "\<exists>A\<subset>B. P" => "EX A. A \<subset> B & P"
- "\<forall>A\<subseteq>B. P" => "ALL A. A \<subseteq> B --> P"
- "\<exists>A\<subseteq>B. P" => "EX A. A \<subseteq> B & P"
- "\<exists>!A\<subseteq>B. P" => "EX! A. A \<subseteq> B & P"
+ "\<forall>A\<subset>B. P" => "ALL A. A \<subset> B --> P"
+ "\<exists>A\<subset>B. P" => "EX A. A \<subset> B & P"
+ "\<forall>A\<subseteq>B. P" => "ALL A. A \<subseteq> B --> P"
+ "\<exists>A\<subseteq>B. P" => "EX A. A \<subseteq> B & P"
+ "\<exists>!A\<subseteq>B. P" => "EX! A. A \<subseteq> B & P"
print_translation {*
let
@@ -758,22 +287,13 @@
end
*}
+
text {*
\medskip Translate between @{text "{e | x1...xn. P}"} and @{text
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
only translated if @{text "[0..n] subset bvs(e)"}.
*}
-syntax
- "@SetCompr" :: "'a => idts => bool => 'a set" ("(1{_ |/_./ _})")
- "@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ :/ _./ _})")
-
-syntax (xsymbols)
- "@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ \<in>/ _./ _})")
-
-translations
- "{x:A. P}" => "{x. x:A & P}"
-
parse_translation {*
let
val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
@@ -791,6 +311,18 @@
in [("@SetCompr", setcompr_tr)] end;
*}
+(* 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 Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex"),
+ (@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")]
+end
+*}
+
print_translation {*
let
val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
@@ -820,6 +352,90 @@
in [("Collect", setcompr_tr')] end;
*}
+
+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)"
+
+instantiation "fun" :: (type, minus) minus
+begin
+
+definition
+ fun_diff_def: "A - B = (%x. A x - B x)"
+
+instance ..
+
+end
+
+instantiation bool :: minus
+begin
+
+definition
+ bool_diff_def: "A - B = (A & ~ B)"
+
+instance ..
+
+end
+
+instantiation "fun" :: (type, uminus) uminus
+begin
+
+definition
+ fun_Compl_def: "- A = (%x. - A x)"
+
+instance ..
+
+end
+
+instantiation bool :: uminus
+begin
+
+definition
+ bool_Compl_def: "- A = (~ A)"
+
+instance ..
+
+end
+
+defs
+ Pow_def: "Pow A == {B. B <= A}"
+ insert_def: "insert a B == {x. x=a} Un B"
+ image_def: "f`A == {y. EX x:A. y = f(x)}"
+
+
+subsection {* Lemmas and proof tool setup *}
+
+subsubsection {* Relating predicates and sets *}
+
+lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
+ by (simp add: Collect_def mem_def)
+
+lemma Collect_mem_eq [simp]: "{x. x:A} = A"
+ by (simp add: Collect_def mem_def)
+
+lemma CollectI: "P(a) ==> a : {x. P(x)}"
+ by simp
+
+lemma CollectD: "a : {x. P(x)} ==> P(a)"
+ by simp
+
+lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
+ by simp
+
+lemmas CollectE = CollectD [elim_format]
+
+
+subsubsection {* Bounded quantifiers *}
+
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
by (simp add: Ball_def)
@@ -910,25 +526,8 @@
Addsimprocs [defBALL_regroup, defBEX_regroup];
*}
-text {*
- \medskip Eta-contracting these four rules (to remove @{text P})
- causes them to be ignored because of their interaction with
- congruence rules.
-*}
-
-lemma ball_UNIV [simp]: "Ball UNIV P = All P"
- by (simp add: Ball_def)
-
-lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
- by (simp add: Bex_def)
-
-lemma ball_empty [simp]: "Ball {} P = True"
- by (simp add: Ball_def)
-
-lemma bex_empty [simp]: "Bex {} P = False"
- by (simp add: Bex_def)
-
-text {* Congruence rules *}
+
+subsubsection {* Congruence rules *}
lemma ball_cong:
"A = B ==> (!!x. x:B ==> P x = Q x) ==>
@@ -950,423 +549,347 @@
(EX x:A. P x) = (EX x:B. Q x)"
by (simp add: simp_implies_def Bex_def cong: conj_cong)
-lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
-
-lemma atomize_ball:
- "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
- by (simp only: Ball_def atomize_all atomize_imp)
-
-lemmas [symmetric, rulify] = atomize_ball
- and [symmetric, defn] = atomize_ball
-
-
-subsubsection {* Image of a set under a function. *}
+
+subsubsection {* Subsets *}
+
+lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"
+ by (auto simp add: mem_def intro: predicate1I)
text {*
- Frequently @{term b} does not have the syntactic form of @{term "f x"}.
+ \medskip Map the type @{text "'a set => anything"} to just @{typ
+ 'a}; for overloading constants whose first argument has type @{typ
+ "'a set"}.
*}
-global
-
-consts
- image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90)
-
-local
-
-defs
- image_def [noatp]: "f`A == {y. EX x:A. y = f(x)}"
-
-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
-
-lemma image_empty [simp]: "f`{} = {}"
- by blast
-
-lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
- by blast
-
-lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
- by auto
-
-lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})"
-by auto
-
-lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
- by blast
-
-lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
- by blast
-
-lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
- by blast
-
-
-lemma image_Collect [noatp]: "f ` {x. P x} = {f x | x. P x}"
- -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
- with its implicit quantifier and conjunction. Also image enjoys better
- equational properties than does the RHS. *}
- by blast
-
-lemma if_image_distrib [simp]:
- "(\<lambda>x. if P x then f x else g x) ` S
- = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
- by (auto simp add: image_def)
-
-lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
- by (simp add: image_def)
-
-
-subsection {* Set reasoning tools *}
+lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
+ -- {* Rule in Modus Ponens style. *}
+ by (unfold mem_def) blast
+
+declare subsetD [intro?] -- FIXME
+
+lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
+ -- {* The same, with reversed premises for use with @{text erule} --
+ cf @{text rev_mp}. *}
+ by (rule subsetD)
+
+declare rev_subsetD [intro?] -- FIXME
text {*
- Rewrite rules for boolean case-splitting: faster than @{text
- "split_if [split]"}.
+ \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
+*}
+
+ML {*
+ fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
*}
-lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
- by (rule split_if)
-
-lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
- by (rule split_if)
+lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
+ -- {* Classical elimination rule. *}
+ by (unfold mem_def) blast
+
+lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
text {*
- Split ifs on either side of the membership relation. Not for @{text
- "[simp]"} -- can cause goals to blow up!
+ \medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
+ creates the assumption @{prop "c \<in> B"}.
*}
-lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
- by (rule split_if)
-
-lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
- by (rule split_if [where P="%S. a : S"])
-
-lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
-
-(*Would like to add these, but the existing code only searches for the
- outer-level constant, which in this case is just "op :"; we instead need
- to use term-nets to associate patterns with rules. Also, if a rule fails to
- apply, then the formula should be kept.
- [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
- ("Int", [IntD1,IntD2]),
- ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
- *)
-
ML {*
- val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
-*}
-declaration {* fn _ =>
- Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
+ fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i
*}
-text {* Transitivity rules for calculational reasoning *}
-
-lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
- by (rule subsetD)
-
-lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
- by (rule subsetD)
-
-lemmas basic_trans_rules [trans] =
- order_trans_rules set_rev_mp set_mp
-
-
-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: antisym Inf_greatest Inf_lower)
-
-lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
- by (auto intro: 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
+lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
+ by blast
+
+lemma subset_refl [simp,atp]: "A \<subseteq> A"
+ by fast
+
+lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
+ by blast
+
+
+subsubsection {* Equality *}
+
+lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
+ apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
+ apply (rule Collect_mem_eq)
+ apply (rule Collect_mem_eq)
+ done
+
+(* Due to Brian Huffman *)
+lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
+by(auto intro:set_ext)
+
+lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
+ -- {* Anti-symmetry of the subset relation. *}
+ by (iprover intro: set_ext subsetD)
+
+lemmas equalityI [intro!] = subset_antisym
+
+text {*
+ \medskip Equality rules from ZF set theory -- are they appropriate
+ here?
+*}
+
+lemma equalityD1: "A = B ==> A \<subseteq> B"
+ by (simp add: subset_refl)
+
+lemma equalityD2: "A = B ==> B \<subseteq> A"
+ by (simp add: subset_refl)
+
+text {*
+ \medskip Be careful when adding this to the claset as @{text
+ subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
+ \<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
*}
-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 {* 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> Sup {}"
- unfolding Sup_bool_def by auto
-
-
-subsubsection {* 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)
-
-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 {* Further operations *}
-
-subsubsection {* Big families as specialisation of lattice operations *}
-
-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)"
+lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
+ by (simp add: subset_refl)
+
+lemma equalityCE [elim]:
+ "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
+ by blast
+
+lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
+ by simp
+
+lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
+ by simp
+
+
+subsubsection {* The universal set -- UNIV *}
+
+lemma UNIV_I [simp]: "x : UNIV"
+ by (simp add: UNIV_def)
+
+declare UNIV_I [intro] -- {* unsafe makes it less likely to cause problems *}
+
+lemma UNIV_witness [intro?]: "EX x. x : UNIV"
+ by simp
+
+lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
+ by (rule subsetI) (rule UNIV_I)
+
+text {*
+ \medskip Eta-contracting these two rules (to remove @{text P})
+ causes them to be ignored because of their interaction with
+ congruence rules.
+*}
+
+lemma ball_UNIV [simp]: "Ball UNIV P = All P"
+ by (simp add: Ball_def)
+
+lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
+ by (simp add: Bex_def)
+
+lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
+ by auto
+
+
+subsubsection {* The empty set *}
+
+lemma empty_iff [simp]: "(c : {}) = False"
+ by (simp add: empty_def)
+
+lemma emptyE [elim!]: "a : {} ==> P"
+ by simp
+
+lemma empty_subsetI [iff]: "{} \<subseteq> A"
+ -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
+ by blast
+
+lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
+ by blast
+
+lemma equals0D: "A = {} ==> a \<notin> A"
+ -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
+ by blast
+
+lemma ball_empty [simp]: "Ball {} P = True"
+ by (simp add: Ball_def)
+
+lemma bex_empty [simp]: "Bex {} P = False"
+ by (simp add: Bex_def)
+
+lemma UNIV_not_empty [iff]: "UNIV ~= {}"
+ by (blast elim: equalityE)
+
+
+subsubsection {* The Powerset operator -- Pow *}
+
+lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
+ by (simp add: Pow_def)
+
+lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
+ by (simp add: Pow_def)
+
+lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
+ by (simp add: Pow_def)
+
+lemma Pow_bottom: "{} \<in> Pow B"
+ by simp
+
+lemma Pow_top: "A \<in> Pow A"
+ by (simp add: subset_refl)
+
+
+subsubsection {* Set complement *}
+
+lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
+ by (simp add: mem_def fun_Compl_def bool_Compl_def)
+
+lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
+ by (unfold mem_def fun_Compl_def bool_Compl_def) blast
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.
+ \medskip This form, with negated conclusion, works well with the
+ Classical prover. Negated assumptions behave like formulae on the
+ right side of the notional turnstile ... *}
+
+lemma ComplD [dest!]: "c : -A ==> c~:A"
+ by (simp add: mem_def fun_Compl_def bool_Compl_def)
+
+lemmas ComplE = ComplD [elim_format]
+
+lemma Compl_eq: "- A = {x. ~ x : A}" by blast
+
+
+subsubsection {* Binary union -- Un *}
+
+lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
+ by (unfold Un_def) blast
+
+lemma UnI1 [elim?]: "c:A ==> c : A Un B"
+ by simp
+
+lemma UnI2 [elim?]: "c:B ==> c : A Un B"
+ by simp
+
+text {*
+ \medskip Classical introduction rule: no commitment to @{prop A} vs
+ @{prop B}.
*}
-(* To avoid eta-contraction of body: *)
-(*FIXME integrate with / factor out from similar situations*)
-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 Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex"),
- (@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")]
-end
-*}
+lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
+ by auto
+
+lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
+ by (unfold Un_def) blast
+
+
+subsubsection {* Binary intersection -- Int *}
+
+lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
+ by (unfold Int_def) blast
+
+lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
+ by simp
+
+lemma IntD1: "c : A Int B ==> c:A"
+ by simp
+
+lemma IntD2: "c : A Int B ==> c:B"
+ by simp
+
+lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
+ by simp
+
+
+subsubsection {* Set difference *}
+
+lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
+ by (simp add: mem_def fun_diff_def bool_diff_def)
+
+lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
+ by simp
+
+lemma DiffD1: "c : A - B ==> c : A"
+ by simp
+
+lemma DiffD2: "c : A - B ==> c : B ==> P"
+ by simp
+
+lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
+ by simp
+
+lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
+
+lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
+by blast
+
+
+subsubsection {* Augmenting a set -- insert *}
+
+lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
+ by (unfold insert_def) blast
+
+lemma insertI1: "a : insert a B"
+ by simp
+
+lemma insertI2: "a : B ==> a : insert b B"
+ by simp
+
+lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
+ by (unfold insert_def) blast
+
+lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
+ -- {* Classical introduction rule. *}
+ by auto
+
+lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
+ by auto
+
+lemma set_insert:
+ assumes "x \<in> A"
+ obtains B where "A = insert x B" and "x \<notin> B"
+proof
+ from assms show "A = insert x (A - {x})" by blast
+next
+ show "x \<notin> A - {x}" by blast
+qed
+
+lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
+by auto
+
+subsubsection {* Singletons, using insert *}
+
+lemma singletonI [intro!,noatp]: "a : {a}"
+ -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
+ by (rule insertI1)
+
+lemma singletonD [dest!,noatp]: "b : {a} ==> b = a"
+ by blast
+
+lemmas singletonE = singletonD [elim_format]
+
+lemma singleton_iff: "(b : {a}) = (b = a)"
+ by blast
+
+lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
+ by blast
+
+lemma singleton_insert_inj_eq [iff,noatp]:
+ "({b} = insert a A) = (a = b & A \<subseteq> {b})"
+ by blast
+
+lemma singleton_insert_inj_eq' [iff,noatp]:
+ "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
+ by blast
+
+lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
+ by fast
+
+lemma singleton_conv [simp]: "{x. x = a} = {a}"
+ by blast
+
+lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
+ by blast
+
+lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
+ by blast
+
+lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
+ by (blast elim: equalityE)
+
subsubsection {* Unions of families *}
@@ -1395,9 +918,6 @@
"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 *}
@@ -1457,6 +977,175 @@
@{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 *}
+
+text {*
+ Rewrite rules for boolean case-splitting: faster than @{text
+ "split_if [split]"}.
+*}
+
+lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
+ by (rule split_if)
+
+lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
+ by (rule split_if)
+
+text {*
+ Split ifs on either side of the membership relation. Not for @{text
+ "[simp]"} -- can cause goals to blow up!
+*}
+
+lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
+ by (rule split_if)
+
+lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
+ by (rule split_if [where P="%S. a : S"])
+
+lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
+
+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. *}
+
+(*Would like to add these, but the existing code only searches for the
+ outer-level constant, which in this case is just "op :"; we instead need
+ to use term-nets to associate patterns with rules. Also, if a rule fails to
+ apply, then the formula should be kept.
+ [("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
+ ("Int", [IntD1,IntD2]),
+ ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
+ *)
+
+ML {*
+ val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
+*}
+declaration {* fn _ =>
+ Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
+*}
+
+
+subsubsection {* The ``proper subset'' relation *}
+
+lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
+ by (unfold less_le) blast
+
+lemma psubsetE [elim!,noatp]:
+ "[|A \<subset> B; [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
+ by (unfold less_le) blast
+
+lemma psubset_insert_iff:
+ "(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
+ by (auto simp add: less_le subset_insert_iff)
+
+lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
+ by (simp only: less_le)
+
+lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"
+ by (simp add: psubset_eq)
+
+lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
+apply (unfold less_le)
+apply (auto dest: subset_antisym)
+done
+
+lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
+apply (unfold less_le)
+apply (auto dest: subsetD)
+done
+
+lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
+ by (auto simp add: psubset_eq)
+
+lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"
+ by (auto simp add: psubset_eq)
+
+lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
+ by (unfold less_le) blast
+
+lemma atomize_ball:
+ "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
+ by (simp only: Ball_def atomize_all atomize_imp)
+
+lemmas [symmetric, rulify] = atomize_ball
+ and [symmetric, defn] = atomize_ball
+
+
+subsection {* Further set-theory lemmas *}
+
+subsubsection {* Derived rules involving subsets. *}
+
+text {* @{text insert}. *}
+
+lemma subset_insertI: "B \<subseteq> insert a B"
+ by (rule subsetI) (erule insertI2)
+
+lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
+ by blast
+
+lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
+ by blast
text {* \medskip Big Union -- least upper bound of a set. *}
@@ -1467,14 +1156,6 @@
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
by (iprover intro: subsetI elim: UnionE dest: subsetD)
-lemma Sup_set_eq: "Sup S = \<Union>S"
- apply (rule subset_antisym)
- apply (rule Sup_least)
- apply (erule Union_upper)
- apply (rule Union_least)
- apply (erule Sup_upper)
- done
-
text {* \medskip General union. *}
@@ -1497,21 +1178,76 @@
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
by (iprover intro: InterI subsetI dest: subsetD)
-lemma Inf_set_eq: "Inf 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 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 UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
+
+text {* \medskip Finite Union -- the least upper bound of two sets. *}
+
+lemma Un_upper1: "A \<subseteq> A \<union> B"
+ by blast
+
+lemma Un_upper2: "B \<subseteq> A \<union> B"
+ by blast
+
+lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
+ by blast
+
+
+text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
+
+lemma Int_lower1: "A \<inter> B \<subseteq> A"
+ by blast
+
+lemma Int_lower2: "A \<inter> B \<subseteq> B"
+ by blast
+
+lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
+ by blast
+
+
+text {* \medskip Set difference. *}
+
+lemma Diff_subset: "A - B \<subseteq> A"
+ by blast
+
+lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
+by blast
+
+
+subsubsection {* Equalities involving union, intersection, inclusion, etc. *}
+
+text {* @{text "{}"}. *}
+
+lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
+ -- {* supersedes @{text "Collect_False_empty"} *}
+ by auto
+
+lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
+ by blast
+
+lemma not_psubset_empty [iff]: "\<not> (A < {})"
+ by (unfold less_le) blast
+
+lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
+by blast
+
+lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
+by blast
+
+lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
+ by blast
+
+lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
+ by blast
+
+lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
+ by blast
+
+lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
by blast
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
@@ -1527,59 +1263,97 @@
by blast
-subsubsection {* The Powerset operator -- Pow *}
-
-global
-
-consts
- Pow :: "'a set => 'a set set"
-
-local
-
-defs
- Pow_def: "Pow A == {B. B <= A}"
-
-lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
- by (simp add: Pow_def)
-
-lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"
- by (simp add: Pow_def)
-
-lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"
- by (simp add: Pow_def)
-
-lemma Pow_bottom: "{} \<in> Pow B"
- by simp
-
-lemma Pow_top: "A \<in> Pow A"
- by (simp add: subset_refl)
-
-
-
-subsubsection {* Getting the Contents of a Singleton Set *}
-
-definition contents :: "'a set \<Rightarrow> 'a" where
- [code del]: "contents X = (THE x. X = {x})"
-
-lemma contents_eq [simp]: "contents {x} = x"
- by (simp add: contents_def)
-
-
-subsubsection {* Range of a function -- just a translation for image! *}
-
-abbreviation
- range :: "('a => 'b) => 'b set" where -- "of function"
- "range f == f ` UNIV"
-
-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"
+text {* \medskip @{text insert}. *}
+
+lemma insert_is_Un: "insert a A = {a} Un A"
+ -- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
+ by blast
+
+lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
+ by blast
+
+lemmas empty_not_insert = insert_not_empty [symmetric, standard]
+declare empty_not_insert [simp]
+
+lemma insert_absorb: "a \<in> A ==> insert a A = A"
+ -- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
+ -- {* with \emph{quadratic} running time *}
+ by blast
+
+lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
+ by blast
+
+lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
+ by blast
+
+lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
+ by blast
+
+lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
+ -- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
+ apply (rule_tac x = "A - {a}" in exI, blast)
+ done
+
+lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
+ by auto
+
+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 insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
by blast
+lemma insert_disjoint [simp,noatp]:
+ "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
+ "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
+ by auto
+
+lemma disjoint_insert [simp,noatp]:
+ "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
+ "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
+ by auto
+
+text {* \medskip @{text image}. *}
+
+lemma image_empty [simp]: "f`{} = {}"
+ by blast
+
+lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)"
+ by blast
+
+lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}"
+ by auto
+
+lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})"
+by auto
+
+lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
+ by blast
+
+lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A"
+ by blast
+
+lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
+ by blast
+
+
+lemma image_Collect [noatp]: "f ` {x. P x} = {f x | x. P x}"
+ -- {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS,
+ with its implicit quantifier and conjunction. Also image enjoys better
+ equational properties than does the RHS. *}
+ by blast
+
+lemma if_image_distrib [simp]:
+ "(\<lambda>x. if P x then f x else g x) ` S
+ = (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))"
+ by (auto simp add: image_def)
+
+lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
+ by (simp add: image_def)
+
+
+text {* \medskip @{text range}. *}
+
lemma full_SetCompr_eq [noatp]: "{u. \<exists>x. u = f x} = range f"
by auto
@@ -1587,8 +1361,6 @@
by (subst image_image, simp)
-subsection {* Further rules and properties *}
-
text {* \medskip @{text Int} *}
lemma Int_absorb [simp]: "A \<inter> A = A"
@@ -2276,16 +2048,6 @@
lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"
by blast
-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
-
text {* \medskip Monotonicity of implications. *}
lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"
@@ -2328,12 +2090,15 @@
by iprover
-subsubsection {* Inverse image of a function *}
+subsection {* Inverse image of a function *}
constdefs
vimage :: "('a => 'b) => 'b set => 'a set" (infixr "-`" 90)
[code del]: "f -` B == {x. f x : B}"
+
+subsubsection {* Basic rules *}
+
lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)"
by (unfold vimage_def) blast
@@ -2352,6 +2117,9 @@
lemma vimageD: "a : f -` A ==> f a : A"
by (unfold vimage_def) fast
+
+subsubsection {* Equations *}
+
lemma vimage_empty [simp]: "f -` {} = {}"
by blast
@@ -2416,7 +2184,28 @@
by blast
-subsubsection {* Least value operator *}
+subsection {* Getting the Contents of a Singleton Set *}
+
+definition contents :: "'a set \<Rightarrow> 'a" where
+ [code del]: "contents X = (THE x. X = {x})"
+
+lemma contents_eq [simp]: "contents {x} = x"
+ by (simp add: contents_def)
+
+
+subsection {* Transitivity rules for calculational reasoning *}
+
+lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
+ by (rule subsetD)
+
+lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
+ by (rule subsetD)
+
+lemmas basic_trans_rules [trans] =
+ order_trans_rules set_rev_mp set_mp
+
+
+subsection {* Least value operator *}
lemma Least_mono:
"mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
@@ -2429,7 +2218,7 @@
done
-subsubsection {* Rudimentary code generation *}
+subsection {* Rudimentary code generation *}
lemma empty_code [code]: "{} x \<longleftrightarrow> False"
unfolding empty_def Collect_def ..
@@ -2450,13 +2239,257 @@
unfolding vimage_def Collect_def mem_def ..
-subsection {* Misc theorem and ML bindings *}
-
-lemmas equalityI = subset_antisym
-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. *}
+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: antisym Inf_greatest Inf_lower)
+
+lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
+ by (auto intro: 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> Sup {}"
+ 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 {* Basic ML bindings *}
ML {*
val Ball_def = @{thm Ball_def}