author  nipkow 
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parent 24730  a87d8d31abc0 
child 25360  b8251517f508 
permissions  rwrr 
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(* Title: HOL/Set.thy 
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ID: $Id$ 

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Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel 
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*) 
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header {* Set theory for higherorder logic *} 
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theory Set 
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imports Code_Setup 
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begin 
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text {* A set in HOL is simply a predicate. *} 

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subsection {* Basic syntax *} 
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global 
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typedecl 'a set 
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arities set :: (type) type 
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consts 
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"{}" :: "'a set" ("{}") 
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UNIV :: "'a set" 

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insert :: "'a => 'a set => 'a set" 

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Collect :: "('a => bool) => 'a set"  "comprehension" 

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"op Int" :: "'a set => 'a set => 'a set" (infixl "Int" 70) 
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"op Un" :: "'a set => 'a set => 'a set" (infixl "Un" 65) 

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UNION :: "'a set => ('a => 'b set) => 'b set"  "general union" 
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INTER :: "'a set => ('a => 'b set) => 'b set"  "general intersection" 

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Union :: "'a set set => 'a set"  "union of a set" 

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Inter :: "'a set set => 'a set"  "intersection of a set" 

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Pow :: "'a set => 'a set set"  "powerset" 

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Ball :: "'a set => ('a => bool) => bool"  "bounded universal quantifiers" 

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Bex :: "'a set => ('a => bool) => bool"  "bounded existential quantifiers" 

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Bex1 :: "'a set => ('a => bool) => bool"  "bounded unique existential quantifiers" 
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image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) 
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"op :" :: "'a => 'a set => bool"  "membership" 
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notation 
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"op :" ("op :") and 
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"op :" ("(_/ : _)" [50, 51] 50) 
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local 

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subsection {* Additional concrete syntax *} 
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abbreviation 
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range :: "('a => 'b) => 'b set" where  "of function" 
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"range f == f ` UNIV" 
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abbreviation 
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"not_mem x A == ~ (x : A)"  "nonmembership" 
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notation 
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not_mem ("op ~:") and 
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not_mem ("(_/ ~: _)" [50, 51] 50) 
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notation (xsymbols) 
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"op Int" (infixl "\<inter>" 70) and 
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"op Un" (infixl "\<union>" 65) and 
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"op :" ("op \<in>") and 
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"op :" ("(_/ \<in> _)" [50, 51] 50) and 
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not_mem ("op \<notin>") and 
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not_mem ("(_/ \<notin> _)" [50, 51] 50) and 
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Union ("\<Union>_" [90] 90) and 
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Inter ("\<Inter>_" [90] 90) 
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notation (HTML output) 
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"op Int" (infixl "\<inter>" 70) and 
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"op Un" (infixl "\<union>" 65) and 
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"op :" ("op \<in>") and 
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"op :" ("(_/ \<in> _)" [50, 51] 50) and 
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not_mem ("op \<notin>") and 
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not_mem ("(_/ \<notin> _)" [50, 51] 50) 
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syntax 
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"@Finset" :: "args => 'a set" ("{(_)}") 
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"@Coll" :: "pttrn => bool => 'a set" ("(1{_./ _})") 

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"@SetCompr" :: "'a => idts => bool => 'a set" ("(1{_ /_./ _})") 

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"@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ :/ _./ _})") 
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"@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) 
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"@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10) 

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"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 10] 10) 

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"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 10] 10) 

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"_Ball" :: "pttrn => 'a set => bool => bool" ("(3ALL _:_./ _)" [0, 0, 10] 10) 
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"_Bex" :: "pttrn => 'a set => bool => bool" ("(3EX _:_./ _)" [0, 0, 10] 10) 

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"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3EX! _:_./ _)" [0, 0, 10] 10) 
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"_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST _:_./ _)" [0, 0, 10] 10) 
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syntax (HOL) 
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"_Ball" :: "pttrn => 'a set => bool => bool" ("(3! _:_./ _)" [0, 0, 10] 10) 
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"_Bex" :: "pttrn => 'a set => bool => bool" ("(3? _:_./ _)" [0, 0, 10] 10) 

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"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3?! _:_./ _)" [0, 0, 10] 10) 
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translations 

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"{x, xs}" == "insert x {xs}" 

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"{x}" == "insert x {}" 

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"{x. P}" == "Collect (%x. P)" 
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"{x:A. P}" => "{x. x:A & P}" 
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"UN x y. B" == "UN x. UN y. B" 
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"UN x. B" == "UNION UNIV (%x. B)" 
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"UN x. B" == "UN x:UNIV. B" 
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"INT x y. B" == "INT x. INT y. B" 
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"INT x. B" == "INTER UNIV (%x. B)" 
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"INT x. B" == "INT x:UNIV. B" 
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"UN x:A. B" == "UNION A (%x. B)" 
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"INT x:A. B" == "INTER A (%x. B)" 

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"ALL x:A. P" == "Ball A (%x. P)" 

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"EX x:A. P" == "Bex A (%x. P)" 

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"EX! x:A. P" == "Bex1 A (%x. P)" 
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"LEAST x:A. P" => "LEAST x. x:A & P" 
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syntax (xsymbols) 
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"_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10) 
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"_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10) 
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"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10) 
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"_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10) 
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syntax (HTML output) 
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"_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10) 

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"_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10) 

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"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10) 
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syntax (xsymbols) 
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"@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ \<in>/ _./ _})") 
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"@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) 
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"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) 

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"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 10] 10) 

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"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 10] 10) 

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syntax (latex output) 
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"@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 
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"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 

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"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) 

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"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) 

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text{* 
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Note the difference between ordinary xsymbol syntax of indexed 
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unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}) 
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and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The 
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former does not make the index expression a subscript of the 
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union/intersection symbol because this leads to problems with nested 
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subscripts in Proof General. *} 
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instance set :: (type) ord 
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subset_def: "A \<le> B \<equiv> \<forall>x\<in>A. x \<in> B" 
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psubset_def: "A < B \<equiv> A \<le> B \<and> A \<noteq> B" .. 
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lemmas [code func del] = subset_def psubset_def 
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abbreviation 

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subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 
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"subset \<equiv> less" 
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abbreviation 
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subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 
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"subset_eq \<equiv> less_eq" 
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notation (output) 

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subset ("op <") and 
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subset ("(_/ < _)" [50, 51] 50) and 
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subset_eq ("op <=") and 
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subset_eq ("(_/ <= _)" [50, 51] 50) 
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notation (xsymbols) 

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subset ("op \<subset>") and 
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subset ("(_/ \<subset> _)" [50, 51] 50) and 
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subset_eq ("op \<subseteq>") and 
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subset_eq ("(_/ \<subseteq> _)" [50, 51] 50) 
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notation (HTML output) 

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subset ("op \<subset>") and 
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subset ("(_/ \<subset> _)" [50, 51] 50) and 
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subset_eq ("op \<subseteq>") and 
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subset_eq ("(_/ \<subseteq> _)" [50, 51] 50) 
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abbreviation (input) 

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supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 
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"supset \<equiv> greater" 

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abbreviation (input) 
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supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 
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"supset_eq \<equiv> greater_eq" 

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notation (xsymbols) 

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supset ("op \<supset>") and 

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supset ("(_/ \<supset> _)" [50, 51] 50) and 

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supset_eq ("op \<supseteq>") and 

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supset_eq ("(_/ \<supseteq> _)" [50, 51] 50) 

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subsubsection "Bounded quantifiers" 
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syntax (output) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3EX! _<=_./ _)" [0, 0, 10] 10) 
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syntax (xsymbols) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) 
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syntax (HOL output) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3?! _<=_./ _)" [0, 0, 10] 10) 
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syntax (HTML output) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) 
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translations 
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"\<forall>A\<subset>B. P" => "ALL A. A \<subset> B > P" 
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"\<exists>A\<subset>B. P" => "EX A. A \<subset> B & P" 
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"\<forall>A\<subseteq>B. P" => "ALL A. A \<subseteq> B > P" 
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"\<exists>A\<subseteq>B. P" => "EX A. A \<subseteq> B & P" 
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"\<exists>!A\<subseteq>B. P" => "EX! A. A \<subseteq> B & P" 
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print_translation {* 
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let 
22377  232 
val Type (set_type, _) = @{typ "'a set"}; 
233 
val All_binder = Syntax.binder_name @{const_syntax "All"}; 

234 
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"}; 

235 
val impl = @{const_syntax "op >"}; 

236 
val conj = @{const_syntax "op &"}; 

237 
val sbset = @{const_syntax "subset"}; 

238 
val sbset_eq = @{const_syntax "subset_eq"}; 

21819  239 

240 
val trans = 

241 
[((All_binder, impl, sbset), "_setlessAll"), 

242 
((All_binder, impl, sbset_eq), "_setleAll"), 

243 
((Ex_binder, conj, sbset), "_setlessEx"), 

244 
((Ex_binder, conj, sbset_eq), "_setleEx")]; 

245 

246 
fun mk v v' c n P = 

247 
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v  _ => false) n) 

248 
then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match; 

249 

250 
fun tr' q = (q, 

251 
fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] => 

252 
if T = (set_type) then case AList.lookup (op =) trans (q, c, d) 

253 
of NONE => raise Match 

254 
 SOME l => mk v v' l n P 

255 
else raise Match 

256 
 _ => raise Match); 

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in 
21819  258 
[tr' All_binder, tr' Ex_binder] 
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end 
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*} 
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11979  263 
text {* 
264 
\medskip Translate between @{text "{e  x1...xn. P}"} and @{text 

265 
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is 

266 
only translated if @{text "[0..n] subset bvs(e)"}. 

267 
*} 

268 

269 
parse_translation {* 

270 
let 

271 
val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); 

3947  272 

11979  273 
fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 
274 
 nvars _ = 1; 

275 

276 
fun setcompr_tr [e, idts, b] = 

277 
let 

278 
val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; 

279 
val P = Syntax.const "op &" $ eq $ b; 

280 
val exP = ex_tr [idts, P]; 

17784  281 
in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end; 
11979  282 

283 
in [("@SetCompr", setcompr_tr)] end; 

284 
*} 

923  285 

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(* To avoid etacontraction of body: *) 
11979  287 
print_translation {* 
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let 
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fun btr' syn [A,Abs abs] = 
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let val (x,t) = atomic_abs_tr' abs 
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in Syntax.const syn $ x $ A $ t end 
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in 
13858  293 
[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"), 
294 
("UNION", btr' "@UNION"),("INTER", btr' "@INTER")] 

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end 
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*} 
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print_translation {* 
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let 
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val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY")); 
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fun setcompr_tr' [Abs (abs as (_, _, P))] = 
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let 
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fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1) 
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 check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) = 
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n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso 
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((0 upto (n  1)) subset add_loose_bnos (e, 0, [])) 
13764  308 
 check _ = false 
923  309 

11979  310 
fun tr' (_ $ abs) = 
311 
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] 

312 
in Syntax.const "@SetCompr" $ e $ idts $ Q end; 

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in if check (P, 0) then tr' P 
15535  314 
else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs 
315 
val M = Syntax.const "@Coll" $ x $ t 

316 
in case t of 

317 
Const("op &",_) 

318 
$ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) 

319 
$ P => 

320 
if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M 

321 
 _ => M 

322 
end 

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end; 
11979  324 
in [("Collect", setcompr_tr')] end; 
325 
*} 

326 

327 

328 
subsection {* Rules and definitions *} 

329 

330 
text {* Isomorphisms between predicates and sets. *} 

923  331 

11979  332 
axioms 
17085  333 
mem_Collect_eq: "(a : {x. P(x)}) = P(a)" 
334 
Collect_mem_eq: "{x. x:A} = A" 

17702  335 
finalconsts 
336 
Collect 

337 
"op :" 

11979  338 

339 
defs 

340 
Ball_def: "Ball A P == ALL x. x:A > P(x)" 

341 
Bex_def: "Bex A P == EX x. x:A & P(x)" 

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Bex1_def: "Bex1 A P == EX! x. x:A & P(x)" 
11979  343 

21333  344 
instance set :: (type) minus 
11979  345 
Compl_def: " A == {x. ~x:A}" 
21333  346 
set_diff_def: "A  B == {x. x:A & ~x:B}" .. 
923  347 

22845  348 
lemmas [code func del] = Compl_def set_diff_def 
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923  350 
defs 
11979  351 
Un_def: "A Un B == {x. x:A  x:B}" 
352 
Int_def: "A Int B == {x. x:A & x:B}" 

353 
INTER_def: "INTER A B == {y. ALL x:A. y: B(x)}" 

354 
UNION_def: "UNION A B == {y. EX x:A. y: B(x)}" 

355 
Inter_def: "Inter S == (INT x:S. x)" 

356 
Union_def: "Union S == (UN x:S. x)" 

357 
Pow_def: "Pow A == {B. B <= A}" 

358 
empty_def: "{} == {x. False}" 

359 
UNIV_def: "UNIV == {x. True}" 

360 
insert_def: "insert a B == {x. x=a} Un B" 

361 
image_def: "f`A == {y. EX x:A. y = f(x)}" 

362 

363 

364 
subsection {* Lemmas and proof tool setup *} 

365 

366 
subsubsection {* Relating predicates and sets *} 

367 

17085  368 
declare mem_Collect_eq [iff] Collect_mem_eq [simp] 
369 

12257  370 
lemma CollectI: "P(a) ==> a : {x. P(x)}" 
11979  371 
by simp 
372 

373 
lemma CollectD: "a : {x. P(x)} ==> P(a)" 

374 
by simp 

375 

376 
lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}" 

377 
by simp 

378 

12257  379 
lemmas CollectE = CollectD [elim_format] 
11979  380 

381 

382 
subsubsection {* Bounded quantifiers *} 

383 

384 
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" 

385 
by (simp add: Ball_def) 

386 

387 
lemmas strip = impI allI ballI 

388 

389 
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" 

390 
by (simp add: Ball_def) 

391 

392 
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" 

393 
by (unfold Ball_def) blast 

22139  394 

395 
ML {* bind_thm ("rev_ballE", permute_prems 1 1 @{thm ballE}) *} 

11979  396 

397 
text {* 

398 
\medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and 

399 
@{prop "a:A"}; creates assumption @{prop "P a"}. 

400 
*} 

401 

402 
ML {* 

22139  403 
fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1) 
11979  404 
*} 
405 

406 
text {* 

407 
Gives better instantiation for bound: 

408 
*} 

409 

410 
ML_setup {* 

22139  411 
change_claset (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1)) 
11979  412 
*} 
413 

414 
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" 

415 
 {* Normally the best argument order: @{prop "P x"} constrains the 

416 
choice of @{prop "x:A"}. *} 

417 
by (unfold Bex_def) blast 

418 

13113  419 
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" 
11979  420 
 {* The best argument order when there is only one @{prop "x:A"}. *} 
421 
by (unfold Bex_def) blast 

422 

423 
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" 

424 
by (unfold Bex_def) blast 

425 

426 
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" 

427 
by (unfold Bex_def) blast 

428 

429 
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) > P)" 

430 
 {* Trival rewrite rule. *} 

431 
by (simp add: Ball_def) 

432 

433 
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" 

434 
 {* Dual form for existentials. *} 

435 
by (simp add: Bex_def) 

436 

437 
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" 

438 
by blast 

439 

440 
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" 

441 
by blast 

442 

443 
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" 

444 
by blast 

445 

446 
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" 

447 
by blast 

448 

449 
lemma ball_one_point1 [simp]: "(ALL x:A. x = a > P x) = (a:A > P a)" 

450 
by blast 

451 

452 
lemma ball_one_point2 [simp]: "(ALL x:A. a = x > P x) = (a:A > P a)" 

453 
by blast 

454 

455 
ML_setup {* 

13462  456 
local 
22139  457 
val unfold_bex_tac = unfold_tac @{thms "Bex_def"}; 
18328  458 
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; 
11979  459 
val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; 
460 

22139  461 
val unfold_ball_tac = unfold_tac @{thms "Ball_def"}; 
18328  462 
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; 
11979  463 
val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; 
464 
in 

18328  465 
val defBEX_regroup = Simplifier.simproc (the_context ()) 
13462  466 
"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; 
18328  467 
val defBALL_regroup = Simplifier.simproc (the_context ()) 
13462  468 
"defined BALL" ["ALL x:A. P x > Q x"] rearrange_ball; 
11979  469 
end; 
13462  470 

471 
Addsimprocs [defBALL_regroup, defBEX_regroup]; 

11979  472 
*} 
473 

474 

475 
subsubsection {* Congruence rules *} 

476 

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lemma ball_cong: 
11979  478 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
479 
(ALL x:A. P x) = (ALL x:B. Q x)" 

480 
by (simp add: Ball_def) 

481 

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482 
lemma strong_ball_cong [cong]: 
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"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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484 
(ALL x:A. P x) = (ALL x:B. Q x)" 
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485 
by (simp add: simp_implies_def Ball_def) 
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486 

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487 
lemma bex_cong: 
11979  488 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
489 
(EX x:A. P x) = (EX x:B. Q x)" 

490 
by (simp add: Bex_def cong: conj_cong) 

1273  491 

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492 
lemma strong_bex_cong [cong]: 
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493 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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494 
(EX x:A. P x) = (EX x:B. Q x)" 
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495 
by (simp add: simp_implies_def Bex_def cong: conj_cong) 
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496 

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497 

11979  498 
subsubsection {* Subsets *} 
499 

19295  500 
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B" 
11979  501 
by (simp add: subset_def) 
502 

503 
text {* 

504 
\medskip Map the type @{text "'a set => anything"} to just @{typ 

505 
'a}; for overloading constants whose first argument has type @{typ 

506 
"'a set"}. 

507 
*} 

508 

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509 
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" 
11979  510 
 {* Rule in Modus Ponens style. *} 
511 
by (unfold subset_def) blast 

512 

513 
declare subsetD [intro?]  FIXME 

514 

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515 
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B" 
11979  516 
 {* The same, with reversed premises for use with @{text erule}  
517 
cf @{text rev_mp}. *} 

518 
by (rule subsetD) 

519 

520 
declare rev_subsetD [intro?]  FIXME 

521 

522 
text {* 

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523 
\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}. 
11979  524 
*} 
525 

526 
ML {* 

22139  527 
fun impOfSubs th = th RSN (2, @{thm rev_subsetD}) 
11979  528 
*} 
529 

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530 
lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P" 
11979  531 
 {* Classical elimination rule. *} 
532 
by (unfold subset_def) blast 

533 

534 
text {* 

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535 
\medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and 
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536 
creates the assumption @{prop "c \<in> B"}. 
11979  537 
*} 
538 

539 
ML {* 

22139  540 
fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i 
11979  541 
*} 
542 

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543 
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A" 
11979  544 
by blast 
545 

19175  546 
lemma subset_refl [simp,atp]: "A \<subseteq> A" 
11979  547 
by fast 
548 

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549 
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" 
11979  550 
by blast 
923  551 

2261  552 

11979  553 
subsubsection {* Equality *} 
554 

13865  555 
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B" 
556 
apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals]) 

557 
apply (rule Collect_mem_eq) 

558 
apply (rule Collect_mem_eq) 

559 
done 

560 

15554  561 
(* Due to Brian Huffman *) 
562 
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))" 

563 
by(auto intro:set_ext) 

564 

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565 
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B" 
11979  566 
 {* Antisymmetry of the subset relation. *} 
17589  567 
by (iprover intro: set_ext subsetD) 
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568 

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569 
lemmas equalityI [intro!] = subset_antisym 
11979  570 

571 
text {* 

572 
\medskip Equality rules from ZF set theory  are they appropriate 

573 
here? 

574 
*} 

575 

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576 
lemma equalityD1: "A = B ==> A \<subseteq> B" 
11979  577 
by (simp add: subset_refl) 
578 

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579 
lemma equalityD2: "A = B ==> B \<subseteq> A" 
11979  580 
by (simp add: subset_refl) 
581 

582 
text {* 

583 
\medskip Be careful when adding this to the claset as @{text 

584 
subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{} 

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585 
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! 
11979  586 
*} 
587 

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588 
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P" 
11979  589 
by (simp add: subset_refl) 
923  590 

11979  591 
lemma equalityCE [elim]: 
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592 
"A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P" 
11979  593 
by blast 
594 

595 
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)" 

596 
by simp 

597 

13865  598 
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)" 
599 
by simp 

600 

11979  601 

602 
subsubsection {* The universal set  UNIV *} 

603 

604 
lemma UNIV_I [simp]: "x : UNIV" 

605 
by (simp add: UNIV_def) 

606 

607 
declare UNIV_I [intro]  {* unsafe makes it less likely to cause problems *} 

608 

609 
lemma UNIV_witness [intro?]: "EX x. x : UNIV" 

610 
by simp 

611 

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612 
lemma subset_UNIV [simp]: "A \<subseteq> UNIV" 
11979  613 
by (rule subsetI) (rule UNIV_I) 
2388  614 

11979  615 
text {* 
616 
\medskip Etacontracting these two rules (to remove @{text P}) 

617 
causes them to be ignored because of their interaction with 

618 
congruence rules. 

619 
*} 

620 

621 
lemma ball_UNIV [simp]: "Ball UNIV P = All P" 

622 
by (simp add: Ball_def) 

623 

624 
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P" 

625 
by (simp add: Bex_def) 

626 

627 

628 
subsubsection {* The empty set *} 

629 

630 
lemma empty_iff [simp]: "(c : {}) = False" 

631 
by (simp add: empty_def) 

632 

633 
lemma emptyE [elim!]: "a : {} ==> P" 

634 
by simp 

635 

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636 
lemma empty_subsetI [iff]: "{} \<subseteq> A" 
11979  637 
 {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *} 
638 
by blast 

639 

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640 
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}" 
11979  641 
by blast 
2388  642 

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643 
lemma equals0D: "A = {} ==> a \<notin> A" 
11979  644 
 {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *} 
645 
by blast 

646 

647 
lemma ball_empty [simp]: "Ball {} P = True" 

648 
by (simp add: Ball_def) 

649 

650 
lemma bex_empty [simp]: "Bex {} P = False" 

651 
by (simp add: Bex_def) 

652 

653 
lemma UNIV_not_empty [iff]: "UNIV ~= {}" 

654 
by (blast elim: equalityE) 

655 

656 

12023  657 
subsubsection {* The Powerset operator  Pow *} 
11979  658 

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659 
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)" 
11979  660 
by (simp add: Pow_def) 
661 

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662 
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B" 
11979  663 
by (simp add: Pow_def) 
664 

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665 
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B" 
11979  666 
by (simp add: Pow_def) 
667 

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668 
lemma Pow_bottom: "{} \<in> Pow B" 
11979  669 
by simp 
670 

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671 
lemma Pow_top: "A \<in> Pow A" 
11979  672 
by (simp add: subset_refl) 
2684  673 

2388  674 

11979  675 
subsubsection {* Set complement *} 
676 

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677 
lemma Compl_iff [simp]: "(c \<in> A) = (c \<notin> A)" 
11979  678 
by (unfold Compl_def) blast 
679 

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680 
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> A" 
11979  681 
by (unfold Compl_def) blast 
682 

683 
text {* 

684 
\medskip This form, with negated conclusion, works well with the 

685 
Classical prover. Negated assumptions behave like formulae on the 

686 
right side of the notional turnstile ... *} 

687 

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688 
lemma ComplD [dest!]: "c : A ==> c~:A" 
11979  689 
by (unfold Compl_def) blast 
690 

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691 
lemmas ComplE = ComplD [elim_format] 
11979  692 

693 

694 
subsubsection {* Binary union  Un *} 

923  695 

11979  696 
lemma Un_iff [simp]: "(c : A Un B) = (c:A  c:B)" 
697 
by (unfold Un_def) blast 

698 

699 
lemma UnI1 [elim?]: "c:A ==> c : A Un B" 

700 
by simp 

701 

702 
lemma UnI2 [elim?]: "c:B ==> c : A Un B" 

703 
by simp 

923  704 

11979  705 
text {* 
706 
\medskip Classical introduction rule: no commitment to @{prop A} vs 

707 
@{prop B}. 

708 
*} 

709 

710 
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B" 

711 
by auto 

712 

713 
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P" 

714 
by (unfold Un_def) blast 

715 

716 

12023  717 
subsubsection {* Binary intersection  Int *} 
923  718 

11979  719 
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)" 
720 
by (unfold Int_def) blast 

721 

722 
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B" 

723 
by simp 

724 

725 
lemma IntD1: "c : A Int B ==> c:A" 

726 
by simp 

727 

728 
lemma IntD2: "c : A Int B ==> c:B" 

729 
by simp 

730 

731 
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P" 

732 
by simp 

733 

734 

12023  735 
subsubsection {* Set difference *} 
11979  736 

737 
lemma Diff_iff [simp]: "(c : A  B) = (c:A & c~:B)" 

738 
by (unfold set_diff_def) blast 

923  739 

11979  740 
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A  B" 
741 
by simp 

742 

743 
lemma DiffD1: "c : A  B ==> c : A" 

744 
by simp 

745 

746 
lemma DiffD2: "c : A  B ==> c : B ==> P" 

747 
by simp 

748 

749 
lemma DiffE [elim!]: "c : A  B ==> (c:A ==> c~:B ==> P) ==> P" 

750 
by simp 

751 

752 

753 
subsubsection {* Augmenting a set  insert *} 

754 

755 
lemma insert_iff [simp]: "(a : insert b A) = (a = b  a:A)" 

756 
by (unfold insert_def) blast 

757 

758 
lemma insertI1: "a : insert a B" 

759 
by simp 

760 

761 
lemma insertI2: "a : B ==> a : insert b B" 

762 
by simp 

923  763 

11979  764 
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P" 
765 
by (unfold insert_def) blast 

766 

767 
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B" 

768 
 {* Classical introduction rule. *} 

769 
by auto 

770 

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771 
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A  {x} \<subseteq> B else A \<subseteq> B)" 
11979  772 
by auto 
773 

24730  774 
lemma set_insert: 
775 
assumes "x \<in> A" 

776 
obtains B where "A = insert x B" and "x \<notin> B" 

777 
proof 

778 
from assms show "A = insert x (A  {x})" by blast 

779 
next 

780 
show "x \<notin> A  {x}" by blast 

781 
qed 

782 

25287  783 
lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)" 
784 
by auto 

11979  785 

786 
subsubsection {* Singletons, using insert *} 

787 

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788 
lemma singletonI [intro!,noatp]: "a : {a}" 
11979  789 
 {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *} 
790 
by (rule insertI1) 

791 

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792 
lemma singletonD [dest!,noatp]: "b : {a} ==> b = a" 
11979  793 
by blast 
794 

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795 
lemmas singletonE = singletonD [elim_format] 
11979  796 

797 
lemma singleton_iff: "(b : {a}) = (b = a)" 

798 
by blast 

799 

800 
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b" 

801 
by blast 

802 

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803 
lemma singleton_insert_inj_eq [iff,noatp]: 
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804 
"({b} = insert a A) = (a = b & A \<subseteq> {b})" 
11979  805 
by blast 
806 

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807 
lemma singleton_insert_inj_eq' [iff,noatp]: 
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808 
"(insert a A = {b}) = (a = b & A \<subseteq> {b})" 
11979  809 
by blast 
810 

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811 
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {}  A = {x}" 
11979  812 
by fast 
813 

814 
lemma singleton_conv [simp]: "{x. x = a} = {a}" 

815 
by blast 

816 

817 
lemma singleton_conv2 [simp]: "{x. a = x} = {a}" 

818 
by blast 

923  819 

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820 
lemma diff_single_insert: "A  {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B" 
11979  821 
by blast 
822 

19870  823 
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d  a=d & b=c)" 
824 
by (blast elim: equalityE) 

825 

11979  826 

827 
subsubsection {* Unions of families *} 

828 

829 
text {* 

830 
@{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}. 

831 
*} 

832 

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833 
declare UNION_def [noatp] 
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834 

11979  835 
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" 
836 
by (unfold UNION_def) blast 

837 

838 
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" 

839 
 {* The order of the premises presupposes that @{term A} is rigid; 

840 
@{term b} may be flexible. *} 

841 
by auto 

842 

843 
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" 

844 
by (unfold UNION_def) blast 

923  845 

11979  846 
lemma UN_cong [cong]: 
847 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

848 
by (simp add: UNION_def) 

849 

850 

851 
subsubsection {* Intersections of families *} 

852 

853 
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *} 

854 

855 
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" 

856 
by (unfold INTER_def) blast 

923  857 

11979  858 
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" 
859 
by (unfold INTER_def) blast 

860 

861 
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" 

862 
by auto 

863 

864 
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" 

865 
 {* "Classical" elimination  by the Excluded Middle on @{prop "a:A"}. *} 

866 
by (unfold INTER_def) blast 

867 

868 
lemma INT_cong [cong]: 

869 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" 

870 
by (simp add: INTER_def) 

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871 

923  872 

11979  873 
subsubsection {* Union *} 
874 

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875 
lemma Union_iff [simp,noatp]: "(A : Union C) = (EX X:C. A:X)" 
11979  876 
by (unfold Union_def) blast 
877 

878 
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C" 

879 
 {* The order of the premises presupposes that @{term C} is rigid; 

880 
@{term A} may be flexible. *} 

881 
by auto 

882 

883 
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R" 

884 
by (unfold Union_def) blast 

885 

886 

887 
subsubsection {* Inter *} 

888 

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889 
lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)" 
11979  890 
by (unfold Inter_def) blast 
891 

892 
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C" 

893 
by (simp add: Inter_def) 

894 

895 
text {* 

896 
\medskip A ``destruct'' rule  every @{term X} in @{term C} 

897 
contains @{term A} as an element, but @{prop "A:X"} can hold when 

898 
@{prop "X:C"} does not! This rule is analogous to @{text spec}. 

899 
*} 

900 

901 
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X" 

902 
by auto 

903 

904 
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R" 

905 
 {* ``Classical'' elimination rule  does not require proving 

906 
@{prop "X:C"}. *} 

907 
by (unfold Inter_def) blast 

908 

909 
text {* 

910 
\medskip Image of a set under a function. Frequently @{term b} does 

911 
not have the syntactic form of @{term "f x"}. 

912 
*} 

913 

24286
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ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

914 
declare image_def [noatp] 
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

915 

11979  916 
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A" 
917 
by (unfold image_def) blast 

918 

919 
lemma imageI: "x : A ==> f x : f ` A" 

920 
by (rule image_eqI) (rule refl) 

921 

922 
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A" 

923 
 {* This version's more effective when we already have the 

924 
required @{term x}. *} 

925 
by (unfold image_def) blast 

926 

927 
lemma imageE [elim!]: 

928 
"b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P" 

929 
 {* The etaexpansion gives variablename preservation. *} 

930 
by (unfold image_def) blast 

931 

932 
lemma image_Un: "f`(A Un B) = f`A Un f`B" 

933 
by blast 

934 

935 
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)" 

936 
by blast 

937 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

938 
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)" 
11979  939 
 {* This rewrite rule would confuse users if made default. *} 
940 
by blast 

941 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

942 
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)" 
11979  943 
apply safe 
944 
prefer 2 apply fast 

14208  945 
apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast) 
11979  946 
done 
947 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

948 
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B" 
11979  949 
 {* Replaces the three steps @{text subsetI}, @{text imageE}, 
950 
@{text hypsubst}, but breaks too many existing proofs. *} 

951 
by blast 

952 

953 
text {* 

954 
\medskip Range of a function  just a translation for image! 

955 
*} 

956 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

957 
lemma range_eqI: "b = f x ==> b \<in> range f" 
11979  958 
by simp 
959 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

960 
lemma rangeI: "f x \<in> range f" 
11979  961 
by simp 
962 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

963 
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P" 
11979  964 
by blast 
965 

966 

967 
subsubsection {* Set reasoning tools *} 

968 

969 
text {* 

970 
Rewrite rules for boolean casesplitting: faster than @{text 

971 
"split_if [split]"}. 

972 
*} 

973 

974 
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q > x = b) & (~ Q > y = b))" 

975 
by (rule split_if) 

976 

977 
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q > a = x) & (~ Q > a = y))" 

978 
by (rule split_if) 

979 

980 
text {* 

981 
Split ifs on either side of the membership relation. Not for @{text 

982 
"[simp]"}  can cause goals to blow up! 

983 
*} 

984 

985 
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q > x : b) & (~ Q > y : b))" 

986 
by (rule split_if) 

987 

988 
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q > a : x) & (~ Q > a : y))" 

989 
by (rule split_if) 

990 

991 
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 

992 

993 
lemmas mem_simps = 

994 
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff 

995 
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff 

996 
 {* Each of these has ALREADY been added @{text "[simp]"} above. *} 

997 

998 
(*Would like to add these, but the existing code only searches for the 

999 
outerlevel constant, which in this case is just "op :"; we instead need 

1000 
to use termnets to associate patterns with rules. Also, if a rule fails to 

1001 
apply, then the formula should be kept. 

19233
77ca20b0ed77
renamed HOL +  * etc. to HOL.plus HOL.minus HOL.times etc.
haftmann
parents:
19175
diff
changeset

1002 
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]), 
11979  1003 
("op Int", [IntD1,IntD2]), 
1004 
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] 

1005 
*) 

1006 

1007 
ML_setup {* 

22139  1008 
val mksimps_pairs = [("Ball", @{thms bspec})] @ mksimps_pairs; 
17875  1009 
change_simpset (fn ss => ss setmksimps (mksimps mksimps_pairs)); 
11979  1010 
*} 
1011 

1012 

1013 
subsubsection {* The ``proper subset'' relation *} 

1014 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1015 
lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B" 
11979  1016 
by (unfold psubset_def) blast 
1017 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1018 
lemma psubsetE [elim!,noatp]: 
13624  1019 
"[A \<subset> B; [A \<subseteq> B; ~ (B\<subseteq>A)] ==> R] ==> R" 
1020 
by (unfold psubset_def) blast 

1021 

11979  1022 
lemma psubset_insert_iff: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1023 
"(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)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1024 
by (auto simp add: psubset_def subset_insert_iff) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1025 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1026 
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)" 
11979  1027 
by (simp only: psubset_def) 
1028 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1029 
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B" 
11979  1030 
by (simp add: psubset_eq) 
1031 

14335  1032 
lemma psubset_trans: "[ A \<subset> B; B \<subset> C ] ==> A \<subset> C" 
1033 
apply (unfold psubset_def) 

1034 
apply (auto dest: subset_antisym) 

1035 
done 

1036 

1037 
lemma psubsetD: "[ A \<subset> B; c \<in> A ] ==> c \<in> B" 

1038 
apply (unfold psubset_def) 

1039 
apply (auto dest: subsetD) 

1040 
done 

1041 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1042 
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C" 
11979  1043 
by (auto simp add: psubset_eq) 
1044 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1045 
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C" 
11979  1046 
by (auto simp add: psubset_eq) 
1047 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1048 
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B  A)" 
11979  1049 
by (unfold psubset_def) blast 
1050 

1051 
lemma atomize_ball: 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1052 
"(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)" 
11979  1053 
by (simp only: Ball_def atomize_all atomize_imp) 
1054 

18832  1055 
lemmas [symmetric, rulify] = atomize_ball 
1056 
and [symmetric, defn] = atomize_ball 

11979  1057 

1058 

22455  1059 
subsection {* Further settheory lemmas *} 
1060 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1061 
subsubsection {* Derived rules involving subsets. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1062 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1063 
text {* @{text insert}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1064 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1065 
lemma subset_insertI: "B \<subseteq> insert a B" 
23878  1066 
by (rule subsetI) (erule insertI2) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1067 

14302  1068 
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B" 
23878  1069 
by blast 
14302  1070 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1071 
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1072 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1073 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1074 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1075 
text {* \medskip Big Union  least upper bound of a set. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1076 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1077 
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A" 
17589  1078 
by (iprover intro: subsetI UnionI) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1079 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1080 
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C" 
17589  1081 
by (iprover intro: subsetI elim: UnionE dest: subsetD) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1082 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1083 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1084 
text {* \medskip General union. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1085 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1086 
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1087 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1088 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1089 
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C" 
17589  1090 
by (iprover intro: subsetI elim: UN_E dest: subsetD) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1091 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1092 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1093 
text {* \medskip Big Intersection  greatest lower bound of a set. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1094 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1095 
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1096 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1097 

14551  1098 
lemma Inter_subset: 
1099 
"[ !!X. X \<in> A ==> X \<subseteq> B; A ~= {} ] ==> \<Inter>A \<subseteq> B" 

1100 
by blast 

1101 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1102 
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A" 
17589  1103 
by (iprover intro: InterI subsetI dest: subsetD) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1104 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1105 
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1106 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1107 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1108 
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)" 
17589  1109 
by (iprover intro: INT_I subsetI dest: subsetD) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1110 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1111 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1112 
text {* \medskip Finite Union  the least upper bound of two sets. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1113 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1114 
lemma Un_upper1: "A \<subseteq> A \<union> B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1115 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1116 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1117 
lemma Un_upper2: "B \<subseteq> A \<union> B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1118 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1119 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1120 
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1121 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1122 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1123 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1124 
text {* \medskip Finite Intersection  the greatest lower bound of two sets. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1125 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1126 
lemma Int_lower1: "A \<inter> B \<subseteq> A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1127 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1128 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1129 
lemma Int_lower2: "A \<inter> B \<subseteq> B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1130 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1131 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1132 
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1133 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1134 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1135 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1136 
text {* \medskip Set difference. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1137 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1138 
lemma Diff_subset: "A  B \<subseteq> A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1139 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1140 

14302  1141 
lemma Diff_subset_conv: "(A  B \<subseteq> C) = (A \<subseteq> B \<union> C)" 
1142 
by blast 

1143 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1144 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1145 
subsubsection {* Equalities involving union, intersection, inclusion, etc. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1146 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1147 
text {* @{text "{}"}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1148 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1149 
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1150 
 {* supersedes @{text "Collect_False_empty"} *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1151 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1152 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1153 
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1154 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1155 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1156 
lemma not_psubset_empty [iff]: "\<not> (A < {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1157 
by (unfold psubset_def) blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1158 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1159 
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)" 
18423  1160 
by blast 
1161 

1162 
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)" 

1163 
by blast 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1164 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1165 
lemma Collect_neg_eq: "{x. \<not> P x} =  {x. P x}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1166 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1167 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1168 
lemma Collect_disj_eq: "{x. P x  Q x} = {x. P x} \<union> {x. Q x}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1169 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1170 

14812  1171 
lemma Collect_imp_eq: "{x. P x > Q x} = {x. P x} \<union> {x. Q x}" 
1172 
by blast 

1173 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1174 
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1175 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1176 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1177 
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1178 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1179 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1180 
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1181 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1182 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1183 
lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1184 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1185 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1186 
lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1187 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1188 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1189 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1190 
text {* \medskip @{text insert}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1191 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1192 
lemma insert_is_Un: "insert a A = {a} Un A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1193 
 {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1194 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1195 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1196 
lemma insert_not_empty [simp]: "insert a A \<noteq> {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1197 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1198 

17715  1199 
lemmas empty_not_insert = insert_not_empty [symmetric, standard] 
1200 
declare empty_not_insert [simp] 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1201 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1202 
lemma insert_absorb: "a \<in> A ==> insert a A = A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1203 
 {* @{text "[simp]"} causes recursive calls when there are nested inserts *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1204 
 {* with \emph{quadratic} running time *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1205 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1206 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1207 
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1208 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1209 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1210 
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1211 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1212 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1213 
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1214 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1215 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1216 
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1217 
 {* use new @{text B} rather than @{text "A  {a}"} to avoid infinite unfolding *} 
14208  1218 
apply (rule_tac x = "A  {a}" in exI, blast) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1219 
done 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1220 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1221 
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a > P u}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1222 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1223 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1224 
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1225 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1226 

14302  1227 
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)" 
14742  1228 
by blast 
14302  1229 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1230 
lemma insert_disjoint [simp,noatp]: 
13103
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1231 
"(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})" 
14742  1232 
"({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)" 
16773  1233 
by auto 
13103
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1234 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1235 
lemma disjoint_insert [simp,noatp]: 
13103
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1236 
"(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})" 
14742  1237 
"({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)" 
16773  1238 
by auto 
14742  1239 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1240 
text {* \medskip @{text image}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1241 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1242 
lemma image_empty [simp]: "f`{} = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1243 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1244 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1245 
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f`B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1246 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1247 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1248 
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}" 
16773  1249 
by auto 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1250 

21316  1251 
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})" 
21312  1252 
by auto 
1253 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1254 
lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1255 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1256 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1257 
lemma insert_image [simp]: "x \<in> A ==> insert (f x) (f`A) = f`A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1258 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1259 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1260 
lemma image_is_empty [iff]: "(f`A = {}) = (A = {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1261 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1262 

16773  1263 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1264 
lemma image_Collect [noatp]: "f ` {x. P x} = {f x  x. P x}" 
16773  1265 
 {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, 
1266 
with its implicit quantifier and conjunction. Also image enjoys better 

1267 
equational properties than does the RHS. *} 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1268 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1269 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1270 
lemma if_image_distrib [simp]: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1271 
"(\<lambda>x. if P x then f x else g x) ` S 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1272 
= (f ` (S \<inter> {x. P x})) \<union> (g ` (S \<inter> {x. \<not> P x}))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1273 
by (auto simp add: image_def) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1274 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1275 
lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1276 
by (simp add: image_def) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1277 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1278 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1279 
text {* \medskip @{text range}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1280 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1281 
lemma full_SetCompr_eq [noatp]: "{u. \<exists>x. u = f x} = range f" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1282 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1283 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1284 
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g" 
14208  1285 
by (subst image_image, simp) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1286 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1287 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1288 
text {* \medskip @{text Int} *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1289 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1290 
lemma Int_absorb [simp]: "A \<inter> A = A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1291 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1292 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1293 
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1294 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1295 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1296 
lemma Int_commute: "A \<inter> B = B \<inter> A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1297 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1298 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1299 
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1300 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1301 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1302 
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1303 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1304 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1305 
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1306 
 {* Intersection is an ACoperator *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1307 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1308 
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1309 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1310 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1311 
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1312 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1313 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1314 
lemma Int_empty_left [simp]: "{} \<inter> B = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1315 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1316 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1317 
lemma Int_empty_right [simp]: "A \<inter> {} = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1318 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1319 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1320 
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1321 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1322 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1323 
lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1324 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1325 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1326 
lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1327 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1328 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1329 
lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1330 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1331 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1332 
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1333 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1334 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1335 
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1336 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1337 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1338 
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1339 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1340 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

1341 
lemma Int_UNIV [simp,noatp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1342 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1343 

15102  1344 
lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)" 
12897 