author  haftmann 
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parent 21316  4d913b8bccf1 
child 21384  2b58b308300c 
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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15131  8 
theory Set 
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imports Lattices 
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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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Int :: "'a set => 'a set => 'a set" (infixl 70) 

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Un :: "'a set => 'a set => 'a set" (infixl 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 :") 
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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"  "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 ~:") 
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not_mem ("(_/ ~: _)" [50, 51] 50) 
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notation (xsymbols) 
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"op Int" (infixl "\<inter>" 70) 
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"op Un" (infixl "\<union>" 65) 
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"op :" ("op \<in>") 
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"op :" ("(_/ \<in> _)" [50, 51] 50) 
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not_mem ("op \<notin>") 
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not_mem ("(_/ \<notin> _)" [50, 51] 50) 
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Union ("\<Union>_" [90] 90) 
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Inter ("\<Inter>_" [90] 90) 
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notation (HTML output) 
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"op Int" (infixl "\<inter>" 70) 
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"op Un" (infixl "\<union>" 65) 
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"op :" ("op \<in>") 
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"op :" ("(_/ \<in> _)" [50, 51] 50) 
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not_mem ("op \<notin>") 
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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 _./ _)" 10) 
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"@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" 10) 

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

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"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" 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>_./ _)" 10) 
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"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" 10) 

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

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

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

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"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" 10) 

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

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"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" 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 <= B == \<forall>x\<in>A. x \<in> B" 

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psubset_def: "A < B == (A::'a set) <= B & ~ A=B" .. 

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abbreviation 

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subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" 

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"subset == less" 

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subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" 

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"subset_eq == less_eq" 

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notation (output) 

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subset ("op <") 

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subset ("(_/ < _)" [50, 51] 50) 

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subset_eq ("op <=") 

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subset_eq ("(_/ <= _)" [50, 51] 50) 

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notation (xsymbols) 

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subset ("op \<subset>") 

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

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subset_eq ("op \<subseteq>") 

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subset_eq ("(_/ \<subseteq> _)" [50, 51] 50) 

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notation (HTML output) 

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subset ("op \<subset>") 

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

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subset_eq ("op \<subseteq>") 

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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" (infixl "\<supset>" 50) 

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"supset == greater" 

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supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "\<supseteq>" 50) 

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"supset_eq == greater_eq" 

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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 
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fun 
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all_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
19637  223 
Const("op >",_) $ (Const ("less",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
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(if v=v' andalso T="set" 
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then Syntax.const "_setlessAll" $ Syntax.mark_bound v' $ n $ P 
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else raise Match) 
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 all_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
19637  229 
Const("op >",_) $ (Const ("less_eq",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
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(if v=v' andalso T="set" 
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then Syntax.const "_setleAll" $ Syntax.mark_bound v' $ n $ P 
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else raise Match); 
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fun 
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ex_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
19637  236 
Const("op &",_) $ (Const ("less",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
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(if v=v' andalso T="set" 
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then Syntax.const "_setlessEx" $ Syntax.mark_bound v' $ n $ P 
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else raise Match) 
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 ex_tr' [Const ("_bound",_) $ Free (v,Type(T,_)), 
19637  242 
Const("op &",_) $ (Const ("less_eq",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
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(if v=v' andalso T="set" 
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then Syntax.const "_setleEx" $ Syntax.mark_bound v' $ n $ P 
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else raise Match) 
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in 
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[("ALL ", all_tr'), ("EX ", ex_tr')] 
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end 
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*} 
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11979  253 
text {* 
254 
\medskip Translate between @{text "{e  x1...xn. P}"} and @{text 

255 
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is 

256 
only translated if @{text "[0..n] subset bvs(e)"}. 

257 
*} 

258 

259 
parse_translation {* 

260 
let 

261 
val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); 

3947  262 

11979  263 
fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 
264 
 nvars _ = 1; 

265 

266 
fun setcompr_tr [e, idts, b] = 

267 
let 

268 
val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; 

269 
val P = Syntax.const "op &" $ eq $ b; 

270 
val exP = ex_tr [idts, P]; 

17784  271 
in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end; 
11979  272 

273 
in [("@SetCompr", setcompr_tr)] end; 

274 
*} 

923  275 

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(* To avoid etacontraction of body: *) 
11979  277 
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  283 
[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"), 
284 
("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  298 
 check _ = false 
923  299 

11979  300 
fun tr' (_ $ abs) = 
301 
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] 

302 
in Syntax.const "@SetCompr" $ e $ idts $ Q end; 

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in if check (P, 0) then tr' P 
15535  304 
else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs 
305 
val M = Syntax.const "@Coll" $ x $ t 

306 
in case t of 

307 
Const("op &",_) 

308 
$ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) 

309 
$ P => 

310 
if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M 

311 
 _ => M 

312 
end 

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end; 
11979  314 
in [("Collect", setcompr_tr')] end; 
315 
*} 

316 

317 

318 
subsection {* Rules and definitions *} 

319 

320 
text {* Isomorphisms between predicates and sets. *} 

923  321 

11979  322 
axioms 
17085  323 
mem_Collect_eq: "(a : {x. P(x)}) = P(a)" 
324 
Collect_mem_eq: "{x. x:A} = A" 

17702  325 
finalconsts 
326 
Collect 

327 
"op :" 

11979  328 

329 
defs 

330 
Ball_def: "Ball A P == ALL x. x:A > P(x)" 

331 
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  333 

21333  334 
instance set :: (type) minus 
11979  335 
Compl_def: " A == {x. ~x:A}" 
21333  336 
set_diff_def: "A  B == {x. x:A & ~x:B}" .. 
923  337 

338 
defs 

11979  339 
Un_def: "A Un B == {x. x:A  x:B}" 
340 
Int_def: "A Int B == {x. x:A & x:B}" 

341 
INTER_def: "INTER A B == {y. ALL x:A. y: B(x)}" 

342 
UNION_def: "UNION A B == {y. EX x:A. y: B(x)}" 

343 
Inter_def: "Inter S == (INT x:S. x)" 

344 
Union_def: "Union S == (UN x:S. x)" 

345 
Pow_def: "Pow A == {B. B <= A}" 

346 
empty_def: "{} == {x. False}" 

347 
UNIV_def: "UNIV == {x. True}" 

348 
insert_def: "insert a B == {x. x=a} Un B" 

349 
image_def: "f`A == {y. EX x:A. y = f(x)}" 

350 

351 

352 
subsection {* Lemmas and proof tool setup *} 

353 

354 
subsubsection {* Relating predicates and sets *} 

355 

17085  356 
declare mem_Collect_eq [iff] Collect_mem_eq [simp] 
357 

12257  358 
lemma CollectI: "P(a) ==> a : {x. P(x)}" 
11979  359 
by simp 
360 

361 
lemma CollectD: "a : {x. P(x)} ==> P(a)" 

362 
by simp 

363 

364 
lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}" 

365 
by simp 

366 

12257  367 
lemmas CollectE = CollectD [elim_format] 
11979  368 

369 

370 
subsubsection {* Bounded quantifiers *} 

371 

372 
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" 

373 
by (simp add: Ball_def) 

374 

375 
lemmas strip = impI allI ballI 

376 

377 
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" 

378 
by (simp add: Ball_def) 

379 

380 
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" 

381 
by (unfold Ball_def) blast 

14098  382 
ML {* bind_thm("rev_ballE",permute_prems 1 1 (thm "ballE")) *} 
11979  383 

384 
text {* 

385 
\medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and 

386 
@{prop "a:A"}; creates assumption @{prop "P a"}. 

387 
*} 

388 

389 
ML {* 

390 
local val ballE = thm "ballE" 

391 
in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end; 

392 
*} 

393 

394 
text {* 

395 
Gives better instantiation for bound: 

396 
*} 

397 

398 
ML_setup {* 

17875  399 
change_claset (fn cs => cs addbefore ("bspec", datac (thm "bspec") 1)); 
11979  400 
*} 
401 

402 
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" 

403 
 {* Normally the best argument order: @{prop "P x"} constrains the 

404 
choice of @{prop "x:A"}. *} 

405 
by (unfold Bex_def) blast 

406 

13113  407 
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" 
11979  408 
 {* The best argument order when there is only one @{prop "x:A"}. *} 
409 
by (unfold Bex_def) blast 

410 

411 
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" 

412 
by (unfold Bex_def) blast 

413 

414 
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" 

415 
by (unfold Bex_def) blast 

416 

417 
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) > P)" 

418 
 {* Trival rewrite rule. *} 

419 
by (simp add: Ball_def) 

420 

421 
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" 

422 
 {* Dual form for existentials. *} 

423 
by (simp add: Bex_def) 

424 

425 
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" 

426 
by blast 

427 

428 
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" 

429 
by blast 

430 

431 
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" 

432 
by blast 

433 

434 
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" 

435 
by blast 

436 

437 
lemma ball_one_point1 [simp]: "(ALL x:A. x = a > P x) = (a:A > P a)" 

438 
by blast 

439 

440 
lemma ball_one_point2 [simp]: "(ALL x:A. a = x > P x) = (a:A > P a)" 

441 
by blast 

442 

443 
ML_setup {* 

13462  444 
local 
18328  445 
val unfold_bex_tac = unfold_tac [thm "Bex_def"]; 
446 
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; 

11979  447 
val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; 
448 

18328  449 
val unfold_ball_tac = unfold_tac [thm "Ball_def"]; 
450 
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; 

11979  451 
val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; 
452 
in 

18328  453 
val defBEX_regroup = Simplifier.simproc (the_context ()) 
13462  454 
"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; 
18328  455 
val defBALL_regroup = Simplifier.simproc (the_context ()) 
13462  456 
"defined BALL" ["ALL x:A. P x > Q x"] rearrange_ball; 
11979  457 
end; 
13462  458 

459 
Addsimprocs [defBALL_regroup, defBEX_regroup]; 

11979  460 
*} 
461 

462 

463 
subsubsection {* Congruence rules *} 

464 

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lemma ball_cong: 
11979  466 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
467 
(ALL x:A. P x) = (ALL x:B. Q x)" 

468 
by (simp add: Ball_def) 

469 

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470 
lemma strong_ball_cong [cong]: 
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"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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472 
(ALL x:A. P x) = (ALL x:B. Q x)" 
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473 
by (simp add: simp_implies_def Ball_def) 
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474 

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475 
lemma bex_cong: 
11979  476 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
477 
(EX x:A. P x) = (EX x:B. Q x)" 

478 
by (simp add: Bex_def cong: conj_cong) 

1273  479 

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480 
lemma strong_bex_cong [cong]: 
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481 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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482 
(EX x:A. P x) = (EX x:B. Q x)" 
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483 
by (simp add: simp_implies_def Bex_def cong: conj_cong) 
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484 

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485 

11979  486 
subsubsection {* Subsets *} 
487 

19295  488 
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B" 
11979  489 
by (simp add: subset_def) 
490 

491 
text {* 

492 
\medskip Map the type @{text "'a set => anything"} to just @{typ 

493 
'a}; for overloading constants whose first argument has type @{typ 

494 
"'a set"}. 

495 
*} 

496 

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497 
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" 
11979  498 
 {* Rule in Modus Ponens style. *} 
499 
by (unfold subset_def) blast 

500 

501 
declare subsetD [intro?]  FIXME 

502 

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503 
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B" 
11979  504 
 {* The same, with reversed premises for use with @{text erule}  
505 
cf @{text rev_mp}. *} 

506 
by (rule subsetD) 

507 

508 
declare rev_subsetD [intro?]  FIXME 

509 

510 
text {* 

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511 
\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}. 
11979  512 
*} 
513 

514 
ML {* 

515 
local val rev_subsetD = thm "rev_subsetD" 

516 
in fun impOfSubs th = th RSN (2, rev_subsetD) end; 

517 
*} 

518 

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519 
lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P" 
11979  520 
 {* Classical elimination rule. *} 
521 
by (unfold subset_def) blast 

522 

523 
text {* 

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524 
\medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and 
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525 
creates the assumption @{prop "c \<in> B"}. 
11979  526 
*} 
527 

528 
ML {* 

529 
local val subsetCE = thm "subsetCE" 

530 
in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end; 

531 
*} 

532 

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533 
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A" 
11979  534 
by blast 
535 

19175  536 
lemma subset_refl [simp,atp]: "A \<subseteq> A" 
11979  537 
by fast 
538 

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539 
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" 
11979  540 
by blast 
923  541 

2261  542 

11979  543 
subsubsection {* Equality *} 
544 

13865  545 
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B" 
546 
apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals]) 

547 
apply (rule Collect_mem_eq) 

548 
apply (rule Collect_mem_eq) 

549 
done 

550 

15554  551 
(* Due to Brian Huffman *) 
552 
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))" 

553 
by(auto intro:set_ext) 

554 

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555 
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B" 
11979  556 
 {* Antisymmetry of the subset relation. *} 
17589  557 
by (iprover intro: set_ext subsetD) 
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558 

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559 
lemmas equalityI [intro!] = subset_antisym 
11979  560 

561 
text {* 

562 
\medskip Equality rules from ZF set theory  are they appropriate 

563 
here? 

564 
*} 

565 

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566 
lemma equalityD1: "A = B ==> A \<subseteq> B" 
11979  567 
by (simp add: subset_refl) 
568 

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569 
lemma equalityD2: "A = B ==> B \<subseteq> A" 
11979  570 
by (simp add: subset_refl) 
571 

572 
text {* 

573 
\medskip Be careful when adding this to the claset as @{text 

574 
subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{} 

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575 
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! 
11979  576 
*} 
577 

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578 
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P" 
11979  579 
by (simp add: subset_refl) 
923  580 

11979  581 
lemma equalityCE [elim]: 
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582 
"A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P" 
11979  583 
by blast 
584 

585 
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)" 

586 
by simp 

587 

13865  588 
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)" 
589 
by simp 

590 

11979  591 

592 
subsubsection {* The universal set  UNIV *} 

593 

594 
lemma UNIV_I [simp]: "x : UNIV" 

595 
by (simp add: UNIV_def) 

596 

597 
declare UNIV_I [intro]  {* unsafe makes it less likely to cause problems *} 

598 

599 
lemma UNIV_witness [intro?]: "EX x. x : UNIV" 

600 
by simp 

601 

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602 
lemma subset_UNIV [simp]: "A \<subseteq> UNIV" 
11979  603 
by (rule subsetI) (rule UNIV_I) 
2388  604 

11979  605 
text {* 
606 
\medskip Etacontracting these two rules (to remove @{text P}) 

607 
causes them to be ignored because of their interaction with 

608 
congruence rules. 

609 
*} 

610 

611 
lemma ball_UNIV [simp]: "Ball UNIV P = All P" 

612 
by (simp add: Ball_def) 

613 

614 
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P" 

615 
by (simp add: Bex_def) 

616 

617 

618 
subsubsection {* The empty set *} 

619 

620 
lemma empty_iff [simp]: "(c : {}) = False" 

621 
by (simp add: empty_def) 

622 

623 
lemma emptyE [elim!]: "a : {} ==> P" 

624 
by simp 

625 

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626 
lemma empty_subsetI [iff]: "{} \<subseteq> A" 
11979  627 
 {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *} 
628 
by blast 

629 

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630 
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}" 
11979  631 
by blast 
2388  632 

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633 
lemma equals0D: "A = {} ==> a \<notin> A" 
11979  634 
 {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *} 
635 
by blast 

636 

637 
lemma ball_empty [simp]: "Ball {} P = True" 

638 
by (simp add: Ball_def) 

639 

640 
lemma bex_empty [simp]: "Bex {} P = False" 

641 
by (simp add: Bex_def) 

642 

643 
lemma UNIV_not_empty [iff]: "UNIV ~= {}" 

644 
by (blast elim: equalityE) 

645 

646 

12023  647 
subsubsection {* The Powerset operator  Pow *} 
11979  648 

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649 
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)" 
11979  650 
by (simp add: Pow_def) 
651 

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652 
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B" 
11979  653 
by (simp add: Pow_def) 
654 

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655 
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B" 
11979  656 
by (simp add: Pow_def) 
657 

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658 
lemma Pow_bottom: "{} \<in> Pow B" 
11979  659 
by simp 
660 

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661 
lemma Pow_top: "A \<in> Pow A" 
11979  662 
by (simp add: subset_refl) 
2684  663 

2388  664 

11979  665 
subsubsection {* Set complement *} 
666 

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667 
lemma Compl_iff [simp]: "(c \<in> A) = (c \<notin> A)" 
11979  668 
by (unfold Compl_def) blast 
669 

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670 
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> A" 
11979  671 
by (unfold Compl_def) blast 
672 

673 
text {* 

674 
\medskip This form, with negated conclusion, works well with the 

675 
Classical prover. Negated assumptions behave like formulae on the 

676 
right side of the notional turnstile ... *} 

677 

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678 
lemma ComplD [dest!]: "c : A ==> c~:A" 
11979  679 
by (unfold Compl_def) blast 
680 

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681 
lemmas ComplE = ComplD [elim_format] 
11979  682 

683 

684 
subsubsection {* Binary union  Un *} 

923  685 

11979  686 
lemma Un_iff [simp]: "(c : A Un B) = (c:A  c:B)" 
687 
by (unfold Un_def) blast 

688 

689 
lemma UnI1 [elim?]: "c:A ==> c : A Un B" 

690 
by simp 

691 

692 
lemma UnI2 [elim?]: "c:B ==> c : A Un B" 

693 
by simp 

923  694 

11979  695 
text {* 
696 
\medskip Classical introduction rule: no commitment to @{prop A} vs 

697 
@{prop B}. 

698 
*} 

699 

700 
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B" 

701 
by auto 

702 

703 
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P" 

704 
by (unfold Un_def) blast 

705 

706 

12023  707 
subsubsection {* Binary intersection  Int *} 
923  708 

11979  709 
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)" 
710 
by (unfold Int_def) blast 

711 

712 
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B" 

713 
by simp 

714 

715 
lemma IntD1: "c : A Int B ==> c:A" 

716 
by simp 

717 

718 
lemma IntD2: "c : A Int B ==> c:B" 

719 
by simp 

720 

721 
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P" 

722 
by simp 

723 

724 

12023  725 
subsubsection {* Set difference *} 
11979  726 

727 
lemma Diff_iff [simp]: "(c : A  B) = (c:A & c~:B)" 

728 
by (unfold set_diff_def) blast 

923  729 

11979  730 
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A  B" 
731 
by simp 

732 

733 
lemma DiffD1: "c : A  B ==> c : A" 

734 
by simp 

735 

736 
lemma DiffD2: "c : A  B ==> c : B ==> P" 

737 
by simp 

738 

739 
lemma DiffE [elim!]: "c : A  B ==> (c:A ==> c~:B ==> P) ==> P" 

740 
by simp 

741 

742 

743 
subsubsection {* Augmenting a set  insert *} 

744 

745 
lemma insert_iff [simp]: "(a : insert b A) = (a = b  a:A)" 

746 
by (unfold insert_def) blast 

747 

748 
lemma insertI1: "a : insert a B" 

749 
by simp 

750 

751 
lemma insertI2: "a : B ==> a : insert b B" 

752 
by simp 

923  753 

11979  754 
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P" 
755 
by (unfold insert_def) blast 

756 

757 
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B" 

758 
 {* Classical introduction rule. *} 

759 
by auto 

760 

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761 
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A  {x} \<subseteq> B else A \<subseteq> B)" 
11979  762 
by auto 
763 

764 

765 
subsubsection {* Singletons, using insert *} 

766 

767 
lemma singletonI [intro!]: "a : {a}" 

768 
 {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *} 

769 
by (rule insertI1) 

770 

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771 
lemma singletonD [dest!]: "b : {a} ==> b = a" 
11979  772 
by blast 
773 

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774 
lemmas singletonE = singletonD [elim_format] 
11979  775 

776 
lemma singleton_iff: "(b : {a}) = (b = a)" 

777 
by blast 

778 

779 
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b" 

780 
by blast 

781 

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782 
lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \<subseteq> {b})" 
11979  783 
by blast 
784 

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785 
lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})" 
11979  786 
by blast 
787 

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788 
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {}  A = {x}" 
11979  789 
by fast 
790 

791 
lemma singleton_conv [simp]: "{x. x = a} = {a}" 

792 
by blast 

793 

794 
lemma singleton_conv2 [simp]: "{x. a = x} = {a}" 

795 
by blast 

923  796 

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797 
lemma diff_single_insert: "A  {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B" 
11979  798 
by blast 
799 

19870  800 
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d  a=d & b=c)" 
801 
by (blast elim: equalityE) 

802 

11979  803 

804 
subsubsection {* Unions of families *} 

805 

806 
text {* 

807 
@{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}. 

808 
*} 

809 

810 
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" 

811 
by (unfold UNION_def) blast 

812 

813 
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" 

814 
 {* The order of the premises presupposes that @{term A} is rigid; 

815 
@{term b} may be flexible. *} 

816 
by auto 

817 

818 
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" 

819 
by (unfold UNION_def) blast 

923  820 

11979  821 
lemma UN_cong [cong]: 
822 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

823 
by (simp add: UNION_def) 

824 

825 

826 
subsubsection {* Intersections of families *} 

827 

828 
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *} 

829 

830 
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" 

831 
by (unfold INTER_def) blast 

923  832 

11979  833 
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" 
834 
by (unfold INTER_def) blast 

835 

836 
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" 

837 
by auto 

838 

839 
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" 

840 
 {* "Classical" elimination  by the Excluded Middle on @{prop "a:A"}. *} 

841 
by (unfold INTER_def) blast 

842 

843 
lemma INT_cong [cong]: 

844 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" 

845 
by (simp add: INTER_def) 

7238
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846 

923  847 

11979  848 
subsubsection {* Union *} 
849 

850 
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)" 

851 
by (unfold Union_def) blast 

852 

853 
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C" 

854 
 {* The order of the premises presupposes that @{term C} is rigid; 

855 
@{term A} may be flexible. *} 

856 
by auto 

857 

858 
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R" 

859 
by (unfold Union_def) blast 

860 

861 

862 
subsubsection {* Inter *} 

863 

864 
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)" 

865 
by (unfold Inter_def) blast 

866 

867 
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C" 

868 
by (simp add: Inter_def) 

869 

870 
text {* 

871 
\medskip A ``destruct'' rule  every @{term X} in @{term C} 

872 
contains @{term A} as an element, but @{prop "A:X"} can hold when 

873 
@{prop "X:C"} does not! This rule is analogous to @{text spec}. 

874 
*} 

875 

876 
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X" 

877 
by auto 

878 

879 
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R" 

880 
 {* ``Classical'' elimination rule  does not require proving 

881 
@{prop "X:C"}. *} 

882 
by (unfold Inter_def) blast 

883 

884 
text {* 

885 
\medskip Image of a set under a function. Frequently @{term b} does 

886 
not have the syntactic form of @{term "f x"}. 

887 
*} 

888 

889 
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A" 

890 
by (unfold image_def) blast 

891 

892 
lemma imageI: "x : A ==> f x : f ` A" 

893 
by (rule image_eqI) (rule refl) 

894 

895 
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A" 

896 
 {* This version's more effective when we already have the 

897 
required @{term x}. *} 

898 
by (unfold image_def) blast 

899 

900 
lemma imageE [elim!]: 

901 
"b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P" 

902 
 {* The etaexpansion gives variablename preservation. *} 

903 
by (unfold image_def) blast 

904 

905 
lemma image_Un: "f`(A Un B) = f`A Un f`B" 

906 
by blast 

907 

908 
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)" 

909 
by blast 

910 

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

911 
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)" 
11979  912 
 {* This rewrite rule would confuse users if made default. *} 
913 
by blast 

914 

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

915 
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)" 
11979  916 
apply safe 
917 
prefer 2 apply fast 

14208  918 
apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast) 
11979  919 
done 
920 

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

921 
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B" 
11979  922 
 {* Replaces the three steps @{text subsetI}, @{text imageE}, 
923 
@{text hypsubst}, but breaks too many existing proofs. *} 

924 
by blast 

925 

926 
text {* 

927 
\medskip Range of a function  just a translation for image! 

928 
*} 

929 

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

930 
lemma range_eqI: "b = f x ==> b \<in> range f" 
11979  931 
by simp 
932 

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

933 
lemma rangeI: "f x \<in> range f" 
11979  934 
by simp 
935 

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

936 
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P" 
11979  937 
by blast 
938 

939 

940 
subsubsection {* Set reasoning tools *} 

941 

942 
text {* 

943 
Rewrite rules for boolean casesplitting: faster than @{text 

944 
"split_if [split]"}. 

945 
*} 

946 

947 
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q > x = b) & (~ Q > y = b))" 

948 
by (rule split_if) 

949 

950 
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q > a = x) & (~ Q > a = y))" 

951 
by (rule split_if) 

952 

953 
text {* 

954 
Split ifs on either side of the membership relation. Not for @{text 

955 
"[simp]"}  can cause goals to blow up! 

956 
*} 

957 

958 
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q > x : b) & (~ Q > y : b))" 

959 
by (rule split_if) 

960 

961 
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q > a : x) & (~ Q > a : y))" 

962 
by (rule split_if) 

963 

964 
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 

965 

966 
lemmas mem_simps = 

967 
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff 

968 
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff 

969 
 {* Each of these has ALREADY been added @{text "[simp]"} above. *} 

970 

971 
(*Would like to add these, but the existing code only searches for the 

972 
outerlevel constant, which in this case is just "op :"; we instead need 

973 
to use termnets to associate patterns with rules. Also, if a rule fails to 

974 
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

975 
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]), 
11979  976 
("op Int", [IntD1,IntD2]), 
977 
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] 

978 
*) 

979 

980 
ML_setup {* 

981 
val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs; 

17875  982 
change_simpset (fn ss => ss setmksimps (mksimps mksimps_pairs)); 
11979  983 
*} 
984 

985 

986 
subsubsection {* The ``proper subset'' relation *} 

987 

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

988 
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B" 
11979  989 
by (unfold psubset_def) blast 
990 

13624  991 
lemma psubsetE [elim!]: 
992 
"[A \<subset> B; [A \<subseteq> B; ~ (B\<subseteq>A)] ==> R] ==> R" 

993 
by (unfold psubset_def) blast 

994 

11979  995 
lemma psubset_insert_iff: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

996 
"(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

997 
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

998 

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

999 
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)" 
11979  1000 
by (simp only: psubset_def) 
1001 

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

1002 
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B" 
11979  1003 
by (simp add: psubset_eq) 
1004 

14335  1005 
lemma psubset_trans: "[ A \<subset> B; B \<subset> C ] ==> A \<subset> C" 
1006 
apply (unfold psubset_def) 

1007 
apply (auto dest: subset_antisym) 

1008 
done 

1009 

1010 
lemma psubsetD: "[ A \<subset> B; c \<in> A ] ==> c \<in> B" 

1011 
apply (unfold psubset_def) 

1012 
apply (auto dest: subsetD) 

1013 
done 

1014 

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

1015 
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C" 
11979  1016 
by (auto simp add: psubset_eq) 
1017 

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

1018 
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C" 
11979  1019 
by (auto simp add: psubset_eq) 
1020 

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

1021 
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B  A)" 
11979  1022 
by (unfold psubset_def) blast 
1023 

1024 
lemma atomize_ball: 

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

1025 
"(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)" 
11979  1026 
by (simp only: Ball_def atomize_all atomize_imp) 
1027 

18832  1028 
lemmas [symmetric, rulify] = atomize_ball 
1029 
and [symmetric, defn] = atomize_ball 

11979  1030 

1031 

1032 
subsection {* Further settheory lemmas *} 

1033 

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

1034 
subsubsection {* Derived rules involving subsets. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1035 

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

1036 
text {* @{text insert}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1037 

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

1038 
lemma subset_insertI: "B \<subseteq> insert a B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1039 
apply (rule subsetI) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1040 
apply (erule insertI2) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1041 
done 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1042 

14302  1043 
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B" 
1044 
by blast 

1045 

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

1046 
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

1047 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1048 

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

1049 

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

1050 
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

1051 

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

1052 
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A" 
17589  1053 
by (iprover intro: subsetI UnionI) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1054 

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

1055 
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C" 
17589  1056 
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

1057 

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

1058 

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

1059 
text {* \medskip General union. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1060 

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

1061 
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

1062 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1063 

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

1064 
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C" 
17589  1065 
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

1066 

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

1067 

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

1068 
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

1069 

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

1070 
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

1071 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1072 

14551  1073 
lemma Inter_subset: 
1074 
"[ !!X. X \<in> A ==> X \<subseteq> B; A ~= {} ] ==> \<Inter>A \<subseteq> B" 

1075 
by blast 

1076 

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

1077 
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A" 
17589  1078 
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

1079 

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

1080 
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

1081 
by blast 
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 
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)" 
17589  1084 
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

1085 

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

1086 

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

1087 
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

1088 

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

1089 
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

1090 
by blast 
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 
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

1093 
by blast 
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 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

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

1097 

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

1098 

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

1099 
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

1100 

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

1101 
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

1102 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1103 

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

1104 
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

1105 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1106 

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

1107 
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

1108 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1109 

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

1112 

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

1113 
lemma Diff_subset: "A  B \<subseteq> A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1114 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1115 

14302  1116 
lemma Diff_subset_conv: "(A  B \<subseteq> C) = (A \<subseteq> B \<union> C)" 
1117 
by blast 

1118 

12897
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 
text {* \medskip Monotonicity. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1121 

15206
09d78ec709c7
Modified locales: improved implementation of "includes".
ballarin
parents:
15140
diff
changeset

1122 
lemma mono_Un: "mono f ==> f A \<union> f B \<subseteq> f (A \<union> B)" 
16773  1123 
by (auto simp add: mono_def) 
15206
09d78ec709c7
Modified locales: improved implementation of "includes".
ballarin
parents:
15140
diff
changeset

1124 

09d78ec709c7
Modified locales: improved implementation of "includes".
ballarin
parents:
15140
diff
changeset

1125 
lemma mono_Int: "mono f ==> f (A \<inter> B) \<subseteq> f A \<inter> f B" 
16773  1126 
by (auto simp add: mono_def) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1127 

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

1128 
subsubsection {* Equalities involving union, intersection, inclusion, etc. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1129 

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

1130 
text {* @{text "{}"}. *} 
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 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

1133 
 {* supersedes @{text "Collect_False_empty"} *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1134 
by auto 
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 
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1137 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1138 

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

1139 
lemma not_psubset_empty [iff]: "\<not> (A < {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1140 
by (unfold psubset_def) blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1141 

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

1142 
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)" 
18423  1143 
by blast 
1144 

1145 
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)" 

1146 
by blast 

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

1147 

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

1148 
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

1149 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1150 

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

1151 
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

1152 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1153 

14812  1154 
lemma Collect_imp_eq: "{x. P x > Q x} = {x. P x} \<union> {x. Q x}" 
1155 
by blast 

1156 

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

1157 
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

1158 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1159 

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

1160 
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

1161 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1162 

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

1163 
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

1164 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1165 

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

1166 
lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1167 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1168 

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

1169 
lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1170 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1171 

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

1172 

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

1173 
text {* \medskip @{text insert}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1174 

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

1175 
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

1176 
 {* 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

1177 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1178 

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

1179 
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

1180 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1181 

17715  1182 
lemmas empty_not_insert = insert_not_empty [symmetric, standard] 
1183 
declare empty_not_insert [simp] 

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

1184 

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

1185 
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

1186 
 {* @{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

1187 
 {* with \emph{quadratic} running time *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1188 
by blast 
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 
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

1191 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1192 

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

1193 
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

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_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

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

1198 

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

1199 
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

1200 
 {* use new @{text B} rather than @{text "A  {a}"} to avoid infinite unfolding *} 
14208  1201 
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

1202 
done 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1203 

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

1204 
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

1205 
by auto 
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 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

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

1209 

14302  1210 
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)" 
14742  1211 
by blast 
14302  1212 

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

1213 
lemma insert_disjoint[simp]: 
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1214 
"(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})" 
14742  1215 
"({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)" 
16773  1216 
by auto 
13103
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1217 

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

1218 
lemma disjoint_insert[simp]: 
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1219 
"(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})" 
14742  1220 
"({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)" 
16773  1221 
by auto 
14742  1222 

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

1223 
text {* \medskip @{text image}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1224 

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

1225 
lemma image_empty [simp]: "f`{} = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1226 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1227 

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

1228 
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

1229 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1230 

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

1231 
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}" 
16773  1232 
by auto 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1233 

21316  1234 
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})" 
21312  1235 
by auto 
1236 

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

1237 
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

1238 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1239 

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

1240 
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

1241 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1242 

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

1243 
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

1244 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1245 

16773  1246 

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

1247 
lemma image_Collect: "f ` {x. P x} = {f x  x. P x}" 
16773  1248 
 {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, 
1249 
with its implicit quantifier and conjunction. Also image enjoys better 

1250 
equational properties than does the RHS. *} 

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

1251 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1252 

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

1253 
lemma if_image_distrib [simp]: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1254 
"(\<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

1255 
= (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

1256 
by (auto simp add: image_def) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1257 

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

1258 
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

1259 
by (simp add: image_def) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1260 

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

1261 

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

1262 
text {* \medskip @{text range}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1263 

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

1264 
lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1265 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1266 

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

1267 
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g" 
14208  1268 
by (subst image_image, simp) 
12897
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 

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

1271 
text {* \medskip @{text Int} *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1272 

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

1273 
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

1274 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1275 

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

1276 
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

1277 
by blast 
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 
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

1280 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1281 

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

1282 
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

1283 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1284 

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

1285 
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

1286 
by blast 
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 
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

1289 
 {* Intersection is an ACoperator *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1290 

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

1291 
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

1292 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1293 

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

1294 
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

1295 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1296 

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

1297 
lemma Int_empty_left [simp]: "{} \<inter> B = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1298 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1299 

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

1300 
lemma Int_empty_right [simp]: "A \<inter> {} = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1301 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1302 

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

1303 
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

1304 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1305 

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

1306 
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

1307 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1308 

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

1309 
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

1310 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1311 

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

1312 
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

1313 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1314 

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

1315 
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

1316 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1317 

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

1318 
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

1319 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1320 

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

1321 
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

1322 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1323 

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

1324 
lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1325 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1326 

15102  1327 
lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1328 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1329 

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

1330 
lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1331 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1332 

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

1333 

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

1334 
text {* \medskip @{text Un}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1335 

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

1336 
lemma Un_absorb [simp]: "A \<union> A = A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1337 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1338 

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

1339 
lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1340 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1341 

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

1342 
lemma Un_commute: "A \<union> B = B \<union> A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1343 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1344 

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

1345 
lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1346 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1347 

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

1348 
lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1349 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1350 
