author  wenzelm 
Tue, 16 May 2006 21:33:01 +0200  
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parent 19637  d33a71ffb9e3 
child 19870  ef037d1b32d1 
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 LOrder 
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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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image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) 

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"op :" :: "'a => 'a set => bool"  "membership" 
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const_syntax 
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"op :" ("op :") 
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"op :" ("(_/ : _)" [50, 51] 50) 
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local 

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instance set :: (type) "{ord, minus}" .. 
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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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const_syntax 
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not_mem ("op ~:") 
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not_mem ("(_/ ~: _)" [50, 51] 50) 
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const_syntax (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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const_syntax (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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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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const_syntax (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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const_syntax (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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const_syntax (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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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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"_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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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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"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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"_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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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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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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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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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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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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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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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  210 
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  216 
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  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 "_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  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 "_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  240 
text {* 
241 
\medskip Translate between @{text "{e  x1...xn. P}"} and @{text 

242 
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is 

243 
only translated if @{text "[0..n] subset bvs(e)"}. 

244 
*} 

245 

246 
parse_translation {* 

247 
let 

248 
val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); 

3947  249 

11979  250 
fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 
251 
 nvars _ = 1; 

252 

253 
fun setcompr_tr [e, idts, b] = 

254 
let 

255 
val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; 

256 
val P = Syntax.const "op &" $ eq $ b; 

257 
val exP = ex_tr [idts, P]; 

17784  258 
in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end; 
11979  259 

260 
in [("@SetCompr", setcompr_tr)] end; 

261 
*} 

923  262 

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(* To avoid etacontraction of body: *) 
11979  264 
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  270 
[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"), 
271 
("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  285 
 check _ = false 
923  286 

11979  287 
fun tr' (_ $ abs) = 
288 
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] 

289 
in Syntax.const "@SetCompr" $ e $ idts $ Q end; 

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in if check (P, 0) then tr' P 
15535  291 
else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs 
292 
val M = Syntax.const "@Coll" $ x $ t 

293 
in case t of 

294 
Const("op &",_) 

295 
$ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) 

296 
$ P => 

297 
if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M 

298 
 _ => M 

299 
end 

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end; 
11979  301 
in [("Collect", setcompr_tr')] end; 
302 
*} 

303 

304 

305 
subsection {* Rules and definitions *} 

306 

307 
text {* Isomorphisms between predicates and sets. *} 

923  308 

11979  309 
axioms 
17085  310 
mem_Collect_eq: "(a : {x. P(x)}) = P(a)" 
311 
Collect_mem_eq: "{x. x:A} = A" 

17702  312 
finalconsts 
313 
Collect 

314 
"op :" 

11979  315 

316 
defs 

317 
Ball_def: "Ball A P == ALL x. x:A > P(x)" 

318 
Bex_def: "Bex A P == EX x. x:A & P(x)" 

319 

320 
defs (overloaded) 

321 
subset_def: "A <= B == ALL x:A. x:B" 

322 
psubset_def: "A < B == (A::'a set) <= B & ~ A=B" 

323 
Compl_def: " A == {x. ~x:A}" 

12257  324 
set_diff_def: "A  B == {x. x:A & ~x:B}" 
923  325 

326 
defs 

11979  327 
Un_def: "A Un B == {x. x:A  x:B}" 
328 
Int_def: "A Int B == {x. x:A & x:B}" 

329 
INTER_def: "INTER A B == {y. ALL x:A. y: B(x)}" 

330 
UNION_def: "UNION A B == {y. EX x:A. y: B(x)}" 

331 
Inter_def: "Inter S == (INT x:S. x)" 

332 
Union_def: "Union S == (UN x:S. x)" 

333 
Pow_def: "Pow A == {B. B <= A}" 

334 
empty_def: "{} == {x. False}" 

335 
UNIV_def: "UNIV == {x. True}" 

336 
insert_def: "insert a B == {x. x=a} Un B" 

337 
image_def: "f`A == {y. EX x:A. y = f(x)}" 

338 

339 

340 
subsection {* Lemmas and proof tool setup *} 

341 

342 
subsubsection {* Relating predicates and sets *} 

343 

17085  344 
declare mem_Collect_eq [iff] Collect_mem_eq [simp] 
345 

12257  346 
lemma CollectI: "P(a) ==> a : {x. P(x)}" 
11979  347 
by simp 
348 

349 
lemma CollectD: "a : {x. P(x)} ==> P(a)" 

350 
by simp 

351 

352 
lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}" 

353 
by simp 

354 

12257  355 
lemmas CollectE = CollectD [elim_format] 
11979  356 

357 

358 
subsubsection {* Bounded quantifiers *} 

359 

360 
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" 

361 
by (simp add: Ball_def) 

362 

363 
lemmas strip = impI allI ballI 

364 

365 
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" 

366 
by (simp add: Ball_def) 

367 

368 
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" 

369 
by (unfold Ball_def) blast 

14098  370 
ML {* bind_thm("rev_ballE",permute_prems 1 1 (thm "ballE")) *} 
11979  371 

372 
text {* 

373 
\medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and 

374 
@{prop "a:A"}; creates assumption @{prop "P a"}. 

375 
*} 

376 

377 
ML {* 

378 
local val ballE = thm "ballE" 

379 
in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end; 

380 
*} 

381 

382 
text {* 

383 
Gives better instantiation for bound: 

384 
*} 

385 

386 
ML_setup {* 

17875  387 
change_claset (fn cs => cs addbefore ("bspec", datac (thm "bspec") 1)); 
11979  388 
*} 
389 

390 
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" 

391 
 {* Normally the best argument order: @{prop "P x"} constrains the 

392 
choice of @{prop "x:A"}. *} 

393 
by (unfold Bex_def) blast 

394 

13113  395 
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" 
11979  396 
 {* The best argument order when there is only one @{prop "x:A"}. *} 
397 
by (unfold Bex_def) blast 

398 

399 
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" 

400 
by (unfold Bex_def) blast 

401 

402 
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" 

403 
by (unfold Bex_def) blast 

404 

405 
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) > P)" 

406 
 {* Trival rewrite rule. *} 

407 
by (simp add: Ball_def) 

408 

409 
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" 

410 
 {* Dual form for existentials. *} 

411 
by (simp add: Bex_def) 

412 

413 
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" 

414 
by blast 

415 

416 
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" 

417 
by blast 

418 

419 
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" 

420 
by blast 

421 

422 
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" 

423 
by blast 

424 

425 
lemma ball_one_point1 [simp]: "(ALL x:A. x = a > P x) = (a:A > P a)" 

426 
by blast 

427 

428 
lemma ball_one_point2 [simp]: "(ALL x:A. a = x > P x) = (a:A > P a)" 

429 
by blast 

430 

431 
ML_setup {* 

13462  432 
local 
18328  433 
val unfold_bex_tac = unfold_tac [thm "Bex_def"]; 
434 
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; 

11979  435 
val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; 
436 

18328  437 
val unfold_ball_tac = unfold_tac [thm "Ball_def"]; 
438 
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; 

11979  439 
val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; 
440 
in 

18328  441 
val defBEX_regroup = Simplifier.simproc (the_context ()) 
13462  442 
"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; 
18328  443 
val defBALL_regroup = Simplifier.simproc (the_context ()) 
13462  444 
"defined BALL" ["ALL x:A. P x > Q x"] rearrange_ball; 
11979  445 
end; 
13462  446 

447 
Addsimprocs [defBALL_regroup, defBEX_regroup]; 

11979  448 
*} 
449 

450 

451 
subsubsection {* Congruence rules *} 

452 

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lemma ball_cong: 
11979  454 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
455 
(ALL x:A. P x) = (ALL x:B. Q x)" 

456 
by (simp add: Ball_def) 

457 

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458 
lemma strong_ball_cong [cong]: 
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"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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(ALL x:A. P x) = (ALL x:B. Q x)" 
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by (simp add: simp_implies_def Ball_def) 
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462 

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463 
lemma bex_cong: 
11979  464 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
465 
(EX x:A. P x) = (EX x:B. Q x)" 

466 
by (simp add: Bex_def cong: conj_cong) 

1273  467 

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468 
lemma strong_bex_cong [cong]: 
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"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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(EX x:A. P x) = (EX x:B. Q x)" 
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471 
by (simp add: simp_implies_def Bex_def cong: conj_cong) 
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472 

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473 

11979  474 
subsubsection {* Subsets *} 
475 

19295  476 
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B" 
11979  477 
by (simp add: subset_def) 
478 

479 
text {* 

480 
\medskip Map the type @{text "'a set => anything"} to just @{typ 

481 
'a}; for overloading constants whose first argument has type @{typ 

482 
"'a set"}. 

483 
*} 

484 

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485 
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" 
11979  486 
 {* Rule in Modus Ponens style. *} 
487 
by (unfold subset_def) blast 

488 

489 
declare subsetD [intro?]  FIXME 

490 

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491 
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B" 
11979  492 
 {* The same, with reversed premises for use with @{text erule}  
493 
cf @{text rev_mp}. *} 

494 
by (rule subsetD) 

495 

496 
declare rev_subsetD [intro?]  FIXME 

497 

498 
text {* 

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499 
\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}. 
11979  500 
*} 
501 

502 
ML {* 

503 
local val rev_subsetD = thm "rev_subsetD" 

504 
in fun impOfSubs th = th RSN (2, rev_subsetD) end; 

505 
*} 

506 

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507 
lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P" 
11979  508 
 {* Classical elimination rule. *} 
509 
by (unfold subset_def) blast 

510 

511 
text {* 

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512 
\medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and 
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513 
creates the assumption @{prop "c \<in> B"}. 
11979  514 
*} 
515 

516 
ML {* 

517 
local val subsetCE = thm "subsetCE" 

518 
in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end; 

519 
*} 

520 

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521 
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A" 
11979  522 
by blast 
523 

19175  524 
lemma subset_refl [simp,atp]: "A \<subseteq> A" 
11979  525 
by fast 
526 

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527 
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" 
11979  528 
by blast 
923  529 

2261  530 

11979  531 
subsubsection {* Equality *} 
532 

13865  533 
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B" 
534 
apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals]) 

535 
apply (rule Collect_mem_eq) 

536 
apply (rule Collect_mem_eq) 

537 
done 

538 

15554  539 
(* Due to Brian Huffman *) 
540 
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))" 

541 
by(auto intro:set_ext) 

542 

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543 
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B" 
11979  544 
 {* Antisymmetry of the subset relation. *} 
17589  545 
by (iprover intro: set_ext subsetD) 
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546 

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547 
lemmas equalityI [intro!] = subset_antisym 
11979  548 

549 
text {* 

550 
\medskip Equality rules from ZF set theory  are they appropriate 

551 
here? 

552 
*} 

553 

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554 
lemma equalityD1: "A = B ==> A \<subseteq> B" 
11979  555 
by (simp add: subset_refl) 
556 

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557 
lemma equalityD2: "A = B ==> B \<subseteq> A" 
11979  558 
by (simp add: subset_refl) 
559 

560 
text {* 

561 
\medskip Be careful when adding this to the claset as @{text 

562 
subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{} 

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563 
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! 
11979  564 
*} 
565 

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566 
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P" 
11979  567 
by (simp add: subset_refl) 
923  568 

11979  569 
lemma equalityCE [elim]: 
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570 
"A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P" 
11979  571 
by blast 
572 

573 
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)" 

574 
by simp 

575 

13865  576 
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)" 
577 
by simp 

578 

11979  579 

580 
subsubsection {* The universal set  UNIV *} 

581 

582 
lemma UNIV_I [simp]: "x : UNIV" 

583 
by (simp add: UNIV_def) 

584 

585 
declare UNIV_I [intro]  {* unsafe makes it less likely to cause problems *} 

586 

587 
lemma UNIV_witness [intro?]: "EX x. x : UNIV" 

588 
by simp 

589 

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590 
lemma subset_UNIV [simp]: "A \<subseteq> UNIV" 
11979  591 
by (rule subsetI) (rule UNIV_I) 
2388  592 

11979  593 
text {* 
594 
\medskip Etacontracting these two rules (to remove @{text P}) 

595 
causes them to be ignored because of their interaction with 

596 
congruence rules. 

597 
*} 

598 

599 
lemma ball_UNIV [simp]: "Ball UNIV P = All P" 

600 
by (simp add: Ball_def) 

601 

602 
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P" 

603 
by (simp add: Bex_def) 

604 

605 

606 
subsubsection {* The empty set *} 

607 

608 
lemma empty_iff [simp]: "(c : {}) = False" 

609 
by (simp add: empty_def) 

610 

611 
lemma emptyE [elim!]: "a : {} ==> P" 

612 
by simp 

613 

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614 
lemma empty_subsetI [iff]: "{} \<subseteq> A" 
11979  615 
 {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *} 
616 
by blast 

617 

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618 
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}" 
11979  619 
by blast 
2388  620 

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621 
lemma equals0D: "A = {} ==> a \<notin> A" 
11979  622 
 {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *} 
623 
by blast 

624 

625 
lemma ball_empty [simp]: "Ball {} P = True" 

626 
by (simp add: Ball_def) 

627 

628 
lemma bex_empty [simp]: "Bex {} P = False" 

629 
by (simp add: Bex_def) 

630 

631 
lemma UNIV_not_empty [iff]: "UNIV ~= {}" 

632 
by (blast elim: equalityE) 

633 

634 

12023  635 
subsubsection {* The Powerset operator  Pow *} 
11979  636 

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637 
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)" 
11979  638 
by (simp add: Pow_def) 
639 

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640 
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B" 
11979  641 
by (simp add: Pow_def) 
642 

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643 
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B" 
11979  644 
by (simp add: Pow_def) 
645 

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646 
lemma Pow_bottom: "{} \<in> Pow B" 
11979  647 
by simp 
648 

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649 
lemma Pow_top: "A \<in> Pow A" 
11979  650 
by (simp add: subset_refl) 
2684  651 

2388  652 

11979  653 
subsubsection {* Set complement *} 
654 

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655 
lemma Compl_iff [simp]: "(c \<in> A) = (c \<notin> A)" 
11979  656 
by (unfold Compl_def) blast 
657 

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658 
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> A" 
11979  659 
by (unfold Compl_def) blast 
660 

661 
text {* 

662 
\medskip This form, with negated conclusion, works well with the 

663 
Classical prover. Negated assumptions behave like formulae on the 

664 
right side of the notional turnstile ... *} 

665 

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666 
lemma ComplD [dest!]: "c : A ==> c~:A" 
11979  667 
by (unfold Compl_def) blast 
668 

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669 
lemmas ComplE = ComplD [elim_format] 
11979  670 

671 

672 
subsubsection {* Binary union  Un *} 

923  673 

11979  674 
lemma Un_iff [simp]: "(c : A Un B) = (c:A  c:B)" 
675 
by (unfold Un_def) blast 

676 

677 
lemma UnI1 [elim?]: "c:A ==> c : A Un B" 

678 
by simp 

679 

680 
lemma UnI2 [elim?]: "c:B ==> c : A Un B" 

681 
by simp 

923  682 

11979  683 
text {* 
684 
\medskip Classical introduction rule: no commitment to @{prop A} vs 

685 
@{prop B}. 

686 
*} 

687 

688 
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B" 

689 
by auto 

690 

691 
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P" 

692 
by (unfold Un_def) blast 

693 

694 

12023  695 
subsubsection {* Binary intersection  Int *} 
923  696 

11979  697 
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)" 
698 
by (unfold Int_def) blast 

699 

700 
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B" 

701 
by simp 

702 

703 
lemma IntD1: "c : A Int B ==> c:A" 

704 
by simp 

705 

706 
lemma IntD2: "c : A Int B ==> c:B" 

707 
by simp 

708 

709 
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P" 

710 
by simp 

711 

712 

12023  713 
subsubsection {* Set difference *} 
11979  714 

715 
lemma Diff_iff [simp]: "(c : A  B) = (c:A & c~:B)" 

716 
by (unfold set_diff_def) blast 

923  717 

11979  718 
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A  B" 
719 
by simp 

720 

721 
lemma DiffD1: "c : A  B ==> c : A" 

722 
by simp 

723 

724 
lemma DiffD2: "c : A  B ==> c : B ==> P" 

725 
by simp 

726 

727 
lemma DiffE [elim!]: "c : A  B ==> (c:A ==> c~:B ==> P) ==> P" 

728 
by simp 

729 

730 

731 
subsubsection {* Augmenting a set  insert *} 

732 

733 
lemma insert_iff [simp]: "(a : insert b A) = (a = b  a:A)" 

734 
by (unfold insert_def) blast 

735 

736 
lemma insertI1: "a : insert a B" 

737 
by simp 

738 

739 
lemma insertI2: "a : B ==> a : insert b B" 

740 
by simp 

923  741 

11979  742 
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P" 
743 
by (unfold insert_def) blast 

744 

745 
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B" 

746 
 {* Classical introduction rule. *} 

747 
by auto 

748 

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749 
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A  {x} \<subseteq> B else A \<subseteq> B)" 
11979  750 
by auto 
751 

752 

753 
subsubsection {* Singletons, using insert *} 

754 

755 
lemma singletonI [intro!]: "a : {a}" 

756 
 {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *} 

757 
by (rule insertI1) 

758 

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759 
lemma singletonD [dest!]: "b : {a} ==> b = a" 
11979  760 
by blast 
761 

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762 
lemmas singletonE = singletonD [elim_format] 
11979  763 

764 
lemma singleton_iff: "(b : {a}) = (b = a)" 

765 
by blast 

766 

767 
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b" 

768 
by blast 

769 

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770 
lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \<subseteq> {b})" 
11979  771 
by blast 
772 

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773 
lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})" 
11979  774 
by blast 
775 

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776 
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {}  A = {x}" 
11979  777 
by fast 
778 

779 
lemma singleton_conv [simp]: "{x. x = a} = {a}" 

780 
by blast 

781 

782 
lemma singleton_conv2 [simp]: "{x. a = x} = {a}" 

783 
by blast 

923  784 

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785 
lemma diff_single_insert: "A  {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B" 
11979  786 
by blast 
787 

788 

789 
subsubsection {* Unions of families *} 

790 

791 
text {* 

792 
@{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}. 

793 
*} 

794 

795 
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" 

796 
by (unfold UNION_def) blast 

797 

798 
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" 

799 
 {* The order of the premises presupposes that @{term A} is rigid; 

800 
@{term b} may be flexible. *} 

801 
by auto 

802 

803 
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" 

804 
by (unfold UNION_def) blast 

923  805 

11979  806 
lemma UN_cong [cong]: 
807 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

808 
by (simp add: UNION_def) 

809 

810 

811 
subsubsection {* Intersections of families *} 

812 

813 
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *} 

814 

815 
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" 

816 
by (unfold INTER_def) blast 

923  817 

11979  818 
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" 
819 
by (unfold INTER_def) blast 

820 

821 
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" 

822 
by auto 

823 

824 
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" 

825 
 {* "Classical" elimination  by the Excluded Middle on @{prop "a:A"}. *} 

826 
by (unfold INTER_def) blast 

827 

828 
lemma INT_cong [cong]: 

829 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" 

830 
by (simp add: INTER_def) 

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831 

923  832 

11979  833 
subsubsection {* Union *} 
834 

835 
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)" 

836 
by (unfold Union_def) blast 

837 

838 
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C" 

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

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

841 
by auto 

842 

843 
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R" 

844 
by (unfold Union_def) blast 

845 

846 

847 
subsubsection {* Inter *} 

848 

849 
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)" 

850 
by (unfold Inter_def) blast 

851 

852 
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C" 

853 
by (simp add: Inter_def) 

854 

855 
text {* 

856 
\medskip A ``destruct'' rule  every @{term X} in @{term C} 

857 
contains @{term A} as an element, but @{prop "A:X"} can hold when 

858 
@{prop "X:C"} does not! This rule is analogous to @{text spec}. 

859 
*} 

860 

861 
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X" 

862 
by auto 

863 

864 
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R" 

865 
 {* ``Classical'' elimination rule  does not require proving 

866 
@{prop "X:C"}. *} 

867 
by (unfold Inter_def) blast 

868 

869 
text {* 

870 
\medskip Image of a set under a function. Frequently @{term b} does 

871 
not have the syntactic form of @{term "f x"}. 

872 
*} 

873 

874 
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A" 

875 
by (unfold image_def) blast 

876 

877 
lemma imageI: "x : A ==> f x : f ` A" 

878 
by (rule image_eqI) (rule refl) 

879 

880 
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A" 

881 
 {* This version's more effective when we already have the 

882 
required @{term x}. *} 

883 
by (unfold image_def) blast 

884 

885 
lemma imageE [elim!]: 

886 
"b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P" 

887 
 {* The etaexpansion gives variablename preservation. *} 

888 
by (unfold image_def) blast 

889 

890 
lemma image_Un: "f`(A Un B) = f`A Un f`B" 

891 
by blast 

892 

893 
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)" 

894 
by blast 

895 

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

896 
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)" 
11979  897 
 {* This rewrite rule would confuse users if made default. *} 
898 
by blast 

899 

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

900 
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)" 
11979  901 
apply safe 
902 
prefer 2 apply fast 

14208  903 
apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast) 
11979  904 
done 
905 

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

906 
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B" 
11979  907 
 {* Replaces the three steps @{text subsetI}, @{text imageE}, 
908 
@{text hypsubst}, but breaks too many existing proofs. *} 

909 
by blast 

910 

911 
text {* 

912 
\medskip Range of a function  just a translation for image! 

913 
*} 

914 

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

915 
lemma range_eqI: "b = f x ==> b \<in> range f" 
11979  916 
by simp 
917 

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

918 
lemma rangeI: "f x \<in> range f" 
11979  919 
by simp 
920 

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

921 
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P" 
11979  922 
by blast 
923 

924 

925 
subsubsection {* Set reasoning tools *} 

926 

927 
text {* 

928 
Rewrite rules for boolean casesplitting: faster than @{text 

929 
"split_if [split]"}. 

930 
*} 

931 

932 
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q > x = b) & (~ Q > y = b))" 

933 
by (rule split_if) 

934 

935 
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q > a = x) & (~ Q > a = y))" 

936 
by (rule split_if) 

937 

938 
text {* 

939 
Split ifs on either side of the membership relation. Not for @{text 

940 
"[simp]"}  can cause goals to blow up! 

941 
*} 

942 

943 
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q > x : b) & (~ Q > y : b))" 

944 
by (rule split_if) 

945 

946 
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q > a : x) & (~ Q > a : y))" 

947 
by (rule split_if) 

948 

949 
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 

950 

951 
lemmas mem_simps = 

952 
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff 

953 
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff 

954 
 {* Each of these has ALREADY been added @{text "[simp]"} above. *} 

955 

956 
(*Would like to add these, but the existing code only searches for the 

957 
outerlevel constant, which in this case is just "op :"; we instead need 

958 
to use termnets to associate patterns with rules. Also, if a rule fails to 

959 
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

960 
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]), 
11979  961 
("op Int", [IntD1,IntD2]), 
962 
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] 

963 
*) 

964 

965 
ML_setup {* 

966 
val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs; 

17875  967 
change_simpset (fn ss => ss setmksimps (mksimps mksimps_pairs)); 
11979  968 
*} 
969 

970 

971 
subsubsection {* The ``proper subset'' relation *} 

972 

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

973 
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B" 
11979  974 
by (unfold psubset_def) blast 
975 

13624  976 
lemma psubsetE [elim!]: 
977 
"[A \<subset> B; [A \<subseteq> B; ~ (B\<subseteq>A)] ==> R] ==> R" 

978 
by (unfold psubset_def) blast 

979 

11979  980 
lemma psubset_insert_iff: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

981 
"(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

982 
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

983 

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

984 
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)" 
11979  985 
by (simp only: psubset_def) 
986 

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

987 
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B" 
11979  988 
by (simp add: psubset_eq) 
989 

14335  990 
lemma psubset_trans: "[ A \<subset> B; B \<subset> C ] ==> A \<subset> C" 
991 
apply (unfold psubset_def) 

992 
apply (auto dest: subset_antisym) 

993 
done 

994 

995 
lemma psubsetD: "[ A \<subset> B; c \<in> A ] ==> c \<in> B" 

996 
apply (unfold psubset_def) 

997 
apply (auto dest: subsetD) 

998 
done 

999 

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

1000 
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C" 
11979  1001 
by (auto simp add: psubset_eq) 
1002 

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

1003 
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C" 
11979  1004 
by (auto simp add: psubset_eq) 
1005 

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

1006 
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B  A)" 
11979  1007 
by (unfold psubset_def) blast 
1008 

1009 
lemma atomize_ball: 

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

1010 
"(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)" 
11979  1011 
by (simp only: Ball_def atomize_all atomize_imp) 
1012 

18832  1013 
lemmas [symmetric, rulify] = atomize_ball 
1014 
and [symmetric, defn] = atomize_ball 

11979  1015 

1016 

1017 
subsection {* Further settheory lemmas *} 

1018 

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

1019 
subsubsection {* Derived rules involving subsets. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1020 

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

1021 
text {* @{text insert}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1022 

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

1023 
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

1024 
apply (rule subsetI) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1025 
apply (erule insertI2) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1026 
done 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1027 

14302  1028 
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B" 
1029 
by blast 

1030 

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

1031 
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

1032 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1033 

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

1034 

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

1035 
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

1036 

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

1037 
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A" 
17589  1038 
by (iprover intro: subsetI UnionI) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1039 

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

1040 
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C" 
17589  1041 
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

1042 

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

1043 

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

1044 
text {* \medskip General union. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1045 

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

1046 
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

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 
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C" 
17589  1050 
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

1051 

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

1052 

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

1053 
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

1054 

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

1055 
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

1056 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1057 

14551  1058 
lemma Inter_subset: 
1059 
"[ !!X. X \<in> A ==> X \<subseteq> B; A ~= {} ] ==> \<Inter>A \<subseteq> B" 

1060 
by blast 

1061 

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

1062 
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A" 
17589  1063 
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

1064 

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

1065 
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

1066 
by blast 
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 
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)" 
17589  1069 
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

1070 

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

1071 

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

1072 
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

1073 

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

1074 
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

1075 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1076 

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

1077 
lemma Un_upper2: "B \<subseteq> A \<union> B" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1078 
by blast 
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 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

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 

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

1084 
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

1085 

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

1086 
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

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

1088 

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

1089 
lemma Int_lower2: "A \<inter> B \<subseteq> 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 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

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 

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

1096 
text {* \medskip Set difference. *} 
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 
lemma Diff_subset: "A  B \<subseteq> A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1099 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1100 

14302  1101 
lemma Diff_subset_conv: "(A  B \<subseteq> C) = (A \<subseteq> B \<union> C)" 
1102 
by blast 

1103 

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

1104 

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

1105 
text {* \medskip Monotonicity. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1106 

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

1107 
lemma mono_Un: "mono f ==> f A \<union> f B \<subseteq> f (A \<union> B)" 
16773  1108 
by (auto simp add: mono_def) 
15206
09d78ec709c7
Modified locales: improved implementation of "includes".
ballarin
parents:
15140
diff
changeset

1109 

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

1110 
lemma mono_Int: "mono f ==> f (A \<inter> B) \<subseteq> f A \<inter> f B" 
16773  1111 
by (auto simp add: mono_def) 
12897
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 
subsubsection {* Equalities involving union, intersection, inclusion, etc. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1114 

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

1115 
text {* @{text "{}"}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1116 

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

1117 
lemma 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

1118 
 {* supersedes @{text "Collect_False_empty"} *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1119 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1120 

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

1121 
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1122 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1123 

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

1124 
lemma not_psubset_empty [iff]: "\<not> (A < {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1125 
by (unfold psubset_def) blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1126 

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

1127 
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)" 
18423  1128 
by blast 
1129 

1130 
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)" 

1131 
by blast 

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

1132 

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

1133 
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

1134 
by blast 
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 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

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

1138 

14812  1139 
lemma Collect_imp_eq: "{x. P x > Q x} = {x. P x} \<union> {x. Q x}" 
1140 
by blast 

1141 

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

1142 
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

1143 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1144 

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

1145 
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

1146 
by blast 
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_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

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_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

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

1153 

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

1154 
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

1155 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1156 

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

1157 

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

1158 
text {* \medskip @{text insert}. *} 
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 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

1161 
 {* 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

1162 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1163 

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

1164 
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

1165 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1166 

17715  1167 
lemmas empty_not_insert = insert_not_empty [symmetric, standard] 
1168 
declare empty_not_insert [simp] 

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

1169 

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

1170 
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

1171 
 {* @{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

1172 
 {* with \emph{quadratic} running time *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1173 
by blast 
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_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

1176 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1177 

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

1178 
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

1179 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1180 

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

1181 
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

1182 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1183 

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

1184 
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

1185 
 {* use new @{text B} rather than @{text "A  {a}"} to avoid infinite unfolding *} 
14208  1186 
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

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

1188 

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

1189 
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

1190 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1191 

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

1192 
lemma 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

1193 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1194 

14302  1195 
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)" 
14742  1196 
by blast 
14302  1197 

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

1198 
lemma insert_disjoint[simp]: 
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1199 
"(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})" 
14742  1200 
"({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)" 
16773  1201 
by auto 
13103
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1202 

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

1203 
lemma disjoint_insert[simp]: 
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1204 
"(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})" 
14742  1205 
"({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)" 
16773  1206 
by auto 
14742  1207 

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

1208 
text {* \medskip @{text image}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1209 

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

1210 
lemma image_empty [simp]: "f`{} = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

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

1212 

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

1213 
lemma 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

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

1215 

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

1216 
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}" 
16773  1217 
by auto 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1218 

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

1219 
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

1220 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1221 

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

1222 
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

1223 
by blast 
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_is_empty [iff]: "(f`A = {}) = (A = {})" 
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 

16773  1228 

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

1229 
lemma image_Collect: "f ` {x. P x} = {f x  x. P x}" 
16773  1230 
 {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, 
1231 
with its implicit quantifier and conjunction. Also image enjoys better 

1232 
equational properties than does the RHS. *} 

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

1233 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1234 

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

1235 
lemma if_image_distrib [simp]: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1236 
"(\<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

1237 
= (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

1238 
by (auto simp add: image_def) 
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 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

1241 
by (simp add: image_def) 
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 

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

1244 
text {* \medskip @{text range}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1245 

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

1246 
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

1247 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1248 

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

1249 
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g" 
14208  1250 
by (subst image_image, simp) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1251 

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

1254 

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

1255 
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

1256 
by blast 
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 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

1259 
by blast 
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 
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

1262 
by blast 
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 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

1265 
by blast 
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 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

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

1269 

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

1270 
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

1271 
 {* Intersection is an ACoperator *} 
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_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

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_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

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_empty_left [simp]: "{} \<inter> B = {}" 
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_empty_right [simp]: "A \<inter> {} = {}" 
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 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

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 
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

1289 
by blast 
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_UNIV_left [simp]: "UNIV \<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_UNIV_right [simp]: "A \<inter> UNIV = 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_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

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_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

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 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

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 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

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

1308 

15102  1309 
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

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_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

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 

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

1316 
text {* \medskip @{text Un}. *} 
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 Un_absorb [simp]: "A \<union> A = A" 
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 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

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 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

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

1326 

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

1327 
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

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 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

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 
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1334 
 {* Union is an ACoperator *} 
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_absorb1: "A \<subseteq> B ==> A \<union> B = B" 
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_absorb2: "B \<subseteq> A ==> A \<union> B = A" 
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_empty_left [simp]: "{} \<union> B = B" 
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_empty_right [simp]: "A \<union> {} = A" 
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_UNIV_left [simp]: "UNIV \<union> B = UNIV" 
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 

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

1351 
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1352 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1353 

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