author  berghofe 
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parent 28562  4e74209f113e 
child 29901  f4b3f8fbf599 
permissions  rwrr 
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(* Title: HOL/Set.thy 
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ID: $Id$ 

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Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel 
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*) 
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header {* Set theory for higherorder logic *} 
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theory Set 
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imports Orderings 
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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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types 'a set = "'a => bool" 
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consts 
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"{}" :: "'a set" ("{}") 
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UNIV :: "'a set" 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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text{* 
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Note the difference between ordinary xsymbol syntax of indexed 
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unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}) 
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and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The 
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former does not make the index expression a subscript of the 
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union/intersection symbol because this leads to problems with nested 
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subscripts in Proof General. *} 
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abbreviation 
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subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 
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"subset \<equiv> less" 
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abbreviation 
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subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where 
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"subset_eq \<equiv> less_eq" 
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notation (output) 

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

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

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

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

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

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notation (xsymbols) 

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supset ("op \<supset>") and 

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

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supset_eq ("op \<supseteq>") and 

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

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subsubsection "Bounded quantifiers" 
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syntax (output) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3EX! _<=_./ _)" [0, 0, 10] 10) 
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syntax (xsymbols) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) 
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syntax (HOL output) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3?! _<=_./ _)" [0, 0, 10] 10) 
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syntax (HTML output) 
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"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) 
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"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) 
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"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) 
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translations 
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"\<forall>A\<subset>B. P" => "ALL A. A \<subset> B > P" 
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"\<exists>A\<subset>B. P" => "EX A. A \<subset> B & P" 
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"\<forall>A\<subseteq>B. P" => "ALL A. A \<subseteq> B > P" 
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"\<exists>A\<subseteq>B. P" => "EX A. A \<subseteq> B & P" 
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"\<exists>!A\<subseteq>B. P" => "EX! A. A \<subseteq> B & P" 
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print_translation {* 
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let 
22377  226 
val Type (set_type, _) = @{typ "'a set"}; 
227 
val All_binder = Syntax.binder_name @{const_syntax "All"}; 

228 
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"}; 

229 
val impl = @{const_syntax "op >"}; 

230 
val conj = @{const_syntax "op &"}; 

231 
val sbset = @{const_syntax "subset"}; 

232 
val sbset_eq = @{const_syntax "subset_eq"}; 

21819  233 

234 
val trans = 

235 
[((All_binder, impl, sbset), "_setlessAll"), 

236 
((All_binder, impl, sbset_eq), "_setleAll"), 

237 
((Ex_binder, conj, sbset), "_setlessEx"), 

238 
((Ex_binder, conj, sbset_eq), "_setleEx")]; 

239 

240 
fun mk v v' c n P = 

241 
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v  _ => false) n) 

242 
then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match; 

243 

244 
fun tr' q = (q, 

245 
fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] => 

246 
if T = (set_type) then case AList.lookup (op =) trans (q, c, d) 

247 
of NONE => raise Match 

248 
 SOME l => mk v v' l n P 

249 
else raise Match 

250 
 _ => raise Match); 

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in 
21819  252 
[tr' All_binder, tr' Ex_binder] 
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end 
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*} 
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11979  257 
text {* 
258 
\medskip Translate between @{text "{e  x1...xn. P}"} and @{text 

259 
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is 

260 
only translated if @{text "[0..n] subset bvs(e)"}. 

261 
*} 

262 

263 
parse_translation {* 

264 
let 

265 
val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); 

3947  266 

11979  267 
fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 
268 
 nvars _ = 1; 

269 

270 
fun setcompr_tr [e, idts, b] = 

271 
let 

272 
val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; 

273 
val P = Syntax.const "op &" $ eq $ b; 

274 
val exP = ex_tr [idts, P]; 

17784  275 
in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end; 
11979  276 

277 
in [("@SetCompr", setcompr_tr)] end; 

278 
*} 

923  279 

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(* To avoid etacontraction of body: *) 
11979  281 
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  287 
[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"), 
288 
("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  302 
 check _ = false 
923  303 

11979  304 
fun tr' (_ $ abs) = 
305 
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] 

306 
in Syntax.const "@SetCompr" $ e $ idts $ Q end; 

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in if check (P, 0) then tr' P 
15535  308 
else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs 
309 
val M = Syntax.const "@Coll" $ x $ t 

310 
in case t of 

311 
Const("op &",_) 

312 
$ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) 

313 
$ P => 

314 
if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M 

315 
 _ => M 

316 
end 

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end; 
11979  318 
in [("Collect", setcompr_tr')] end; 
319 
*} 

320 

321 

322 
subsection {* Rules and definitions *} 

323 

324 
text {* Isomorphisms between predicates and sets. *} 

923  325 

26800  326 
defs 
28562  327 
mem_def [code]: "x : S == S x" 
328 
Collect_def [code]: "Collect P == P" 

11979  329 

330 
defs 

331 
Ball_def: "Ball A P == ALL x. x:A > P(x)" 

332 
Bex_def: "Bex A P == EX x. x:A & P(x)" 

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Bex1_def: "Bex1 A P == EX! x. x:A & P(x)" 
11979  334 

26800  335 
instantiation "fun" :: (type, minus) minus 
25510  336 
begin 
337 

338 
definition 

26800  339 
fun_diff_def: "A  B = (%x. A x  B x)" 
25762  340 

341 
instance .. 

342 

343 
end 

344 

26800  345 
instantiation bool :: minus 
25762  346 
begin 
25510  347 

348 
definition 

26800  349 
bool_diff_def: "A  B = (A & ~ B)" 
350 

351 
instance .. 

352 

353 
end 

354 

355 
instantiation "fun" :: (type, uminus) uminus 

356 
begin 

357 

358 
definition 

359 
fun_Compl_def: " A = (%x.  A x)" 

360 

361 
instance .. 

362 

363 
end 

364 

365 
instantiation bool :: uminus 

366 
begin 

367 

368 
definition 

369 
bool_Compl_def: " A = (~ A)" 

25510  370 

371 
instance .. 

372 

373 
end 

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923  375 
defs 
11979  376 
Un_def: "A Un B == {x. x:A  x:B}" 
377 
Int_def: "A Int B == {x. x:A & x:B}" 

378 
INTER_def: "INTER A B == {y. ALL x:A. y: B(x)}" 

379 
UNION_def: "UNION A B == {y. EX x:A. y: B(x)}" 

380 
Inter_def: "Inter S == (INT x:S. x)" 

381 
Union_def: "Union S == (UN x:S. x)" 

382 
Pow_def: "Pow A == {B. B <= A}" 

383 
empty_def: "{} == {x. False}" 

384 
UNIV_def: "UNIV == {x. True}" 

385 
insert_def: "insert a B == {x. x=a} Un B" 

386 
image_def: "f`A == {y. EX x:A. y = f(x)}" 

387 

388 

389 
subsection {* Lemmas and proof tool setup *} 

390 

391 
subsubsection {* Relating predicates and sets *} 

392 

26800  393 
lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)" 
394 
by (simp add: Collect_def mem_def) 

395 

396 
lemma Collect_mem_eq [simp]: "{x. x:A} = A" 

397 
by (simp add: Collect_def mem_def) 

17085  398 

12257  399 
lemma CollectI: "P(a) ==> a : {x. P(x)}" 
11979  400 
by simp 
401 

402 
lemma CollectD: "a : {x. P(x)} ==> P(a)" 

403 
by simp 

404 

405 
lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}" 

406 
by simp 

407 

12257  408 
lemmas CollectE = CollectD [elim_format] 
11979  409 

410 

411 
subsubsection {* Bounded quantifiers *} 

412 

413 
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" 

414 
by (simp add: Ball_def) 

415 

416 
lemmas strip = impI allI ballI 

417 

418 
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" 

419 
by (simp add: Ball_def) 

420 

421 
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" 

422 
by (unfold Ball_def) blast 

22139  423 

424 
ML {* bind_thm ("rev_ballE", permute_prems 1 1 @{thm ballE}) *} 

11979  425 

426 
text {* 

427 
\medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and 

428 
@{prop "a:A"}; creates assumption @{prop "P a"}. 

429 
*} 

430 

431 
ML {* 

22139  432 
fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1) 
11979  433 
*} 
434 

435 
text {* 

436 
Gives better instantiation for bound: 

437 
*} 

438 

26339  439 
declaration {* fn _ => 
440 
Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1)) 

11979  441 
*} 
442 

443 
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" 

444 
 {* Normally the best argument order: @{prop "P x"} constrains the 

445 
choice of @{prop "x:A"}. *} 

446 
by (unfold Bex_def) blast 

447 

13113  448 
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" 
11979  449 
 {* The best argument order when there is only one @{prop "x:A"}. *} 
450 
by (unfold Bex_def) blast 

451 

452 
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" 

453 
by (unfold Bex_def) blast 

454 

455 
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" 

456 
by (unfold Bex_def) blast 

457 

458 
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) > P)" 

459 
 {* Trival rewrite rule. *} 

460 
by (simp add: Ball_def) 

461 

462 
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" 

463 
 {* Dual form for existentials. *} 

464 
by (simp add: Bex_def) 

465 

466 
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" 

467 
by blast 

468 

469 
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" 

470 
by blast 

471 

472 
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" 

473 
by blast 

474 

475 
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" 

476 
by blast 

477 

478 
lemma ball_one_point1 [simp]: "(ALL x:A. x = a > P x) = (a:A > P a)" 

479 
by blast 

480 

481 
lemma ball_one_point2 [simp]: "(ALL x:A. a = x > P x) = (a:A > P a)" 

482 
by blast 

483 

26480  484 
ML {* 
13462  485 
local 
22139  486 
val unfold_bex_tac = unfold_tac @{thms "Bex_def"}; 
18328  487 
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; 
11979  488 
val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; 
489 

22139  490 
val unfold_ball_tac = unfold_tac @{thms "Ball_def"}; 
18328  491 
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; 
11979  492 
val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; 
493 
in 

18328  494 
val defBEX_regroup = Simplifier.simproc (the_context ()) 
13462  495 
"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; 
18328  496 
val defBALL_regroup = Simplifier.simproc (the_context ()) 
13462  497 
"defined BALL" ["ALL x:A. P x > Q x"] rearrange_ball; 
11979  498 
end; 
13462  499 

500 
Addsimprocs [defBALL_regroup, defBEX_regroup]; 

11979  501 
*} 
502 

503 

504 
subsubsection {* Congruence rules *} 

505 

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lemma ball_cong: 
11979  507 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
508 
(ALL x:A. P x) = (ALL x:B. Q x)" 

509 
by (simp add: Ball_def) 

510 

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511 
lemma strong_ball_cong [cong]: 
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512 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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513 
(ALL x:A. P x) = (ALL x:B. Q x)" 
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514 
by (simp add: simp_implies_def Ball_def) 
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515 

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516 
lemma bex_cong: 
11979  517 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
518 
(EX x:A. P x) = (EX x:B. Q x)" 

519 
by (simp add: Bex_def cong: conj_cong) 

1273  520 

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521 
lemma strong_bex_cong [cong]: 
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522 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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523 
(EX x:A. P x) = (EX x:B. Q x)" 
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524 
by (simp add: simp_implies_def Bex_def cong: conj_cong) 
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525 

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526 

11979  527 
subsubsection {* Subsets *} 
528 

19295  529 
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B" 
26800  530 
by (auto simp add: mem_def intro: predicate1I) 
11979  531 

532 
text {* 

533 
\medskip Map the type @{text "'a set => anything"} to just @{typ 

534 
'a}; for overloading constants whose first argument has type @{typ 

535 
"'a set"}. 

536 
*} 

537 

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538 
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" 
11979  539 
 {* Rule in Modus Ponens style. *} 
26800  540 
by (unfold mem_def) blast 
11979  541 

542 
declare subsetD [intro?]  FIXME 

543 

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544 
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B" 
11979  545 
 {* The same, with reversed premises for use with @{text erule}  
546 
cf @{text rev_mp}. *} 

547 
by (rule subsetD) 

548 

549 
declare rev_subsetD [intro?]  FIXME 

550 

551 
text {* 

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552 
\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}. 
11979  553 
*} 
554 

555 
ML {* 

22139  556 
fun impOfSubs th = th RSN (2, @{thm rev_subsetD}) 
11979  557 
*} 
558 

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559 
lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P" 
11979  560 
 {* Classical elimination rule. *} 
26800  561 
by (unfold mem_def) blast 
562 

563 
lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast 

11979  564 

565 
text {* 

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566 
\medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and 
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567 
creates the assumption @{prop "c \<in> B"}. 
11979  568 
*} 
569 

570 
ML {* 

22139  571 
fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i 
11979  572 
*} 
573 

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574 
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A" 
11979  575 
by blast 
576 

19175  577 
lemma subset_refl [simp,atp]: "A \<subseteq> A" 
11979  578 
by fast 
579 

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580 
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" 
11979  581 
by blast 
923  582 

2261  583 

11979  584 
subsubsection {* Equality *} 
585 

13865  586 
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B" 
587 
apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals]) 

588 
apply (rule Collect_mem_eq) 

589 
apply (rule Collect_mem_eq) 

590 
done 

591 

15554  592 
(* Due to Brian Huffman *) 
593 
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))" 

594 
by(auto intro:set_ext) 

595 

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596 
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B" 
11979  597 
 {* Antisymmetry of the subset relation. *} 
17589  598 
by (iprover intro: set_ext subsetD) 
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599 

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600 
lemmas equalityI [intro!] = subset_antisym 
11979  601 

602 
text {* 

603 
\medskip Equality rules from ZF set theory  are they appropriate 

604 
here? 

605 
*} 

606 

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607 
lemma equalityD1: "A = B ==> A \<subseteq> B" 
11979  608 
by (simp add: subset_refl) 
609 

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610 
lemma equalityD2: "A = B ==> B \<subseteq> A" 
11979  611 
by (simp add: subset_refl) 
612 

613 
text {* 

614 
\medskip Be careful when adding this to the claset as @{text 

615 
subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{} 

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616 
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! 
11979  617 
*} 
618 

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619 
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P" 
11979  620 
by (simp add: subset_refl) 
923  621 

11979  622 
lemma equalityCE [elim]: 
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623 
"A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P" 
11979  624 
by blast 
625 

626 
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)" 

627 
by simp 

628 

13865  629 
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)" 
630 
by simp 

631 

11979  632 

633 
subsubsection {* The universal set  UNIV *} 

634 

635 
lemma UNIV_I [simp]: "x : UNIV" 

636 
by (simp add: UNIV_def) 

637 

638 
declare UNIV_I [intro]  {* unsafe makes it less likely to cause problems *} 

639 

640 
lemma UNIV_witness [intro?]: "EX x. x : UNIV" 

641 
by simp 

642 

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643 
lemma subset_UNIV [simp]: "A \<subseteq> UNIV" 
11979  644 
by (rule subsetI) (rule UNIV_I) 
2388  645 

11979  646 
text {* 
647 
\medskip Etacontracting these two rules (to remove @{text P}) 

648 
causes them to be ignored because of their interaction with 

649 
congruence rules. 

650 
*} 

651 

652 
lemma ball_UNIV [simp]: "Ball UNIV P = All P" 

653 
by (simp add: Ball_def) 

654 

655 
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P" 

656 
by (simp add: Bex_def) 

657 

26150  658 
lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A" 
659 
by auto 

660 

11979  661 

662 
subsubsection {* The empty set *} 

663 

664 
lemma empty_iff [simp]: "(c : {}) = False" 

665 
by (simp add: empty_def) 

666 

667 
lemma emptyE [elim!]: "a : {} ==> P" 

668 
by simp 

669 

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670 
lemma empty_subsetI [iff]: "{} \<subseteq> A" 
11979  671 
 {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *} 
672 
by blast 

673 

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674 
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}" 
11979  675 
by blast 
2388  676 

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677 
lemma equals0D: "A = {} ==> a \<notin> A" 
11979  678 
 {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *} 
679 
by blast 

680 

681 
lemma ball_empty [simp]: "Ball {} P = True" 

682 
by (simp add: Ball_def) 

683 

684 
lemma bex_empty [simp]: "Bex {} P = False" 

685 
by (simp add: Bex_def) 

686 

687 
lemma UNIV_not_empty [iff]: "UNIV ~= {}" 

688 
by (blast elim: equalityE) 

689 

690 

12023  691 
subsubsection {* The Powerset operator  Pow *} 
11979  692 

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693 
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)" 
11979  694 
by (simp add: Pow_def) 
695 

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696 
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B" 
11979  697 
by (simp add: Pow_def) 
698 

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699 
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B" 
11979  700 
by (simp add: Pow_def) 
701 

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702 
lemma Pow_bottom: "{} \<in> Pow B" 
11979  703 
by simp 
704 

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705 
lemma Pow_top: "A \<in> Pow A" 
11979  706 
by (simp add: subset_refl) 
2684  707 

2388  708 

11979  709 
subsubsection {* Set complement *} 
710 

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711 
lemma Compl_iff [simp]: "(c \<in> A) = (c \<notin> A)" 
26800  712 
by (simp add: mem_def fun_Compl_def bool_Compl_def) 
11979  713 

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714 
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> A" 
26800  715 
by (unfold mem_def fun_Compl_def bool_Compl_def) blast 
11979  716 

717 
text {* 

718 
\medskip This form, with negated conclusion, works well with the 

719 
Classical prover. Negated assumptions behave like formulae on the 

720 
right side of the notional turnstile ... *} 

721 

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722 
lemma ComplD [dest!]: "c : A ==> c~:A" 
26800  723 
by (simp add: mem_def fun_Compl_def bool_Compl_def) 
11979  724 

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725 
lemmas ComplE = ComplD [elim_format] 
11979  726 

26800  727 
lemma Compl_eq: " A = {x. ~ x : A}" by blast 
728 

11979  729 

730 
subsubsection {* Binary union  Un *} 

923  731 

11979  732 
lemma Un_iff [simp]: "(c : A Un B) = (c:A  c:B)" 
733 
by (unfold Un_def) blast 

734 

735 
lemma UnI1 [elim?]: "c:A ==> c : A Un B" 

736 
by simp 

737 

738 
lemma UnI2 [elim?]: "c:B ==> c : A Un B" 

739 
by simp 

923  740 

11979  741 
text {* 
742 
\medskip Classical introduction rule: no commitment to @{prop A} vs 

743 
@{prop B}. 

744 
*} 

745 

746 
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B" 

747 
by auto 

748 

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

750 
by (unfold Un_def) blast 

751 

752 

12023  753 
subsubsection {* Binary intersection  Int *} 
923  754 

11979  755 
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)" 
756 
by (unfold Int_def) blast 

757 

758 
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B" 

759 
by simp 

760 

761 
lemma IntD1: "c : A Int B ==> c:A" 

762 
by simp 

763 

764 
lemma IntD2: "c : A Int B ==> c:B" 

765 
by simp 

766 

767 
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P" 

768 
by simp 

769 

770 

12023  771 
subsubsection {* Set difference *} 
11979  772 

773 
lemma Diff_iff [simp]: "(c : A  B) = (c:A & c~:B)" 

26800  774 
by (simp add: mem_def fun_diff_def bool_diff_def) 
923  775 

11979  776 
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A  B" 
777 
by simp 

778 

779 
lemma DiffD1: "c : A  B ==> c : A" 

780 
by simp 

781 

782 
lemma DiffD2: "c : A  B ==> c : B ==> P" 

783 
by simp 

784 

785 
lemma DiffE [elim!]: "c : A  B ==> (c:A ==> c~:B ==> P) ==> P" 

786 
by simp 

787 

26800  788 
lemma set_diff_eq: "A  B = {x. x : A & ~ x : B}" by blast 
789 

11979  790 

791 
subsubsection {* Augmenting a set  insert *} 

792 

793 
lemma insert_iff [simp]: "(a : insert b A) = (a = b  a:A)" 

794 
by (unfold insert_def) blast 

795 

796 
lemma insertI1: "a : insert a B" 

797 
by simp 

798 

799 
lemma insertI2: "a : B ==> a : insert b B" 

800 
by simp 

923  801 

11979  802 
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P" 
803 
by (unfold insert_def) blast 

804 

805 
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B" 

806 
 {* Classical introduction rule. *} 

807 
by auto 

808 

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809 
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A  {x} \<subseteq> B else A \<subseteq> B)" 
11979  810 
by auto 
811 

24730  812 
lemma set_insert: 
813 
assumes "x \<in> A" 

814 
obtains B where "A = insert x B" and "x \<notin> B" 

815 
proof 

816 
from assms show "A = insert x (A  {x})" by blast 

817 
next 

818 
show "x \<notin> A  {x}" by blast 

819 
qed 

820 

25287  821 
lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)" 
822 
by auto 

11979  823 

824 
subsubsection {* Singletons, using insert *} 

825 

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826 
lemma singletonI [intro!,noatp]: "a : {a}" 
11979  827 
 {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *} 
828 
by (rule insertI1) 

829 

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830 
lemma singletonD [dest!,noatp]: "b : {a} ==> b = a" 
11979  831 
by blast 
832 

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833 
lemmas singletonE = singletonD [elim_format] 
11979  834 

835 
lemma singleton_iff: "(b : {a}) = (b = a)" 

836 
by blast 

837 

838 
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b" 

839 
by blast 

840 

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841 
lemma singleton_insert_inj_eq [iff,noatp]: 
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842 
"({b} = insert a A) = (a = b & A \<subseteq> {b})" 
11979  843 
by blast 
844 

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845 
lemma singleton_insert_inj_eq' [iff,noatp]: 
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846 
"(insert a A = {b}) = (a = b & A \<subseteq> {b})" 
11979  847 
by blast 
848 

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849 
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {}  A = {x}" 
11979  850 
by fast 
851 

852 
lemma singleton_conv [simp]: "{x. x = a} = {a}" 

853 
by blast 

854 

855 
lemma singleton_conv2 [simp]: "{x. a = x} = {a}" 

856 
by blast 

923  857 

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858 
lemma diff_single_insert: "A  {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B" 
11979  859 
by blast 
860 

19870  861 
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d  a=d & b=c)" 
862 
by (blast elim: equalityE) 

863 

11979  864 

865 
subsubsection {* Unions of families *} 

866 

867 
text {* 

868 
@{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}. 

869 
*} 

870 

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871 
declare UNION_def [noatp] 
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872 

11979  873 
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" 
874 
by (unfold UNION_def) blast 

875 

876 
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" 

877 
 {* The order of the premises presupposes that @{term A} is rigid; 

878 
@{term b} may be flexible. *} 

879 
by auto 

880 

881 
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" 

882 
by (unfold UNION_def) blast 

923  883 

11979  884 
lemma UN_cong [cong]: 
885 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

886 
by (simp add: UNION_def) 

887 

29691  888 
lemma strong_UN_cong: 
889 
"A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

890 
by (simp add: UNION_def simp_implies_def) 

891 

11979  892 

893 
subsubsection {* Intersections of families *} 

894 

895 
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *} 

896 

897 
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" 

898 
by (unfold INTER_def) blast 

923  899 

11979  900 
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" 
901 
by (unfold INTER_def) blast 

902 

903 
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" 

904 
by auto 

905 

906 
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" 

907 
 {* "Classical" elimination  by the Excluded Middle on @{prop "a:A"}. *} 

908 
by (unfold INTER_def) blast 

909 

910 
lemma INT_cong [cong]: 

911 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" 

912 
by (simp add: INTER_def) 

7238
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913 

923  914 

11979  915 
subsubsection {* Union *} 
916 

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parents:
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diff
changeset

917 
lemma Union_iff [simp,noatp]: "(A : Union C) = (EX X:C. A:X)" 
11979  918 
by (unfold Union_def) blast 
919 

920 
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C" 

921 
 {* The order of the premises presupposes that @{term C} is rigid; 

922 
@{term A} may be flexible. *} 

923 
by auto 

924 

925 
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R" 

926 
by (unfold Union_def) blast 

927 

928 

929 
subsubsection {* Inter *} 

930 

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paulson
parents:
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diff
changeset

931 
lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)" 
11979  932 
by (unfold Inter_def) blast 
933 

934 
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C" 

935 
by (simp add: Inter_def) 

936 

937 
text {* 

938 
\medskip A ``destruct'' rule  every @{term X} in @{term C} 

939 
contains @{term A} as an element, but @{prop "A:X"} can hold when 

940 
@{prop "X:C"} does not! This rule is analogous to @{text spec}. 

941 
*} 

942 

943 
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X" 

944 
by auto 

945 

946 
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R" 

947 
 {* ``Classical'' elimination rule  does not require proving 

948 
@{prop "X:C"}. *} 

949 
by (unfold Inter_def) blast 

950 

951 
text {* 

952 
\medskip Image of a set under a function. Frequently @{term b} does 

953 
not have the syntactic form of @{term "f x"}. 

954 
*} 

955 

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

956 
declare image_def [noatp] 
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset

957 

11979  958 
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A" 
959 
by (unfold image_def) blast 

960 

961 
lemma imageI: "x : A ==> f x : f ` A" 

962 
by (rule image_eqI) (rule refl) 

963 

964 
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A" 

965 
 {* This version's more effective when we already have the 

966 
required @{term x}. *} 

967 
by (unfold image_def) blast 

968 

969 
lemma imageE [elim!]: 

970 
"b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P" 

971 
 {* The etaexpansion gives variablename preservation. *} 

972 
by (unfold image_def) blast 

973 

974 
lemma image_Un: "f`(A Un B) = f`A Un f`B" 

975 
by blast 

976 

26150  977 
lemma image_eq_UN: "f`A = (UN x:A. {f x})" 
978 
by blast 

979 

11979  980 
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)" 
981 
by blast 

982 

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

983 
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)" 
11979  984 
 {* This rewrite rule would confuse users if made default. *} 
985 
by blast 

986 

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

987 
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)" 
11979  988 
apply safe 
989 
prefer 2 apply fast 

14208  990 
apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast) 
11979  991 
done 
992 

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

993 
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B" 
11979  994 
 {* Replaces the three steps @{text subsetI}, @{text imageE}, 
995 
@{text hypsubst}, but breaks too many existing proofs. *} 

996 
by blast 

997 

998 
text {* 

999 
\medskip Range of a function  just a translation for image! 

1000 
*} 

1001 

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

1002 
lemma range_eqI: "b = f x ==> b \<in> range f" 
11979  1003 
by simp 
1004 

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

1005 
lemma rangeI: "f x \<in> range f" 
11979  1006 
by simp 
1007 

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

1008 
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P" 
11979  1009 
by blast 
1010 

1011 

1012 
subsubsection {* Set reasoning tools *} 

1013 

1014 
text {* 

1015 
Rewrite rules for boolean casesplitting: faster than @{text 

1016 
"split_if [split]"}. 

1017 
*} 

1018 

1019 
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q > x = b) & (~ Q > y = b))" 

1020 
by (rule split_if) 

1021 

1022 
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q > a = x) & (~ Q > a = y))" 

1023 
by (rule split_if) 

1024 

1025 
text {* 

1026 
Split ifs on either side of the membership relation. Not for @{text 

1027 
"[simp]"}  can cause goals to blow up! 

1028 
*} 

1029 

1030 
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q > x : b) & (~ Q > y : b))" 

1031 
by (rule split_if) 

1032 

1033 
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q > a : x) & (~ Q > a : y))" 

26800  1034 
by (rule split_if [where P="%S. a : S"]) 
11979  1035 

1036 
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 

1037 

1038 
lemmas mem_simps = 

1039 
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff 

1040 
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff 

1041 
 {* Each of these has ALREADY been added @{text "[simp]"} above. *} 

1042 

1043 
(*Would like to add these, but the existing code only searches for the 

1044 
outerlevel constant, which in this case is just "op :"; we instead need 

1045 
to use termnets to associate patterns with rules. Also, if a rule fails to 

1046 
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

1047 
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]), 
11979  1048 
("op Int", [IntD1,IntD2]), 
1049 
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] 

1050 
*) 

1051 

26339  1052 
ML {* 
22139  1053 
val mksimps_pairs = [("Ball", @{thms bspec})] @ mksimps_pairs; 
26339  1054 
*} 
1055 
declaration {* fn _ => 

1056 
Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs)) 

11979  1057 
*} 
1058 

1059 

1060 
subsubsection {* The ``proper subset'' relation *} 

1061 

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

1062 
lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B" 
26800  1063 
by (unfold less_le) blast 
11979  1064 

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

1065 
lemma psubsetE [elim!,noatp]: 
13624  1066 
"[A \<subset> B; [A \<subseteq> B; ~ (B\<subseteq>A)] ==> R] ==> R" 
26800  1067 
by (unfold less_le) blast 
13624  1068 

11979  1069 
lemma psubset_insert_iff: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1070 
"(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)" 
26800  1071 
by (auto simp add: less_le subset_insert_iff) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1072 

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

1073 
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)" 
26800  1074 
by (simp only: less_le) 
11979  1075 

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

1076 
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B" 
11979  1077 
by (simp add: psubset_eq) 
1078 

14335  1079 
lemma psubset_trans: "[ A \<subset> B; B \<subset> C ] ==> A \<subset> C" 
26800  1080 
apply (unfold less_le) 
14335  1081 
apply (auto dest: subset_antisym) 
1082 
done 

1083 

1084 
lemma psubsetD: "[ A \<subset> B; c \<in> A ] ==> c \<in> B" 

26800  1085 
apply (unfold less_le) 
14335  1086 
apply (auto dest: subsetD) 
1087 
done 

1088 

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

1089 
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C" 
11979  1090 
by (auto simp add: psubset_eq) 
1091 

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

1092 
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C" 
11979  1093 
by (auto simp add: psubset_eq) 
1094 

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

1095 
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B  A)" 
26800  1096 
by (unfold less_le) blast 
11979  1097 

1098 
lemma atomize_ball: 

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

1099 
"(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)" 
11979  1100 
by (simp only: Ball_def atomize_all atomize_imp) 
1101 

18832  1102 
lemmas [symmetric, rulify] = atomize_ball 
1103 
and [symmetric, defn] = atomize_ball 

11979  1104 

1105 

22455  1106 
subsection {* Further settheory lemmas *} 
1107 

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

1108 
subsubsection {* Derived rules involving subsets. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1109 

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

1110 
text {* @{text insert}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1111 

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

1112 
lemma subset_insertI: "B \<subseteq> insert a B" 
23878  1113 
by (rule subsetI) (erule insertI2) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1114 

14302  1115 
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B" 
23878  1116 
by blast 
14302  1117 

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

1118 
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

1119 
by blast 
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 

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

1122 
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

1123 

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

1124 
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A" 
17589  1125 
by (iprover intro: subsetI UnionI) 
12897
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 Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C" 
17589  1128 
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

1129 

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

1130 

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

1131 
text {* \medskip General union. *} 
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 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

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 UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C" 
17589  1137 
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

1138 

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

1139 

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

1140 
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

1141 

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

1142 
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

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

1144 

14551  1145 
lemma Inter_subset: 
1146 
"[ !!X. X \<in> A ==> X \<subseteq> B; A ~= {} ] ==> \<Inter>A \<subseteq> B" 

1147 
by blast 

1148 

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

1149 
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A" 
17589  1150 
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

1151 

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

1152 
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

1153 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1154 

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

1155 
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)" 
17589  1156 
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

1157 

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

1158 

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

1159 
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

1160 

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

1161 
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

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 Un_upper2: "B \<subseteq> A \<union> B" 
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 

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

1167 
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

1168 
by blast 
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 

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

1171 
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

1172 

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

1173 
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

1174 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1175 

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

1176 
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

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

1178 

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

1179 
lemma 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

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

1181 

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

1182 

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

1183 
text {* \medskip Set difference. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1184 

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

1185 
lemma Diff_subset: "A  B \<subseteq> A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1186 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1187 

14302  1188 
lemma Diff_subset_conv: "(A  B \<subseteq> C) = (A \<subseteq> B \<union> C)" 
1189 
by blast 

1190 

12897
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 
subsubsection {* Equalities involving union, intersection, inclusion, etc. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1193 

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

1194 
text {* @{text "{}"}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1195 

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

1196 
lemma 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

1197 
 {* supersedes @{text "Collect_False_empty"} *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1198 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1199 

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

1200 
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1201 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1202 

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

1203 
lemma not_psubset_empty [iff]: "\<not> (A < {})" 
26800  1204 
by (unfold less_le) blast 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1205 

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

1206 
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)" 
18423  1207 
by blast 
1208 

1209 
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)" 

1210 
by blast 

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

1211 

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

1212 
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

1213 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1214 

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

1215 
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

1216 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1217 

14812  1218 
lemma Collect_imp_eq: "{x. P x > Q x} = {x. P x} \<union> {x. Q x}" 
1219 
by blast 

1220 

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

1221 
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

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

1223 

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

1224 
lemma 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

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

1226 

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

1227 
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

1228 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1229 

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

1230 
lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1231 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1232 

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

1233 
lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1234 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1235 

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

1236 

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

1237 
text {* \medskip @{text insert}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1238 

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

1239 
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

1240 
 {* 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

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

1242 

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

1243 
lemma 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

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

1245 

17715  1246 
lemmas empty_not_insert = insert_not_empty [symmetric, standard] 
1247 
declare empty_not_insert [simp] 

12897
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 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

1250 
 {* @{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

1251 
 {* with \emph{quadratic} running time *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1252 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1253 

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

1254 
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

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

1256 

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

1257 
lemma insert_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

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

1259 

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

1260 
lemma 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

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

1262 

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

1263 
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

1264 
 {* use new @{text B} rather than @{text "A  {a}"} to avoid infinite unfolding *} 
14208  1265 
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

1266 
done 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1267 

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

1268 
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

1269 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1270 

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

1271 
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

1272 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1273 

14302  1274 
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)" 
14742  1275 
by blast 
14302  1276 

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

1277 
lemma insert_disjoint [simp,noatp]: 
13103
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1278 
"(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})" 
14742  1279 
"({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)" 
16773  1280 
by auto 
13103
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1281 

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

1282 
lemma disjoint_insert [simp,noatp]: 
13103
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1283 
"(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})" 
14742  1284 
"({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)" 
16773  1285 
by auto 
14742  1286 

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

1287 
text {* \medskip @{text image}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1288 

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

1289 
lemma image_empty [simp]: "f`{} = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1290 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1291 

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

1292 
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

1293 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1294 

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

1295 
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}" 
16773  1296 
by auto 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1297 

21316  1298 
lemma image_constant_conv: "(%x. c) ` A = (if A = {} then {} else {c})" 
21312  1299 
by auto 
1300 

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

1301 
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

1302 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1303 

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

1304 
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

1305 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1306 

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

1307 
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

1308 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1309 

16773  1310 

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

1311 
lemma image_Collect [noatp]: "f ` {x. P x} = {f x  x. P x}" 
16773  1312 
 {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, 
1313 
with its implicit quantifier and conjunction. Also image enjoys better 

1314 
equational properties than does the RHS. *} 

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

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

1316 

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

1317 
lemma if_image_distrib [simp]: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1318 
"(\<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

1319 
= (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

1320 
by (auto simp add: image_def) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1321 

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

1322 
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

1323 
by (simp add: image_def) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1324 

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

1325 

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

1326 
text {* \medskip @{text range}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1327 

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

1328 
lemma full_SetCompr_eq [noatp]: "{u. \<exists>x. u = f x} = range f" 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1329 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1330 

27418  1331 
lemma range_composition: "range (\<lambda>x. f (g x)) = f`range g" 
14208  1332 
by (subst image_image, simp) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1333 

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

1334 

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

1335 
text {* \medskip @{text Int} *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1336 

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

1337 
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

1338 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1339 

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

1340 
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

1341 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1342 

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

1343 
lemma Int_commute: "A \<inter> B = B \<inter> A" 