author  paulson 
Mon, 20 Mar 2006 17:38:22 +0100  
changeset 19295  c5d236fe9668 
parent 19277  f7602e74d948 
child 19323  ec5cd5b1804c 
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 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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syntax 

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"op :" :: "'a => 'a set => bool" ("op :") 

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consts 

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

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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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syntax 
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range :: "('a => 'b) => 'b set"  "of function" 
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"op ~:" :: "'a => 'a set => bool" ("op ~:")  "nonmembership" 
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"op ~:" :: "'a => 'a set => bool" ("(_/ ~: _)" [50, 51] 50) 

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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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"range f" == "f`UNIV" 
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"x ~: y" == "~ (x : y)" 
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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 (output) 
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"_setle" :: "'a set => 'a set => bool" ("op <=") 
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"_setle" :: "'a set => 'a set => bool" ("(_/ <= _)" [50, 51] 50) 

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"_setless" :: "'a set => 'a set => bool" ("op <") 

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"_setless" :: "'a set => 'a set => bool" ("(_/ < _)" [50, 51] 50) 

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syntax (xsymbols) 
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"_setle" :: "'a set => 'a set => bool" ("op \<subseteq>") 
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"_setle" :: "'a set => 'a set => bool" ("(_/ \<subseteq> _)" [50, 51] 50) 

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"_setless" :: "'a set => 'a set => bool" ("op \<subset>") 

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"_setless" :: "'a set => 'a set => bool" ("(_/ \<subset> _)" [50, 51] 50) 

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"op Int" :: "'a set => 'a set => 'a set" (infixl "\<inter>" 70) 

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"op Un" :: "'a set => 'a set => 'a set" (infixl "\<union>" 65) 

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"op :" :: "'a => 'a set => bool" ("op \<in>") 

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"op :" :: "'a => 'a set => bool" ("(_/ \<in> _)" [50, 51] 50) 

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"op ~:" :: "'a => 'a set => bool" ("op \<notin>") 

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"op ~:" :: "'a => 'a set => bool" ("(_/ \<notin> _)" [50, 51] 50) 

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Union :: "'a set set => 'a set" ("\<Union>_" [90] 90) 
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Inter :: "'a set set => 'a set" ("\<Inter>_" [90] 90) 
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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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"_setle" :: "'a set => 'a set => bool" ("op \<subseteq>") 

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"_setle" :: "'a set => 'a set => bool" ("(_/ \<subseteq> _)" [50, 51] 50) 

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"_setless" :: "'a set => 'a set => bool" ("op \<subset>") 

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"_setless" :: "'a set => 'a set => bool" ("(_/ \<subset> _)" [50, 51] 50) 

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"op Int" :: "'a set => 'a set => 'a set" (infixl "\<inter>" 70) 

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"op Un" :: "'a set => 'a set => 'a set" (infixl "\<union>" 65) 

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"op :" :: "'a => 'a set => bool" ("op \<in>") 

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"op :" :: "'a => 'a set => bool" ("(_/ \<in> _)" [50, 51] 50) 

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"op ~:" :: "'a => 'a set => bool" ("op \<notin>") 

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"op ~:" :: "'a => 'a set => bool" ("(_/ \<notin> _)" [50, 51] 50) 

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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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(* 
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syntax (xsymbols) 
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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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*) 
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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{* 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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translations 
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"op \<subseteq>" => "op <= :: _ set => _ set => bool" 
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"op \<subset>" => "op < :: _ set => _ set => bool" 

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typed_print_translation {* 
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let 

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fun le_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts = 

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list_comb (Syntax.const "_setle", ts) 

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 le_tr' _ _ _ = raise Match; 

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fun less_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts = 

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list_comb (Syntax.const "_setless", ts) 

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 less_tr' _ _ _ = raise Match; 

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in [("Orderings.less_eq", le_tr'), ("Orderings.less", less_tr')] end 
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*} 
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subsubsection "Bounded quantifiers" 
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syntax 
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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) 
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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,_)), 
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Const("op >",_) $ (Const ("Orderings.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,_)), 
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Const("op >",_) $ (Const ("Orderings.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,_)), 
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Const("op &",_) $ (Const ("Orderings.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,_)), 
19277  230 
Const("op &",_) $ (Const ("Orderings.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  241 
text {* 
242 
\medskip Translate between @{text "{e  x1...xn. P}"} and @{text 

243 
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is 

244 
only translated if @{text "[0..n] subset bvs(e)"}. 

245 
*} 

246 

247 
parse_translation {* 

248 
let 

249 
val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); 

3947  250 

11979  251 
fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 
252 
 nvars _ = 1; 

253 

254 
fun setcompr_tr [e, idts, b] = 

255 
let 

256 
val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; 

257 
val P = Syntax.const "op &" $ eq $ b; 

258 
val exP = ex_tr [idts, P]; 

17784  259 
in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end; 
11979  260 

261 
in [("@SetCompr", setcompr_tr)] end; 

262 
*} 

923  263 

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(* To avoid etacontraction of body: *) 
11979  265 
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  271 
[("Ball", btr' "_Ball"),("Bex", btr' "_Bex"), 
272 
("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  286 
 check _ = false 
923  287 

11979  288 
fun tr' (_ $ abs) = 
289 
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] 

290 
in Syntax.const "@SetCompr" $ e $ idts $ Q end; 

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in if check (P, 0) then tr' P 
15535  292 
else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs 
293 
val M = Syntax.const "@Coll" $ x $ t 

294 
in case t of 

295 
Const("op &",_) 

296 
$ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) 

297 
$ P => 

298 
if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M 

299 
 _ => M 

300 
end 

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end; 
11979  302 
in [("Collect", setcompr_tr')] end; 
303 
*} 

304 

305 

306 
subsection {* Rules and definitions *} 

307 

308 
text {* Isomorphisms between predicates and sets. *} 

923  309 

11979  310 
axioms 
17085  311 
mem_Collect_eq: "(a : {x. P(x)}) = P(a)" 
312 
Collect_mem_eq: "{x. x:A} = A" 

17702  313 
finalconsts 
314 
Collect 

315 
"op :" 

11979  316 

317 
defs 

318 
Ball_def: "Ball A P == ALL x. x:A > P(x)" 

319 
Bex_def: "Bex A P == EX x. x:A & P(x)" 

320 

321 
defs (overloaded) 

322 
subset_def: "A <= B == ALL x:A. x:B" 

323 
psubset_def: "A < B == (A::'a set) <= B & ~ A=B" 

324 
Compl_def: " A == {x. ~x:A}" 

12257  325 
set_diff_def: "A  B == {x. x:A & ~x:B}" 
923  326 

327 
defs 

11979  328 
Un_def: "A Un B == {x. x:A  x:B}" 
329 
Int_def: "A Int B == {x. x:A & x:B}" 

330 
INTER_def: "INTER A B == {y. ALL x:A. y: B(x)}" 

331 
UNION_def: "UNION A B == {y. EX x:A. y: B(x)}" 

332 
Inter_def: "Inter S == (INT x:S. x)" 

333 
Union_def: "Union S == (UN x:S. x)" 

334 
Pow_def: "Pow A == {B. B <= A}" 

335 
empty_def: "{} == {x. False}" 

336 
UNIV_def: "UNIV == {x. True}" 

337 
insert_def: "insert a B == {x. x=a} Un B" 

338 
image_def: "f`A == {y. EX x:A. y = f(x)}" 

339 

340 

341 
subsection {* Lemmas and proof tool setup *} 

342 

343 
subsubsection {* Relating predicates and sets *} 

344 

17085  345 
declare mem_Collect_eq [iff] Collect_mem_eq [simp] 
346 

12257  347 
lemma CollectI: "P(a) ==> a : {x. P(x)}" 
11979  348 
by simp 
349 

350 
lemma CollectD: "a : {x. P(x)} ==> P(a)" 

351 
by simp 

352 

353 
lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}" 

354 
by simp 

355 

12257  356 
lemmas CollectE = CollectD [elim_format] 
11979  357 

358 

359 
subsubsection {* Bounded quantifiers *} 

360 

361 
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" 

362 
by (simp add: Ball_def) 

363 

364 
lemmas strip = impI allI ballI 

365 

366 
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" 

367 
by (simp add: Ball_def) 

368 

369 
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" 

370 
by (unfold Ball_def) blast 

14098  371 
ML {* bind_thm("rev_ballE",permute_prems 1 1 (thm "ballE")) *} 
11979  372 

373 
text {* 

374 
\medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and 

375 
@{prop "a:A"}; creates assumption @{prop "P a"}. 

376 
*} 

377 

378 
ML {* 

379 
local val ballE = thm "ballE" 

380 
in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end; 

381 
*} 

382 

383 
text {* 

384 
Gives better instantiation for bound: 

385 
*} 

386 

387 
ML_setup {* 

17875  388 
change_claset (fn cs => cs addbefore ("bspec", datac (thm "bspec") 1)); 
11979  389 
*} 
390 

391 
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" 

392 
 {* Normally the best argument order: @{prop "P x"} constrains the 

393 
choice of @{prop "x:A"}. *} 

394 
by (unfold Bex_def) blast 

395 

13113  396 
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" 
11979  397 
 {* The best argument order when there is only one @{prop "x:A"}. *} 
398 
by (unfold Bex_def) blast 

399 

400 
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" 

401 
by (unfold Bex_def) blast 

402 

403 
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" 

404 
by (unfold Bex_def) blast 

405 

406 
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) > P)" 

407 
 {* Trival rewrite rule. *} 

408 
by (simp add: Ball_def) 

409 

410 
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" 

411 
 {* Dual form for existentials. *} 

412 
by (simp add: Bex_def) 

413 

414 
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" 

415 
by blast 

416 

417 
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" 

418 
by blast 

419 

420 
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" 

421 
by blast 

422 

423 
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" 

424 
by blast 

425 

426 
lemma ball_one_point1 [simp]: "(ALL x:A. x = a > P x) = (a:A > P a)" 

427 
by blast 

428 

429 
lemma ball_one_point2 [simp]: "(ALL x:A. a = x > P x) = (a:A > P a)" 

430 
by blast 

431 

432 
ML_setup {* 

13462  433 
local 
18328  434 
val unfold_bex_tac = unfold_tac [thm "Bex_def"]; 
435 
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; 

11979  436 
val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; 
437 

18328  438 
val unfold_ball_tac = unfold_tac [thm "Ball_def"]; 
439 
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; 

11979  440 
val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; 
441 
in 

18328  442 
val defBEX_regroup = Simplifier.simproc (the_context ()) 
13462  443 
"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; 
18328  444 
val defBALL_regroup = Simplifier.simproc (the_context ()) 
13462  445 
"defined BALL" ["ALL x:A. P x > Q x"] rearrange_ball; 
11979  446 
end; 
13462  447 

448 
Addsimprocs [defBALL_regroup, defBEX_regroup]; 

11979  449 
*} 
450 

451 

452 
subsubsection {* Congruence rules *} 

453 

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lemma ball_cong: 
11979  455 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
456 
(ALL x:A. P x) = (ALL x:B. Q x)" 

457 
by (simp add: Ball_def) 

458 

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459 
lemma strong_ball_cong [cong]: 
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460 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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461 
(ALL x:A. P x) = (ALL x:B. Q x)" 
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462 
by (simp add: simp_implies_def Ball_def) 
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463 

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464 
lemma bex_cong: 
11979  465 
"A = B ==> (!!x. x:B ==> P x = Q x) ==> 
466 
(EX x:A. P x) = (EX x:B. Q x)" 

467 
by (simp add: Bex_def cong: conj_cong) 

1273  468 

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469 
lemma strong_bex_cong [cong]: 
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470 
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> 
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471 
(EX x:A. P x) = (EX x:B. Q x)" 
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472 
by (simp add: simp_implies_def Bex_def cong: conj_cong) 
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473 

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474 

11979  475 
subsubsection {* Subsets *} 
476 

19295  477 
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B" 
11979  478 
by (simp add: subset_def) 
479 

480 
text {* 

481 
\medskip Map the type @{text "'a set => anything"} to just @{typ 

482 
'a}; for overloading constants whose first argument has type @{typ 

483 
"'a set"}. 

484 
*} 

485 

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486 
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" 
11979  487 
 {* Rule in Modus Ponens style. *} 
488 
by (unfold subset_def) blast 

489 

490 
declare subsetD [intro?]  FIXME 

491 

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492 
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B" 
11979  493 
 {* The same, with reversed premises for use with @{text erule}  
494 
cf @{text rev_mp}. *} 

495 
by (rule subsetD) 

496 

497 
declare rev_subsetD [intro?]  FIXME 

498 

499 
text {* 

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500 
\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}. 
11979  501 
*} 
502 

503 
ML {* 

504 
local val rev_subsetD = thm "rev_subsetD" 

505 
in fun impOfSubs th = th RSN (2, rev_subsetD) end; 

506 
*} 

507 

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508 
lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P" 
11979  509 
 {* Classical elimination rule. *} 
510 
by (unfold subset_def) blast 

511 

512 
text {* 

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513 
\medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and 
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514 
creates the assumption @{prop "c \<in> B"}. 
11979  515 
*} 
516 

517 
ML {* 

518 
local val subsetCE = thm "subsetCE" 

519 
in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end; 

520 
*} 

521 

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522 
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A" 
11979  523 
by blast 
524 

19175  525 
lemma subset_refl [simp,atp]: "A \<subseteq> A" 
11979  526 
by fast 
527 

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528 
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" 
11979  529 
by blast 
923  530 

2261  531 

11979  532 
subsubsection {* Equality *} 
533 

13865  534 
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B" 
535 
apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals]) 

536 
apply (rule Collect_mem_eq) 

537 
apply (rule Collect_mem_eq) 

538 
done 

539 

15554  540 
(* Due to Brian Huffman *) 
541 
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))" 

542 
by(auto intro:set_ext) 

543 

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544 
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B" 
11979  545 
 {* Antisymmetry of the subset relation. *} 
17589  546 
by (iprover intro: set_ext subsetD) 
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547 

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548 
lemmas equalityI [intro!] = subset_antisym 
11979  549 

550 
text {* 

551 
\medskip Equality rules from ZF set theory  are they appropriate 

552 
here? 

553 
*} 

554 

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555 
lemma equalityD1: "A = B ==> A \<subseteq> B" 
11979  556 
by (simp add: subset_refl) 
557 

12897
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558 
lemma equalityD2: "A = B ==> B \<subseteq> A" 
11979  559 
by (simp add: subset_refl) 
560 

561 
text {* 

562 
\medskip Be careful when adding this to the claset as @{text 

563 
subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{} 

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564 
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! 
11979  565 
*} 
566 

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567 
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P" 
11979  568 
by (simp add: subset_refl) 
923  569 

11979  570 
lemma equalityCE [elim]: 
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571 
"A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P" 
11979  572 
by blast 
573 

574 
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)" 

575 
by simp 

576 

13865  577 
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)" 
578 
by simp 

579 

11979  580 

581 
subsubsection {* The universal set  UNIV *} 

582 

583 
lemma UNIV_I [simp]: "x : UNIV" 

584 
by (simp add: UNIV_def) 

585 

586 
declare UNIV_I [intro]  {* unsafe makes it less likely to cause problems *} 

587 

588 
lemma UNIV_witness [intro?]: "EX x. x : UNIV" 

589 
by simp 

590 

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591 
lemma subset_UNIV [simp]: "A \<subseteq> UNIV" 
11979  592 
by (rule subsetI) (rule UNIV_I) 
2388  593 

11979  594 
text {* 
595 
\medskip Etacontracting these two rules (to remove @{text P}) 

596 
causes them to be ignored because of their interaction with 

597 
congruence rules. 

598 
*} 

599 

600 
lemma ball_UNIV [simp]: "Ball UNIV P = All P" 

601 
by (simp add: Ball_def) 

602 

603 
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P" 

604 
by (simp add: Bex_def) 

605 

606 

607 
subsubsection {* The empty set *} 

608 

609 
lemma empty_iff [simp]: "(c : {}) = False" 

610 
by (simp add: empty_def) 

611 

612 
lemma emptyE [elim!]: "a : {} ==> P" 

613 
by simp 

614 

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615 
lemma empty_subsetI [iff]: "{} \<subseteq> A" 
11979  616 
 {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *} 
617 
by blast 

618 

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619 
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}" 
11979  620 
by blast 
2388  621 

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622 
lemma equals0D: "A = {} ==> a \<notin> A" 
11979  623 
 {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *} 
624 
by blast 

625 

626 
lemma ball_empty [simp]: "Ball {} P = True" 

627 
by (simp add: Ball_def) 

628 

629 
lemma bex_empty [simp]: "Bex {} P = False" 

630 
by (simp add: Bex_def) 

631 

632 
lemma UNIV_not_empty [iff]: "UNIV ~= {}" 

633 
by (blast elim: equalityE) 

634 

635 

12023  636 
subsubsection {* The Powerset operator  Pow *} 
11979  637 

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638 
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)" 
11979  639 
by (simp add: Pow_def) 
640 

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641 
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B" 
11979  642 
by (simp add: Pow_def) 
643 

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644 
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B" 
11979  645 
by (simp add: Pow_def) 
646 

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647 
lemma Pow_bottom: "{} \<in> Pow B" 
11979  648 
by simp 
649 

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650 
lemma Pow_top: "A \<in> Pow A" 
11979  651 
by (simp add: subset_refl) 
2684  652 

2388  653 

11979  654 
subsubsection {* Set complement *} 
655 

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656 
lemma Compl_iff [simp]: "(c \<in> A) = (c \<notin> A)" 
11979  657 
by (unfold Compl_def) blast 
658 

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659 
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> A" 
11979  660 
by (unfold Compl_def) blast 
661 

662 
text {* 

663 
\medskip This form, with negated conclusion, works well with the 

664 
Classical prover. Negated assumptions behave like formulae on the 

665 
right side of the notional turnstile ... *} 

666 

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667 
lemma ComplD [dest!]: "c : A ==> c~:A" 
11979  668 
by (unfold Compl_def) blast 
669 

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670 
lemmas ComplE = ComplD [elim_format] 
11979  671 

672 

673 
subsubsection {* Binary union  Un *} 

923  674 

11979  675 
lemma Un_iff [simp]: "(c : A Un B) = (c:A  c:B)" 
676 
by (unfold Un_def) blast 

677 

678 
lemma UnI1 [elim?]: "c:A ==> c : A Un B" 

679 
by simp 

680 

681 
lemma UnI2 [elim?]: "c:B ==> c : A Un B" 

682 
by simp 

923  683 

11979  684 
text {* 
685 
\medskip Classical introduction rule: no commitment to @{prop A} vs 

686 
@{prop B}. 

687 
*} 

688 

689 
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B" 

690 
by auto 

691 

692 
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P" 

693 
by (unfold Un_def) blast 

694 

695 

12023  696 
subsubsection {* Binary intersection  Int *} 
923  697 

11979  698 
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)" 
699 
by (unfold Int_def) blast 

700 

701 
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B" 

702 
by simp 

703 

704 
lemma IntD1: "c : A Int B ==> c:A" 

705 
by simp 

706 

707 
lemma IntD2: "c : A Int B ==> c:B" 

708 
by simp 

709 

710 
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P" 

711 
by simp 

712 

713 

12023  714 
subsubsection {* Set difference *} 
11979  715 

716 
lemma Diff_iff [simp]: "(c : A  B) = (c:A & c~:B)" 

717 
by (unfold set_diff_def) blast 

923  718 

11979  719 
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A  B" 
720 
by simp 

721 

722 
lemma DiffD1: "c : A  B ==> c : A" 

723 
by simp 

724 

725 
lemma DiffD2: "c : A  B ==> c : B ==> P" 

726 
by simp 

727 

728 
lemma DiffE [elim!]: "c : A  B ==> (c:A ==> c~:B ==> P) ==> P" 

729 
by simp 

730 

731 

732 
subsubsection {* Augmenting a set  insert *} 

733 

734 
lemma insert_iff [simp]: "(a : insert b A) = (a = b  a:A)" 

735 
by (unfold insert_def) blast 

736 

737 
lemma insertI1: "a : insert a B" 

738 
by simp 

739 

740 
lemma insertI2: "a : B ==> a : insert b B" 

741 
by simp 

923  742 

11979  743 
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P" 
744 
by (unfold insert_def) blast 

745 

746 
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B" 

747 
 {* Classical introduction rule. *} 

748 
by auto 

749 

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750 
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A  {x} \<subseteq> B else A \<subseteq> B)" 
11979  751 
by auto 
752 

753 

754 
subsubsection {* Singletons, using insert *} 

755 

756 
lemma singletonI [intro!]: "a : {a}" 

757 
 {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *} 

758 
by (rule insertI1) 

759 

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760 
lemma singletonD [dest!]: "b : {a} ==> b = a" 
11979  761 
by blast 
762 

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763 
lemmas singletonE = singletonD [elim_format] 
11979  764 

765 
lemma singleton_iff: "(b : {a}) = (b = a)" 

766 
by blast 

767 

768 
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b" 

769 
by blast 

770 

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771 
lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A \<subseteq> {b})" 
11979  772 
by blast 
773 

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774 
lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})" 
11979  775 
by blast 
776 

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777 
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {}  A = {x}" 
11979  778 
by fast 
779 

780 
lemma singleton_conv [simp]: "{x. x = a} = {a}" 

781 
by blast 

782 

783 
lemma singleton_conv2 [simp]: "{x. a = x} = {a}" 

784 
by blast 

923  785 

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786 
lemma diff_single_insert: "A  {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B" 
11979  787 
by blast 
788 

789 

790 
subsubsection {* Unions of families *} 

791 

792 
text {* 

793 
@{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}. 

794 
*} 

795 

796 
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" 

797 
by (unfold UNION_def) blast 

798 

799 
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" 

800 
 {* The order of the premises presupposes that @{term A} is rigid; 

801 
@{term b} may be flexible. *} 

802 
by auto 

803 

804 
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" 

805 
by (unfold UNION_def) blast 

923  806 

11979  807 
lemma UN_cong [cong]: 
808 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

809 
by (simp add: UNION_def) 

810 

811 

812 
subsubsection {* Intersections of families *} 

813 

814 
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *} 

815 

816 
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" 

817 
by (unfold INTER_def) blast 

923  818 

11979  819 
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" 
820 
by (unfold INTER_def) blast 

821 

822 
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" 

823 
by auto 

824 

825 
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" 

826 
 {* "Classical" elimination  by the Excluded Middle on @{prop "a:A"}. *} 

827 
by (unfold INTER_def) blast 

828 

829 
lemma INT_cong [cong]: 

830 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" 

831 
by (simp add: INTER_def) 

7238
36e58620ffc8
replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
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5931
diff
changeset

832 

923  833 

11979  834 
subsubsection {* Union *} 
835 

836 
lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)" 

837 
by (unfold Union_def) blast 

838 

839 
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C" 

840 
 {* The order of the premises presupposes that @{term C} is rigid; 

841 
@{term A} may be flexible. *} 

842 
by auto 

843 

844 
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R" 

845 
by (unfold Union_def) blast 

846 

847 

848 
subsubsection {* Inter *} 

849 

850 
lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)" 

851 
by (unfold Inter_def) blast 

852 

853 
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C" 

854 
by (simp add: Inter_def) 

855 

856 
text {* 

857 
\medskip A ``destruct'' rule  every @{term X} in @{term C} 

858 
contains @{term A} as an element, but @{prop "A:X"} can hold when 

859 
@{prop "X:C"} does not! This rule is analogous to @{text spec}. 

860 
*} 

861 

862 
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X" 

863 
by auto 

864 

865 
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R" 

866 
 {* ``Classical'' elimination rule  does not require proving 

867 
@{prop "X:C"}. *} 

868 
by (unfold Inter_def) blast 

869 

870 
text {* 

871 
\medskip Image of a set under a function. Frequently @{term b} does 

872 
not have the syntactic form of @{term "f x"}. 

873 
*} 

874 

875 
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A" 

876 
by (unfold image_def) blast 

877 

878 
lemma imageI: "x : A ==> f x : f ` A" 

879 
by (rule image_eqI) (rule refl) 

880 

881 
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A" 

882 
 {* This version's more effective when we already have the 

883 
required @{term x}. *} 

884 
by (unfold image_def) blast 

885 

886 
lemma imageE [elim!]: 

887 
"b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P" 

888 
 {* The etaexpansion gives variablename preservation. *} 

889 
by (unfold image_def) blast 

890 

891 
lemma image_Un: "f`(A Un B) = f`A Un f`B" 

892 
by blast 

893 

894 
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)" 

895 
by blast 

896 

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897 
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)" 
11979  898 
 {* This rewrite rule would confuse users if made default. *} 
899 
by blast 

900 

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901 
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)" 
11979  902 
apply safe 
903 
prefer 2 apply fast 

14208  904 
apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast) 
11979  905 
done 
906 

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907 
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B" 
11979  908 
 {* Replaces the three steps @{text subsetI}, @{text imageE}, 
909 
@{text hypsubst}, but breaks too many existing proofs. *} 

910 
by blast 

911 

912 
text {* 

913 
\medskip Range of a function  just a translation for image! 

914 
*} 

915 

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916 
lemma range_eqI: "b = f x ==> b \<in> range f" 
11979  917 
by simp 
918 

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919 
lemma rangeI: "f x \<in> range f" 
11979  920 
by simp 
921 

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922 
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P" 
11979  923 
by blast 
924 

925 

926 
subsubsection {* Set reasoning tools *} 

927 

928 
text {* 

929 
Rewrite rules for boolean casesplitting: faster than @{text 

930 
"split_if [split]"}. 

931 
*} 

932 

933 
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q > x = b) & (~ Q > y = b))" 

934 
by (rule split_if) 

935 

936 
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q > a = x) & (~ Q > a = y))" 

937 
by (rule split_if) 

938 

939 
text {* 

940 
Split ifs on either side of the membership relation. Not for @{text 

941 
"[simp]"}  can cause goals to blow up! 

942 
*} 

943 

944 
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q > x : b) & (~ Q > y : b))" 

945 
by (rule split_if) 

946 

947 
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q > a : x) & (~ Q > a : y))" 

948 
by (rule split_if) 

949 

950 
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 

951 

952 
lemmas mem_simps = 

953 
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff 

954 
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff 

955 
 {* Each of these has ALREADY been added @{text "[simp]"} above. *} 

956 

957 
(*Would like to add these, but the existing code only searches for the 

958 
outerlevel constant, which in this case is just "op :"; we instead need 

959 
to use termnets to associate patterns with rules. Also, if a rule fails to 

960 
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

961 
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]), 
11979  962 
("op Int", [IntD1,IntD2]), 
963 
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] 

964 
*) 

965 

966 
ML_setup {* 

967 
val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs; 

17875  968 
change_simpset (fn ss => ss setmksimps (mksimps mksimps_pairs)); 
11979  969 
*} 
970 

971 

972 
subsubsection {* The ``proper subset'' relation *} 

973 

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974 
lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B" 
11979  975 
by (unfold psubset_def) blast 
976 

13624  977 
lemma psubsetE [elim!]: 
978 
"[A \<subset> B; [A \<subseteq> B; ~ (B\<subseteq>A)] ==> R] ==> R" 

979 
by (unfold psubset_def) blast 

980 

11979  981 
lemma psubset_insert_iff: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

982 
"(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

983 
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

984 

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

985 
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)" 
11979  986 
by (simp only: psubset_def) 
987 

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

988 
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B" 
11979  989 
by (simp add: psubset_eq) 
990 

14335  991 
lemma psubset_trans: "[ A \<subset> B; B \<subset> C ] ==> A \<subset> C" 
992 
apply (unfold psubset_def) 

993 
apply (auto dest: subset_antisym) 

994 
done 

995 

996 
lemma psubsetD: "[ A \<subset> B; c \<in> A ] ==> c \<in> B" 

997 
apply (unfold psubset_def) 

998 
apply (auto dest: subsetD) 

999 
done 

1000 

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

1001 
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C" 
11979  1002 
by (auto simp add: psubset_eq) 
1003 

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

1004 
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C" 
11979  1005 
by (auto simp add: psubset_eq) 
1006 

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

1007 
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B  A)" 
11979  1008 
by (unfold psubset_def) blast 
1009 

1010 
lemma atomize_ball: 

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

1011 
"(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)" 
11979  1012 
by (simp only: Ball_def atomize_all atomize_imp) 
1013 

18832  1014 
lemmas [symmetric, rulify] = atomize_ball 
1015 
and [symmetric, defn] = atomize_ball 

11979  1016 

1017 

1018 
subsection {* Further settheory lemmas *} 

1019 

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

1020 
subsubsection {* Derived rules involving subsets. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1021 

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

1022 
text {* @{text insert}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1023 

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

1024 
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

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

1026 
apply (erule insertI2) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1027 
done 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1028 

14302  1029 
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B" 
1030 
by blast 

1031 

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

1032 
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

1033 
by blast 
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 

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

1036 
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

1037 

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

1038 
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A" 
17589  1039 
by (iprover intro: subsetI UnionI) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1040 

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

1041 
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C" 
17589  1042 
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

1043 

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

1044 

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

1045 
text {* \medskip General union. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1046 

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

1047 
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

1048 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1049 

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

1050 
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C" 
17589  1051 
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

1052 

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

1053 

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

1054 
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

1055 

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

1056 
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

1057 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1058 

14551  1059 
lemma Inter_subset: 
1060 
"[ !!X. X \<in> A ==> X \<subseteq> B; A ~= {} ] ==> \<Inter>A \<subseteq> B" 

1061 
by blast 

1062 

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

1063 
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A" 
17589  1064 
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

1065 

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

1066 
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

1067 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1068 

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

1069 
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)" 
17589  1070 
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

1071 

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 
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

1074 

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

1075 
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

1076 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1077 

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

1078 
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

1079 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1080 

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

1081 
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

1082 
by blast 
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 

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

1085 
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

1086 

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

1087 
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

1088 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1089 

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

1090 
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

1091 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1092 

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

1093 
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

1094 
by blast 
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 

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

1097 
text {* \medskip Set difference. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1098 

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

1099 
lemma Diff_subset: "A  B \<subseteq> A" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1100 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1101 

14302  1102 
lemma Diff_subset_conv: "(A  B \<subseteq> C) = (A \<subseteq> B \<union> C)" 
1103 
by blast 

1104 

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

1105 

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

1106 
text {* \medskip Monotonicity. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1107 

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

1108 
lemma mono_Un: "mono f ==> f A \<union> f B \<subseteq> f (A \<union> B)" 
16773  1109 
by (auto simp add: mono_def) 
15206
09d78ec709c7
Modified locales: improved implementation of "includes".
ballarin
parents:
15140
diff
changeset

1110 

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

1111 
lemma mono_Int: "mono f ==> f (A \<inter> B) \<subseteq> f A \<inter> f B" 
16773  1112 
by (auto simp add: mono_def) 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1113 

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

1114 
subsubsection {* Equalities involving union, intersection, inclusion, etc. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1115 

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

1116 
text {* @{text "{}"}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1117 

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

1118 
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

1119 
 {* supersedes @{text "Collect_False_empty"} *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1120 
by auto 
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 
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1123 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1124 

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

1125 
lemma not_psubset_empty [iff]: "\<not> (A < {})" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1126 
by (unfold psubset_def) blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1127 

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

1128 
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)" 
18423  1129 
by blast 
1130 

1131 
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)" 

1132 
by blast 

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

1133 

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

1134 
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

1135 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1136 

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

1137 
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

1138 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1139 

14812  1140 
lemma Collect_imp_eq: "{x. P x > Q x} = {x. P x} \<union> {x. Q x}" 
1141 
by blast 

1142 

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

1143 
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

1144 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1145 

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

1146 
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

1147 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1148 

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

1149 
lemma Collect_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

1150 
by blast 
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 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

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 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

1156 
by blast 
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 @{text insert}. *} 
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 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

1162 
 {* 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

1163 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1164 

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

1165 
lemma 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

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

1167 

17715  1168 
lemmas empty_not_insert = insert_not_empty [symmetric, standard] 
1169 
declare empty_not_insert [simp] 

12897
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 
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

1172 
 {* @{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

1173 
 {* with \emph{quadratic} running time *} 
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 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

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

1178 

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

1179 
lemma insert_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

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 
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

1183 
by blast 
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 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

1186 
 {* use new @{text B} rather than @{text "A  {a}"} to avoid infinite unfolding *} 
14208  1187 
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

1188 
done 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1189 

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

1190 
lemma insert_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

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

1192 

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

1193 
lemma 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

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

1195 

14302  1196 
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)" 
14742  1197 
by blast 
14302  1198 

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

1199 
lemma insert_disjoint[simp]: 
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1200 
"(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})" 
14742  1201 
"({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)" 
16773  1202 
by auto 
13103
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1203 

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

1204 
lemma disjoint_insert[simp]: 
66659a4b16f6
Added insert_disjoint and disjoint_insert [simp], and simplified proofs
nipkow
parents:
12937
diff
changeset

1205 
"(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})" 
14742  1206 
"({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)" 
16773  1207 
by auto 
14742  1208 

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

1209 
text {* \medskip @{text image}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1210 

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

1211 
lemma image_empty [simp]: "f`{} = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1212 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1213 

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

1214 
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

1215 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1216 

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

1217 
lemma image_constant: "x \<in> A ==> (\<lambda>x. c) ` A = {c}" 
16773  1218 
by auto 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1219 

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

1220 
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

1221 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1222 

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

1223 
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

1224 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1225 

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

1226 
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

1227 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1228 

16773  1229 

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

1230 
lemma image_Collect: "f ` {x. P x} = {f x  x. P x}" 
16773  1231 
 {* NOT suitable as a default simprule: the RHS isn't simpler than the LHS, 
1232 
with its implicit quantifier and conjunction. Also image enjoys better 

1233 
equational properties than does the RHS. *} 

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

1237 
"(\<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

1238 
= (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

1239 
by (auto simp add: image_def) 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1240 

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

1241 
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

1242 
by (simp add: image_def) 
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 

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

1245 
text {* \medskip @{text range}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1246 

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

1247 
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

1248 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1249 

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

1250 
lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = f`range g" 
14208  1251 
by (subst image_image, simp) 
12897
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 

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

1254 
text {* \medskip @{text Int} *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1255 

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

1256 
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

1257 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1258 

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

1259 
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

1260 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1261 

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

1262 
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

1263 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1264 

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

1265 
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

1266 
by blast 
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 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

1269 
by blast 
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 
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

1272 
 {* Intersection is an ACoperator *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1273 

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

1274 
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

1275 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1276 

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

1277 
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

1278 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1279 

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

1280 
lemma Int_empty_left [simp]: "{} \<inter> B = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1281 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1282 

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

1283 
lemma Int_empty_right [simp]: "A \<inter> {} = {}" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1284 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1285 

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

1286 
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

1287 
by blast 
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 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

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 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

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 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

1296 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1297 

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

1298 
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

1299 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1300 

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

1301 
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

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 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

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 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

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

1309 

15102  1310 
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

1311 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1312 

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

1313 
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

1314 
by blast 
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 

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

1317 
text {* \medskip @{text Un}. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1318 

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

1319 
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

1320 
by blast 
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 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

1323 
by blast 
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 
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

1326 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1327 

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

1328 
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

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

1330 

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

1331 
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

1332 
by blast 
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 
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

1335 
 {* Union is an ACoperator *} 
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 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

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 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

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

1344 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1345 

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

1346 
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

1347 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1348 

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

1349 
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

1350 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

1351 

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

1352 
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV&quo 