src/ZF/ZF.thy
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(*  Title:      ZF/ZF.thy
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    Author:     Lawrence C Paulson and Martin D Coen, CU Computer Laboratory
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    Copyright   1993  University of Cambridge
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*)
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header{*Zermelo-Fraenkel Set Theory*}
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theory ZF
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imports "~~/src/FOL/FOL"
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begin
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declare [[eta_contract = false]]
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typedecl i
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arities  i :: "term"
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axiomatization
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  zero :: "i"  ("0")   --{*the empty set*}  and
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  Pow :: "i => i"  --{*power sets*}  and
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  Inf :: "i"  --{*infinite set*}
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text {*Bounded Quantifiers *}
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consts
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  Ball   :: "[i, i => o] => o"
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  Bex   :: "[i, i => o] => o"
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text {*General Union and Intersection *}
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axiomatization Union :: "i => i"
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consts Inter :: "i => i"
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text {*Variations on Replacement *}
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axiomatization PrimReplace :: "[i, [i, i] => o] => i"
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consts
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  Replace     :: "[i, [i, i] => o] => i"
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  RepFun      :: "[i, i => i] => i"
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  Collect     :: "[i, i => o] => i"
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text{*Definite descriptions -- via Replace over the set "1"*}
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consts
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  The         :: "(i => o) => i"      (binder "THE " 10)
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  If          :: "[o, i, i] => i"     ("(if (_)/ then (_)/ else (_))" [10] 10)
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abbreviation (input)
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  old_if      :: "[o, i, i] => i"   ("if '(_,_,_')") where
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  "if(P,a,b) == If(P,a,b)"
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text {*Finite Sets *}
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consts
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  Upair :: "[i, i] => i"
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  cons  :: "[i, i] => i"
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  succ  :: "i => i"
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text {*Ordered Pairing *}
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consts
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  Pair  :: "[i, i] => i"
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  fst   :: "i => i"
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  snd   :: "i => i"
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  split :: "[[i, i] => 'a, i] => 'a::{}"  --{*for pattern-matching*}
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text {*Sigma and Pi Operators *}
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consts
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  Sigma :: "[i, i => i] => i"
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  Pi    :: "[i, i => i] => i"
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text {*Relations and Functions *}
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consts
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  "domain"    :: "i => i"
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  range       :: "i => i"
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  field       :: "i => i"
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  converse    :: "i => i"
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  relation    :: "i => o"        --{*recognizes sets of pairs*}
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  "function"  :: "i => o"        --{*recognizes functions; can have non-pairs*}
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  Lambda      :: "[i, i => i] => i"
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  restrict    :: "[i, i] => i"
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text {*Infixes in order of decreasing precedence *}
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consts
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  Image       :: "[i, i] => i"    (infixl "``" 90) --{*image*}
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  vimage      :: "[i, i] => i"    (infixl "-``" 90) --{*inverse image*}
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  "apply"     :: "[i, i] => i"    (infixl "`" 90) --{*function application*}
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  "Int"       :: "[i, i] => i"    (infixl "Int" 70) --{*binary intersection*}
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  "Un"        :: "[i, i] => i"    (infixl "Un" 65) --{*binary union*}
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  Diff        :: "[i, i] => i"    (infixl "-" 65) --{*set difference*}
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  Subset      :: "[i, i] => o"    (infixl "<=" 50) --{*subset relation*}
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axiomatization
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  mem         :: "[i, i] => o"    (infixl ":" 50) --{*membership relation*}
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abbreviation
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  not_mem :: "[i, i] => o"  (infixl "~:" 50)  --{*negated membership relation*}
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  where "x ~: y == ~ (x : y)"
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abbreviation
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  cart_prod :: "[i, i] => i"    (infixr "*" 80) --{*Cartesian product*}
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  where "A * B == Sigma(A, %_. B)"
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abbreviation
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  function_space :: "[i, i] => i"  (infixr "->" 60) --{*function space*}
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  where "A -> B == Pi(A, %_. B)"
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nonterminal "is" and patterns
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syntax
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  ""          :: "i => is"                   ("_")
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  "_Enum"     :: "[i, is] => is"             ("_,/ _")
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  "_Finset"   :: "is => i"                   ("{(_)}")
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  "_Tuple"    :: "[i, is] => i"              ("<(_,/ _)>")
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  "_Collect"  :: "[pttrn, i, o] => i"        ("(1{_: _ ./ _})")
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  "_Replace"  :: "[pttrn, pttrn, i, o] => i" ("(1{_ ./ _: _, _})")
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  "_RepFun"   :: "[i, pttrn, i] => i"        ("(1{_ ./ _: _})" [51,0,51])
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  "_INTER"    :: "[pttrn, i, i] => i"        ("(3INT _:_./ _)" 10)
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  "_UNION"    :: "[pttrn, i, i] => i"        ("(3UN _:_./ _)" 10)
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  "_PROD"     :: "[pttrn, i, i] => i"        ("(3PROD _:_./ _)" 10)
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  "_SUM"      :: "[pttrn, i, i] => i"        ("(3SUM _:_./ _)" 10)
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  "_lam"      :: "[pttrn, i, i] => i"        ("(3lam _:_./ _)" 10)
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  "_Ball"     :: "[pttrn, i, o] => o"        ("(3ALL _:_./ _)" 10)
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  "_Bex"      :: "[pttrn, i, o] => o"        ("(3EX _:_./ _)" 10)
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  (** Patterns -- extends pre-defined type "pttrn" used in abstractions **)
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  "_pattern"  :: "patterns => pttrn"         ("<_>")
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  ""          :: "pttrn => patterns"         ("_")
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  "_patterns" :: "[pttrn, patterns] => patterns"  ("_,/_")
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translations
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  "{x, xs}"     == "CONST cons(x, {xs})"
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  "{x}"         == "CONST cons(x, 0)"
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  "{x:A. P}"    == "CONST Collect(A, %x. P)"
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  "{y. x:A, Q}" == "CONST Replace(A, %x y. Q)"
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  "{b. x:A}"    == "CONST RepFun(A, %x. b)"
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  "INT x:A. B"  == "CONST Inter({B. x:A})"
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  "UN x:A. B"   == "CONST Union({B. x:A})"
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  "PROD x:A. B" == "CONST Pi(A, %x. B)"
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  "SUM x:A. B"  == "CONST Sigma(A, %x. B)"
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  "lam x:A. f"  == "CONST Lambda(A, %x. f)"
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  "ALL x:A. P"  == "CONST Ball(A, %x. P)"
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  "EX x:A. P"   == "CONST Bex(A, %x. P)"
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  "<x, y, z>"   == "<x, <y, z>>"
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  "<x, y>"      == "CONST Pair(x, y)"
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  "%<x,y,zs>.b" == "CONST split(%x <y,zs>.b)"
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  "%<x,y>.b"    == "CONST split(%x y. b)"
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notation (xsymbols)
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  cart_prod       (infixr "\<times>" 80) and
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  Int             (infixl "\<inter>" 70) and
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  Un              (infixl "\<union>" 65) and
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  function_space  (infixr "\<rightarrow>" 60) and
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  Subset          (infixl "\<subseteq>" 50) and
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  mem             (infixl "\<in>" 50) and
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  not_mem         (infixl "\<notin>" 50) and
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  Union           ("\<Union>_" [90] 90) and
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  Inter           ("\<Inter>_" [90] 90)
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syntax (xsymbols)
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  "_Collect"  :: "[pttrn, i, o] => i"        ("(1{_ \<in> _ ./ _})")
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  "_Replace"  :: "[pttrn, pttrn, i, o] => i" ("(1{_ ./ _ \<in> _, _})")
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  "_RepFun"   :: "[i, pttrn, i] => i"        ("(1{_ ./ _ \<in> _})" [51,0,51])
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  "_UNION"    :: "[pttrn, i, i] => i"        ("(3\<Union>_\<in>_./ _)" 10)
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  "_INTER"    :: "[pttrn, i, i] => i"        ("(3\<Inter>_\<in>_./ _)" 10)
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  "_PROD"     :: "[pttrn, i, i] => i"        ("(3\<Pi>_\<in>_./ _)" 10)
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  "_SUM"      :: "[pttrn, i, i] => i"        ("(3\<Sigma>_\<in>_./ _)" 10)
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  "_lam"      :: "[pttrn, i, i] => i"        ("(3\<lambda>_\<in>_./ _)" 10)
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  "_Ball"     :: "[pttrn, i, o] => o"        ("(3\<forall>_\<in>_./ _)" 10)
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  "_Bex"      :: "[pttrn, i, o] => o"        ("(3\<exists>_\<in>_./ _)" 10)
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  "_Tuple"    :: "[i, is] => i"              ("\<langle>(_,/ _)\<rangle>")
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  "_pattern"  :: "patterns => pttrn"         ("\<langle>_\<rangle>")
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notation (HTML output)
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  cart_prod       (infixr "\<times>" 80) and
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  Int             (infixl "\<inter>" 70) and
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  Un              (infixl "\<union>" 65) and
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  Subset          (infixl "\<subseteq>" 50) and
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  mem             (infixl "\<in>" 50) and
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  not_mem         (infixl "\<notin>" 50) and
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  Union           ("\<Union>_" [90] 90) and
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  Inter           ("\<Inter>_" [90] 90)
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syntax (HTML output)
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  "_Collect"  :: "[pttrn, i, o] => i"        ("(1{_ \<in> _ ./ _})")
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  "_Replace"  :: "[pttrn, pttrn, i, o] => i" ("(1{_ ./ _ \<in> _, _})")
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  "_RepFun"   :: "[i, pttrn, i] => i"        ("(1{_ ./ _ \<in> _})" [51,0,51])
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  "_UNION"    :: "[pttrn, i, i] => i"        ("(3\<Union>_\<in>_./ _)" 10)
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  "_INTER"    :: "[pttrn, i, i] => i"        ("(3\<Inter>_\<in>_./ _)" 10)
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  "_PROD"     :: "[pttrn, i, i] => i"        ("(3\<Pi>_\<in>_./ _)" 10)
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  "_SUM"      :: "[pttrn, i, i] => i"        ("(3\<Sigma>_\<in>_./ _)" 10)
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  "_lam"      :: "[pttrn, i, i] => i"        ("(3\<lambda>_\<in>_./ _)" 10)
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  "_Ball"     :: "[pttrn, i, o] => o"        ("(3\<forall>_\<in>_./ _)" 10)
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  "_Bex"      :: "[pttrn, i, o] => o"        ("(3\<exists>_\<in>_./ _)" 10)
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  "_Tuple"    :: "[i, is] => i"              ("\<langle>(_,/ _)\<rangle>")
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  "_pattern"  :: "patterns => pttrn"         ("\<langle>_\<rangle>")
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defs  (* Bounded Quantifiers *)
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  Ball_def:      "Ball(A, P) == \<forall>x. x\<in>A \<longrightarrow> P(x)"
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  Bex_def:       "Bex(A, P) == \<exists>x. x\<in>A & P(x)"
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  subset_def:    "A \<subseteq> B == \<forall>x\<in>A. x\<in>B"
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axiomatization where
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  (* ZF axioms -- see Suppes p.238
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     Axioms for Union, Pow and Replace state existence only,
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     uniqueness is derivable using extensionality. *)
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  extension:     "A = B <-> A \<subseteq> B & B \<subseteq> A" and
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  Union_iff:     "A \<in> \<Union>(C) <-> (\<exists>B\<in>C. A\<in>B)" and
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  Pow_iff:       "A \<in> Pow(B) <-> A \<subseteq> B" and
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  (*We may name this set, though it is not uniquely defined.*)
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  infinity:      "0\<in>Inf & (\<forall>y\<in>Inf. succ(y): Inf)" and
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  (*This formulation facilitates case analysis on A.*)
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  foundation:    "A=0 | (\<exists>x\<in>A. \<forall>y\<in>x. y\<notin>A)" and
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  (*Schema axiom since predicate P is a higher-order variable*)
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  replacement:   "(\<forall>x\<in>A. \<forall>y z. P(x,y) & P(x,z) \<longrightarrow> y=z) ==>
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                         b \<in> PrimReplace(A,P) <-> (\<exists>x\<in>A. P(x,b))"
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defs
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  (* Derived form of replacement, restricting P to its functional part.
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     The resulting set (for functional P) is the same as with
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     PrimReplace, but the rules are simpler. *)
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  Replace_def:  "Replace(A,P) == PrimReplace(A, %x y. (EX!z. P(x,z)) & P(x,y))"
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  (* Functional form of replacement -- analgous to ML's map functional *)
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  RepFun_def:   "RepFun(A,f) == {y . x\<in>A, y=f(x)}"
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  (* Separation and Pairing can be derived from the Replacement
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     and Powerset Axioms using the following definitions. *)
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  Collect_def:  "Collect(A,P) == {y . x\<in>A, x=y & P(x)}"
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  (*Unordered pairs (Upair) express binary union/intersection and cons;
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    set enumerations translate as {a,...,z} = cons(a,...,cons(z,0)...)*)
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  Upair_def: "Upair(a,b) == {y. x\<in>Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b)}"
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  cons_def:  "cons(a,A) == Upair(a,a) \<union> A"
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  succ_def:  "succ(i) == cons(i, i)"
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  (* Difference, general intersection, binary union and small intersection *)
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  Diff_def:      "A - B    == { x\<in>A . ~(x\<in>B) }"
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  Inter_def:     "\<Inter>(A) == { x\<in>\<Union>(A) . \<forall>y\<in>A. x\<in>y}"
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  Un_def:        "A \<union>  B  == \<Union>(Upair(A,B))"
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  Int_def:      "A \<inter> B  == \<Inter>(Upair(A,B))"
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  (* definite descriptions *)
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  the_def:      "The(P)    == \<Union>({y . x \<in> {0}, P(y)})"
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  if_def:       "if(P,a,b) == THE z. P & z=a | ~P & z=b"
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  (* this "symmetric" definition works better than {{a}, {a,b}} *)
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  Pair_def:     "<a,b>  == {{a,a}, {a,b}}"
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  fst_def:      "fst(p) == THE a. \<exists>b. p=<a,b>"
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  snd_def:      "snd(p) == THE b. \<exists>a. p=<a,b>"
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  split_def:    "split(c) == %p. c(fst(p), snd(p))"
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  Sigma_def:    "Sigma(A,B) == \<Union>x\<in>A. \<Union>y\<in>B(x). {<x,y>}"
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  (* Operations on relations *)
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  (*converse of relation r, inverse of function*)
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  converse_def: "converse(r) == {z. w\<in>r, \<exists>x y. w=<x,y> & z=<y,x>}"
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  domain_def:   "domain(r) == {x. w\<in>r, \<exists>y. w=<x,y>}"
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  range_def:    "range(r) == domain(converse(r))"
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  field_def:    "field(r) == domain(r) \<union> range(r)"
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  relation_def: "relation(r) == \<forall>z\<in>r. \<exists>x y. z = <x,y>"
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  function_def: "function(r) ==
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                    \<forall>x y. <x,y>:r \<longrightarrow> (\<forall>y'. <x,y'>:r \<longrightarrow> y=y')"
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  image_def:    "r `` A  == {y \<in> range(r) . \<exists>x\<in>A. <x,y> \<in> r}"
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  vimage_def:   "r -`` A == converse(r)``A"
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  (* Abstraction, application and Cartesian product of a family of sets *)
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  lam_def:      "Lambda(A,b) == {<x,b(x)> . x\<in>A}"
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  apply_def:    "f`a == \<Union>(f``{a})"
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  Pi_def:       "Pi(A,B)  == {f\<in>Pow(Sigma(A,B)). A<=domain(f) & function(f)}"
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  (* Restrict the relation r to the domain A *)
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  restrict_def: "restrict(r,A) == {z \<in> r. \<exists>x\<in>A. \<exists>y. z = <x,y>}"
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subsection {* Substitution*}
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(*Useful examples:  singletonI RS subst_elem,  subst_elem RSN (2,IntI) *)
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lemma subst_elem: "[| b\<in>A;  a=b |] ==> a\<in>A"
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by (erule ssubst, assumption)
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subsection{*Bounded universal quantifier*}
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lemma ballI [intro!]: "[| !!x. x\<in>A ==> P(x) |] ==> \<forall>x\<in>A. P(x)"
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by (simp add: Ball_def)
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lemmas strip = impI allI ballI
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lemma bspec [dest?]: "[| \<forall>x\<in>A. P(x);  x: A |] ==> P(x)"
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by (simp add: Ball_def)
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(*Instantiates x first: better for automatic theorem proving?*)
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lemma rev_ballE [elim]:
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    "[| \<forall>x\<in>A. P(x);  x\<notin>A ==> Q;  P(x) ==> Q |] ==> Q"
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by (simp add: Ball_def, blast)
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lemma ballE: "[| \<forall>x\<in>A. P(x);  P(x) ==> Q;  x\<notin>A ==> Q |] ==> Q"
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by blast
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(*Used in the datatype package*)
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lemma rev_bspec: "[| x: A;  \<forall>x\<in>A. P(x) |] ==> P(x)"
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by (simp add: Ball_def)
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(*Trival rewrite rule;   @{term"(\<forall>x\<in>A.P)<->P"} holds only if A is nonempty!*)
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lemma ball_triv [simp]: "(\<forall>x\<in>A. P) <-> ((\<exists>x. x\<in>A) \<longrightarrow> P)"
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by (simp add: Ball_def)
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(*Congruence rule for rewriting*)
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lemma ball_cong [cong]:
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    "[| A=A';  !!x. x\<in>A' ==> P(x) <-> P'(x) |] ==> (\<forall>x\<in>A. P(x)) <-> (\<forall>x\<in>A'. P'(x))"
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by (simp add: Ball_def)
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lemma atomize_ball:
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    "(!!x. x \<in> A ==> P(x)) == Trueprop (\<forall>x\<in>A. P(x))"
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  by (simp only: Ball_def atomize_all atomize_imp)
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lemmas [symmetric, rulify] = atomize_ball
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  and [symmetric, defn] = atomize_ball
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subsection{*Bounded existential quantifier*}
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lemma bexI [intro]: "[| P(x);  x: A |] ==> \<exists>x\<in>A. P(x)"
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by (simp add: Bex_def, blast)
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(*The best argument order when there is only one @{term"x\<in>A"}*)
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lemma rev_bexI: "[| x\<in>A;  P(x) |] ==> \<exists>x\<in>A. P(x)"
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by blast
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(*Not of the general form for such rules. The existential quanitifer becomes universal. *)
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lemma bexCI: "[| \<forall>x\<in>A. ~P(x) ==> P(a);  a: A |] ==> \<exists>x\<in>A. P(x)"
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by blast
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lemma bexE [elim!]: "[| \<exists>x\<in>A. P(x);  !!x. [| x\<in>A; P(x) |] ==> Q |] ==> Q"
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by (simp add: Bex_def, blast)
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(*We do not even have @{term"(\<exists>x\<in>A. True) <-> True"} unless @{term"A" is nonempty!!*)
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lemma bex_triv [simp]: "(\<exists>x\<in>A. P) <-> ((\<exists>x. x\<in>A) & P)"
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by (simp add: Bex_def)
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lemma bex_cong [cong]:
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    "[| A=A';  !!x. x\<in>A' ==> P(x) <-> P'(x) |]
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     ==> (\<exists>x\<in>A. P(x)) <-> (\<exists>x\<in>A'. P'(x))"
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by (simp add: Bex_def cong: conj_cong)
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subsection{*Rules for subsets*}
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lemma subsetI [intro!]:
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    "(!!x. x\<in>A ==> x\<in>B) ==> A \<subseteq> B"
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by (simp add: subset_def)
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(*Rule in Modus Ponens style [was called subsetE] *)
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lemma subsetD [elim]: "[| A \<subseteq> B;  c\<in>A |] ==> c\<in>B"
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apply (unfold subset_def)
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apply (erule bspec, assumption)
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done
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(*Classical elimination rule*)
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lemma subsetCE [elim]:
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    "[| A \<subseteq> B;  c\<notin>A ==> P;  c\<in>B ==> P |] ==> P"
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by (simp add: subset_def, blast)
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(*Sometimes useful with premises in this order*)
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lemma rev_subsetD: "[| c\<in>A; A<=B |] ==> c\<in>B"
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by blast
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lemma contra_subsetD: "[| A \<subseteq> B; c \<notin> B |] ==> c \<notin> A"
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by blast
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lemma rev_contra_subsetD: "[| c \<notin> B;  A \<subseteq> B |] ==> c \<notin> A"
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by blast
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lemma subset_refl [simp]: "A \<subseteq> A"
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by blast
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   395
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lemma subset_trans: "[| A<=B;  B<=C |] ==> A<=C"
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by blast
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(*Useful for proving A<=B by rewriting in some cases*)
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lemma subset_iff:
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     "A<=B <-> (\<forall>x. x\<in>A \<longrightarrow> x\<in>B)"
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apply (unfold subset_def Ball_def)
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apply (rule iff_refl)
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done
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text{*For calculations*}
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declare subsetD [trans] rev_subsetD [trans] subset_trans [trans]
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   408
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subsection{*Rules for equality*}
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   411
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(*Anti-symmetry of the subset relation*)
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lemma equalityI [intro]: "[| A \<subseteq> B;  B \<subseteq> A |] ==> A = B"
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by (rule extension [THEN iffD2], rule conjI)
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   415
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lemma equality_iffI: "(!!x. x\<in>A <-> x\<in>B) ==> A = B"
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by (rule equalityI, blast+)
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lemmas equalityD1 = extension [THEN iffD1, THEN conjunct1]
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lemmas equalityD2 = extension [THEN iffD1, THEN conjunct2]
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lemma equalityE: "[| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P"
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by (blast dest: equalityD1 equalityD2)
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   425
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lemma equalityCE:
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    "[| A = B;  [| c\<in>A; c\<in>B |] ==> P;  [| c\<notin>A; c\<notin>B |] ==> P |]  ==>  P"
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by (erule equalityE, blast)
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lemma equality_iffD:
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   431
  "A = B ==> (!!x. x \<in> A <-> x \<in> B)"
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   432
  by auto
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diff changeset
   433
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   434
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subsection{*Rules for Replace -- the derived form of replacement*}
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   436
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lemma Replace_iff:
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   438
    "b \<in> {y. x\<in>A, P(x,y)}  <->  (\<exists>x\<in>A. P(x,b) & (\<forall>y. P(x,y) \<longrightarrow> y=b))"
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apply (unfold Replace_def)
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apply (rule replacement [THEN iff_trans], blast+)
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done
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   442
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(*Introduction; there must be a unique y such that P(x,y), namely y=b. *)
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   444
lemma ReplaceI [intro]:
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   445
    "[| P(x,b);  x: A;  !!y. P(x,y) ==> y=b |] ==>
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   446
     b \<in> {y. x\<in>A, P(x,y)}"
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by (rule Replace_iff [THEN iffD2], blast)
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   448
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(*Elimination; may asssume there is a unique y such that P(x,y), namely y=b. *)
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lemma ReplaceE:
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   451
    "[| b \<in> {y. x\<in>A, P(x,y)};
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        !!x. [| x: A;  P(x,b);  \<forall>y. P(x,y)\<longrightarrow>y=b |] ==> R
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     |] ==> R"
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   454
by (rule Replace_iff [THEN iffD1, THEN bexE], simp+)
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   455
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(*As above but without the (generally useless) 3rd assumption*)
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   457
lemma ReplaceE2 [elim!]:
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   458
    "[| b \<in> {y. x\<in>A, P(x,y)};
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   459
        !!x. [| x: A;  P(x,b) |] ==> R
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   460
     |] ==> R"
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   461
by (erule ReplaceE, blast)
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   462
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   463
lemma Replace_cong [cong]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   464
    "[| A=B;  !!x y. x\<in>B ==> P(x,y) <-> Q(x,y) |] ==>
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   465
     Replace(A,P) = Replace(B,Q)"
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   466
apply (rule equality_iffI)
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   467
apply (simp add: Replace_iff)
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   468
done
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   469
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   470
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   471
subsection{*Rules for RepFun*}
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   472
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   473
lemma RepFunI: "a \<in> A ==> f(a) \<in> {f(x). x\<in>A}"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   474
by (simp add: RepFun_def Replace_iff, blast)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   475
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   476
(*Useful for coinduction proofs*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   477
lemma RepFun_eqI [intro]: "[| b=f(a);  a \<in> A |] ==> b \<in> {f(x). x\<in>A}"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   478
apply (erule ssubst)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   479
apply (erule RepFunI)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   480
done
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   481
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   482
lemma RepFunE [elim!]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   483
    "[| b \<in> {f(x). x\<in>A};
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   484
        !!x.[| x\<in>A;  b=f(x) |] ==> P |] ==>
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   485
     P"
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   486
by (simp add: RepFun_def Replace_iff, blast)
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   487
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   488
lemma RepFun_cong [cong]:
14227
0356666744ec finalconsts
paulson
parents: 14095
diff changeset
   489
    "[| A=B;  !!x. x\<in>B ==> f(x)=g(x) |] ==> RepFun(A,f) = RepFun(B,g)"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   490
by (simp add: RepFun_def)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   491
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   492
lemma RepFun_iff [simp]: "b \<in> {f(x). x\<in>A} <-> (\<exists>x\<in>A. b=f(x))"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   493
by (unfold Bex_def, blast)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   494
14227
0356666744ec finalconsts
paulson
parents: 14095
diff changeset
   495
lemma triv_RepFun [simp]: "{x. x\<in>A} = A"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   496
by blast
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   497
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   498
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   499
subsection{*Rules for Collect -- forming a subset by separation*}
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   500
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   501
(*Separation is derivable from Replacement*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   502
lemma separation [simp]: "a \<in> {x\<in>A. P(x)} <-> a\<in>A & P(a)"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   503
by (unfold Collect_def, blast)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   504
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   505
lemma CollectI [intro!]: "[| a\<in>A;  P(a) |] ==> a \<in> {x\<in>A. P(x)}"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   506
by simp
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   507
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   508
lemma CollectE [elim!]: "[| a \<in> {x\<in>A. P(x)};  [| a\<in>A; P(a) |] ==> R |] ==> R"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   509
by simp
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   510
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   511
lemma CollectD1: "a \<in> {x\<in>A. P(x)} ==> a\<in>A"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   512
by (erule CollectE, assumption)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   513
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   514
lemma CollectD2: "a \<in> {x\<in>A. P(x)} ==> P(a)"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   515
by (erule CollectE, assumption)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   516
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   517
lemma Collect_cong [cong]:
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   518
    "[| A=B;  !!x. x\<in>B ==> P(x) <-> Q(x) |]
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   519
     ==> Collect(A, %x. P(x)) = Collect(B, %x. Q(x))"
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   520
by (simp add: Collect_def)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   521
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   522
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   523
subsection{*Rules for Unions*}
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   524
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   525
declare Union_iff [simp]
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   526
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   527
(*The order of the premises presupposes that C is rigid; A may be flexible*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   528
lemma UnionI [intro]: "[| B: C;  A: B |] ==> A: \<Union>(C)"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   529
by (simp, blast)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   530
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   531
lemma UnionE [elim!]: "[| A \<in> \<Union>(C);  !!B.[| A: B;  B: C |] ==> R |] ==> R"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   532
by (simp, blast)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   533
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   534
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   535
subsection{*Rules for Unions of families*}
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   536
(* @{term"\<Union>x\<in>A. B(x)"} abbreviates @{term"\<Union>({B(x). x\<in>A})"} *)
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   537
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   538
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B(x)) <-> (\<exists>x\<in>A. b \<in> B(x))"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   539
by (simp add: Bex_def, blast)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   540
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   541
(*The order of the premises presupposes that A is rigid; b may be flexible*)
14227
0356666744ec finalconsts
paulson
parents: 14095
diff changeset
   542
lemma UN_I: "[| a: A;  b: B(a) |] ==> b: (\<Union>x\<in>A. B(x))"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   543
by (simp, blast)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   544
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   545
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   546
lemma UN_E [elim!]:
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   547
    "[| b \<in> (\<Union>x\<in>A. B(x));  !!x.[| x: A;  b: B(x) |] ==> R |] ==> R"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   548
by blast
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   549
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   550
lemma UN_cong:
14227
0356666744ec finalconsts
paulson
parents: 14095
diff changeset
   551
    "[| A=B;  !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Union>x\<in>A. C(x)) = (\<Union>x\<in>B. D(x))"
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   552
by simp
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   553
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   554
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   555
(*No "Addcongs [UN_cong]" because @{term\<Union>} is a combination of constants*)
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   556
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   557
(* UN_E appears before UnionE so that it is tried first, to avoid expensive
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   558
  calls to hyp_subst_tac.  Cannot include UN_I as it is unsafe: would enlarge
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   559
  the search space.*)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   560
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   561
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   562
subsection{*Rules for the empty set*}
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   563
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   564
(*The set @{term"{x\<in>0. False}"} is empty; by foundation it equals 0
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   565
  See Suppes, page 21.*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   566
lemma not_mem_empty [simp]: "a \<notin> 0"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   567
apply (cut_tac foundation)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   568
apply (best dest: equalityD2)
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   569
done
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   570
45602
2a858377c3d2 eliminated obsolete "standard";
wenzelm
parents: 44147
diff changeset
   571
lemmas emptyE [elim!] = not_mem_empty [THEN notE]
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   572
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   573
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   574
lemma empty_subsetI [simp]: "0 \<subseteq> A"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   575
by blast
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   576
14227
0356666744ec finalconsts
paulson
parents: 14095
diff changeset
   577
lemma equals0I: "[| !!y. y\<in>A ==> False |] ==> A=0"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   578
by blast
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   579
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   580
lemma equals0D [dest]: "A=0 ==> a \<notin> A"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   581
by blast
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   582
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   583
declare sym [THEN equals0D, dest]
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   584
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   585
lemma not_emptyI: "a\<in>A ==> A \<noteq> 0"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   586
by blast
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   587
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   588
lemma not_emptyE:  "[| A \<noteq> 0;  !!x. x\<in>A ==> R |] ==> R"
13780
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   589
by blast
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   590
af7b79271364 more new-style theories
paulson
parents: 13175
diff changeset
   591
14095
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   592
subsection{*Rules for Inter*}
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   593
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   594
(*Not obviously useful for proving InterI, InterD, InterE*)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   595
lemma Inter_iff: "A \<in> \<Inter>(C) <-> (\<forall>x\<in>C. A: x) & C\<noteq>0"
14095
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   596
by (simp add: Inter_def Ball_def, blast)
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   597
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   598
(* Intersection is well-behaved only if the family is non-empty! *)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   599
lemma InterI [intro!]:
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   600
    "[| !!x. x: C ==> A: x;  C\<noteq>0 |] ==> A \<in> \<Inter>(C)"
14095
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   601
by (simp add: Inter_iff)
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   602
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   603
(*A "destruct" rule -- every B in C contains A as an element, but
14227
0356666744ec finalconsts
paulson
parents: 14095
diff changeset
   604
  A\<in>B can hold when B\<in>C does not!  This rule is analogous to "spec". *)
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   605
lemma InterD [elim, Pure.elim]: "[| A \<in> \<Inter>(C);  B \<in> C |] ==> A \<in> B"
14095
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   606
by (unfold Inter_def, blast)
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   607
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   608
(*"Classical" elimination rule -- does not require exhibiting @{term"B\<in>C"} *)
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   609
lemma InterE [elim]:
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   610
    "[| A \<in> \<Inter>(C);  B\<notin>C ==> R;  A\<in>B ==> R |] ==> R"
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   611
by (simp add: Inter_def, blast)
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   612
14095
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   613
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   614
subsection{*Rules for Intersections of families*}
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   615
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   616
(* @{term"\<Inter>x\<in>A. B(x)"} abbreviates @{term"\<Inter>({B(x). x\<in>A})"} *)
14095
a1ba833d6b61 Changed many Intersection rules from i:I to I~=0 to avoid introducing a new
paulson
parents: 14076
diff changeset
   617
46820
c656222c4dc1 mathematical symbols instead of ASCII
paulson
parents: 46751
diff changeset
   618
lemma INT_iff: "b \<in> (\<Inter>x\<in>A. B(x)) <-> (\<forall>x\<in>A. b \<in> B(x)) & A\<noteq>0"
14095
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by (force simp add: Inter_def)
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lemma INT_I: "[| !!x. x: A ==> b: B(x);  A\<noteq>0 |] ==> b: (\<Inter>x\<in>A. B(x))"
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   622
by blast
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   623
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   624
lemma INT_E: "[| b \<in> (\<Inter>x\<in>A. B(x));  a: A |] ==> b \<in> B(a)"
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   625
by blast
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   626
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   627
lemma INT_cong:
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   628
    "[| A=B;  !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Inter>x\<in>A. C(x)) = (\<Inter>x\<in>B. D(x))"
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   629
by simp
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   630
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(*No "Addcongs [INT_cong]" because @{term\<Inter>} is a combination of constants*)
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   632
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   633
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   634
subsection{*Rules for Powersets*}
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   635
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lemma PowI: "A \<subseteq> B ==> A \<in> Pow(B)"
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   637
by (erule Pow_iff [THEN iffD2])
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   638
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   639
lemma PowD: "A \<in> Pow(B)  ==>  A<=B"
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   640
by (erule Pow_iff [THEN iffD1])
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   641
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   642
declare Pow_iff [iff]
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   643
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lemmas Pow_bottom = empty_subsetI [THEN PowI]    --{* @{term"0 \<in> Pow(B)"} *}
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lemmas Pow_top = subset_refl [THEN PowI]         --{* @{term"A \<in> Pow(A)"} *}
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   646
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   647
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   648
subsection{*Cantor's Theorem: There is no surjection from a set to its powerset.*}
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   649
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   650
(*The search is undirected.  Allowing redundant introduction rules may
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   651
  make it diverge.  Variable b represents ANY map, such as
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   652
  (lam x\<in>A.b(x)): A->Pow(A). *)
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   653
lemma cantor: "\<exists>S \<in> Pow(A). \<forall>x\<in>A. b(x) \<noteq> S"
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   654
by (best elim!: equalityCE del: ReplaceI RepFun_eqI)
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   655
0
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end
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