src/ZF/ZF.thy
author wenzelm
Thu Sep 02 00:48:07 2010 +0200 (2010-09-02)
changeset 38980 af73cf0dc31f
parent 38798 89f273ab1d42
child 39128 93a7365fb4ee
permissions -rw-r--r--
turned show_question_marks into proper configuration option;
show_question_marks only affects regular type/term pretty printing, not raw Term.string_of_vname;
tuned;
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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 FOL
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uses "~~/src/Tools/misc_legacy.ML"
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begin
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ML {* eta_contract := false *}
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typedecl i
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arities  i :: "term"
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consts
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  "0"         :: "i"                  ("0")   --{*the empty set*}
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  Pow         :: "i => i"                     --{*power sets*}
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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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consts
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  Union :: "i => i"
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  Inter :: "i => i"
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text {*Variations on Replacement *}
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consts
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  PrimReplace :: "[i, [i, i] => o] => i"
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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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  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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nonterminals "is" 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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finalconsts
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  0 Pow Inf Union PrimReplace mem
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defs  (* Bounded Quantifiers *)
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  Ball_def:      "Ball(A, P) == \<forall>x. x\<in>A --> 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 <= B == \<forall>x\<in>A. x\<in>B"
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axioms
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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 <= B & B <= A"
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  Union_iff:     "A \<in> Union(C) <-> (\<exists>B\<in>C. A\<in>B)"
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  Pow_iff:       "A \<in> Pow(B) <-> A <= B"
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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)"
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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~:A)"
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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) --> 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) Un 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 Un  B  == Union(Upair(A,B))"
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  Int_def:      "A Int 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) Un 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 --> (\<forall>y'. <x,y'>:r --> y=y')"
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   284
  image_def:    "r `` A  == {y : range(r) . \<exists>x\<in>A. <x,y> : r}"
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   285
  vimage_def:   "r -`` A == converse(r)``A"
clasohm@0
   286
wenzelm@615
   287
  (* Abstraction, application and Cartesian product of a family of sets *)
clasohm@0
   288
paulson@14227
   289
  lam_def:      "Lambda(A,b) == {<x,b(x)> . x\<in>A}"
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   290
  apply_def:    "f`a == Union(f``{a})"
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   291
  Pi_def:       "Pi(A,B)  == {f\<in>Pow(Sigma(A,B)). A<=domain(f) & function(f)}"
clasohm@0
   292
paulson@12891
   293
  (* Restrict the relation r to the domain A *)
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   294
  restrict_def: "restrict(r,A) == {z : r. \<exists>x\<in>A. \<exists>y. z = <x,y>}"
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   295
paulson@13780
   296
paulson@13780
   297
subsection {* Substitution*}
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   298
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   299
(*Useful examples:  singletonI RS subst_elem,  subst_elem RSN (2,IntI) *)
paulson@14227
   300
lemma subst_elem: "[| b\<in>A;  a=b |] ==> a\<in>A"
paulson@13780
   301
by (erule ssubst, assumption)
paulson@13780
   302
paulson@13780
   303
paulson@13780
   304
subsection{*Bounded universal quantifier*}
paulson@13780
   305
paulson@14227
   306
lemma ballI [intro!]: "[| !!x. x\<in>A ==> P(x) |] ==> \<forall>x\<in>A. P(x)"
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   307
by (simp add: Ball_def)
paulson@13780
   308
paulson@15481
   309
lemmas strip = impI allI ballI
paulson@15481
   310
paulson@14227
   311
lemma bspec [dest?]: "[| \<forall>x\<in>A. P(x);  x: A |] ==> P(x)"
paulson@13780
   312
by (simp add: Ball_def)
paulson@13780
   313
paulson@13780
   314
(*Instantiates x first: better for automatic theorem proving?*)
paulson@13780
   315
lemma rev_ballE [elim]: 
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   316
    "[| \<forall>x\<in>A. P(x);  x~:A ==> Q;  P(x) ==> Q |] ==> Q"
paulson@13780
   317
by (simp add: Ball_def, blast) 
paulson@13780
   318
paulson@14227
   319
lemma ballE: "[| \<forall>x\<in>A. P(x);  P(x) ==> Q;  x~:A ==> Q |] ==> Q"
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   320
by blast
paulson@13780
   321
paulson@13780
   322
(*Used in the datatype package*)
paulson@14227
   323
lemma rev_bspec: "[| x: A;  \<forall>x\<in>A. P(x) |] ==> P(x)"
paulson@13780
   324
by (simp add: Ball_def)
paulson@13780
   325
paulson@14227
   326
(*Trival rewrite rule;   (\<forall>x\<in>A.P)<->P holds only if A is nonempty!*)
paulson@14227
   327
lemma ball_triv [simp]: "(\<forall>x\<in>A. P) <-> ((\<exists>x. x\<in>A) --> P)"
paulson@13780
   328
by (simp add: Ball_def)
paulson@13780
   329
paulson@13780
   330
(*Congruence rule for rewriting*)
paulson@13780
   331
lemma ball_cong [cong]:
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   332
    "[| A=A';  !!x. x\<in>A' ==> P(x) <-> P'(x) |] ==> (\<forall>x\<in>A. P(x)) <-> (\<forall>x\<in>A'. P'(x))"
paulson@13780
   333
by (simp add: Ball_def)
paulson@13780
   334
wenzelm@18845
   335
lemma atomize_ball:
wenzelm@18845
   336
    "(!!x. x \<in> A ==> P(x)) == Trueprop (\<forall>x\<in>A. P(x))"
wenzelm@18845
   337
  by (simp only: Ball_def atomize_all atomize_imp)
wenzelm@18845
   338
wenzelm@18845
   339
lemmas [symmetric, rulify] = atomize_ball
wenzelm@18845
   340
  and [symmetric, defn] = atomize_ball
wenzelm@18845
   341
paulson@13780
   342
paulson@13780
   343
subsection{*Bounded existential quantifier*}
paulson@13780
   344
paulson@14227
   345
lemma bexI [intro]: "[| P(x);  x: A |] ==> \<exists>x\<in>A. P(x)"
paulson@13780
   346
by (simp add: Bex_def, blast)
paulson@13780
   347
paulson@14227
   348
(*The best argument order when there is only one x\<in>A*)
paulson@14227
   349
lemma rev_bexI: "[| x\<in>A;  P(x) |] ==> \<exists>x\<in>A. P(x)"
paulson@13780
   350
by blast
paulson@13780
   351
paulson@14227
   352
(*Not of the general form for such rules; ~\<exists>has become ALL~ *)
paulson@14227
   353
lemma bexCI: "[| \<forall>x\<in>A. ~P(x) ==> P(a);  a: A |] ==> \<exists>x\<in>A. P(x)"
paulson@13780
   354
by blast
paulson@13780
   355
paulson@14227
   356
lemma bexE [elim!]: "[| \<exists>x\<in>A. P(x);  !!x. [| x\<in>A; P(x) |] ==> Q |] ==> Q"
paulson@13780
   357
by (simp add: Bex_def, blast)
paulson@13780
   358
paulson@14227
   359
(*We do not even have (\<exists>x\<in>A. True) <-> True unless A is nonempty!!*)
paulson@14227
   360
lemma bex_triv [simp]: "(\<exists>x\<in>A. P) <-> ((\<exists>x. x\<in>A) & P)"
paulson@13780
   361
by (simp add: Bex_def)
paulson@13780
   362
paulson@13780
   363
lemma bex_cong [cong]:
paulson@14227
   364
    "[| A=A';  !!x. x\<in>A' ==> P(x) <-> P'(x) |] 
paulson@14227
   365
     ==> (\<exists>x\<in>A. P(x)) <-> (\<exists>x\<in>A'. P'(x))"
paulson@13780
   366
by (simp add: Bex_def cong: conj_cong)
paulson@13780
   367
paulson@13780
   368
paulson@13780
   369
paulson@13780
   370
subsection{*Rules for subsets*}
paulson@13780
   371
paulson@13780
   372
lemma subsetI [intro!]:
paulson@14227
   373
    "(!!x. x\<in>A ==> x\<in>B) ==> A <= B"
paulson@13780
   374
by (simp add: subset_def) 
paulson@13780
   375
paulson@13780
   376
(*Rule in Modus Ponens style [was called subsetE] *)
paulson@14227
   377
lemma subsetD [elim]: "[| A <= B;  c\<in>A |] ==> c\<in>B"
paulson@13780
   378
apply (unfold subset_def)
paulson@13780
   379
apply (erule bspec, assumption)
paulson@13780
   380
done
paulson@13780
   381
paulson@13780
   382
(*Classical elimination rule*)
paulson@13780
   383
lemma subsetCE [elim]:
paulson@14227
   384
    "[| A <= B;  c~:A ==> P;  c\<in>B ==> P |] ==> P"
paulson@13780
   385
by (simp add: subset_def, blast) 
paulson@13780
   386
paulson@13780
   387
(*Sometimes useful with premises in this order*)
paulson@14227
   388
lemma rev_subsetD: "[| c\<in>A; A<=B |] ==> c\<in>B"
paulson@13780
   389
by blast
paulson@13780
   390
paulson@13780
   391
lemma contra_subsetD: "[| A <= B; c ~: B |] ==> c ~: A"
paulson@13780
   392
by blast
paulson@13780
   393
paulson@13780
   394
lemma rev_contra_subsetD: "[| c ~: B;  A <= B |] ==> c ~: A"
paulson@13780
   395
by blast
paulson@13780
   396
paulson@13780
   397
lemma subset_refl [simp]: "A <= A"
paulson@13780
   398
by blast
paulson@13780
   399
paulson@13780
   400
lemma subset_trans: "[| A<=B;  B<=C |] ==> A<=C"
paulson@13780
   401
by blast
paulson@13780
   402
paulson@13780
   403
(*Useful for proving A<=B by rewriting in some cases*)
paulson@13780
   404
lemma subset_iff: 
paulson@14227
   405
     "A<=B <-> (\<forall>x. x\<in>A --> x\<in>B)"
paulson@13780
   406
apply (unfold subset_def Ball_def)
paulson@13780
   407
apply (rule iff_refl)
paulson@13780
   408
done
paulson@13780
   409
paulson@13780
   410
paulson@13780
   411
subsection{*Rules for equality*}
paulson@13780
   412
paulson@13780
   413
(*Anti-symmetry of the subset relation*)
paulson@13780
   414
lemma equalityI [intro]: "[| A <= B;  B <= A |] ==> A = B"
paulson@13780
   415
by (rule extension [THEN iffD2], rule conjI) 
paulson@13780
   416
paulson@13780
   417
paulson@14227
   418
lemma equality_iffI: "(!!x. x\<in>A <-> x\<in>B) ==> A = B"
paulson@13780
   419
by (rule equalityI, blast+)
paulson@13780
   420
paulson@13780
   421
lemmas equalityD1 = extension [THEN iffD1, THEN conjunct1, standard]
paulson@13780
   422
lemmas equalityD2 = extension [THEN iffD1, THEN conjunct2, standard]
paulson@13780
   423
paulson@13780
   424
lemma equalityE: "[| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P"
paulson@13780
   425
by (blast dest: equalityD1 equalityD2) 
paulson@13780
   426
paulson@13780
   427
lemma equalityCE:
paulson@14227
   428
    "[| A = B;  [| c\<in>A; c\<in>B |] ==> P;  [| c~:A; c~:B |] ==> P |]  ==>  P"
paulson@13780
   429
by (erule equalityE, blast) 
paulson@13780
   430
ballarin@27702
   431
lemma equality_iffD:
ballarin@27702
   432
  "A = B ==> (!!x. x : A <-> x : B)"
ballarin@27702
   433
  by auto
ballarin@27702
   434
paulson@13780
   435
paulson@13780
   436
subsection{*Rules for Replace -- the derived form of replacement*}
paulson@13780
   437
paulson@13780
   438
lemma Replace_iff: 
paulson@14227
   439
    "b : {y. x\<in>A, P(x,y)}  <->  (\<exists>x\<in>A. P(x,b) & (\<forall>y. P(x,y) --> y=b))"
paulson@13780
   440
apply (unfold Replace_def)
paulson@13780
   441
apply (rule replacement [THEN iff_trans], blast+)
paulson@13780
   442
done
paulson@13780
   443
paulson@13780
   444
(*Introduction; there must be a unique y such that P(x,y), namely y=b. *)
paulson@13780
   445
lemma ReplaceI [intro]: 
paulson@13780
   446
    "[| P(x,b);  x: A;  !!y. P(x,y) ==> y=b |] ==>  
paulson@14227
   447
     b : {y. x\<in>A, P(x,y)}"
paulson@13780
   448
by (rule Replace_iff [THEN iffD2], blast) 
paulson@13780
   449
paulson@13780
   450
(*Elimination; may asssume there is a unique y such that P(x,y), namely y=b. *)
paulson@13780
   451
lemma ReplaceE: 
paulson@14227
   452
    "[| b : {y. x\<in>A, P(x,y)};   
paulson@14227
   453
        !!x. [| x: A;  P(x,b);  \<forall>y. P(x,y)-->y=b |] ==> R  
paulson@13780
   454
     |] ==> R"
paulson@13780
   455
by (rule Replace_iff [THEN iffD1, THEN bexE], simp+)
paulson@13780
   456
paulson@13780
   457
(*As above but without the (generally useless) 3rd assumption*)
paulson@13780
   458
lemma ReplaceE2 [elim!]: 
paulson@14227
   459
    "[| b : {y. x\<in>A, P(x,y)};   
paulson@13780
   460
        !!x. [| x: A;  P(x,b) |] ==> R  
paulson@13780
   461
     |] ==> R"
paulson@13780
   462
by (erule ReplaceE, blast) 
paulson@13780
   463
paulson@13780
   464
lemma Replace_cong [cong]:
paulson@14227
   465
    "[| A=B;  !!x y. x\<in>B ==> P(x,y) <-> Q(x,y) |] ==>  
paulson@13780
   466
     Replace(A,P) = Replace(B,Q)"
paulson@13780
   467
apply (rule equality_iffI) 
paulson@13780
   468
apply (simp add: Replace_iff) 
paulson@13780
   469
done
paulson@13780
   470
paulson@13780
   471
paulson@13780
   472
subsection{*Rules for RepFun*}
paulson@13780
   473
paulson@14227
   474
lemma RepFunI: "a \<in> A ==> f(a) : {f(x). x\<in>A}"
paulson@13780
   475
by (simp add: RepFun_def Replace_iff, blast)
paulson@13780
   476
paulson@13780
   477
(*Useful for coinduction proofs*)
paulson@14227
   478
lemma RepFun_eqI [intro]: "[| b=f(a);  a \<in> A |] ==> b : {f(x). x\<in>A}"
paulson@13780
   479
apply (erule ssubst)
paulson@13780
   480
apply (erule RepFunI)
paulson@13780
   481
done
paulson@13780
   482
paulson@13780
   483
lemma RepFunE [elim!]:
paulson@14227
   484
    "[| b : {f(x). x\<in>A};   
paulson@14227
   485
        !!x.[| x\<in>A;  b=f(x) |] ==> P |] ==>  
paulson@13780
   486
     P"
paulson@13780
   487
by (simp add: RepFun_def Replace_iff, blast) 
paulson@13780
   488
paulson@13780
   489
lemma RepFun_cong [cong]: 
paulson@14227
   490
    "[| A=B;  !!x. x\<in>B ==> f(x)=g(x) |] ==> RepFun(A,f) = RepFun(B,g)"
paulson@13780
   491
by (simp add: RepFun_def)
paulson@13780
   492
paulson@14227
   493
lemma RepFun_iff [simp]: "b : {f(x). x\<in>A} <-> (\<exists>x\<in>A. b=f(x))"
paulson@13780
   494
by (unfold Bex_def, blast)
paulson@13780
   495
paulson@14227
   496
lemma triv_RepFun [simp]: "{x. x\<in>A} = A"
paulson@13780
   497
by blast
paulson@13780
   498
paulson@13780
   499
paulson@13780
   500
subsection{*Rules for Collect -- forming a subset by separation*}
paulson@13780
   501
paulson@13780
   502
(*Separation is derivable from Replacement*)
paulson@14227
   503
lemma separation [simp]: "a : {x\<in>A. P(x)} <-> a\<in>A & P(a)"
paulson@13780
   504
by (unfold Collect_def, blast)
paulson@13780
   505
paulson@14227
   506
lemma CollectI [intro!]: "[| a\<in>A;  P(a) |] ==> a : {x\<in>A. P(x)}"
paulson@13780
   507
by simp
paulson@13780
   508
paulson@14227
   509
lemma CollectE [elim!]: "[| a : {x\<in>A. P(x)};  [| a\<in>A; P(a) |] ==> R |] ==> R"
paulson@13780
   510
by simp
paulson@13780
   511
paulson@14227
   512
lemma CollectD1: "a : {x\<in>A. P(x)} ==> a\<in>A"
paulson@13780
   513
by (erule CollectE, assumption)
paulson@13780
   514
paulson@14227
   515
lemma CollectD2: "a : {x\<in>A. P(x)} ==> P(a)"
paulson@13780
   516
by (erule CollectE, assumption)
paulson@13780
   517
paulson@13780
   518
lemma Collect_cong [cong]:
paulson@14227
   519
    "[| A=B;  !!x. x\<in>B ==> P(x) <-> Q(x) |]  
paulson@13780
   520
     ==> Collect(A, %x. P(x)) = Collect(B, %x. Q(x))"
paulson@13780
   521
by (simp add: Collect_def)
paulson@13780
   522
paulson@13780
   523
paulson@13780
   524
subsection{*Rules for Unions*}
paulson@13780
   525
paulson@13780
   526
declare Union_iff [simp]
paulson@13780
   527
paulson@13780
   528
(*The order of the premises presupposes that C is rigid; A may be flexible*)
paulson@13780
   529
lemma UnionI [intro]: "[| B: C;  A: B |] ==> A: Union(C)"
paulson@13780
   530
by (simp, blast)
paulson@13780
   531
paulson@14227
   532
lemma UnionE [elim!]: "[| A \<in> Union(C);  !!B.[| A: B;  B: C |] ==> R |] ==> R"
paulson@13780
   533
by (simp, blast)
paulson@13780
   534
paulson@13780
   535
paulson@13780
   536
subsection{*Rules for Unions of families*}
paulson@14227
   537
(* \<Union>x\<in>A. B(x) abbreviates Union({B(x). x\<in>A}) *)
paulson@13780
   538
paulson@14227
   539
lemma UN_iff [simp]: "b : (\<Union>x\<in>A. B(x)) <-> (\<exists>x\<in>A. b \<in> B(x))"
paulson@13780
   540
by (simp add: Bex_def, blast)
paulson@13780
   541
paulson@13780
   542
(*The order of the premises presupposes that A is rigid; b may be flexible*)
paulson@14227
   543
lemma UN_I: "[| a: A;  b: B(a) |] ==> b: (\<Union>x\<in>A. B(x))"
paulson@13780
   544
by (simp, blast)
paulson@13780
   545
paulson@13780
   546
paulson@13780
   547
lemma UN_E [elim!]: 
paulson@14227
   548
    "[| b : (\<Union>x\<in>A. B(x));  !!x.[| x: A;  b: B(x) |] ==> R |] ==> R"
paulson@13780
   549
by blast 
paulson@13780
   550
paulson@13780
   551
lemma UN_cong: 
paulson@14227
   552
    "[| A=B;  !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Union>x\<in>A. C(x)) = (\<Union>x\<in>B. D(x))"
paulson@13780
   553
by simp 
paulson@13780
   554
paulson@13780
   555
paulson@14227
   556
(*No "Addcongs [UN_cong]" because \<Union>is a combination of constants*)
paulson@13780
   557
paulson@13780
   558
(* UN_E appears before UnionE so that it is tried first, to avoid expensive
paulson@13780
   559
  calls to hyp_subst_tac.  Cannot include UN_I as it is unsafe: would enlarge
paulson@13780
   560
  the search space.*)
paulson@13780
   561
paulson@13780
   562
paulson@13780
   563
subsection{*Rules for the empty set*}
paulson@13780
   564
paulson@14227
   565
(*The set {x\<in>0. False} is empty; by foundation it equals 0 
paulson@13780
   566
  See Suppes, page 21.*)
paulson@13780
   567
lemma not_mem_empty [simp]: "a ~: 0"
paulson@13780
   568
apply (cut_tac foundation)
paulson@13780
   569
apply (best dest: equalityD2)
paulson@13780
   570
done
paulson@13780
   571
paulson@13780
   572
lemmas emptyE [elim!] = not_mem_empty [THEN notE, standard]
paulson@13780
   573
paulson@13780
   574
paulson@13780
   575
lemma empty_subsetI [simp]: "0 <= A"
paulson@13780
   576
by blast 
paulson@13780
   577
paulson@14227
   578
lemma equals0I: "[| !!y. y\<in>A ==> False |] ==> A=0"
paulson@13780
   579
by blast
paulson@13780
   580
paulson@13780
   581
lemma equals0D [dest]: "A=0 ==> a ~: A"
paulson@13780
   582
by blast
paulson@13780
   583
paulson@13780
   584
declare sym [THEN equals0D, dest]
paulson@13780
   585
paulson@14227
   586
lemma not_emptyI: "a\<in>A ==> A ~= 0"
paulson@13780
   587
by blast
paulson@13780
   588
paulson@14227
   589
lemma not_emptyE:  "[| A ~= 0;  !!x. x\<in>A ==> R |] ==> R"
paulson@13780
   590
by blast
paulson@13780
   591
paulson@13780
   592
paulson@14095
   593
subsection{*Rules for Inter*}
paulson@14095
   594
paulson@14095
   595
(*Not obviously useful for proving InterI, InterD, InterE*)
paulson@14227
   596
lemma Inter_iff: "A \<in> Inter(C) <-> (\<forall>x\<in>C. A: x) & C\<noteq>0"
paulson@14095
   597
by (simp add: Inter_def Ball_def, blast)
paulson@14095
   598
paulson@14095
   599
(* Intersection is well-behaved only if the family is non-empty! *)
paulson@14095
   600
lemma InterI [intro!]: 
paulson@14227
   601
    "[| !!x. x: C ==> A: x;  C\<noteq>0 |] ==> A \<in> Inter(C)"
paulson@14095
   602
by (simp add: Inter_iff)
paulson@14095
   603
paulson@14095
   604
(*A "destruct" rule -- every B in C contains A as an element, but
paulson@14227
   605
  A\<in>B can hold when B\<in>C does not!  This rule is analogous to "spec". *)
paulson@14227
   606
lemma InterD [elim]: "[| A \<in> Inter(C);  B \<in> C |] ==> A \<in> B"
paulson@14095
   607
by (unfold Inter_def, blast)
paulson@14095
   608
paulson@14227
   609
(*"Classical" elimination rule -- does not require exhibiting B\<in>C *)
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   610
lemma InterE [elim]: 
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   611
    "[| A \<in> Inter(C);  B~:C ==> R;  A\<in>B ==> R |] ==> R"
paulson@14095
   612
by (simp add: Inter_def, blast) 
paulson@14095
   613
  
paulson@14095
   614
paulson@14095
   615
subsection{*Rules for Intersections of families*}
paulson@14095
   616
paulson@14227
   617
(* \<Inter>x\<in>A. B(x) abbreviates Inter({B(x). x\<in>A}) *)
paulson@14095
   618
paulson@14227
   619
lemma INT_iff: "b : (\<Inter>x\<in>A. B(x)) <-> (\<forall>x\<in>A. b \<in> B(x)) & A\<noteq>0"
paulson@14095
   620
by (force simp add: Inter_def)
paulson@14095
   621
paulson@14227
   622
lemma INT_I: "[| !!x. x: A ==> b: B(x);  A\<noteq>0 |] ==> b: (\<Inter>x\<in>A. B(x))"
paulson@14095
   623
by blast
paulson@14095
   624
paulson@14227
   625
lemma INT_E: "[| b : (\<Inter>x\<in>A. B(x));  a: A |] ==> b \<in> B(a)"
paulson@14095
   626
by blast
paulson@14095
   627
paulson@14095
   628
lemma INT_cong:
paulson@14227
   629
    "[| A=B;  !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Inter>x\<in>A. C(x)) = (\<Inter>x\<in>B. D(x))"
paulson@14095
   630
by simp
paulson@14095
   631
paulson@14227
   632
(*No "Addcongs [INT_cong]" because \<Inter>is a combination of constants*)
paulson@14095
   633
paulson@14095
   634
paulson@13780
   635
subsection{*Rules for Powersets*}
paulson@13780
   636
paulson@14227
   637
lemma PowI: "A <= B ==> A \<in> Pow(B)"
paulson@13780
   638
by (erule Pow_iff [THEN iffD2])
paulson@13780
   639
paulson@14227
   640
lemma PowD: "A \<in> Pow(B)  ==>  A<=B"
paulson@13780
   641
by (erule Pow_iff [THEN iffD1])
paulson@13780
   642
paulson@13780
   643
declare Pow_iff [iff]
paulson@13780
   644
paulson@14227
   645
lemmas Pow_bottom = empty_subsetI [THEN PowI] (* 0 \<in> Pow(B) *)
paulson@14227
   646
lemmas Pow_top = subset_refl [THEN PowI] (* A \<in> Pow(A) *)
paulson@13780
   647
paulson@13780
   648
paulson@13780
   649
subsection{*Cantor's Theorem: There is no surjection from a set to its powerset.*}
paulson@13780
   650
paulson@13780
   651
(*The search is undirected.  Allowing redundant introduction rules may 
paulson@13780
   652
  make it diverge.  Variable b represents ANY map, such as
paulson@14227
   653
  (lam x\<in>A.b(x)): A->Pow(A). *)
paulson@14227
   654
lemma cantor: "\<exists>S \<in> Pow(A). \<forall>x\<in>A. b(x) ~= S"
paulson@13780
   655
by (best elim!: equalityCE del: ReplaceI RepFun_eqI)
paulson@13780
   656
paulson@13780
   657
(*Functions for ML scripts*)
paulson@13780
   658
ML
paulson@13780
   659
{*
paulson@14227
   660
(*Converts A<=B to x\<in>A ==> x\<in>B*)
wenzelm@24893
   661
fun impOfSubs th = th RSN (2, @{thm rev_subsetD});
paulson@13780
   662
paulson@14227
   663
(*Takes assumptions \<forall>x\<in>A.P(x) and a\<in>A; creates assumption P(a)*)
wenzelm@24893
   664
val ball_tac = dtac @{thm bspec} THEN' assume_tac
paulson@13780
   665
*}
clasohm@0
   666
clasohm@0
   667
end
clasohm@0
   668