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(* Title: HOL/Fun.thy

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ID: $Id$

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Author: Tobias Nipkow, Cambridge University Computer Laboratory

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Copyright 1994 University of Cambridge


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Notions about functions.

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*)


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Fun = Vimage + equalities +

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instance set :: (term) order


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(subset_refl,subset_trans,subset_antisym,psubset_eq)

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consts


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fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"


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nonterminals


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updbinds updbind

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syntax


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"_updbind" :: ['a, 'a] => updbind ("(2_ :=/ _)")


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"" :: updbind => updbinds ("_")


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"_updbinds" :: [updbind, updbinds] => updbinds ("_,/ _")

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"_Update" :: ['a, updbinds] => 'a ("_/'((_)')" [1000,0] 900)

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translations


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"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"


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"f(x:=y)" == "fun_upd f x y"

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defs

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fun_upd_def "f(a:=b) == % x. if x=a then b else f x"

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constdefs


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id :: 'a => 'a


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"id == %x. x"


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o :: ['b => 'c, 'a => 'b, 'a] => 'c (infixl 55)


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"f o g == %x. f(g(x))"

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inv :: ('a => 'b) => ('b => 'a)


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"inv(f::'a=>'b) == % y. @x. f(x)=y"

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inj_on :: ['a => 'b, 'a set] => bool


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"inj_on f A == ! x:A. ! y:A. f(x)=f(y) > x=y"

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syntax


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inj :: ('a => 'b) => bool (*injective*)


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translations


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"inj f" == "inj_on f UNIV"

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constdefs


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surj :: ('a => 'b) => bool (*surjective*)


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"surj f == ! y. ? x. y=f(x)"


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bij :: ('a => 'b) => bool (*bijective*)


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"bij f == inj f & surj f"


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(*The Pioperator, by Florian Kammueller*)


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constdefs


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Pi :: "['a set, 'a => 'b set] => ('a => 'b) set"


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"Pi A B == {f. ! x. if x:A then f(x) : B(x) else f(x) = (@ y. True)}"


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restrict :: "['a => 'b, 'a set] => ('a => 'b)"


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"restrict f A == (%x. if x : A then f x else (@ y. True))"


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syntax


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"@Pi" :: "[idt, 'a set, 'b set] => ('a => 'b) set" ("(3PI _:_./ _)" 10)


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funcset :: "['a set, 'b set] => ('a => 'b) set" (infixr 60)


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


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(*Giving funcset the nice arrow syntax > clashes with existing theories*)


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translations


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"PI x:A. B" => "Pi A (%x. B)"


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"A funcset B" => "Pi A (_K B)"


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"lam x:A. f" == "restrict (%x. f) A"


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constdefs


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Applyall :: "[('a => 'b) set, 'a]=> 'b set"


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"Applyall F a == (%f. f a) `` F"


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compose :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"

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"compose A g f == lam x : A. g(f x)"


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Inv :: "['a set, 'a => 'b] => ('b => 'a)"


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"Inv A f == (% x. (@ y. y : A & f y = x))"


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end

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ML


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val print_translation = [("Pi", dependent_tr' ("@Pi", "op funcset"))];
