src/HOL/Fun.thy
author nipkow
Fri, 24 Nov 2000 16:49:27 +0100
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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 = Inverse_Image + 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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(* Hint: to define the sum of two functions (or maps), use sum_case.
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         A nice infix syntax could be defined (in Datatype.thy or below) by
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consts
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  fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
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translations
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 "fun_sum" == "sum_case"
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*)
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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 (symbols)
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  "op o"        :: "['b => 'c, 'a => 'b, 'a] => 'c"      (infixl "\\<circ>" 55)
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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 Pi-operator, 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"))];