src/CCL/Set.thy
author paulson
Mon Dec 07 18:26:25 1998 +0100 (1998-12-07)
changeset 6019 0e55c2fb2ebb
parent 3935 52c14fe8f16b
child 17456 bcf7544875b2
permissions -rw-r--r--
tidying
     1 (*  Title:      CCL/set.thy
     2     ID:         $Id$
     3 
     4 Modified version of HOL/set.thy that extends FOL
     5 
     6 *)
     7 
     8 Set = FOL +
     9 
    10 global
    11 
    12 types
    13   'a set
    14 
    15 arities
    16   set :: (term) term
    17 
    18 consts
    19   Collect       :: "['a => o] => 'a set"                    (*comprehension*)
    20   Compl         :: "('a set) => 'a set"                     (*complement*)
    21   Int           :: "['a set, 'a set] => 'a set"         (infixl 70)
    22   Un            :: "['a set, 'a set] => 'a set"         (infixl 65)
    23   Union, Inter  :: "(('a set)set) => 'a set"                (*...of a set*)
    24   UNION, INTER  :: "['a set, 'a => 'b set] => 'b set"       (*general*)
    25   Ball, Bex     :: "['a set, 'a => o] => o"                 (*bounded quants*)
    26   mono          :: "['a set => 'b set] => o"                (*monotonicity*)
    27   ":"           :: "['a, 'a set] => o"                  (infixl 50) (*membership*)
    28   "<="          :: "['a set, 'a set] => o"              (infixl 50)
    29   singleton     :: "'a => 'a set"                       ("{_}")
    30   empty         :: "'a set"                             ("{}")
    31   "oo"          :: "['b => 'c, 'a => 'b, 'a] => 'c"     (infixr 50) (*composition*)
    32 
    33 syntax
    34   "@Coll"       :: "[idt, o] => 'a set"                 ("(1{_./ _})") (*collection*)
    35 
    36   (* Big Intersection / Union *)
    37 
    38   "@INTER"      :: "[idt, 'a set, 'b set] => 'b set"    ("(INT _:_./ _)" [0, 0, 0] 10)
    39   "@UNION"      :: "[idt, 'a set, 'b set] => 'b set"    ("(UN _:_./ _)" [0, 0, 0] 10)
    40 
    41   (* Bounded Quantifiers *)
    42 
    43   "@Ball"       :: "[idt, 'a set, o] => o"              ("(ALL _:_./ _)" [0, 0, 0] 10)
    44   "@Bex"        :: "[idt, 'a set, o] => o"              ("(EX _:_./ _)" [0, 0, 0] 10)
    45 
    46 
    47 translations
    48   "{x. P}"      == "Collect(%x. P)"
    49   "INT x:A. B"  == "INTER(A, %x. B)"
    50   "UN x:A. B"   == "UNION(A, %x. B)"
    51   "ALL x:A. P"  == "Ball(A, %x. P)"
    52   "EX x:A. P"   == "Bex(A, %x. P)"
    53 
    54 local
    55 
    56 rules
    57   mem_Collect_iff       "(a : {x. P(x)}) <-> P(a)"
    58   set_extension         "A=B <-> (ALL x. x:A <-> x:B)"
    59 
    60   Ball_def      "Ball(A, P)  == ALL x. x:A --> P(x)"
    61   Bex_def       "Bex(A, P)   == EX x. x:A & P(x)"
    62   mono_def      "mono(f)     == (ALL A B. A <= B --> f(A) <= f(B))"
    63   subset_def    "A <= B      == ALL x:A. x:B"
    64   singleton_def "{a}         == {x. x=a}"
    65   empty_def     "{}          == {x. False}"
    66   Un_def        "A Un B      == {x. x:A | x:B}"
    67   Int_def       "A Int B     == {x. x:A & x:B}"
    68   Compl_def     "Compl(A)    == {x. ~x:A}"
    69   INTER_def     "INTER(A, B) == {y. ALL x:A. y: B(x)}"
    70   UNION_def     "UNION(A, B) == {y. EX x:A. y: B(x)}"
    71   Inter_def     "Inter(S)    == (INT x:S. x)"
    72   Union_def     "Union(S)    == (UN x:S. x)"
    73 
    74 end
    75