src/HOL/HOL.thy
author nipkow
Fri Apr 04 16:33:28 1997 +0200 (1997-04-04)
changeset 2912 3fac3e8d5d3e
parent 2762 2ade3a141934
child 3066 3c548f92e032
permissions -rw-r--r--
moved inj and surj from Set to Fun and Inv -> inv.
     1 (*  Title:      HOL/HOL.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1993  University of Cambridge
     5 
     6 Higher-Order Logic.
     7 *)
     8 
     9 HOL = CPure +
    10 
    11 
    12 (** Core syntax **)
    13 
    14 classes
    15   term < logic
    16 
    17 default
    18   term
    19 
    20 types
    21   bool
    22 
    23 arities
    24   fun :: (term, term) term
    25   bool :: term
    26 
    27 
    28 consts
    29 
    30   (* Constants *)
    31 
    32   Trueprop      :: bool => prop                     ("(_)" 5)
    33   Not           :: bool => bool                     ("~ _" [40] 40)
    34   True, False   :: bool
    35   If            :: [bool, 'a, 'a] => 'a   ("(if (_)/ then (_)/ else (_))" 10)
    36 
    37   (* Binders *)
    38 
    39   Eps           :: ('a => bool) => 'a
    40   All           :: ('a => bool) => bool             (binder "! " 10)
    41   Ex            :: ('a => bool) => bool             (binder "? " 10)
    42   Ex1           :: ('a => bool) => bool             (binder "?! " 10)
    43   Let           :: ['a, 'a => 'b] => 'b
    44 
    45   (* Infixes *)
    46 
    47   o             :: ['b => 'c, 'a => 'b, 'a] => 'c   (infixl 55)
    48   "="           :: ['a, 'a] => bool                 (infixl 50)
    49   "&"           :: [bool, bool] => bool             (infixr 35)
    50   "|"           :: [bool, bool] => bool             (infixr 30)
    51   "-->"         :: [bool, bool] => bool             (infixr 25)
    52 
    53 
    54 (* Overloaded Constants *)
    55 
    56 axclass
    57   plus < term
    58 
    59 axclass
    60   minus < term
    61 
    62 axclass
    63   times < term
    64 
    65 consts
    66   "+"           :: ['a::plus, 'a] => 'a             (infixl 65)
    67   "-"           :: ['a::minus, 'a] => 'a            (infixl 65)
    68   "*"           :: ['a::times, 'a] => 'a            (infixl 70)
    69 
    70 
    71 
    72 (** Additional concrete syntax **)
    73 
    74 types
    75   letbinds  letbind
    76   case_syn  cases_syn
    77 
    78 syntax
    79 
    80   "~="          :: ['a, 'a] => bool                 (infixl 50)
    81 
    82   "@Eps"        :: [pttrn, bool] => 'a              ("(3@ _./ _)" [0, 10] 10)
    83 
    84   (* Alternative Quantifiers *)
    85 
    86   "*All"        :: [idts, bool] => bool             ("(3ALL _./ _)" [0, 10] 10)
    87   "*Ex"         :: [idts, bool] => bool             ("(3EX _./ _)" [0, 10] 10)
    88   "*Ex1"        :: [idts, bool] => bool             ("(3EX! _./ _)" [0, 10] 10)
    89 
    90   (* Let expressions *)
    91 
    92   "_bind"       :: [pttrn, 'a] => letbind           ("(2_ =/ _)" 10)
    93   ""            :: letbind => letbinds              ("_")
    94   "_binds"      :: [letbind, letbinds] => letbinds  ("_;/ _")
    95   "_Let"        :: [letbinds, 'a] => 'a             ("(let (_)/ in (_))" 10)
    96 
    97   (* Case expressions *)
    98 
    99   "@case"       :: ['a, cases_syn] => 'b            ("(case _ of/ _)" 10)
   100   "@case1"      :: ['a, 'b] => case_syn             ("(2_ =>/ _)" 10)
   101   ""            :: case_syn => cases_syn            ("_")
   102   "@case2"      :: [case_syn, cases_syn] => cases_syn   ("_/ | _")
   103 
   104 translations
   105   "x ~= y"      == "~ (x = y)"
   106   "@ x.b"       == "Eps (%x. b)"
   107   "ALL xs. P"   => "! xs. P"
   108   "EX xs. P"    => "? xs. P"
   109   "EX! xs. P"   => "?! xs. P"
   110   "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
   111   "let x = a in e"        == "Let a (%x. e)"
   112 
   113 
   114 syntax (symbols)
   115   Not           :: bool => bool                     ("\\<not> _" [40] 40)
   116   "op &"        :: [bool, bool] => bool             (infixr "\\<and>" 35)
   117   "op |"        :: [bool, bool] => bool             (infixr "\\<or>" 30)
   118   "op -->"      :: [bool, bool] => bool             (infixr "\\<midarrow>\\<rightarrow>" 25)
   119   "op o"        :: ['b => 'c, 'a => 'b, 'a] => 'c   (infixl "\\<circ>" 55)
   120   "op ~="       :: ['a, 'a] => bool                 (infixl "\\<noteq>" 50)
   121   "@Eps"        :: [pttrn, bool] => 'a              ("(3\\<epsilon>_./ _)" [0, 10] 10)
   122   "! "          :: [idts, bool] => bool             ("(3\\<forall>_./ _)" [0, 10] 10)
   123   "? "          :: [idts, bool] => bool             ("(3\\<exists>_./ _)" [0, 10] 10)
   124   "?! "         :: [idts, bool] => bool             ("(3\\<exists>!_./ _)" [0, 10] 10)
   125   "@case1"      :: ['a, 'b] => case_syn             ("(2_ \\<Rightarrow>/ _)" 10)
   126 
   127 syntax (symbols output)
   128   "*All"        :: [idts, bool] => bool             ("(3\\<forall>_./ _)" [0, 10] 10)
   129   "*Ex"         :: [idts, bool] => bool             ("(3\\<exists>_./ _)" [0, 10] 10)
   130   "*Ex1"        :: [idts, bool] => bool             ("(3\\<exists>!_./ _)" [0, 10] 10)
   131 
   132 
   133 
   134 
   135 (** Rules and definitions **)
   136 
   137 rules
   138 
   139   eq_reflection "(x=y) ==> (x==y)"
   140 
   141   (* Basic Rules *)
   142 
   143   refl          "t = (t::'a)"
   144   subst         "[| s = t; P(s) |] ==> P(t::'a)"
   145   ext           "(!!x::'a. (f(x)::'b) = g(x)) ==> (%x.f(x)) = (%x.g(x))"
   146   selectI       "P(x::'a) ==> P(@x.P(x))"
   147 
   148   impI          "(P ==> Q) ==> P-->Q"
   149   mp            "[| P-->Q;  P |] ==> Q"
   150 
   151 defs
   152 
   153   True_def      "True      == ((%x::bool.x)=(%x.x))"
   154   All_def       "All(P)    == (P = (%x.True))"
   155   Ex_def        "Ex(P)     == P(@x.P(x))"
   156   False_def     "False     == (!P.P)"
   157   not_def       "~ P       == P-->False"
   158   and_def       "P & Q     == !R. (P-->Q-->R) --> R"
   159   or_def        "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
   160   Ex1_def       "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
   161 
   162 rules
   163   (* Axioms *)
   164 
   165   iff           "(P-->Q) --> (Q-->P) --> (P=Q)"
   166   True_or_False "(P=True) | (P=False)"
   167 
   168 defs
   169   (* Misc Definitions *)
   170 
   171   Let_def       "Let s f == f(s)"
   172   o_def         "(f::'b=>'c) o g == (%(x::'a). f(g(x)))"
   173   if_def        "If P x y == @z::'a. (P=True --> z=x) & (P=False --> z=y)"
   174 
   175 end
   176 
   177 
   178 ML
   179 
   180 (** Choice between the HOL and Isabelle style of quantifiers **)
   181 
   182 val HOL_quantifiers = ref true;
   183 
   184 fun alt_ast_tr' (name, alt_name) =
   185   let
   186     fun ast_tr' (*name*) args =
   187       if ! HOL_quantifiers then raise Match
   188       else Syntax.mk_appl (Syntax.Constant alt_name) args;
   189   in
   190     (name, ast_tr')
   191   end;
   192 
   193 
   194 val print_ast_translation =
   195   map alt_ast_tr' [("! ", "*All"), ("? ", "*Ex"), ("?! ", "*Ex1")];