HOL.thy

minor cleanup, added 'axclass', 'instance', 'syntax', 'defs' sections;

(*  Title:      HOL/HOL.thy
    ID:         $Id$
    Author:     Tobias Nipkow
    Copyright   1993  University of Cambridge

Higher-Order Logic
*)

HOL = Pure +

classes
  term < logic

axclass
  plus < term

axclass
  minus < term

axclass
  times < term

default
  term

types
  bool

arities
  fun :: (term, term) term
  bool :: term


consts

  (* Constants *)

  Trueprop      :: "bool => prop"                     ("(_)" 5)
  not           :: "bool => bool"                     ("~ _" [40] 40)
  True, False   :: "bool"
  if            :: "[bool, 'a, 'a] => 'a"
  Inv           :: "('a => 'b) => ('b => 'a)"

  (* Binders *)

  Eps           :: "('a => bool) => 'a"               (binder "@" 10)
  All           :: "('a => bool) => bool"             (binder "! " 10)
  Ex            :: "('a => bool) => bool"             (binder "? " 10)
  Ex1           :: "('a => bool) => bool"             (binder "?! " 10)
  Let           :: "['a, 'a => 'b] => 'b"

  (* Infixes *)

  o             :: "['b => 'c, 'a => 'b, 'a] => 'c"   (infixr 50)
  "="           :: "['a, 'a] => bool"                 (infixl 50)
(*"~="          :: "['a, 'a] => bool"                 (infixl 50)*)
  "&"           :: "[bool, bool] => bool"             (infixr 35)
  "|"           :: "[bool, bool] => bool"             (infixr 30)
  "-->"         :: "[bool, bool] => bool"             (infixr 25)

  (* Overloaded Constants *)

  "+"           :: "['a::plus, 'a] => 'a"             (infixl 65)
  "-"           :: "['a::minus, 'a] => 'a"            (infixl 65)
  "*"           :: "['a::times, 'a] => 'a"            (infixl 70)


types
  letbinds  letbind
  case_syn  cases_syn

syntax

  "~="          :: "['a, 'a] => bool"                 (infixl 50)

  (* Alternative Quantifiers *)

  "*All"        :: "[idts, bool] => bool"             ("(3ALL _./ _)" 10)
  "*Ex"         :: "[idts, bool] => bool"             ("(3EX _./ _)" 10)
  "*Ex1"        :: "[idts, bool] => bool"             ("(3EX! _./ _)" 10)

  (* Let expressions *)

  "_bind"       :: "[idt, 'a] => letbind"             ("(2_ =/ _)" 10)
  ""            :: "letbind => letbinds"              ("_")
  "_binds"      :: "[letbind, letbinds] => letbinds"  ("_;/ _")
  "_Let"        :: "[letbinds, 'a] => 'a"             ("(let (_)/ in (_))" 10)

  (* Case expressions *)

  "@case"       :: "['a, cases_syn] => 'b"            ("(case _ of/ _)" 10)
  "@case1"      :: "['a, 'b] => case_syn"             ("(2_ =>/ _)" 10)
  ""            :: "case_syn => cases_syn"            ("_")
  "@case2"      :: "[case_syn, cases_syn] => cases_syn"   ("_/ | _")

translations
  "x ~= y"      == "~ (x = y)"
  "ALL xs. P"   => "! xs. P"
  "EX xs. P"    => "? xs. P"
  "EX! xs. P"   => "?! xs. P"
  "_Let(_binds(b, bs), e)"  == "_Let(b, _Let(bs, e))"
  "let x = a in e"          == "Let(a, %x. e)"


rules

  eq_reflection "(x=y) ==> (x==y)"

  (* Basic Rules *)

  refl          "t = (t::'a)"
  subst         "[| s = t; P(s) |] ==> P(t::'a)"
  ext           "(!!x::'a. (f(x)::'b) = g(x)) ==> (%x.f(x)) = (%x.g(x))"
  selectI       "P(x::'a) ==> P(@x.P(x))"

  impI          "(P ==> Q) ==> P-->Q"
  mp            "[| P-->Q;  P |] ==> Q"

  (* Definitions *)

  True_def      "True      == ((%x::bool.x)=(%x.x))"
  All_def       "All(P)    == (P = (%x.True))"
  Ex_def        "Ex(P)     == P(@x.P(x))"
  False_def     "False     == (!P.P)"
  not_def       "~ P       == P-->False"
  and_def       "P & Q     == !R. (P-->Q-->R) --> R"
  or_def        "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
  Ex1_def       "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
  Let_def       "Let(s, f) == f(s)"

  (* Axioms *)

  iff           "(P-->Q) --> (Q-->P) --> (P=Q)"
  True_or_False "(P=True) | (P=False)"

  (* Misc Definitions *)

  Inv_def       "Inv(f::'a=>'b)  == (% y. @x. f(x)=y)"
  o_def         "(f::'b=>'c) o g == (%(x::'a). f(g(x)))"
  if_def        "if(P,x,y) == @z::'a. (P=True --> z=x) & (P=False --> z=y)"

end


ML

(** Choice between the HOL and Isabelle style of quantifiers **)

val HOL_quantifiers = ref true;

fun alt_ast_tr' (name, alt_name) =
  let
    fun ast_tr' (*name*) args =
      if ! HOL_quantifiers then raise Match
      else Syntax.mk_appl (Syntax.Constant alt_name) args;
  in
    (name, ast_tr')
  end;


val print_ast_translation =
  map alt_ast_tr' [("! ", "*All"), ("? ", "*Ex"), ("?! ", "*Ex1")];