src/HOL/HOL.thy
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(*  Title:      HOL/HOL.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1993  University of Cambridge
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Higher-Order Logic.
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*)
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HOL = CPure +
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(** Core syntax **)
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classes
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  term < logic
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default
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  term
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types
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  bool
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arities
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  fun :: (term, term) term
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  bool :: term
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syntax ("" output)
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  "op ="        :: ['a, 'a] => bool                 ("(_ =/ _)" [51, 51] 50)
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  "op ~="       :: ['a, 'a] => bool                 ("(_ ~=/ _)" [51, 51] 50)
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consts
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  (* Constants *)
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  Trueprop      :: bool => prop                     ("(_)" 5)
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  Not           :: bool => bool                     ("~ _" [40] 40)
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  True, False   :: bool
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  If            :: [bool, 'a, 'a] => 'a   ("(if (_)/ then (_)/ else (_))" 10)
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  (* Binders *)
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  Eps           :: ('a => bool) => 'a
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  All           :: ('a => bool) => bool             (binder "! " 10)
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  Ex            :: ('a => bool) => bool             (binder "? " 10)
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  Ex1           :: ('a => bool) => bool             (binder "?! " 10)
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  Let           :: ['a, 'a => 'b] => 'b
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  (* Infixes *)
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  o             :: ['b => 'c, 'a => 'b, 'a] => 'c   (infixl 55)
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  "="           :: ['a, 'a] => bool                 (infixl 50)
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  "&"           :: [bool, bool] => bool             (infixr 35)
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  "|"           :: [bool, bool] => bool             (infixr 30)
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  "-->"         :: [bool, bool] => bool             (infixr 25)
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(* Overloaded Constants *)
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axclass
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  plus < term
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axclass
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  minus < term
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  times < term
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consts
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  "+"           :: ['a::plus, 'a] => 'a             (infixl 65)
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  "-"           :: ['a::minus, 'a] => 'a            (infixl 65)
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  "*"           :: ['a::times, 'a] => 'a            (infixl 70)
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(** Additional concrete syntax **)
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types
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  letbinds  letbind
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  case_syn  cases_syn
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syntax
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  "~="          :: ['a, 'a] => bool                 (infixl 50)
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  "@Eps"        :: [pttrn, bool] => 'a              ("(3@ _./ _)" [0, 10] 10)
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  (* Alternative Quantifiers *)
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  "*All"        :: [idts, bool] => bool             ("(3ALL _./ _)" [0, 10] 10)
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  "*Ex"         :: [idts, bool] => bool             ("(3EX _./ _)" [0, 10] 10)
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  "*Ex1"        :: [idts, bool] => bool             ("(3EX! _./ _)" [0, 10] 10)
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  (* Let expressions *)
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  "_bind"       :: [pttrn, 'a] => letbind           ("(2_ =/ _)" 10)
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  ""            :: letbind => letbinds              ("_")
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  "_binds"      :: [letbind, letbinds] => letbinds  ("_;/ _")
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  "_Let"        :: [letbinds, 'a] => 'a             ("(let (_)/ in (_))" 10)
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  (* Case expressions *)
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  "@case"       :: ['a, cases_syn] => 'b            ("(case _ of/ _)" 10)
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  "@case1"      :: ['a, 'b] => case_syn             ("(2_ =>/ _)" 10)
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  ""            :: case_syn => cases_syn            ("_")
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  "@case2"      :: [case_syn, cases_syn] => cases_syn   ("_/ | _")
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translations
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  "x ~= y"      == "~ (x = y)"
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  "@ x.b"       == "Eps (%x. b)"
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  "ALL xs. P"   => "! xs. P"
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  "EX xs. P"    => "? xs. P"
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  "EX! xs. P"   => "?! xs. P"
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  "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
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  "let x = a in e"        == "Let a (%x. e)"
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syntax (symbols output)
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  "op ~="       :: ['a, 'a] => bool                 ("(_ \\<noteq>/ _)" [51, 51] 50)
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syntax (symbols)
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  Not           :: bool => bool                     ("\\<not> _" [40] 40)
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  "op &"        :: [bool, bool] => bool             (infixr "\\<and>" 35)
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  "op |"        :: [bool, bool] => bool             (infixr "\\<or>" 30)
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  "op -->"      :: [bool, bool] => bool             (infixr "\\<midarrow>\\<rightarrow>" 25)
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  "op o"        :: ['b => 'c, 'a => 'b, 'a] => 'c   (infixl "\\<circ>" 55)
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  "op ~="       :: ['a, 'a] => bool                 (infixl "\\<noteq>" 50)
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  "@Eps"        :: [pttrn, bool] => 'a              ("(3\\<epsilon>_./ _)" [0, 10] 10)
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  "! "          :: [idts, bool] => bool             ("(3\\<forall>_./ _)" [0, 10] 10)
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  "? "          :: [idts, bool] => bool             ("(3\\<exists>_./ _)" [0, 10] 10)
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  "?! "         :: [idts, bool] => bool             ("(3\\<exists>!_./ _)" [0, 10] 10)
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  "@case1"      :: ['a, 'b] => case_syn             ("(2_ \\<Rightarrow>/ _)" 10)
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(*"@case2"      :: [case_syn, cases_syn] => cases_syn   ("_/ \\<orelse> _")*)
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syntax (symbols output)
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  "*All"        :: [idts, bool] => bool             ("(3\\<forall>_./ _)" [0, 10] 10)
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  "*Ex"         :: [idts, bool] => bool             ("(3\\<exists>_./ _)" [0, 10] 10)
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  "*Ex1"        :: [idts, bool] => bool             ("(3\\<exists>!_./ _)" [0, 10] 10)
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(** Rules and definitions **)
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rules
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  eq_reflection "(x=y) ==> (x==y)"
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  (* Basic Rules *)
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  refl          "t = (t::'a)"
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  subst         "[| s = t; P(s) |] ==> P(t::'a)"
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  ext           "(!!x::'a. (f(x)::'b) = g(x)) ==> (%x.f(x)) = (%x.g(x))"
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  selectI       "P(x::'a) ==> P(@x.P(x))"
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  impI          "(P ==> Q) ==> P-->Q"
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  mp            "[| P-->Q;  P |] ==> Q"
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defs
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  True_def      "True      == ((%x::bool.x)=(%x.x))"
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  All_def       "All(P)    == (P = (%x.True))"
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  Ex_def        "Ex(P)     == P(@x.P(x))"
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  False_def     "False     == (!P.P)"
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  not_def       "~ P       == P-->False"
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  and_def       "P & Q     == !R. (P-->Q-->R) --> R"
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  or_def        "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
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  Ex1_def       "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
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rules
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  (* Axioms *)
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  iff           "(P-->Q) --> (Q-->P) --> (P=Q)"
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  True_or_False "(P=True) | (P=False)"
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defs
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  (* Misc Definitions *)
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  Let_def       "Let s f == f(s)"
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  o_def         "(f::'b=>'c) o g == (%(x::'a). f(g(x)))"
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  if_def        "If P x y == @z::'a. (P=True --> z=x) & (P=False --> z=y)"
923
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end
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b59781f2b809 added symbols syntax;
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923
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ML
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(** Choice between the HOL and Isabelle style of quantifiers **)
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val HOL_quantifiers = ref true;
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fun alt_ast_tr' (name, alt_name) =
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  let
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    fun ast_tr' (*name*) args =
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      if ! HOL_quantifiers then raise Match
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      else Syntax.mk_appl (Syntax.Constant alt_name) args;
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  in
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    (name, ast_tr')
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  end;
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val print_ast_translation =
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  map alt_ast_tr' [("! ", "*All"), ("? ", "*Ex"), ("?! ", "*Ex1")];