(* Title: HOL/hol.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1993 University of Cambridge
Higher-Order Logic
*)
HOL = Pure +
classes
term < logic
plus < term
minus < term
times < term
default term
types
bool 0
arities
fun :: (term, term) term
bool :: term
consts
(* Constants *)
Trueprop :: "bool => prop" ("(_)" [0] 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"
"@let" :: "[idt,'a,'b] => 'b" ("(let _ = (2_)/ in (2_))" 10)
(* Alternative Quantifiers *)
"*All" :: "[idts, bool] => bool" ("(3ALL _./ _)" [0,0] 10)
"*Ex" :: "[idts, bool] => bool" ("(3EX _./ _)" [0,0] 10)
"*Ex1" :: "[idts, bool] => bool" ("(3EX! _./ _)" [0,0] 10)
(* Infixes *)
o :: "['b => 'c, 'a => 'b, 'a] => 'c" (infixr 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)
(* Rewriting Gadget *)
NORM :: "'a => 'a"
translations
"ALL xs. P" => "! xs. P"
"EX xs. P" => "? xs. P"
"EX! xs. P" => "?! xs. P"
"let x = s in t" == "Let(s,%x.t)"
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.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 "not = (%P. P-->False)"
and_def "op & = (%P Q. !R. (P-->Q-->R) --> R)"
or_def "op | = (%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 "op o = (%(f::'b=>'c) g (x::'a). f(g(x)))"
if_def "if = (%P x y.@z::'a. (P=True --> z=x) & (P=False --> z=y))"
(* Rewriting -- special constant to flag normalized terms *)
NORM_def "NORM(x) = x"
end
ML
(** Choice between the HOL and Isabelle style of quantifiers **)
val HOL_quantifiers = ref true;
fun mk_alt_ast_tr' (name, alt_name) =
let
fun ast_tr' (*name*) args =
if ! HOL_quantifiers then raise Match
else Ast.mk_appl (Ast.Constant alt_name) args;
in
(name, ast_tr')
end;
val print_ast_translation =
map mk_alt_ast_tr' [("! ", "*All"), ("? ", "*Ex"), ("?! ", "*Ex1")];