(* Title: HOL/HOL.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1993 University of Cambridge
Higher-Order Logic.
*)
HOL = CPure +
(** Core syntax **)
global
classes
term < logic
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 ("(if (_)/ then (_)/ else (_))" 10)
arbitrary :: 'a
(* Binders *)
Eps :: ('a => bool) => 'a
All :: ('a => bool) => bool (binder "! " 10)
Ex :: ('a => bool) => bool (binder "? " 10)
Ex1 :: ('a => bool) => bool (binder "?! " 10)
Let :: ['a, 'a => 'b] => 'b
(* Infixes *)
"=" :: ['a, 'a] => bool (infixl 50)
"&" :: [bool, bool] => bool (infixr 35)
"|" :: [bool, bool] => bool (infixr 30)
"-->" :: [bool, bool] => bool (infixr 25)
(* Overloaded Constants *)
axclass
plus < term
axclass
minus < term
axclass
times < term
axclass
power < term
consts
"+" :: ['a::plus, 'a] => 'a (infixl 65)
"-" :: ['a::minus, 'a] => 'a (infixl 65)
uminus :: ['a::minus] => 'a ("- _" [100] 100)
"*" :: ['a::times, 'a] => 'a (infixl 70)
(*See Nat.thy for "^"*)
(** Additional concrete syntax **)
nonterminals
letbinds letbind
case_syn cases_syn
syntax
"~=" :: ['a, 'a] => bool (infixl 50)
"@Eps" :: [pttrn, bool] => 'a ("(3@ _./ _)" [0, 10] 10)
(* Alternative Quantifiers *)
"*All" :: [idts, bool] => bool ("(3ALL _./ _)" [0, 10] 10)
"*Ex" :: [idts, bool] => bool ("(3EX _./ _)" [0, 10] 10)
"*Ex1" :: [idts, bool] => bool ("(3EX! _./ _)" [0, 10] 10)
(* Let expressions *)
"_bind" :: [pttrn, '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)"
"@ x. b" == "Eps (%x. b)"
"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)"
syntax ("" output)
"op =" :: ['a, 'a] => bool ("(_ =/ _)" [51, 51] 50)
"op ~=" :: ['a, 'a] => bool ("(_ ~=/ _)" [51, 51] 50)
syntax (symbols)
Not :: bool => bool ("\\<not> _" [40] 40)
"op &" :: [bool, bool] => bool (infixr "\\<and>" 35)
"op |" :: [bool, bool] => bool (infixr "\\<or>" 30)
"op -->" :: [bool, bool] => bool (infixr "\\<midarrow>\\<rightarrow>" 25)
"op o" :: ['b => 'c, 'a => 'b, 'a] => 'c (infixl "\\<circ>" 55)
"op ~=" :: ['a, 'a] => bool (infixl "\\<noteq>" 50)
"@Eps" :: [pttrn, bool] => 'a ("(3\\<epsilon>_./ _)" [0, 10] 10)
"! " :: [idts, bool] => bool ("(3\\<forall>_./ _)" [0, 10] 10)
"? " :: [idts, bool] => bool ("(3\\<exists>_./ _)" [0, 10] 10)
"?! " :: [idts, bool] => bool ("(3\\<exists>!_./ _)" [0, 10] 10)
"@case1" :: ['a, 'b] => case_syn ("(2_ \\<Rightarrow>/ _)" 10)
(*"@case2" :: [case_syn, cases_syn] => cases_syn ("_/ \\<orelse> _")*)
syntax (symbols output)
"op ~=" :: ['a, 'a] => bool ("(_ \\<noteq>/ _)" [51, 51] 50)
"*All" :: [idts, bool] => bool ("(3\\<forall>_./ _)" [0, 10] 10)
"*Ex" :: [idts, bool] => bool ("(3\\<exists>_./ _)" [0, 10] 10)
"*Ex1" :: [idts, bool] => bool ("(3\\<exists>!_./ _)" [0, 10] 10)
syntax (xsymbols)
"op -->" :: [bool, bool] => bool (infixr "\\<longrightarrow>" 25)
syntax (HTML output)
Not :: bool => bool ("\\<not> _" [40] 40)
(** Rules and definitions **)
local
rules
eq_reflection "(x=y) ==> (x==y)"
(* Basic Rules *)
refl "t = (t::'a)"
subst "[| s = t; P(s) |] ==> P(t::'a)"
(*Extensionality is built into the meta-logic, and this rule expresses
a related property. It is an eta-expanded version of the traditional
rule, and similar to the ABS rule of HOL.*)
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"
defs
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)"
rules
(* Axioms *)
iff "(P-->Q) --> (Q-->P) --> (P=Q)"
True_or_False "(P=True) | (P=False)"
defs
(*misc definitions*)
Let_def "Let s f == f(s)"
if_def "If P x y == @z::'a. (P=True --> z=x) & (P=False --> z=y)"
(*arbitrary is completely unspecified, but is made to appear as a
definition syntactically*)
arbitrary_def "False ==> arbitrary == (@x. False)"
(** initial HOL theory setup **)
setup Simplifier.setup
setup ClasetThyData.setup
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")];