(* Title: HOL/HOL.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1993 University of Cambridge
Higher-Order Logic.
*)
theory HOL = CPure
files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML"):
(** Core syntax **)
global
classes "term" < logic
defaultsort "term"
typedecl bool
arities
bool :: "term"
fun :: ("term", "term") "term"
consts
(* Constants *)
Trueprop :: "bool => prop" ("(_)" 5)
Not :: "bool => bool" ("~ _" [40] 40)
True :: bool
False :: bool
If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10)
arbitrary :: 'a
(* Binders *)
Eps :: "('a => bool) => 'a"
All :: "('a => bool) => bool" (binder "ALL " 10)
Ex :: "('a => bool) => bool" (binder "EX " 10)
Ex1 :: "('a => bool) => bool" (binder "EX! " 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" ("- _" [81] 80)
* :: "['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" ("(3SOME _./ _)" [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)"
"SOME x. P" == "Eps (%x. 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)
"ALL " :: "[idts, bool] => bool" ("(3\\<forall>_./ _)" [0, 10] 10)
"EX " :: "[idts, bool] => bool" ("(3\\<exists>_./ _)" [0, 10] 10)
"EX! " :: "[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)
syntax (xsymbols)
"op -->" :: "[bool, bool] => bool" (infixr "\\<longrightarrow>" 25)
syntax (HTML output)
Not :: "bool => bool" ("\\<not> _" [40] 40)
syntax (HOL)
"_Eps" :: "[pttrn, bool] => 'a" ("(3@ _./ _)" [0, 10] 10)
"ALL " :: "[idts, bool] => bool" ("(3! _./ _)" [0, 10] 10)
"EX " :: "[idts, bool] => bool" ("(3? _./ _)" [0, 10] 10)
"EX! " :: "[idts, bool] => bool" ("(3?! _./ _)" [0, 10] 10)
(** Rules and definitions **)
local
axioms
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)"
axioms
(* 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)"
(* theory and package setup *)
use "HOL_lemmas.ML" setup attrib_setup
use "cladata.ML" setup Classical.setup setup clasetup
use "blastdata.ML" setup Blast.setup
use "simpdata.ML" setup Simplifier.setup setup iff_attrib_setup
setup simpsetup setup Clasimp.setup
end