(* Title: LCF/lcf.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1992 University of Cambridge
Natural Deduction Rules for LCF
*)
LCF = FOL +
classes cpo < term
default cpo
types tr,void 0
"*" 2 (infixl 6)
"+" 2 (infixl 5)
arities fun, "*", "+" :: (cpo,cpo)cpo
tr,void :: cpo
consts
UU :: "'a"
TT,FF :: "tr"
FIX :: "('a => 'a) => 'a"
FST :: "'a*'b => 'a"
SND :: "'a*'b => 'b"
INL :: "'a => 'a+'b"
INR :: "'b => 'a+'b"
WHEN :: "['a=>'c, 'b=>'c, 'a+'b] => 'c"
adm :: "('a => o) => o"
VOID :: "void" ("()")
PAIR :: "['a,'b] => 'a*'b" ("(1<_,/_>)" [0,0] 100)
COND :: "[tr,'a,'a] => 'a" ("(_ =>/ (_ |/ _))" [60,60,60] 60)
"<<" :: "['a,'a] => o" (infixl 50)
rules
(** DOMAIN THEORY **)
eq_def "x=y == x << y & y << x"
less_trans "[| x << y; y << z |] ==> x << z"
less_ext "(ALL x. f(x) << g(x)) ==> f << g"
mono "[| f << g; x << y |] ==> f(x) << g(y)"
minimal "UU << x"
FIX_eq "f(FIX(f)) = FIX(f)"
(** TR **)
tr_cases "p=UU | p=TT | p=FF"
not_TT_less_FF "~ TT << FF"
not_FF_less_TT "~ FF << TT"
not_TT_less_UU "~ TT << UU"
not_FF_less_UU "~ FF << UU"
COND_UU "UU => x | y = UU"
COND_TT "TT => x | y = x"
COND_FF "FF => x | y = y"
(** PAIRS **)
surj_pairing "<FST(z),SND(z)> = z"
FST "FST(<x,y>) = x"
SND "SND(<x,y>) = y"
(*** STRICT SUM ***)
INL_DEF "~x=UU ==> ~INL(x)=UU"
INR_DEF "~x=UU ==> ~INR(x)=UU"
INL_STRICT "INL(UU) = UU"
INR_STRICT "INR(UU) = UU"
WHEN_UU "WHEN(f,g,UU) = UU"
WHEN_INL "~x=UU ==> WHEN(f,g,INL(x)) = f(x)"
WHEN_INR "~x=UU ==> WHEN(f,g,INR(x)) = g(x)"
SUM_EXHAUSTION
"z = UU | (EX x. ~x=UU & z = INL(x)) | (EX y. ~y=UU & z = INR(y))"
(** VOID **)
void_cases "(x::void) = UU"
(** INDUCTION **)
induct "[| adm(P); P(UU); ALL x. P(x) --> P(f(x)) |] ==> P(FIX(f))"
(** Admissibility / Chain Completeness **)
(* All rules can be found on pages 199--200 of Larry's LCF book.
Note that "easiness" of types is not taken into account
because it cannot be expressed schematically; flatness could be. *)
adm_less "adm(%x.t(x) << u(x))"
adm_not_less "adm(%x.~ t(x) << u)"
adm_not_free "adm(%x.A)"
adm_subst "adm(P) ==> adm(%x.P(t(x)))"
adm_conj "[| adm(P); adm(Q) |] ==> adm(%x.P(x)&Q(x))"
adm_disj "[| adm(P); adm(Q) |] ==> adm(%x.P(x)|Q(x))"
adm_imp "[| adm(%x.~P(x)); adm(Q) |] ==> adm(%x.P(x)-->Q(x))"
adm_all "(!!y.adm(P(y))) ==> adm(%x.ALL y.P(y,x))"
end