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