src/LCF/lcf.thy
author clasohm
Thu Sep 16 12:20:38 1993 +0200 (1993-09-16)
changeset 0 a5a9c433f639
child 283 76caebd18756
permissions -rw-r--r--
Initial revision
     1 (*  Title: 	LCF/lcf.thy
     2     ID:         $Id$
     3     Author: 	Tobias Nipkow
     4     Copyright   1992  University of Cambridge
     5 
     6 Natural Deduction Rules for LCF
     7 *)
     8 
     9 LCF = FOL +
    10 
    11 classes cpo < term
    12 
    13 default cpo
    14 
    15 types tr,void 0
    16       "*" 2 (infixl 6)
    17       "+" 2 (infixl 5)
    18 
    19 arities fun, "*", "+" :: (cpo,cpo)cpo
    20         tr,void :: cpo
    21 
    22 consts
    23  UU	:: "'a"
    24  TT,FF	:: "tr"
    25  FIX	:: "('a => 'a) => 'a"
    26  FST	:: "'a*'b => 'a"
    27  SND	:: "'a*'b => 'b"
    28  INL    :: "'a => 'a+'b"
    29  INR    :: "'b => 'a+'b"
    30  WHEN   :: "['a=>'c, 'b=>'c, 'a+'b] => 'c"
    31  adm	:: "('a => o) => o"
    32  VOID	:: "void"		("()")
    33  PAIR	:: "['a,'b] => 'a*'b"	("(1<_,/_>)" [0,0] 100)
    34  COND	:: "[tr,'a,'a] => 'a"	("(_ =>/ (_ |/ _))" [60,60,60] 60)
    35  "<<"	:: "['a,'a] => o"	(infixl 50)
    36 rules
    37   (** DOMAIN THEORY **)
    38 
    39   eq_def	"x=y == x << y & y << x"
    40 
    41   less_trans	"[| x << y; y << z |] ==> x << z"
    42 
    43   less_ext	"(ALL x. f(x) << g(x)) ==> f << g"
    44 
    45   mono		"[| f << g; x << y |] ==> f(x) << g(y)"
    46 
    47   minimal	"UU << x"
    48 
    49   FIX_eq	"f(FIX(f)) = FIX(f)"
    50 
    51   (** TR **)
    52 
    53   tr_cases	"p=UU | p=TT | p=FF"
    54 
    55   not_TT_less_FF "~ TT << FF"
    56   not_FF_less_TT "~ FF << TT"
    57   not_TT_less_UU "~ TT << UU"
    58   not_FF_less_UU "~ FF << UU"
    59 
    60   COND_UU	"UU => x | y  =  UU"
    61   COND_TT	"TT => x | y  =  x"
    62   COND_FF	"FF => x | y  =  y"
    63 
    64   (** PAIRS **)
    65 
    66   surj_pairing	"<FST(z),SND(z)> = z"
    67 
    68   FST	"FST(<x,y>) = x"
    69   SND	"SND(<x,y>) = y"
    70 
    71   (*** STRICT SUM ***)
    72 
    73   INL_DEF "~x=UU ==> ~INL(x)=UU"
    74   INR_DEF "~x=UU ==> ~INR(x)=UU"
    75 
    76   INL_STRICT "INL(UU) = UU"
    77   INR_STRICT "INR(UU) = UU"
    78 
    79   WHEN_UU  "WHEN(f,g,UU) = UU"
    80   WHEN_INL "~x=UU ==> WHEN(f,g,INL(x)) = f(x)"
    81   WHEN_INR "~x=UU ==> WHEN(f,g,INR(x)) = g(x)"
    82 
    83   SUM_EXHAUSTION
    84     "z = UU | (EX x. ~x=UU & z = INL(x)) | (EX y. ~y=UU & z = INR(y))"
    85 
    86   (** VOID **)
    87 
    88   void_cases	"(x::void) = UU"
    89 
    90   (** INDUCTION **)
    91 
    92   induct	"[| adm(P); P(UU); ALL x. P(x) --> P(f(x)) |] ==> P(FIX(f))"
    93 
    94   (** Admissibility / Chain Completeness **)
    95   (* All rules can be found on pages 199--200 of Larry's LCF book.
    96      Note that "easiness" of types is not taken into account
    97      because it cannot be expressed schematically; flatness could be. *)
    98 
    99   adm_less	"adm(%x.t(x) << u(x))"
   100   adm_not_less	"adm(%x.~ t(x) << u)"
   101   adm_not_free  "adm(%x.A)"
   102   adm_subst	"adm(P) ==> adm(%x.P(t(x)))"
   103   adm_conj	"[| adm(P); adm(Q) |] ==> adm(%x.P(x)&Q(x))"
   104   adm_disj	"[| adm(P); adm(Q) |] ==> adm(%x.P(x)|Q(x))"
   105   adm_imp	"[| adm(%x.~P(x)); adm(Q) |] ==> adm(%x.P(x)-->Q(x))"
   106   adm_all	"(!!y.adm(P(y))) ==> adm(%x.ALL y.P(y,x))"
   107 end