src/LCF/lcf.thy
changeset 0 a5a9c433f639
child 283 76caebd18756
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/LCF/lcf.thy	Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,107 @@
+(*  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