--- a/src/LCF/LCF.thy Sat Sep 03 17:54:07 2005 +0200
+++ b/src/LCF/LCF.thy Sat Sep 03 17:54:10 2005 +0200
@@ -2,29 +2,38 @@
ID: $Id$
Author: Tobias Nipkow
Copyright 1992 University of Cambridge
-
-Natural Deduction Rules for LCF
*)
-LCF = FOL +
+header {* LCF on top of First-Order Logic *}
-classes cpo < term
+theory LCF
+imports FOL
+uses ("pair.ML") ("fix.ML")
+begin
-default cpo
+text {* This theory is based on Lawrence Paulson's book Logic and Computation. *}
-types
- tr
- void
- ('a,'b) "*" (infixl 6)
- ('a,'b) "+" (infixl 5)
+subsection {* Natural Deduction Rules for LCF *}
+
+classes cpo < "term"
+defaultsort cpo
+
+typedecl tr
+typedecl void
+typedecl ('a,'b) "*" (infixl 6)
+typedecl ('a,'b) "+" (infixl 5)
arities
- fun, "*", "+" :: (cpo,cpo)cpo
- tr,void :: cpo
+ fun :: (cpo, cpo) cpo
+ "*" :: (cpo, cpo) cpo
+ "+" :: (cpo, cpo) cpo
+ tr :: cpo
+ void :: cpo
consts
UU :: "'a"
- TT,FF :: "tr"
+ TT :: "tr"
+ FF :: "tr"
FIX :: "('a => 'a) => 'a"
FST :: "'a*'b => 'a"
SND :: "'a*'b => 'b"
@@ -36,75 +45,91 @@
PAIR :: "['a,'b] => 'a*'b" ("(1<_,/_>)" [0,0] 100)
COND :: "[tr,'a,'a] => 'a" ("(_ =>/ (_ |/ _))" [60,60,60] 60)
"<<" :: "['a,'a] => o" (infixl 50)
-rules
+
+axioms
(** DOMAIN THEORY **)
- eq_def "x=y == x << y & y << x"
+ eq_def: "x=y == x << y & y << x"
- less_trans "[| x << y; y << z |] ==> x << z"
+ less_trans: "[| x << y; y << z |] ==> x << z"
- less_ext "(ALL x. f(x) << g(x)) ==> f << g"
+ less_ext: "(ALL x. f(x) << g(x)) ==> f << g"
- mono "[| f << g; x << y |] ==> f(x) << g(y)"
+ mono: "[| f << g; x << y |] ==> f(x) << g(y)"
- minimal "UU << x"
+ minimal: "UU << x"
- FIX_eq "f(FIX(f)) = FIX(f)"
+ FIX_eq: "f(FIX(f)) = FIX(f)"
(** TR **)
- tr_cases "p=UU | p=TT | p=FF"
+ 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"
+ 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"
+ 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"
+ surj_pairing: "<FST(z),SND(z)> = z"
- FST "FST(<x,y>) = x"
- SND "SND(<x,y>) = y"
+ 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_DEF: "~x=UU ==> ~INL(x)=UU"
+ INR_DEF: "~x=UU ==> ~INR(x)=UU"
- INL_STRICT "INL(UU) = UU"
- INR_STRICT "INR(UU) = 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)"
+ 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
+ SUM_EXHAUSTION:
"z = UU | (EX x. ~x=UU & z = INL(x)) | (EX y. ~y=UU & z = INR(y))"
(** VOID **)
- void_cases "(x::void) = UU"
+ void_cases: "(x::void) = UU"
(** INDUCTION **)
- induct "[| adm(P); P(UU); ALL x. P(x) --> P(f(x)) |] ==> P(FIX(f))"
+ 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))"
+ 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))"
+
+ML {* use_legacy_bindings (the_context ()) *}
+
+use "LCF_lemmas.ML"
+
+
+subsection {* Ordered pairs and products *}
+
+use "pair.ML"
+
+
+subsection {* Fixedpoint theory *}
+
+use "fix.ML"
+
end