src/LCF/LCF.thy
 changeset 47025 b2b8ae61d6ad parent 41310 65631ca437c9 child 48475 02dd825f5a4e
```--- a/src/LCF/LCF.thy	Mon Mar 19 21:25:15 2012 +0100
+++ b/src/LCF/LCF.thy	Mon Mar 19 21:49:52 2012 +0100
@@ -44,77 +44,83 @@
COND   :: "[tr,'a,'a] => 'a"   ("(_ =>/ (_ |/ _))" [60,60,60] 60)
less   :: "['a,'a] => o"       (infixl "<<" 50)

-axioms
+axiomatization where
(** DOMAIN THEORY **)

-  eq_def:        "x=y == x << y & y << x"
+  eq_def:        "x=y == x << y & y << x" and

-  less_trans:    "[| x << y; y << z |] ==> x << z"
+  less_trans:    "[| x << y; y << z |] ==> x << z" and

-  less_ext:      "(ALL x. f(x) << g(x)) ==> f << g"
+  less_ext:      "(ALL x. f(x) << g(x)) ==> f << g" and

-  mono:          "[| f << g; x << y |] ==> f(x) << g(y)"
+  mono:          "[| f << g; x << y |] ==> f(x) << g(y)" and
+
+  minimal:       "UU << x" and

-  minimal:       "UU << x"
+  FIX_eq:        "\<And>f. f(FIX(f)) = FIX(f)"

-  FIX_eq:        "f(FIX(f)) = FIX(f)"
-
+axiomatization where
(** TR **)

-  tr_cases:      "p=UU | p=TT | p=FF"
+  tr_cases:      "p=UU | p=TT | p=FF" and

-  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" and
+  not_FF_less_TT: "~ FF << TT" and
+  not_TT_less_UU: "~ TT << UU" and
+  not_FF_less_UU: "~ FF << UU" and

-  COND_UU:       "UU => x | y  =  UU"
-  COND_TT:       "TT => x | y  =  x"
+  COND_UU:       "UU => x | y  =  UU" and
+  COND_TT:       "TT => x | y  =  x" and
COND_FF:       "FF => x | y  =  y"

+axiomatization where
(** PAIRS **)

-  surj_pairing:  "<FST(z),SND(z)> = z"
+  surj_pairing:  "<FST(z),SND(z)> = z" and

-  FST:   "FST(<x,y>) = x"
+  FST:   "FST(<x,y>) = x" and
SND:   "SND(<x,y>) = y"

+axiomatization where
(*** STRICT SUM ***)

-  INL_DEF: "~x=UU ==> ~INL(x)=UU"
-  INR_DEF: "~x=UU ==> ~INR(x)=UU"
+  INL_DEF: "~x=UU ==> ~INL(x)=UU" and
+  INR_DEF: "~x=UU ==> ~INR(x)=UU" and

-  INL_STRICT: "INL(UU) = UU"
-  INR_STRICT: "INR(UU) = UU"
+  INL_STRICT: "INL(UU) = UU" and
+  INR_STRICT: "INR(UU) = UU" and

-  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" and
+  WHEN_INL: "~x=UU ==> WHEN(f,g,INL(x)) = f(x)" and
+  WHEN_INR: "~x=UU ==> WHEN(f,g,INR(x)) = g(x)" and

SUM_EXHAUSTION:
"z = UU | (EX x. ~x=UU & z = INL(x)) | (EX y. ~y=UU & z = INR(y))"

+axiomatization where
(** VOID **)

void_cases:    "(x::void) = UU"

(** INDUCTION **)

+axiomatization where
induct:        "[| adm(P); P(UU); ALL x. P(x) --> P(f(x)) |] ==> P(FIX(f))"

+axiomatization where
(** 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. *)