--- a/src/HOLCF/Tr.thy Wed Nov 02 23:59:49 2005 +0100
+++ b/src/HOLCF/Tr.thy Thu Nov 03 00:12:29 2005 +0100
@@ -20,22 +20,24 @@
"tr" <= (type) "bool lift"
consts
- TT :: "tr"
- FF :: "tr"
- Icifte :: "tr -> 'c -> 'c -> 'c"
- trand :: "tr -> tr -> tr"
- tror :: "tr -> tr -> tr"
- neg :: "tr -> tr"
- If2 :: "tr=>'c=>'c=>'c"
+ TT :: "tr"
+ FF :: "tr"
+ Icifte :: "tr \<rightarrow> 'c \<rightarrow> 'c \<rightarrow> 'c"
+ trand :: "tr \<rightarrow> tr \<rightarrow> tr"
+ tror :: "tr \<rightarrow> tr \<rightarrow> tr"
+ neg :: "tr \<rightarrow> tr"
+ If2 :: "[tr, 'c, 'c] \<Rightarrow> 'c"
-syntax "@cifte" :: "tr=>'c=>'c=>'c" ("(3If _/ (then _/ else _) fi)" 60)
- "@andalso" :: "tr => tr => tr" ("_ andalso _" [36,35] 35)
- "@orelse" :: "tr => tr => tr" ("_ orelse _" [31,30] 30)
+syntax
+ "@cifte" :: "[tr, 'c, 'c] \<Rightarrow> 'c" ("(3If _/ (then _/ else _) fi)" 60)
+ "@andalso" :: "tr \<Rightarrow> tr \<Rightarrow> tr" ("_ andalso _" [36,35] 35)
+ "@orelse" :: "tr \<Rightarrow> tr \<Rightarrow> tr" ("_ orelse _" [31,30] 30)
-translations
- "x andalso y" == "trand$x$y"
- "x orelse y" == "tror$x$y"
- "If b then e1 else e2 fi" == "Icifte$b$e1$e2"
+translations
+ "x andalso y" == "trand\<cdot>x\<cdot>y"
+ "x orelse y" == "tror\<cdot>x\<cdot>y"
+ "If b then e1 else e2 fi" == "Icifte\<cdot>b\<cdot>e1\<cdot>e2"
+
defs
TT_def: "TT==Def True"
FF_def: "FF==Def False"
@@ -47,14 +49,14 @@
text {* Exhaustion and Elimination for type @{typ tr} *}
-lemma Exh_tr: "t=UU | t = TT | t = FF"
+lemma Exh_tr: "t = \<bottom> \<or> t = TT \<or> t = FF"
apply (unfold FF_def TT_def)
apply (induct_tac "t")
apply fast
apply fast
done
-lemma trE: "[| p=UU ==> Q; p = TT ==>Q; p = FF ==>Q|] ==>Q"
+lemma trE: "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = TT \<Longrightarrow> Q; p = FF \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
apply (rule Exh_tr [THEN disjE])
apply fast
apply (erule disjE)
@@ -76,16 +78,18 @@
*)
text {* distinctness for type @{typ tr} *}
-lemma dist_less_tr [simp]: "~TT << UU" "~FF << UU" "~TT << FF" "~FF << TT"
+lemma dist_less_tr [simp]:
+ "\<not> TT \<sqsubseteq> \<bottom>" "\<not> FF \<sqsubseteq> \<bottom>" "\<not> TT \<sqsubseteq> FF" "\<not> FF \<sqsubseteq> TT"
by (simp_all add: tr_defs)
-lemma dist_eq_tr [simp]: "TT~=UU" "FF~=UU" "TT~=FF" "UU~=TT" "UU~=FF" "FF~=TT"
+lemma dist_eq_tr [simp]:
+ "TT \<noteq> \<bottom>" "FF \<noteq> \<bottom>" "TT \<noteq> FF" "\<bottom> \<noteq> TT" "\<bottom> \<noteq> FF" "FF \<noteq> TT"
by (simp_all add: tr_defs)
text {* lemmas about andalso, orelse, neg and if *}
lemma ifte_thms [simp]:
- "If UU then e1 else e2 fi = UU"
+ "If \<bottom> then e1 else e2 fi = \<bottom>"
"If FF then e1 else e2 fi = e2"
"If TT then e1 else e2 fi = e1"
by (simp_all add: ifte_def TT_def FF_def)
@@ -93,7 +97,7 @@
lemma andalso_thms [simp]:
"(TT andalso y) = y"
"(FF andalso y) = FF"
- "(UU andalso y) = UU"
+ "(\<bottom> andalso y) = \<bottom>"
"(y andalso TT) = y"
"(y andalso y) = y"
apply (unfold andalso_def, simp_all)
@@ -104,7 +108,7 @@
lemma orelse_thms [simp]:
"(TT orelse y) = TT"
"(FF orelse y) = y"
- "(UU orelse y) = UU"
+ "(\<bottom> orelse y) = \<bottom>"
"(y orelse FF) = y"
"(y orelse y) = y"
apply (unfold orelse_def, simp_all)
@@ -113,15 +117,15 @@
done
lemma neg_thms [simp]:
- "neg$TT = FF"
- "neg$FF = TT"
- "neg$UU = UU"
+ "neg\<cdot>TT = FF"
+ "neg\<cdot>FF = TT"
+ "neg\<cdot>\<bottom> = \<bottom>"
by (simp_all add: neg_def TT_def FF_def)
text {* split-tac for If via If2 because the constant has to be a constant *}
lemma split_If2:
- "P (If2 Q x y ) = ((Q=UU --> P UU) & (Q=TT --> P x) & (Q=FF --> P y))"
+ "P (If2 Q x y) = ((Q = \<bottom> \<longrightarrow> P \<bottom>) \<and> (Q = TT \<longrightarrow> P x) \<and> (Q = FF \<longrightarrow> P y))"
apply (unfold If2_def)
apply (rule_tac p = "Q" in trE)
apply (simp_all)
@@ -136,58 +140,41 @@
subsection "Rewriting of HOLCF operations to HOL functions"
lemma andalso_or:
-"!!t.[|t~=UU|]==> ((t andalso s)=FF)=(t=FF | s=FF)"
+ "t \<noteq> \<bottom> \<Longrightarrow> ((t andalso s) = FF) = (t = FF \<or> s = FF)"
apply (rule_tac p = "t" in trE)
apply simp_all
done
-lemma andalso_and: "[|t~=UU|]==> ((t andalso s)~=FF)=(t~=FF & s~=FF)"
+lemma andalso_and:
+ "t \<noteq> \<bottom> \<Longrightarrow> ((t andalso s) \<noteq> FF) = (t \<noteq> FF \<and> s \<noteq> FF)"
apply (rule_tac p = "t" in trE)
apply simp_all
done
-lemma Def_bool1 [simp]: "(Def x ~= FF) = x"
+lemma Def_bool1 [simp]: "(Def x \<noteq> FF) = x"
by (simp add: FF_def)
-lemma Def_bool2 [simp]: "(Def x = FF) = (~x)"
+lemma Def_bool2 [simp]: "(Def x = FF) = (\<not> x)"
by (simp add: FF_def)
lemma Def_bool3 [simp]: "(Def x = TT) = x"
by (simp add: TT_def)
-lemma Def_bool4 [simp]: "(Def x ~= TT) = (~x)"
+lemma Def_bool4 [simp]: "(Def x \<noteq> TT) = (\<not> x)"
by (simp add: TT_def)
lemma If_and_if:
- "(If Def P then A else B fi)= (if P then A else B)"
+ "(If Def P then A else B fi) = (if P then A else B)"
apply (rule_tac p = "Def P" in trE)
apply (auto simp add: TT_def[symmetric] FF_def[symmetric])
done
-subsection "admissibility"
-
-text {*
- The following rewrite rules for admissibility should in the future be
- replaced by a more general admissibility test that also checks
- chain-finiteness, of which these lemmata are specific examples
-*}
-
-lemma adm_trick_1: "(x~=FF) = (x=TT|x=UU)"
-apply (rule_tac p = "x" in trE)
-apply (simp_all)
-done
+subsection {* Compactness *}
-lemma adm_trick_2: "(x~=TT) = (x=FF|x=UU)"
-apply (rule_tac p = "x" in trE)
-apply (simp_all)
-done
+lemma compact_TT [simp]: "compact TT"
+by (rule compact_chfin)
-lemmas adm_tricks = adm_trick_1 adm_trick_2
-
-lemma adm_nTT [simp]: "cont(f) ==> adm (%x. (f x)~=TT)"
-by (simp add: adm_tricks)
-
-lemma adm_nFF [simp]: "cont(f) ==> adm (%x. (f x)~=FF)"
-by (simp add: adm_tricks)
+lemma compact_FF [simp]: "compact FF"
+by (rule compact_chfin)
end