--- a/src/HOL/MicroJava/BV/Err.thy Sat Mar 02 12:09:23 2002 +0100
+++ b/src/HOL/MicroJava/BV/Err.thy Sun Mar 03 16:59:08 2002 +0100
@@ -12,48 +12,48 @@
datatype 'a err = Err | OK 'a
-types 'a ebinop = "'a => 'a => 'a err"
+types 'a ebinop = "'a \<Rightarrow> 'a \<Rightarrow> 'a err"
'a esl = "'a set * 'a ord * 'a ebinop"
consts
- ok_val :: "'a err => 'a"
+ ok_val :: "'a err \<Rightarrow> 'a"
primrec
"ok_val (OK x) = x"
constdefs
- lift :: "('a => 'b err) => ('a err => 'b err)"
-"lift f e == case e of Err => Err | OK x => f x"
+ lift :: "('a \<Rightarrow> 'b err) \<Rightarrow> ('a err \<Rightarrow> 'b err)"
+"lift f e == case e of Err \<Rightarrow> Err | OK x \<Rightarrow> f x"
- lift2 :: "('a => 'b => 'c err) => 'a err => 'b err => 'c err"
+ lift2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c err) \<Rightarrow> 'a err \<Rightarrow> 'b err \<Rightarrow> 'c err"
"lift2 f e1 e2 ==
- case e1 of Err => Err
- | OK x => (case e2 of Err => Err | OK y => f x y)"
+ case e1 of Err \<Rightarrow> Err
+ | OK x \<Rightarrow> (case e2 of Err \<Rightarrow> Err | OK y \<Rightarrow> f x y)"
- le :: "'a ord => 'a err ord"
+ le :: "'a ord \<Rightarrow> 'a err ord"
"le r e1 e2 ==
- case e2 of Err => True |
- OK y => (case e1 of Err => False | OK x => x <=_r y)"
+ case e2 of Err \<Rightarrow> True |
+ OK y \<Rightarrow> (case e1 of Err \<Rightarrow> False | OK x \<Rightarrow> x <=_r y)"
- sup :: "('a => 'b => 'c) => ('a err => 'b err => 'c err)"
+ sup :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a err \<Rightarrow> 'b err \<Rightarrow> 'c err)"
"sup f == lift2(%x y. OK(x +_f y))"
- err :: "'a set => 'a err set"
+ err :: "'a set \<Rightarrow> 'a err set"
"err A == insert Err {x . ? y:A. x = OK y}"
- esl :: "'a sl => 'a esl"
+ esl :: "'a sl \<Rightarrow> 'a esl"
"esl == %(A,r,f). (A,r, %x y. OK(f x y))"
- sl :: "'a esl => 'a err sl"
+ sl :: "'a esl \<Rightarrow> 'a err sl"
"sl == %(A,r,f). (err A, le r, lift2 f)"
syntax
- err_semilat :: "'a esl => bool"
+ err_semilat :: "'a esl \<Rightarrow> bool"
translations
"err_semilat L" == "semilat(Err.sl L)"
consts
- strict :: "('a => 'b err) => ('a err => 'b err)"
+ strict :: "('a \<Rightarrow> 'b err) \<Rightarrow> ('a err \<Rightarrow> 'b err)"
primrec
"strict f Err = Err"
"strict f (OK x) = f x"
@@ -75,20 +75,20 @@
by (simp add: lesub_def)
lemma le_err_refl:
- "!x. x <=_r x ==> e <=_(Err.le r) e"
+ "!x. x <=_r x \<Longrightarrow> e <=_(Err.le r) e"
apply (unfold lesub_def Err.le_def)
apply (simp split: err.split)
done
lemma le_err_trans [rule_format]:
- "order r ==> e1 <=_(le r) e2 --> e2 <=_(le r) e3 --> e1 <=_(le r) e3"
+ "order r \<Longrightarrow> e1 <=_(le r) e2 \<longrightarrow> e2 <=_(le r) e3 \<longrightarrow> e1 <=_(le r) e3"
apply (unfold unfold_lesub_err le_def)
apply (simp split: err.split)
apply (blast intro: order_trans)
done
lemma le_err_antisym [rule_format]:
- "order r ==> e1 <=_(le r) e2 --> e2 <=_(le r) e1 --> e1=e2"
+ "order r \<Longrightarrow> e1 <=_(le r) e2 \<longrightarrow> e2 <=_(le r) e1 \<longrightarrow> e1=e2"
apply (unfold unfold_lesub_err le_def)
apply (simp split: err.split)
apply (blast intro: order_antisym)
@@ -136,20 +136,20 @@
by (simp add: lesssub_def lesub_def le_def split: err.split)
lemma semilat_errI:
- "semilat(A,r,f) ==> semilat(err A, Err.le r, lift2(%x y. OK(f x y)))"
+ "semilat(A,r,f) \<Longrightarrow> semilat(err A, Err.le r, lift2(%x y. OK(f x y)))"
apply (unfold semilat_Def closed_def plussub_def lesub_def
lift2_def Err.le_def err_def)
apply (simp split: err.split)
done
lemma err_semilat_eslI:
- "!!L. semilat L ==> err_semilat(esl L)"
+ "\<And>L. semilat L \<Longrightarrow> err_semilat(esl L)"
apply (unfold sl_def esl_def)
apply (simp (no_asm_simp) only: split_tupled_all)
apply (simp add: semilat_errI)
done
-lemma acc_err [simp, intro!]: "acc r ==> acc(le r)"
+lemma acc_err [simp, intro!]: "acc r \<Longrightarrow> acc(le r)"
apply (unfold acc_def lesub_def le_def lesssub_def)
apply (simp add: wf_eq_minimal split: err.split)
apply clarify
@@ -170,7 +170,7 @@
section {* lift *}
lemma lift_in_errI:
- "[| e : err S; !x:S. e = OK x --> f x : err S |] ==> lift f e : err S"
+ "\<lbrakk> e : err S; !x:S. e = OK x \<longrightarrow> f x : err S \<rbrakk> \<Longrightarrow> lift f e : err S"
apply (unfold lift_def)
apply (simp split: err.split)
apply blast
@@ -221,42 +221,42 @@
section {* semilat (err A) (le r) f *}
lemma semilat_le_err_Err_plus [simp]:
- "[| x: err A; semilat(err A, le r, f) |] ==> Err +_f x = Err"
+ "\<lbrakk> x: err A; semilat(err A, le r, f) \<rbrakk> \<Longrightarrow> Err +_f x = Err"
by (blast intro: le_iff_plus_unchanged [THEN iffD1] le_iff_plus_unchanged2 [THEN iffD1])
lemma semilat_le_err_plus_Err [simp]:
- "[| x: err A; semilat(err A, le r, f) |] ==> x +_f Err = Err"
+ "\<lbrakk> x: err A; semilat(err A, le r, f) \<rbrakk> \<Longrightarrow> x +_f Err = Err"
by (blast intro: le_iff_plus_unchanged [THEN iffD1] le_iff_plus_unchanged2 [THEN iffD1])
lemma semilat_le_err_OK1:
- "[| x:A; y:A; semilat(err A, le r, f); OK x +_f OK y = OK z |]
- ==> x <=_r z";
+ "\<lbrakk> x:A; y:A; semilat(err A, le r, f); OK x +_f OK y = OK z \<rbrakk>
+ \<Longrightarrow> x <=_r z";
apply (rule OK_le_err_OK [THEN iffD1])
apply (erule subst)
apply simp
done
lemma semilat_le_err_OK2:
- "[| x:A; y:A; semilat(err A, le r, f); OK x +_f OK y = OK z |]
- ==> y <=_r z"
+ "\<lbrakk> x:A; y:A; semilat(err A, le r, f); OK x +_f OK y = OK z \<rbrakk>
+ \<Longrightarrow> y <=_r z"
apply (rule OK_le_err_OK [THEN iffD1])
apply (erule subst)
apply simp
done
lemma eq_order_le:
- "[| x=y; order r |] ==> x <=_r y"
+ "\<lbrakk> x=y; order r \<rbrakk> \<Longrightarrow> x <=_r y"
apply (unfold order_def)
apply blast
done
lemma OK_plus_OK_eq_Err_conv [simp]:
- "[| x:A; y:A; semilat(err A, le r, fe) |] ==>
+ "\<lbrakk> x:A; y:A; semilat(err A, le r, fe) \<rbrakk> \<Longrightarrow>
((OK x) +_fe (OK y) = Err) = (~(? z:A. x <=_r z & y <=_r z))"
proof -
- have plus_le_conv3: "!!A x y z f r.
- [| semilat (A,r,f); x +_f y <=_r z; x:A; y:A; z:A |]
- ==> x <=_r z \<and> y <=_r z"
+ have plus_le_conv3: "\<And>A x y z f r.
+ \<lbrakk> semilat (A,r,f); x +_f y <=_r z; x:A; y:A; z:A \<rbrakk>
+ \<Longrightarrow> x <=_r z \<and> y <=_r z"
by (rule plus_le_conv [THEN iffD1])
case rule_context
thus ?thesis
@@ -287,13 +287,13 @@
(* FIXME? *)
lemma all_bex_swap_lemma [iff]:
- "(!x. (? y:A. x = f y) --> P x) = (!y:A. P(f y))"
+ "(!x. (? y:A. x = f y) \<longrightarrow> P x) = (!y:A. P(f y))"
by blast
lemma closed_err_Union_lift2I:
- "[| !A:AS. closed (err A) (lift2 f); AS ~= {};
- !A:AS.!B:AS. A~=B --> (!a:A.!b:B. a +_f b = Err) |]
- ==> closed (err(Union AS)) (lift2 f)"
+ "\<lbrakk> !A:AS. closed (err A) (lift2 f); AS ~= {};
+ !A:AS.!B:AS. A~=B \<longrightarrow> (!a:A.!b:B. a +_f b = Err) \<rbrakk>
+ \<Longrightarrow> closed (err(Union AS)) (lift2 f)"
apply (unfold closed_def err_def)
apply simp
apply clarify
@@ -307,9 +307,9 @@
which may not hold
*}
lemma err_semilat_UnionI:
- "[| !A:AS. err_semilat(A, r, f); AS ~= {};
- !A:AS.!B:AS. A~=B --> (!a:A.!b:B. ~ a <=_r b & a +_f b = Err) |]
- ==> err_semilat(Union AS, r, f)"
+ "\<lbrakk> !A:AS. err_semilat(A, r, f); AS ~= {};
+ !A:AS.!B:AS. A~=B \<longrightarrow> (!a:A.!b:B. ~ a <=_r b & a +_f b = Err) \<rbrakk>
+ \<Longrightarrow> err_semilat(Union AS, r, f)"
apply (unfold semilat_def sl_def)
apply (simp add: closed_err_Union_lift2I)
apply (rule conjI)