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theory Err = Semilat:(* Title: HOL/MicroJava/BV/Err.thy
ID: $Id: Err.html,v 1.1 2002/11/28 16:11:18 kleing Exp $
Author: Tobias Nipkow
Copyright 2000 TUM
The error type
*)
header {* \isaheader{The Error Type} *}
theory Err = Semilat:
datatype 'a err = Err | OK 'a
types 'a ebinop = "'a \<Rightarrow> 'a \<Rightarrow> 'a err"
'a esl = "'a set * 'a ord * 'a ebinop"
consts
ok_val :: "'a err \<Rightarrow> 'a"
primrec
"ok_val (OK x) = x"
constdefs
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 \<Rightarrow> 'b \<Rightarrow> 'c err) \<Rightarrow> 'a err \<Rightarrow> 'b err \<Rightarrow> 'c err"
"lift2 f e1 e2 ==
case e1 of Err \<Rightarrow> Err
| OK x \<Rightarrow> (case e2 of Err \<Rightarrow> Err | OK y \<Rightarrow> f x y)"
le :: "'a ord \<Rightarrow> 'a err ord"
"le r e1 e2 ==
case e2 of Err \<Rightarrow> True |
OK y \<Rightarrow> (case e1 of Err \<Rightarrow> False | OK x \<Rightarrow> x <=_r y)"
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 \<Rightarrow> 'a err set"
"err A == insert Err {x . ? y:A. x = OK y}"
esl :: "'a sl \<Rightarrow> 'a esl"
"esl == %(A,r,f). (A,r, %x y. OK(f x y))"
sl :: "'a esl \<Rightarrow> 'a err sl"
"sl == %(A,r,f). (err A, le r, lift2 f)"
syntax
err_semilat :: "'a esl \<Rightarrow> bool"
translations
"err_semilat L" == "semilat(Err.sl L)"
consts
strict :: "('a \<Rightarrow> 'b err) \<Rightarrow> ('a err \<Rightarrow> 'b err)"
primrec
"strict f Err = Err"
"strict f (OK x) = f x"
lemma strict_Some [simp]:
"(strict f x = OK y) = (\<exists> z. x = OK z \<and> f z = OK y)"
by (cases x, auto)
lemma not_Err_eq:
"(x \<noteq> Err) = (\<exists>a. x = OK a)"
by (cases x) auto
lemma not_OK_eq:
"(\<forall>y. x \<noteq> OK y) = (x = Err)"
by (cases x) auto
lemma unfold_lesub_err:
"e1 <=_(le r) e2 == le r e1 e2"
by (simp add: lesub_def)
lemma le_err_refl:
"!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 \<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 \<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)
done
lemma OK_le_err_OK:
"(OK x <=_(le r) OK y) = (x <=_r y)"
by (simp add: unfold_lesub_err le_def)
lemma order_le_err [iff]:
"order(le r) = order r"
apply (rule iffI)
apply (subst order_def)
apply (blast dest: order_antisym OK_le_err_OK [THEN iffD2]
intro: order_trans OK_le_err_OK [THEN iffD1])
apply (subst order_def)
apply (blast intro: le_err_refl le_err_trans le_err_antisym
dest: order_refl)
done
lemma le_Err [iff]: "e <=_(le r) Err"
by (simp add: unfold_lesub_err le_def)
lemma Err_le_conv [iff]:
"Err <=_(le r) e = (e = Err)"
by (simp add: unfold_lesub_err le_def split: err.split)
lemma le_OK_conv [iff]:
"e <=_(le r) OK x = (? y. e = OK y & y <=_r x)"
by (simp add: unfold_lesub_err le_def split: err.split)
lemma OK_le_conv:
"OK x <=_(le r) e = (e = Err | (? y. e = OK y & x <=_r y))"
by (simp add: unfold_lesub_err le_def split: err.split)
lemma top_Err [iff]: "top (le r) Err";
by (simp add: top_def)
lemma OK_less_conv [rule_format, iff]:
"OK x <_(le r) e = (e=Err | (? y. e = OK y & x <_r y))"
by (simp add: lesssub_def lesub_def le_def split: err.split)
lemma not_Err_less [rule_format, iff]:
"~(Err <_(le r) x)"
by (simp add: lesssub_def lesub_def le_def split: err.split)
lemma semilat_errI [intro]: includes semilat
shows "semilat(err A, Err.le r, lift2(%x y. OK(f x y)))"
apply(insert semilat)
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_aux:
includes semilat shows "err_semilat(esl(A,r,f))"
apply (unfold sl_def esl_def)
apply (simp add: semilat_errI[OF semilat])
done
lemma err_semilat_eslI [intro, simp]:
"\<And>L. semilat L \<Longrightarrow> err_semilat(esl L)"
by(simp add: err_semilat_eslI_aux split_tupled_all)
lemma Err_in_err [iff]: "Err : err A"
by (simp add: err_def)
lemma Ok_in_err [iff]: "(OK x : err A) = (x:A)"
by (auto simp add: err_def)
lemma OK_subset_err [iff]: "(OK ` A \<subseteq> err B) = (A \<subseteq> B)"
by (unfold err_def) blast
lemma acc_err [simp, intro!]: "acc r A \<Longrightarrow> acc (le r) (err A)"
apply (unfold acc_def lesub_def le_def lesssub_def)
apply (simp add: wf_eq_minimal split: err.split)
apply clarify
apply (case_tac "Err : Q")
apply blast
apply (erule_tac x = "{a . OK a : Q}" in allE)
apply (case_tac "x")
apply fast
apply blast
done
section {* lift *}
lemma lift_in_errI:
"\<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
done
lemma Err_lift2 [simp]:
"Err +_(lift2 f) x = Err"
by (simp add: lift2_def plussub_def)
lemma lift2_Err [simp]:
"x +_(lift2 f) Err = Err"
by (simp add: lift2_def plussub_def split: err.split)
lemma OK_lift2_OK [simp]:
"OK x +_(lift2 f) OK y = x +_f y"
by (simp add: lift2_def plussub_def split: err.split)
section {* sup *}
lemma Err_sup_Err [simp]:
"Err +_(Err.sup f) x = Err"
by (simp add: plussub_def Err.sup_def Err.lift2_def)
lemma Err_sup_Err2 [simp]:
"x +_(Err.sup f) Err = Err"
by (simp add: plussub_def Err.sup_def Err.lift2_def split: err.split)
lemma Err_sup_OK [simp]:
"OK x +_(Err.sup f) OK y = OK(x +_f y)"
by (simp add: plussub_def Err.sup_def Err.lift2_def)
lemma Err_sup_eq_OK_conv [iff]:
"(Err.sup f ex ey = OK z) = (? x y. ex = OK x & ey = OK y & f x y = z)"
apply (unfold Err.sup_def lift2_def plussub_def)
apply (rule iffI)
apply (simp split: err.split_asm)
apply clarify
apply simp
done
lemma Err_sup_eq_Err [iff]:
"(Err.sup f ex ey = Err) = (ex=Err | ey=Err)"
apply (unfold Err.sup_def lift2_def plussub_def)
apply (simp split: err.split)
done
section {* semilat (err A) (le r) f *}
lemma semilat_le_err_Err_plus [simp]:
"\<lbrakk> x: err A; semilat(err A, le r, f) \<rbrakk> \<Longrightarrow> Err +_f x = Err"
by (blast intro: semilat.le_iff_plus_unchanged [THEN iffD1]
semilat.le_iff_plus_unchanged2 [THEN iffD1])
lemma semilat_le_err_plus_Err [simp]:
"\<lbrakk> x: err A; semilat(err A, le r, f) \<rbrakk> \<Longrightarrow> x +_f Err = Err"
by (blast intro: semilat.le_iff_plus_unchanged [THEN iffD1]
semilat.le_iff_plus_unchanged2 [THEN iffD1])
lemma semilat_le_err_OK1:
"\<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 add:semilat.ub1)
done
lemma semilat_le_err_OK2:
"\<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 add:semilat.ub2)
done
lemma eq_order_le:
"\<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]:
"\<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: "\<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 semilat.plus_le_conv [THEN iffD1])
case rule_context
thus ?thesis
apply (rule_tac iffI)
apply clarify
apply (drule OK_le_err_OK [THEN iffD2])
apply (drule OK_le_err_OK [THEN iffD2])
apply (drule semilat.lub[of _ _ _ "OK x" _ "OK y"])
apply assumption
apply assumption
apply simp
apply simp
apply simp
apply simp
apply (case_tac "(OK x) +_fe (OK y)")
apply assumption
apply (rename_tac z)
apply (subgoal_tac "OK z: err A")
apply (drule eq_order_le)
apply (erule semilat.orderI)
apply (blast dest: plus_le_conv3)
apply (erule subst)
apply (blast intro: semilat.closedI closedD)
done
qed
section {* semilat (err(Union AS)) *}
(* FIXME? *)
lemma all_bex_swap_lemma [iff]:
"(!x. (? y:A. x = f y) \<longrightarrow> P x) = (!y:A. P(f y))"
by blast
lemma closed_err_Union_lift2I:
"\<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
apply simp
apply fast
done
text {*
If @{term "AS = {}"} the thm collapses to
@{prop "order r & closed {Err} f & Err +_f Err = Err"}
which may not hold
*}
lemma err_semilat_UnionI:
"\<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 supremum_def sl_def)
apply (simp add: closed_err_Union_lift2I)
apply (rule conjI)
apply blast
apply (simp add: err_def)
apply (rule conjI)
apply clarify
apply (rename_tac A a u B b)
apply (case_tac "A = B")
apply simp
apply simp
apply (rule conjI)
apply clarify
apply (rename_tac A a u B b)
apply (case_tac "A = B")
apply simp
apply simp
apply clarify
apply (rename_tac A ya yb B yd z C c a b)
apply (case_tac "A = B")
apply (case_tac "A = C")
apply simp
apply (rotate_tac -1)
apply simp
apply (rotate_tac -1)
apply (case_tac "B = C")
apply simp
apply (rotate_tac -1)
apply simp
done
end
lemma strict_Some:
(strict f x = OK y) = (EX z. x = OK z & f z = OK y)
lemma not_Err_eq:
(x ~= Err) = (EX a. x = OK a)
lemma not_OK_eq:
(ALL y. x ~= OK y) = (x = Err)
lemma unfold_lesub_err:
e1 <=_(le r) e2 == le r e1 e2
lemma le_err_refl:
ALL x. x <=_r x ==> e <=_(le r) e
lemma le_err_trans:
[| order r; e1 <=_(le r) e2; e2 <=_(le r) e3 |] ==> e1 <=_(le r) e3
lemma le_err_antisym:
[| order r; e1 <=_(le r) e2; e2 <=_(le r) e1 |] ==> e1 = e2
lemma OK_le_err_OK:
(OK x <=_(le r) OK y) = (x <=_r y)
lemma order_le_err:
order (le r) = order r
lemma le_Err:
e <=_(le r) Err
lemma Err_le_conv:
(Err <=_(le r) e) = (e = Err)
lemma le_OK_conv:
(e <=_(le r) OK x) = (EX y. e = OK y & y <=_r x)
lemma OK_le_conv:
(OK x <=_(le r) e) = (e = Err | (EX y. e = OK y & x <=_r y))
lemma top_Err:
top (le r) Err
lemma OK_less_conv:
(OK x <_(le r) e) = (e = Err | (EX y. e = OK y & x <_r y))
lemma not_Err_less:
¬ Err <_(le r) x
lemma
semilat (A, r, f) ==> semilat (err A, le r, lift2 (%x y. OK (f x y)))
lemma
semilat (A, r, f) ==> semilat (sl (esl (A, r, f)))
lemma err_semilat_eslI:
semilat L ==> semilat (sl (esl L))
lemma Err_in_err:
Err : err A
lemma Ok_in_err:
(OK x : err A) = (x : A)
lemma OK_subset_err:
(OK ` A <= err B) = (A <= B)
lemma acc_err:
acc r A ==> acc (le r) (err A)
lemma lift_in_errI:
[| e : err S; ALL x:S. e = OK x --> f x : err S |] ==> lift f e : err S
lemma Err_lift2:
Err +_(lift2 f) x = Err
lemma lift2_Err:
x +_(lift2 f) Err = Err
lemma OK_lift2_OK:
OK x +_(lift2 f) OK y = x +_f y
lemma Err_sup_Err:
Err +_(sup f) x = Err
lemma Err_sup_Err2:
x +_(sup f) Err = Err
lemma Err_sup_OK:
OK x +_(sup f) OK y = OK (x +_f y)
lemma Err_sup_eq_OK_conv:
(sup f ex ey = OK z) = (EX x y. ex = OK x & ey = OK y & f x y = z)
lemma Err_sup_eq_Err:
(sup f ex ey = Err) = (ex = Err | ey = Err)
lemma semilat_le_err_Err_plus:
[| x : err A; semilat (err A, le r, f) |] ==> Err +_f x = Err
lemma semilat_le_err_plus_Err:
[| x : err A; semilat (err A, le r, f) |] ==> x +_f Err = Err
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
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
lemma eq_order_le:
[| x = y; order r |] ==> x <=_r y
lemma OK_plus_OK_eq_Err_conv:
[| x : A; y : A; semilat (err A, le r, fe) |] ==> (OK x +_fe OK y = Err) = (¬ (EX z:A. x <=_r z & y <=_r z))
lemma all_bex_swap_lemma:
(ALL x. (EX y:A. x = f y) --> P x) = (ALL y:A. P (f y))
lemma closed_err_Union_lift2I:
[| ALL A:AS. closed (err A) (lift2 f); AS ~= {};
ALL A:AS. ALL B:AS. A ~= B --> (ALL a:A. ALL b:B. a +_f b = Err) |]
==> closed (err (Union AS)) (lift2 f)
lemma err_semilat_UnionI:
[| ALL A:AS. semilat (sl (A, r, f)); AS ~= {};
ALL A:AS.
ALL B:AS. A ~= B --> (ALL a:A. ALL b:B. ¬ a <=_r b & a +_f b = Err) |]
==> semilat (sl (Union AS, r, f))