author | wenzelm |
Wed, 08 Jun 2016 18:45:44 +0200 | |
changeset 63258 | 576fb8068ba6 |
parent 63170 | eae6549dbea2 |
child 67613 | ce654b0e6d69 |
permissions | -rw-r--r-- |
42150 | 1 |
(* Title: HOL/MicroJava/DFA/Err.thy |
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Author: Tobias Nipkow |
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Copyright 2000 TUM |
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*) |
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section \<open>The Error Type\<close> |
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theory Err |
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imports Semilat |
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begin |
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datatype 'a err = Err | OK 'a |
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type_synonym 'a ebinop = "'a \<Rightarrow> 'a \<Rightarrow> 'a err" |
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type_synonym 'a esl = "'a set * 'a ord * 'a ebinop" |
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primrec ok_val :: "'a err \<Rightarrow> 'a" where |
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"ok_val (OK x) = x" |
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definition lift :: "('a \<Rightarrow> 'b err) \<Rightarrow> ('a err \<Rightarrow> 'b err)" where |
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"lift f e == case e of Err \<Rightarrow> Err | OK x \<Rightarrow> f x" |
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definition lift2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c err) \<Rightarrow> 'a err \<Rightarrow> 'b err \<Rightarrow> 'c err" where |
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"lift2 f e1 e2 == |
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case e1 of Err \<Rightarrow> Err |
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| OK x \<Rightarrow> (case e2 of Err \<Rightarrow> Err | OK y \<Rightarrow> f x y)" |
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definition le :: "'a ord \<Rightarrow> 'a err ord" where |
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"le r e1 e2 == |
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case e2 of Err \<Rightarrow> True | |
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OK y \<Rightarrow> (case e1 of Err \<Rightarrow> False | OK x \<Rightarrow> x <=_r y)" |
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definition sup :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a err \<Rightarrow> 'b err \<Rightarrow> 'c err)" where |
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"sup f == lift2(%x y. OK(x +_f y))" |
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definition err :: "'a set \<Rightarrow> 'a err set" where |
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"err A == insert Err {x . ? y:A. x = OK y}" |
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definition esl :: "'a sl \<Rightarrow> 'a esl" where |
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"esl == %(A,r,f). (A,r, %x y. OK(f x y))" |
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definition sl :: "'a esl \<Rightarrow> 'a err sl" where |
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"sl == %(A,r,f). (err A, le r, lift2 f)" |
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abbreviation |
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err_semilat :: "'a esl \<Rightarrow> bool" |
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where "err_semilat L == semilat(Err.sl L)" |
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primrec strict :: "('a \<Rightarrow> 'b err) \<Rightarrow> ('a err \<Rightarrow> 'b err)" where |
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"strict f Err = Err" |
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| "strict f (OK x) = f x" |
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lemma strict_Some [simp]: |
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"(strict f x = OK y) = (\<exists> z. x = OK z \<and> f z = OK y)" |
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by (cases x, auto) |
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lemma not_Err_eq: |
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"(x \<noteq> Err) = (\<exists>a. x = OK a)" |
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by (cases x) auto |
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lemma not_OK_eq: |
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"(\<forall>y. x \<noteq> OK y) = (x = Err)" |
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by (cases x) auto |
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lemma unfold_lesub_err: |
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"e1 <=_(le r) e2 == le r e1 e2" |
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by (simp add: lesub_def) |
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lemma le_err_refl: |
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"!x. x <=_r x \<Longrightarrow> e <=_(Err.le r) e" |
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apply (unfold lesub_def Err.le_def) |
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apply (simp split: err.split) |
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done |
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lemma le_err_trans [rule_format]: |
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"order r \<Longrightarrow> e1 <=_(le r) e2 \<longrightarrow> e2 <=_(le r) e3 \<longrightarrow> e1 <=_(le r) e3" |
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apply (unfold unfold_lesub_err le_def) |
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apply (simp split: err.split) |
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apply (blast intro: order_trans) |
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done |
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lemma le_err_antisym [rule_format]: |
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"order r \<Longrightarrow> e1 <=_(le r) e2 \<longrightarrow> e2 <=_(le r) e1 \<longrightarrow> e1=e2" |
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apply (unfold unfold_lesub_err le_def) |
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apply (simp split: err.split) |
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apply (blast intro: order_antisym) |
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done |
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lemma OK_le_err_OK: |
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"(OK x <=_(le r) OK y) = (x <=_r y)" |
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by (simp add: unfold_lesub_err le_def) |
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lemma order_le_err [iff]: |
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"order(le r) = order r" |
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apply (rule iffI) |
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apply (subst Semilat.order_def) |
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apply (blast dest: order_antisym OK_le_err_OK [THEN iffD2] |
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intro: order_trans OK_le_err_OK [THEN iffD1]) |
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apply (subst Semilat.order_def) |
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apply (blast intro: le_err_refl le_err_trans le_err_antisym |
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dest: order_refl) |
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done |
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lemma le_Err [iff]: "e <=_(le r) Err" |
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by (simp add: unfold_lesub_err le_def) |
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lemma Err_le_conv [iff]: |
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"Err <=_(le r) e = (e = Err)" |
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by (simp add: unfold_lesub_err le_def split: err.split) |
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lemma le_OK_conv [iff]: |
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"e <=_(le r) OK x = (? y. e = OK y & y <=_r x)" |
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by (simp add: unfold_lesub_err le_def split: err.split) |
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lemma OK_le_conv: |
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"OK x <=_(le r) e = (e = Err | (? y. e = OK y & x <=_r y))" |
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by (simp add: unfold_lesub_err le_def split: err.split) |
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lemma top_Err [iff]: "top (le r) Err" |
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by (simp add: top_def) |
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lemma OK_less_conv [rule_format, iff]: |
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"OK x <_(le r) e = (e=Err | (? y. e = OK y & x <_r y))" |
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by (simp add: lesssub_def lesub_def le_def split: err.split) |
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lemma not_Err_less [rule_format, iff]: |
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"~(Err <_(le r) x)" |
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by (simp add: lesssub_def lesub_def le_def split: err.split) |
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lemma semilat_errI [intro]: |
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assumes semilat: "semilat (A, r, f)" |
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diff
changeset
|
133 |
shows "semilat(err A, Err.le r, lift2(%x y. OK(f x y)))" |
63258 | 134 |
using semilat |
63170 | 135 |
apply (simp only: semilat_Def closed_def plussub_def lesub_def |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
136 |
lift2_def Err.le_def err_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
137 |
apply (simp split: err.split) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
138 |
done |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
139 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
140 |
lemma err_semilat_eslI_aux: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
141 |
assumes semilat: "semilat (A, r, f)" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
142 |
shows "err_semilat(esl(A,r,f))" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
143 |
apply (unfold sl_def esl_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
144 |
apply (simp add: semilat_errI[OF semilat]) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
145 |
done |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
146 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
147 |
lemma err_semilat_eslI [intro, simp]: |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
148 |
"\<And>L. semilat L \<Longrightarrow> err_semilat(esl L)" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
149 |
by(simp add: err_semilat_eslI_aux split_tupled_all) |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
150 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
151 |
lemma acc_err [simp, intro!]: "acc r \<Longrightarrow> acc(le r)" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
152 |
apply (unfold acc_def lesub_def le_def lesssub_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
153 |
apply (simp add: wf_eq_minimal split: err.split) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
154 |
apply clarify |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
155 |
apply (case_tac "Err : Q") |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
156 |
apply blast |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
157 |
apply (erule_tac x = "{a . OK a : Q}" in allE) |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
158 |
apply (case_tac "x") |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
159 |
apply fast |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
160 |
apply blast |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
161 |
done |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
162 |
|
1bc3b688548c
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haftmann
parents:
diff
changeset
|
163 |
lemma Err_in_err [iff]: "Err : err A" |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
164 |
by (simp add: err_def) |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
165 |
|
1bc3b688548c
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haftmann
parents:
diff
changeset
|
166 |
lemma Ok_in_err [iff]: "(OK x : err A) = (x:A)" |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
167 |
by (auto simp add: err_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
168 |
|
61361 | 169 |
subsection \<open>lift\<close> |
33954
1bc3b688548c
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haftmann
parents:
diff
changeset
|
170 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
171 |
lemma lift_in_errI: |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
172 |
"\<lbrakk> e : err S; !x:S. e = OK x \<longrightarrow> f x : err S \<rbrakk> \<Longrightarrow> lift f e : err S" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
173 |
apply (unfold lift_def) |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
174 |
apply (simp split: err.split) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
175 |
apply blast |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
176 |
done |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
177 |
|
1bc3b688548c
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haftmann
parents:
diff
changeset
|
178 |
lemma Err_lift2 [simp]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
179 |
"Err +_(lift2 f) x = Err" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
180 |
by (simp add: lift2_def plussub_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
181 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
182 |
lemma lift2_Err [simp]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
183 |
"x +_(lift2 f) Err = Err" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
184 |
by (simp add: lift2_def plussub_def split: err.split) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
185 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
186 |
lemma OK_lift2_OK [simp]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
187 |
"OK x +_(lift2 f) OK y = x +_f y" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
188 |
by (simp add: lift2_def plussub_def split: err.split) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
189 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
190 |
|
61361 | 191 |
subsection \<open>sup\<close> |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
192 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
193 |
lemma Err_sup_Err [simp]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
194 |
"Err +_(Err.sup f) x = Err" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
195 |
by (simp add: plussub_def Err.sup_def Err.lift2_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
196 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
197 |
lemma Err_sup_Err2 [simp]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
198 |
"x +_(Err.sup f) Err = Err" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
199 |
by (simp add: plussub_def Err.sup_def Err.lift2_def split: err.split) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
200 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
201 |
lemma Err_sup_OK [simp]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
202 |
"OK x +_(Err.sup f) OK y = OK(x +_f y)" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
203 |
by (simp add: plussub_def Err.sup_def Err.lift2_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
204 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
205 |
lemma Err_sup_eq_OK_conv [iff]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
206 |
"(Err.sup f ex ey = OK z) = (? x y. ex = OK x & ey = OK y & f x y = z)" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
207 |
apply (unfold Err.sup_def lift2_def plussub_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
208 |
apply (rule iffI) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
209 |
apply (simp split: err.split_asm) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
210 |
apply clarify |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
211 |
apply simp |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
212 |
done |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
213 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
214 |
lemma Err_sup_eq_Err [iff]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
215 |
"(Err.sup f ex ey = Err) = (ex=Err | ey=Err)" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
216 |
apply (unfold Err.sup_def lift2_def plussub_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
217 |
apply (simp split: err.split) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
218 |
done |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
219 |
|
61361 | 220 |
subsection \<open>semilat (err A) (le r) f\<close> |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
221 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
222 |
lemma semilat_le_err_Err_plus [simp]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
223 |
"\<lbrakk> x: err A; semilat(err A, le r, f) \<rbrakk> \<Longrightarrow> Err +_f x = Err" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
224 |
by (blast intro: Semilat.le_iff_plus_unchanged [OF Semilat.intro, THEN iffD1] |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
225 |
Semilat.le_iff_plus_unchanged2 [OF Semilat.intro, THEN iffD1]) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
226 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
227 |
lemma semilat_le_err_plus_Err [simp]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
228 |
"\<lbrakk> x: err A; semilat(err A, le r, f) \<rbrakk> \<Longrightarrow> x +_f Err = Err" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
229 |
by (blast intro: Semilat.le_iff_plus_unchanged [OF Semilat.intro, THEN iffD1] |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
230 |
Semilat.le_iff_plus_unchanged2 [OF Semilat.intro, THEN iffD1]) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
231 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
232 |
lemma semilat_le_err_OK1: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
233 |
"\<lbrakk> x:A; y:A; semilat(err A, le r, f); OK x +_f OK y = OK z \<rbrakk> |
58860 | 234 |
\<Longrightarrow> x <=_r z" |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
235 |
apply (rule OK_le_err_OK [THEN iffD1]) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
236 |
apply (erule subst) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
237 |
apply (simp add: Semilat.ub1 [OF Semilat.intro]) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
238 |
done |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
239 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
240 |
lemma semilat_le_err_OK2: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
241 |
"\<lbrakk> x:A; y:A; semilat(err A, le r, f); OK x +_f OK y = OK z \<rbrakk> |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
242 |
\<Longrightarrow> y <=_r z" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
243 |
apply (rule OK_le_err_OK [THEN iffD1]) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
244 |
apply (erule subst) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
245 |
apply (simp add: Semilat.ub2 [OF Semilat.intro]) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
246 |
done |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
247 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
248 |
lemma eq_order_le: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
249 |
"\<lbrakk> x=y; order r \<rbrakk> \<Longrightarrow> x <=_r y" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
250 |
apply (unfold Semilat.order_def) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
251 |
apply blast |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
252 |
done |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
253 |
|
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
254 |
lemma OK_plus_OK_eq_Err_conv [simp]: |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
255 |
assumes "x:A" and "y:A" and "semilat(err A, le r, fe)" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
256 |
shows "((OK x) +_fe (OK y) = Err) = (~(? z:A. x <=_r z & y <=_r z))" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
257 |
proof - |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
258 |
have plus_le_conv3: "\<And>A x y z f r. |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
259 |
\<lbrakk> semilat (A,r,f); x +_f y <=_r z; x:A; y:A; z:A \<rbrakk> |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
260 |
\<Longrightarrow> x <=_r z \<and> y <=_r z" |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
261 |
by (rule Semilat.plus_le_conv [OF Semilat.intro, THEN iffD1]) |
41541 | 262 |
from assms show ?thesis |
33954
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
263 |
apply (rule_tac iffI) |
1bc3b688548c
backported parts of abstract byte code verifier from AFP/Jinja
haftmann
parents:
diff
changeset
|
264 |
apply clarify |
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|
265 |
apply (drule OK_le_err_OK [THEN iffD2]) |
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|
266 |
apply (drule OK_le_err_OK [THEN iffD2]) |
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|
267 |
apply (drule Semilat.lub [OF Semilat.intro, of _ _ _ "OK x" _ "OK y"]) |
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|
268 |
apply assumption |
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parents:
diff
changeset
|
269 |
apply assumption |
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parents:
diff
changeset
|
270 |
apply simp |
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parents:
diff
changeset
|
271 |
apply simp |
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parents:
diff
changeset
|
272 |
apply simp |
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parents:
diff
changeset
|
273 |
apply simp |
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|
274 |
apply (case_tac "(OK x) +_fe (OK y)") |
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|
275 |
apply assumption |
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parents:
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changeset
|
276 |
apply (rename_tac z) |
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|
277 |
apply (subgoal_tac "OK z: err A") |
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parents:
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|
278 |
apply (drule eq_order_le) |
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parents:
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|
279 |
apply (erule Semilat.orderI [OF Semilat.intro]) |
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parents:
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changeset
|
280 |
apply (blast dest: plus_le_conv3) |
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parents:
diff
changeset
|
281 |
apply (erule subst) |
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parents:
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changeset
|
282 |
apply (blast intro: Semilat.closedI [OF Semilat.intro] closedD) |
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parents:
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|
283 |
done |
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parents:
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|
284 |
qed |
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|
285 |
|
61952 | 286 |
subsection \<open>semilat (err (Union AS))\<close> |
33954
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|
287 |
|
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|
288 |
(* FIXME? *) |
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|
289 |
lemma all_bex_swap_lemma [iff]: |
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|
290 |
"(!x. (? y:A. x = f y) \<longrightarrow> P x) = (!y:A. P(f y))" |
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parents:
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changeset
|
291 |
by blast |
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parents:
diff
changeset
|
292 |
|
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parents:
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changeset
|
293 |
lemma closed_err_Union_lift2I: |
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parents:
diff
changeset
|
294 |
"\<lbrakk> !A:AS. closed (err A) (lift2 f); AS ~= {}; |
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|
295 |
!A:AS.!B:AS. A~=B \<longrightarrow> (!a:A.!b:B. a +_f b = Err) \<rbrakk> |
61952 | 296 |
\<Longrightarrow> closed (err (\<Union>AS)) (lift2 f)" |
33954
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parents:
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changeset
|
297 |
apply (unfold closed_def err_def) |
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parents:
diff
changeset
|
298 |
apply simp |
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parents:
diff
changeset
|
299 |
apply clarify |
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parents:
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changeset
|
300 |
apply simp |
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parents:
diff
changeset
|
301 |
apply fast |
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parents:
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changeset
|
302 |
done |
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parents:
diff
changeset
|
303 |
|
61361 | 304 |
text \<open> |
33954
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changeset
|
305 |
If @{term "AS = {}"} the thm collapses to |
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changeset
|
306 |
@{prop "order r & closed {Err} f & Err +_f Err = Err"} |
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changeset
|
307 |
which may not hold |
61361 | 308 |
\<close> |
33954
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parents:
diff
changeset
|
309 |
lemma err_semilat_UnionI: |
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parents:
diff
changeset
|
310 |
"\<lbrakk> !A:AS. err_semilat(A, r, f); AS ~= {}; |
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parents:
diff
changeset
|
311 |
!A:AS.!B:AS. A~=B \<longrightarrow> (!a:A.!b:B. ~ a <=_r b & a +_f b = Err) \<rbrakk> |
61952 | 312 |
\<Longrightarrow> err_semilat (\<Union>AS, r, f)" |
33954
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parents:
diff
changeset
|
313 |
apply (unfold semilat_def sl_def) |
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parents:
diff
changeset
|
314 |
apply (simp add: closed_err_Union_lift2I) |
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parents:
diff
changeset
|
315 |
apply (rule conjI) |
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parents:
diff
changeset
|
316 |
apply blast |
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parents:
diff
changeset
|
317 |
apply (simp add: err_def) |
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haftmann
parents:
diff
changeset
|
318 |
apply (rule conjI) |
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haftmann
parents:
diff
changeset
|
319 |
apply clarify |
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parents:
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changeset
|
320 |
apply (rename_tac A a u B b) |
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haftmann
parents:
diff
changeset
|
321 |
apply (case_tac "A = B") |
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haftmann
parents:
diff
changeset
|
322 |
apply simp |
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haftmann
parents:
diff
changeset
|
323 |
apply simp |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
324 |
apply (rule conjI) |
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haftmann
parents:
diff
changeset
|
325 |
apply clarify |
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haftmann
parents:
diff
changeset
|
326 |
apply (rename_tac A a u B b) |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
327 |
apply (case_tac "A = B") |
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haftmann
parents:
diff
changeset
|
328 |
apply simp |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
329 |
apply simp |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
330 |
apply clarify |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
331 |
apply (rename_tac A ya yb B yd z C c a b) |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
332 |
apply (case_tac "A = B") |
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haftmann
parents:
diff
changeset
|
333 |
apply (case_tac "A = C") |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
334 |
apply simp |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
335 |
apply (rotate_tac -1) |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
336 |
apply simp |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
337 |
apply (rotate_tac -1) |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
338 |
apply (case_tac "B = C") |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
339 |
apply simp |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
340 |
apply (rotate_tac -1) |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
341 |
apply simp |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
342 |
done |
1bc3b688548c
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haftmann
parents:
diff
changeset
|
343 |
|
1bc3b688548c
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haftmann
parents:
diff
changeset
|
344 |
end |