src/HOL/Bali/DefiniteAssignment.thy
 author schirmer Thu, 31 Oct 2002 18:27:10 +0100 changeset 13688 a0b16d42d489 child 14030 cd928c0ac225 permissions -rw-r--r--
"Definite Assignment Analysis" included, with proof of correctness. Large adjustments of type safety proof and soundness proof of the axiomatic semantics were necessary. Completeness proof of the loop rule of the axiomatic semantic was altered. So the additional polymorphic variants of some rules could be removed.


theory DefiniteAssignment = WellType:

text {* Definite Assignment Analysis (cf. 16)

The definite assignment analysis approximates the sets of local
variables that will be assigned at a certain point of evaluation, and ensures
that we will only read variables which previously were assigned.
It should conform to the following idea:
If the evaluation of a term completes normally (no abruption (exception,
break, continue, return) appeared) , the set of local variables calculated
by the analysis is a subset of the
variables that were actually assigned during evaluation.

To get more precise information about the sets of assigned variables the
analysis includes the following optimisations:
\begin{itemize}
\item Inside of a while loop we also take care of the variables assigned
before break statements, since the break causes the while loop to
continue normally.
\item For conditional statements we take care of constant conditions to
statically determine the path of evaluation.
\item Inside a distinct path of a conditional statements we know to which
boolean value the condition has evaluated to, and so can retrieve more
information about the variables assigned during evaluation of the
boolean condition.
\end{itemize}

Since in our model of Java the return values of methods are stored in a local
variable we also ensure that every path of (normal) evaluation will assign the
result variable, or in the sense of real Java every path ends up in and
return instruction.

Not covered yet:
\begin{itemize}
\item analysis of definite unassigned
\item special treatment of final fields
\end{itemize}
*}

section {* Correct nesting of jump statements *}

text {* For definite assignment it becomes crucial, that jumps (break,
continue, return) are nested correctly i.e. a continue jump is nested in a
matching while statement, a break jump is nested in a proper label statement,
a class initialiser does not terminate abruptly with a return. With this we
can for example ensure that evaluation of an expression will never end up
with a jump, since no breaks, continues or returns are allowed in an
expression. *}

consts jumpNestingOkS :: "jump set \<Rightarrow> stmt \<Rightarrow> bool"
primrec
"jumpNestingOkS jmps (Skip)   = True"
"jumpNestingOkS jmps (Expr e) = True"
"jumpNestingOkS jmps (j\<bullet> s) = jumpNestingOkS ({j} \<union> jmps) s"
"jumpNestingOkS jmps (c1;;c2) = (jumpNestingOkS jmps c1 \<and>
jumpNestingOkS jmps c2)"
"jumpNestingOkS jmps (If(e) c1 Else c2) = (jumpNestingOkS jmps c1 \<and>
jumpNestingOkS jmps c2)"
"jumpNestingOkS jmps (l\<bullet> While(e) c) = jumpNestingOkS ({Cont l} \<union> jmps) c"
--{* The label of the while loop only handles continue jumps. Breaks are only
handled by @{term Lab} *}
"jumpNestingOkS jmps (Jmp j) = (j \<in> jmps)"
"jumpNestingOkS jmps (Throw e) = True"
"jumpNestingOkS jmps (Try c1 Catch(C vn) c2) = (jumpNestingOkS jmps c1 \<and>
jumpNestingOkS jmps c2)"
"jumpNestingOkS jmps (c1 Finally c2) = (jumpNestingOkS jmps c1 \<and>
jumpNestingOkS jmps c2)"
"jumpNestingOkS jmps (Init C) = True"
--{* wellformedness of the program must enshure that for all initializers
jumpNestingOkS {} holds *}
--{* Dummy analysis for intermediate smallstep term @{term  FinA} *}
"jumpNestingOkS jmps (FinA a c) = False"

constdefs jumpNestingOk :: "jump set \<Rightarrow> term \<Rightarrow> bool"
"jumpNestingOk jmps t \<equiv> (case t of
In1 se \<Rightarrow> (case se of
Inl e \<Rightarrow> True
| Inr s \<Rightarrow> jumpNestingOkS jmps s)
| In2  v \<Rightarrow> True
| In3  es \<Rightarrow> True)"

lemma jumpNestingOk_expr_simp [simp]: "jumpNestingOk jmps (In1l e) = True"

lemma jumpNestingOk_expr_simp1 [simp]: "jumpNestingOk jmps \<langle>e::expr\<rangle> = True"

lemma jumpNestingOk_stmt_simp [simp]:
"jumpNestingOk jmps (In1r s) = jumpNestingOkS jmps s"

lemma jumpNestingOk_stmt_simp1 [simp]:
"jumpNestingOk jmps \<langle>s::stmt\<rangle> = jumpNestingOkS jmps s"

lemma jumpNestingOk_var_simp [simp]: "jumpNestingOk jmps (In2 v) = True"

lemma jumpNestingOk_var_simp1 [simp]: "jumpNestingOk jmps \<langle>v::var\<rangle> = True"

lemma jumpNestingOk_expr_list_simp [simp]: "jumpNestingOk jmps (In3 es) = True"

lemma jumpNestingOk_expr_list_simp1 [simp]:
"jumpNestingOk jmps \<langle>es::expr list\<rangle> = True"

section {* Calculation of assigned variables for boolean expressions*}

subsection {* Very restricted calculation fallback calculation *}

consts the_LVar_name:: "var \<Rightarrow> lname"
primrec
"the_LVar_name (LVar n) = n"

consts assignsE :: "expr      \<Rightarrow> lname set"
assignsV :: "var       \<Rightarrow> lname set"
assignsEs:: "expr list \<Rightarrow> lname set"
text {* *}
primrec
"assignsE (NewC c)            = {}"
"assignsE (NewA t e)          = assignsE e"
"assignsE (Cast t e)          = assignsE e"
"assignsE (e InstOf r)        = assignsE e"
"assignsE (Lit val)           = {}"
"assignsE (UnOp unop e)       = assignsE e"
"assignsE (BinOp binop e1 e2) = (if binop=CondAnd \<or> binop=CondOr
then (assignsE e1)
else (assignsE e1) \<union> (assignsE e2))"
"assignsE (Super)             = {}"
"assignsE (Acc v)             = assignsV v"
"assignsE (v:=e)              = (assignsV v) \<union> (assignsE e) \<union>
(if \<exists> n. v=(LVar n) then {the_LVar_name v}
else {})"
"assignsE (b? e1 : e2) = (assignsE b) \<union> ((assignsE e1) \<inter> (assignsE e2))"
"assignsE ({accC,statT,mode}objRef\<cdot>mn({pTs}args))
= (assignsE objRef) \<union> (assignsEs args)"
-- {* Only dummy analysis for intermediate expressions
@{term Methd}, @{term Body}, @{term InsInitE} and @{term Callee} *}
"assignsE (Methd C sig)   = {}"
"assignsE (Body  C s)     = {}"
"assignsE (InsInitE s e)  = {}"
"assignsE (Callee l e)    = {}"

"assignsV (LVar n)       = {}"
"assignsV ({accC,statDeclC,stat}objRef..fn) = assignsE objRef"
"assignsV (e1.[e2])      = assignsE e1 \<union> assignsE e2"

"assignsEs     [] = {}"
"assignsEs (e#es) = assignsE e \<union> assignsEs es"

constdefs assigns:: "term \<Rightarrow> lname set"
"assigns t \<equiv> (case t of
In1 se \<Rightarrow> (case se of
Inl e \<Rightarrow> assignsE e
| Inr s \<Rightarrow> {})
| In2  v \<Rightarrow> assignsV v
| In3  es \<Rightarrow> assignsEs es)"

lemma assigns_expr_simp [simp]: "assigns (In1l e) = assignsE e"

lemma assigns_expr_simp1 [simp]: "assigns (\<langle>e\<rangle>) = assignsE e"

lemma assigns_stmt_simp [simp]: "assigns (In1r s) = {}"

lemma assigns_stmt_simp1 [simp]: "assigns (\<langle>s::stmt\<rangle>) = {}"

lemma assigns_var_simp [simp]: "assigns (In2 v) = assignsV v"

lemma assigns_var_simp1 [simp]: "assigns (\<langle>v\<rangle>) = assignsV v"

lemma assigns_expr_list_simp [simp]: "assigns (In3 es) = assignsEs es"

lemma assigns_expr_list_simp1 [simp]: "assigns (\<langle>es\<rangle>) = assignsEs es"

subsection "Analysis of constant expressions"

consts constVal :: "expr \<Rightarrow> val option"
primrec
"constVal (NewC c)      = None"
"constVal (NewA t e)    = None"
"constVal (Cast t e)    = None"
"constVal (Inst e r)    = None"
"constVal (Lit val)     = Some val"
"constVal (UnOp unop e) = (case (constVal e) of
None   \<Rightarrow> None
| Some v \<Rightarrow> Some (eval_unop unop v))"
"constVal (BinOp binop e1 e2) = (case (constVal e1) of
None    \<Rightarrow> None
| Some v1 \<Rightarrow> (case (constVal e2) of
None    \<Rightarrow> None
| Some v2 \<Rightarrow> Some (eval_binop
binop v1 v2)))"
"constVal (Super)         = None"
"constVal (Acc v)         = None"
"constVal (Ass v e)       = None"
"constVal (Cond b e1 e2)  = (case (constVal b) of
None   \<Rightarrow> None
| Some bv\<Rightarrow> (case the_Bool bv of
True \<Rightarrow> (case (constVal e2) of
None   \<Rightarrow> None
| Some v \<Rightarrow> constVal e1)
| False\<Rightarrow> (case (constVal e1) of
None   \<Rightarrow> None
| Some v \<Rightarrow> constVal e2)))"
--{* Note that @{text "constVal (Cond b e1 e2)"} is stricter as it could be.
It requires that all tree expressions are constant even if we can decide
which branch to choose, provided the constant value of @{term b} *}
"constVal (Call accC statT mode objRef mn pTs args) = None"
"constVal (Methd C sig)   = None"
"constVal (Body  C s)     = None"
"constVal (InsInitE s e)  = None"
"constVal (Callee l e)    = None"

lemma constVal_Some_induct [consumes 1, case_names Lit UnOp BinOp CondL CondR]:
assumes const: "constVal e = Some v" and
hyp_Lit: "\<And> v. P (Lit v)" and
hyp_UnOp: "\<And> unop e'. P e' \<Longrightarrow> P (UnOp unop e')" and
hyp_BinOp: "\<And> binop e1 e2. \<lbrakk>P e1; P e2\<rbrakk> \<Longrightarrow> P (BinOp binop e1 e2)" and
hyp_CondL: "\<And> b bv e1 e2. \<lbrakk>constVal b = Some bv; the_Bool bv; P b; P e1\<rbrakk>
\<Longrightarrow> P (b? e1 : e2)" and
hyp_CondR: "\<And> b bv e1 e2. \<lbrakk>constVal b = Some bv; \<not>the_Bool bv; P b; P e2\<rbrakk>
\<Longrightarrow> P (b? e1 : e2)"
shows "P e"
proof -
have "True" and "\<And> v. constVal e = Some v \<Longrightarrow> P e" and "True" and "True"
proof (induct "x::var" and e and "s::stmt" and "es::expr list")
case Lit
show ?case by (rule hyp_Lit)
next
case UnOp
thus ?case
by (auto intro: hyp_UnOp)
next
case BinOp
thus ?case
by (auto intro: hyp_BinOp)
next
case (Cond b e1 e2)
then obtain v where   v: "constVal (b ? e1 : e2) = Some v"
by blast
then obtain bv where bv: "constVal b = Some bv"
by simp
show ?case
proof (cases "the_Bool bv")
case True
with Cond show ?thesis using v bv
by (auto intro: hyp_CondL)
next
case False
with Cond show ?thesis using v bv
by (auto intro: hyp_CondR)
qed
qed (simp_all)
with const
show ?thesis
by blast
qed

lemma assignsE_const_simp: "constVal e = Some v \<Longrightarrow> assignsE e = {}"
by (induct rule: constVal_Some_induct) simp_all

subsection {* Main analysis for boolean expressions *}

text {* Assigned local variables after evaluating the expression if it evaluates
to a specific boolean value. If the expression cannot evaluate to a
@{term Boolean} value UNIV is returned. If we expect true/false the opposite
constant false/true will also lead to UNIV. *}
consts assigns_if:: "bool \<Rightarrow> expr \<Rightarrow> lname set"
primrec
"assigns_if b (NewC c)            = UNIV" --{*can never evaluate to Boolean*}
"assigns_if b (NewA t e)          = UNIV" --{*can never evaluate to Boolean*}
"assigns_if b (Cast t e)          = assigns_if b e"
"assigns_if b (Inst e r)          = assignsE e" --{*Inst has type Boolean but
e is a reference type*}
"assigns_if b (Lit val)           = (if val=Bool b then {} else UNIV)"
"assigns_if b (UnOp unop e)       = (case constVal (UnOp unop e) of
None   \<Rightarrow> (if unop = UNot
then assigns_if (\<not>b) e
else UNIV)
| Some v \<Rightarrow> (if v=Bool b
then {}
else UNIV))"
"assigns_if b (BinOp binop e1 e2)
= (case constVal (BinOp binop e1 e2) of
None \<Rightarrow> (if binop=CondAnd then
(case b of
True  \<Rightarrow> assigns_if True  e1 \<union> assigns_if True e2
|  False \<Rightarrow> assigns_if False e1 \<inter>
(assigns_if True e1 \<union> assigns_if False e2))
else
(if binop=CondOr then
(case b of
True  \<Rightarrow> assigns_if True e1 \<inter>
(assigns_if False e1 \<union> assigns_if True e2)
|  False \<Rightarrow> assigns_if False  e1 \<union> assigns_if False e2)
else assignsE e1 \<union> assignsE e2))
| Some v \<Rightarrow> (if v=Bool b then {} else UNIV))"

"assigns_if b (Super)      = UNIV" --{*can never evaluate to Boolean*}
"assigns_if b (Acc v)      = (assignsV v)"
"assigns_if b (v := e)     = (assignsE (Ass v e))"
"assigns_if b (c? e1 : e2) = (assignsE c) \<union>
(case (constVal c) of
None    \<Rightarrow> (assigns_if b e1) \<inter>
(assigns_if b e2)
| Some bv \<Rightarrow> (case the_Bool bv of
True  \<Rightarrow> assigns_if b e1
| False \<Rightarrow> assigns_if b e2))"
"assigns_if b ({accC,statT,mode}objRef\<cdot>mn({pTs}args))
= assignsE ({accC,statT,mode}objRef\<cdot>mn({pTs}args)) "
-- {* Only dummy analysis for intermediate expressions
@{term Methd}, @{term Body}, @{term InsInitE} and @{term Callee} *}
"assigns_if b (Methd C sig)   = {}"
"assigns_if b (Body  C s)     = {}"
"assigns_if b (InsInitE s e)  = {}"
"assigns_if b (Callee l e)    = {}"

lemma assigns_if_const_b_simp:
assumes boolConst: "constVal e = Some (Bool b)" (is "?Const b e")
shows   "assigns_if b e = {}" (is "?Ass b e")
proof -
have "True" and "\<And> b. ?Const b e \<Longrightarrow> ?Ass b e" and "True" and "True"
proof (induct _ and e and _ and _ rule: var_expr_stmt.induct)
case Lit
thus ?case by simp
next
case UnOp
thus ?case by simp
next
case (BinOp binop)
thus ?case
by (cases binop) (simp_all)
next
case (Cond c e1 e2 b)
have hyp_c:  "\<And> b. ?Const b c \<Longrightarrow> ?Ass b c" .
have hyp_e1: "\<And> b. ?Const b e1 \<Longrightarrow> ?Ass b e1" .
have hyp_e2: "\<And> b. ?Const b e2 \<Longrightarrow> ?Ass b e2" .
have const: "constVal (c ? e1 : e2) = Some (Bool b)" .
then obtain bv where bv: "constVal c = Some bv"
by simp
hence emptyC: "assignsE c = {}" by (rule assignsE_const_simp)
show ?case
proof (cases "the_Bool bv")
case True
with const bv
have "?Const b e1" by simp
hence "?Ass b e1" by (rule hyp_e1)
with emptyC bv True
show ?thesis
by simp
next
case False
with const bv
have "?Const b e2" by simp
hence "?Ass b e2" by (rule hyp_e2)
with emptyC bv False
show ?thesis
by simp
qed
qed (simp_all)
with boolConst
show ?thesis
by blast
qed

lemma assigns_if_const_not_b_simp:
assumes boolConst: "constVal e = Some (Bool b)"        (is "?Const b e")
shows "assigns_if (\<not>b) e = UNIV"                    (is "?Ass b e")
proof -
have True and "\<And> b. ?Const b e \<Longrightarrow> ?Ass b e" and True and True
proof (induct _ and e and _ and _ rule: var_expr_stmt.induct)
case Lit
thus ?case by simp
next
case UnOp
thus ?case by simp
next
case (BinOp binop)
thus ?case
by (cases binop) (simp_all)
next
case (Cond c e1 e2 b)
have hyp_c:  "\<And> b. ?Const b c \<Longrightarrow> ?Ass b c" .
have hyp_e1: "\<And> b. ?Const b e1 \<Longrightarrow> ?Ass b e1" .
have hyp_e2: "\<And> b. ?Const b e2 \<Longrightarrow> ?Ass b e2" .
have const: "constVal (c ? e1 : e2) = Some (Bool b)" .
then obtain bv where bv: "constVal c = Some bv"
by simp
show ?case
proof (cases "the_Bool bv")
case True
with const bv
have "?Const b e1" by simp
hence "?Ass b e1" by (rule hyp_e1)
with bv True
show ?thesis
by simp
next
case False
with const bv
have "?Const b e2" by simp
hence "?Ass b e2" by (rule hyp_e2)
with bv False
show ?thesis
by simp
qed
qed (simp_all)
with boolConst
show ?thesis
by blast
qed

subsection {* Lifting set operations to range of tables (map to a set) *}

constdefs
union_ts:: "('a,'b) tables \<Rightarrow> ('a,'b) tables \<Rightarrow> ('a,'b) tables"
("_ \<Rightarrow>\<union> _" [67,67] 65)
"A \<Rightarrow>\<union> B \<equiv> \<lambda> k. A k \<union> B k"

constdefs
intersect_ts:: "('a,'b) tables \<Rightarrow> ('a,'b) tables \<Rightarrow> ('a,'b) tables"
("_ \<Rightarrow>\<inter>  _" [72,72] 71)
"A \<Rightarrow>\<inter>  B \<equiv> \<lambda> k. A k \<inter> B k"

constdefs
all_union_ts:: "('a,'b) tables \<Rightarrow> 'b set \<Rightarrow> ('a,'b) tables"
(infixl "\<Rightarrow>\<union>\<^sub>\<forall>" 40)
"A \<Rightarrow>\<union>\<^sub>\<forall> B \<equiv> \<lambda> k. A k \<union> B"

subsubsection {* Binary union of tables *}

lemma union_ts_iff [simp]: "(c \<in> (A \<Rightarrow>\<union> B) k) = (c \<in> A k \<or>  c \<in> B k)"
by (unfold union_ts_def) blast

lemma union_tsI1 [elim?]: "c \<in> A k \<Longrightarrow> c \<in> (A \<Rightarrow>\<union> B) k"
by simp

lemma union_tsI2 [elim?]: "c \<in> B k \<Longrightarrow> c \<in> (A \<Rightarrow>\<union> B) k"
by simp

lemma union_tsCI [intro!]: "(c \<notin> B k \<Longrightarrow> c \<in> A k) \<Longrightarrow> c \<in> (A \<Rightarrow>\<union> B) k"
by auto

lemma union_tsE [elim!]:
"\<lbrakk>c \<in> (A \<Rightarrow>\<union> B) k; (c \<in> A k \<Longrightarrow> P); (c \<in> B k \<Longrightarrow> P)\<rbrakk> \<Longrightarrow> P"
by (unfold union_ts_def) blast

subsubsection {* Binary intersection of tables *}

lemma intersect_ts_iff [simp]: "c \<in> (A \<Rightarrow>\<inter> B) k = (c \<in> A k \<and> c \<in> B k)"
by (unfold intersect_ts_def) blast

lemma intersect_tsI [intro!]: "\<lbrakk>c \<in> A k; c \<in> B k\<rbrakk> \<Longrightarrow> c \<in>  (A \<Rightarrow>\<inter> B) k"
by simp

lemma intersect_tsD1: "c \<in> (A \<Rightarrow>\<inter> B) k \<Longrightarrow> c \<in> A k"
by simp

lemma intersect_tsD2: "c \<in> (A \<Rightarrow>\<inter> B) k \<Longrightarrow> c \<in> B k"
by simp

lemma intersect_tsE [elim!]:
"\<lbrakk>c \<in> (A \<Rightarrow>\<inter> B) k; \<lbrakk>c \<in> A k; c \<in> B k\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by simp

subsubsection {* All-Union of tables and set *}

lemma all_union_ts_iff [simp]: "(c \<in> (A \<Rightarrow>\<union>\<^sub>\<forall> B) k) = (c \<in> A k \<or>  c \<in> B)"
by (unfold all_union_ts_def) blast

lemma all_union_tsI1 [elim?]: "c \<in> A k \<Longrightarrow> c \<in> (A \<Rightarrow>\<union>\<^sub>\<forall> B) k"
by simp

lemma all_union_tsI2 [elim?]: "c \<in> B \<Longrightarrow> c \<in> (A \<Rightarrow>\<union>\<^sub>\<forall> B) k"
by simp

lemma all_union_tsCI [intro!]: "(c \<notin> B \<Longrightarrow> c \<in> A k) \<Longrightarrow> c \<in> (A \<Rightarrow>\<union>\<^sub>\<forall> B) k"
by auto

lemma all_union_tsE [elim!]:
"\<lbrakk>c \<in> (A \<Rightarrow>\<union>\<^sub>\<forall> B) k; (c \<in> A k \<Longrightarrow> P); (c \<in> B \<Longrightarrow> P)\<rbrakk> \<Longrightarrow> P"
by (unfold all_union_ts_def) blast

section "The rules of definite assignment"

types breakass = "(label, lname) tables"
--{* Mapping from a break label, to the set of variables that will be assigned
if the evaluation terminates with this break *}

record assigned =
nrm :: "lname set" --{* Definetly assigned variables
for normal completion*}
brk :: "breakass" --{* Definetly assigned variables for
abnormal completion with a break *}

consts da :: "(env \<times> lname set \<times> term \<times> assigned) set"
text {* The environment @{term env} is only needed for the
conditional @{text "_ ? _ : _"}.
The definite assignment rules refer to the typing rules here to
distinguish boolean and other expressions.
*}

syntax
da :: "env \<Rightarrow> lname set \<Rightarrow> term \<Rightarrow> assigned \<Rightarrow> bool"
("_\<turnstile> _ \<guillemotright>_\<guillemotright> _" [65,65,65,65] 71)

translations
"E\<turnstile> B \<guillemotright>t\<guillemotright> A" == "(E,B,t,A) \<in> da"

text {* @{text B}: the ''assigned'' variables before evaluating term @{text t};
@{text A}: the ''assigned'' variables after evaluating term @{text t}
*}

constdefs rmlab :: "'a \<Rightarrow> ('a,'b) tables \<Rightarrow> ('a,'b) tables"
"rmlab k A \<equiv> \<lambda> x. if x=k then UNIV else A x"

(*
constdefs setbrk :: "breakass \<Rightarrow> assigned \<Rightarrow> breakass set"
"setbrk b A \<equiv> {b} \<union> {a| a. a\<in> brk A \<and> lab a \<noteq> lab b}"
*)

constdefs range_inter_ts :: "('a,'b) tables \<Rightarrow> 'b set" ("\<Rightarrow>\<Inter>_" 80)
"\<Rightarrow>\<Inter>A \<equiv> {x |x. \<forall> k. x \<in> A k}"

inductive "da" intros

Skip: "Env\<turnstile> B \<guillemotright>\<langle>Skip\<rangle>\<guillemotright> \<lparr>nrm=B,brk=\<lambda> l. UNIV\<rparr>"

Expr: "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>Expr e\<rangle>\<guillemotright> A"
Lab:  "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C; nrm A = nrm C \<inter> (brk C) l; brk A = rmlab l (brk C)\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>Break l\<bullet> c\<rangle>\<guillemotright> A"

Comp: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1; Env\<turnstile> nrm C1 \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2;
nrm A = nrm C2; brk A = (brk C1) \<Rightarrow>\<inter> (brk C2)\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>c1;; c2\<rangle>\<guillemotright> A"

If:   "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E;
Env\<turnstile> (B \<union> assigns_if True  e) \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1;
Env\<turnstile> (B \<union> assigns_if False e) \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2;
nrm A = nrm C1 \<inter> nrm C2;
brk A = brk C1 \<Rightarrow>\<inter> brk C2 \<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>If(e) c1 Else c2\<rangle>\<guillemotright> A"

--{* Note that @{term E} is not further used, because we take the specialized
sets that also consider if the expression evaluates to true or false.
Inside of @{term e} there is no {\tt break} or {\tt finally}, so the break
map of @{term E} will be the trivial one. So
@{term "Env\<turnstile>B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E"} is just used to enshure the definite assignment in
expression @{term e}.
Notice the implicit analysis of a constant boolean expression @{term e}
in this rule. For example, if @{term e} is constantly @{term True} then
@{term "assigns_if False e = UNIV"} and therefor @{term "nrm C2=UNIV"}.
So finally @{term "nrm A = nrm C1"}. For the break maps this trick
workd too, because the trival break map will map all labels to
@{term UNIV}. In the example, if no break occurs in @{term c2} the break
maps will trivially map to @{term UNIV} and if a break occurs it will map
to @{term UNIV} too, because @{term "assigns_if False e = UNIV"}. So
in the intersection of the break maps the path @{term c2} will have no
contribution.
*}

Loop: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E;
Env\<turnstile> (B \<union> assigns_if True e) \<guillemotright>\<langle>c\<rangle>\<guillemotright> C;
nrm A = nrm C \<inter> (B \<union> assigns_if False e);
brk A = brk C\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>l\<bullet> While(e) c\<rangle>\<guillemotright> A"
--{* The @{text Loop} rule resembles some of the ideas of the @{text If} rule.
For the @{term "nrm A"} the set @{term "B \<union> assigns_if False e"}
will be @{term UNIV} if the condition is constantly true. To normally exit
the while loop, we must consider the body @{term c} to be completed
normally (@{term "nrm C"}) or with a break. But in this model,
the label @{term l} of the loop
only handles continue labels, not break labels. The break label will be
handled by an enclosing @{term Lab} statement. So we don't have to
handle the breaks specially.
*}

Jmp: "\<lbrakk>jump=Ret \<longrightarrow> Result \<in> B;
nrm A = UNIV;
brk A = (case jump of
Break l \<Rightarrow> \<lambda> k. if k=l then B else UNIV
| Cont l  \<Rightarrow> \<lambda> k. UNIV
| Ret     \<Rightarrow> \<lambda> k. UNIV)\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>Jmp jump\<rangle>\<guillemotright> A"
--{* In case of a break to label @{term l} the corresponding break set is all
variables assigned before the break. The assigned variables for normal
completion of the @{term Jmp} is @{term UNIV}, because the statement will
never complete normally. For continue and return the break map is the
trivial one. In case of a return we enshure that the result value is
assigned.
*}

Throw: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E; nrm A = UNIV; brk A = (\<lambda> l. UNIV)\<rbrakk>
\<Longrightarrow> Env\<turnstile> B \<guillemotright>\<langle>Throw e\<rangle>\<guillemotright> A"

Try:  "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1;
Env\<lparr>lcl := lcl Env(VName vn\<mapsto>Class C)\<rparr>\<turnstile> (B \<union> {VName vn}) \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2;
nrm A = nrm C1 \<inter> nrm C2;
brk A = brk C1 \<Rightarrow>\<inter> brk C2\<rbrakk>
\<Longrightarrow> Env\<turnstile> B \<guillemotright>\<langle>Try c1 Catch(C vn) c2\<rangle>\<guillemotright> A"

Fin:  "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1;
Env\<turnstile> B \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2;
nrm A = nrm C1 \<union> nrm C2;
brk A = ((brk C1) \<Rightarrow>\<union>\<^sub>\<forall> (nrm C2)) \<Rightarrow>\<inter> (brk C2)\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>c1 Finally c2\<rangle>\<guillemotright> A"
--{* The set of assigned variables before execution @{term c2} are the same
as before execution @{term c1}, because @{term c1} could throw an exception
and so we can't guarantee that any variable will be assigned in @{term c1}.
The @{text Finally} statement completes
normally if both @{term c1} and @{term c2} complete normally. If @{term c1}
completes abnormally with a break, then @{term c2} also will be executed
and may terminate normally or with a break. The overall break map then is
the intersection of the maps of both paths. If @{term c2} terminates
normally we have to extend all break sets in @{term "brk C1"} with
@{term "nrm C2"} (@{text "\<Rightarrow>\<union>\<^sub>\<forall>"}). If @{term c2} exits with a break this
break will appear in the overall result state. We don't know if
@{term c1} completed normally or abruptly (maybe with an exception not only
a break) so @{term c1} has no contribution to the break map following this
path.
*}

--{* Evaluation of expressions and the break sets of definite assignment:
Thinking of a Java expression we assume that we can never have
a break statement inside of a expression. So for all expressions the
break sets could be set to the trivial one: @{term "\<lambda> l. UNIV"}.
But we can't
trivially proof, that evaluating an expression will never result in a
break, allthough Java expressions allready syntactically don't allow
nested stetements in them. The reason are the nested class initialzation
statements which are inserted by the evaluation rules. So to proof the
absence of a break we need to ensure, that the initialization statements
will never end up in a break. In a wellfromed initialization statement,
of course, were breaks are nested correctly inside of @{term Lab}
or @{term Loop} statements evaluation of the whole initialization
statement will never result in a break, because this break will be
handled inside of the statement. But for simplicity we haven't added
the analysis of the correct nesting of breaks in the typing judgments
right now. So we have decided to adjust the rules of definite assignment
to fit to these circumstances. If an initialization is involved during
evaluation of the expression (evaluation rules @{text FVar}, @{text NewC}
and @{text NewA}
*}

Init: "Env\<turnstile> B \<guillemotright>\<langle>Init C\<rangle>\<guillemotright> \<lparr>nrm=B,brk=\<lambda> l. UNIV\<rparr>"
--{* Wellformedness of a program will ensure, that every static initialiser
is definetly assigned and the jumps are nested correctly. The case here
for @{term Init} is just for convenience, to get a proper precondition
for the induction hypothesis in various proofs, so that we don't have to
expand the initialisation on every point where it is triggerred by the
evaluation rules.
*}
NewC: "Env\<turnstile> B \<guillemotright>\<langle>NewC C\<rangle>\<guillemotright> \<lparr>nrm=B,brk=\<lambda> l. UNIV\<rparr>"

NewA: "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>New T[e]\<rangle>\<guillemotright> A"

Cast: "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>Cast T e\<rangle>\<guillemotright> A"

Inst: "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>e InstOf T\<rangle>\<guillemotright> A"

Lit:  "Env\<turnstile> B \<guillemotright>\<langle>Lit v\<rangle>\<guillemotright> \<lparr>nrm=B,brk=\<lambda> l. UNIV\<rparr>"

UnOp: "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>UnOp unop e\<rangle>\<guillemotright> A"

CondAnd: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1; Env\<turnstile> (B \<union> assigns_if True e1) \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2;
nrm A = B \<union> (assigns_if True (BinOp CondAnd e1 e2) \<inter>
assigns_if False (BinOp CondAnd e1 e2));
brk A = (\<lambda> l. UNIV) \<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>BinOp CondAnd e1 e2\<rangle>\<guillemotright> A"

CondOr: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1; Env\<turnstile> (B \<union> assigns_if False e1) \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2;
nrm A = B \<union> (assigns_if True (BinOp CondOr e1 e2) \<inter>
assigns_if False (BinOp CondOr e1 e2));
brk A = (\<lambda> l. UNIV) \<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>BinOp CondOr e1 e2\<rangle>\<guillemotright> A"

BinOp: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1; Env\<turnstile> nrm E1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> A;
binop \<noteq> CondAnd; binop \<noteq> CondOr\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>BinOp binop e1 e2\<rangle>\<guillemotright> A"

Super: "This \<in> B
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>Super\<rangle>\<guillemotright> \<lparr>nrm=B,brk=\<lambda> l. UNIV\<rparr>"

AccLVar: "\<lbrakk>vn \<in> B;
nrm A = B; brk A = (\<lambda> k. UNIV)\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>Acc (LVar vn)\<rangle>\<guillemotright> A"
--{* To properly access a local variable we have to test the definite
assignment here. The variable must occur in the set @{term B}
*}

Acc: "\<lbrakk>\<forall> vn. v \<noteq> LVar vn;
Env\<turnstile> B \<guillemotright>\<langle>v\<rangle>\<guillemotright> A\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>Acc v\<rangle>\<guillemotright> A"

AssLVar: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E; nrm A = nrm E \<union> {vn}; brk A = brk E\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>(LVar vn) := e\<rangle>\<guillemotright> A"

Ass: "\<lbrakk>\<forall> vn. v \<noteq> LVar vn; Env\<turnstile> B \<guillemotright>\<langle>v\<rangle>\<guillemotright> V; Env\<turnstile> nrm V \<guillemotright>\<langle>e\<rangle>\<guillemotright> A\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>v := e\<rangle>\<guillemotright> A"

CondBool: "\<lbrakk>Env\<turnstile>(c ? e1 : e2)\<Colon>-(PrimT Boolean);
Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C;
Env\<turnstile> (B \<union> assigns_if True  c) \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1;
Env\<turnstile> (B \<union> assigns_if False c) \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2;
nrm A = B \<union> (assigns_if True  (c ? e1 : e2) \<inter>
assigns_if False (c ? e1 : e2));
brk A = (\<lambda> l. UNIV)\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>c ? e1 : e2\<rangle>\<guillemotright> A"

Cond: "\<lbrakk>\<not> Env\<turnstile>(c ? e1 : e2)\<Colon>-(PrimT Boolean);
Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C;
Env\<turnstile> (B \<union> assigns_if True  c) \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1;
Env\<turnstile> (B \<union> assigns_if False c) \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2;
nrm A = nrm E1 \<inter> nrm E2; brk A = (\<lambda> l. UNIV)\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>c ? e1 : e2\<rangle>\<guillemotright> A"

Call: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E; Env\<turnstile> nrm E \<guillemotright>\<langle>args\<rangle>\<guillemotright> A\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>{accC,statT,mode}e\<cdot>mn({pTs}args)\<rangle>\<guillemotright> A"

-- {* The interplay of @{term Call}, @{term Methd} and @{term Body}:
Why rules for @{term Methd} and @{term Body} at all? Note that a
Java source program will not include bare  @{term Methd} or @{term Body}
terms. These terms are just introduced during evaluation. So definite
assignment of @{term Call} does not consider @{term Methd} or
@{term Body} at all. So for definite assignment alone we could omit the
rules for @{term Methd} and @{term Body}.
But since evaluation of the method invocation is
split up into three rules we must ensure that we have enough information
about the call even in the @{term Body} term to make sure that we can
proof type safety. Also we must be able transport this information
from @{term Call} to @{term Methd} and then further to @{term Body}
during evaluation to establish the definite assignment of @{term Methd}
during evaluation of @{term Call}, and of @{term Body} during evaluation
of @{term Methd}. This is necessary since definite assignment will be
a precondition for each induction hypothesis coming out of the evaluation
rules, and therefor we have to establish the definite assignment of the
sub-evaluation during the type-safety proof. Note that well-typedness is
also a precondition for type-safety and so we can omit some assertion
that are already ensured by well-typedness.
*}
Methd: "\<lbrakk>methd (prg Env) D sig = Some m;
Env\<turnstile> B \<guillemotright>\<langle>Body (declclass m) (stmt (mbody (mthd m)))\<rangle>\<guillemotright> A
\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>Methd D sig\<rangle>\<guillemotright> A"

Body: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C; jumpNestingOkS {Ret} c; Result \<in> nrm C;
nrm A = B; brk A = (\<lambda> l. UNIV)\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>Body D c\<rangle>\<guillemotright> A"
-- {* Note that @{term A} is not correlated to  @{term C}. If the body
statement returns abruptly with return, evaluation of  @{term Body}
will absorb this return and complete normally. So we cannot trivially
get the assigned variables of the body statement since it has not
completed normally or with a break.
If the body completes normally we guarantee that the result variable
is set with this rule. But if the body completes abruptly with a return
we can't guarantee that the result variable is set here, since
definite assignment only talks about normal completion and breaks. So
for a return the @{term Jump} rule ensures that the result variable is
set and then this information must be carried over to the @{term Body}
rule by the conformance predicate of the state.
*}
LVar: "Env\<turnstile> B \<guillemotright>\<langle>LVar vn\<rangle>\<guillemotright> \<lparr>nrm=B, brk=\<lambda> l. UNIV\<rparr>"

FVar: "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>{accC,statDeclC,stat}e..fn\<rangle>\<guillemotright> A"

AVar: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1; Env\<turnstile> nrm E1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> A\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>e1.[e2]\<rangle>\<guillemotright> A"

Nil: "Env\<turnstile> B \<guillemotright>\<langle>[]::expr list\<rangle>\<guillemotright> \<lparr>nrm=B, brk=\<lambda> l. UNIV\<rparr>"

Cons: "\<lbrakk>Env\<turnstile> B \<guillemotright>\<langle>e::expr\<rangle>\<guillemotright> E; Env\<turnstile> nrm E \<guillemotright>\<langle>es\<rangle>\<guillemotright> A\<rbrakk>
\<Longrightarrow>
Env\<turnstile> B \<guillemotright>\<langle>e#es\<rangle>\<guillemotright> A"

declare inj_term_sym_simps [simp]
declare assigns_if.simps [simp del]
declare split_paired_All [simp del] split_paired_Ex [simp del]
ML_setup {*
simpset_ref() := simpset() delloop "split_all_tac"
*}
inductive_cases da_elim_cases [cases set]:
"Env\<turnstile> B \<guillemotright>\<langle>Skip\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r Skip\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Expr e\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r (Expr e)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>l\<bullet> c\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r (l\<bullet> c)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>c1;; c2\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r (c1;; c2)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>If(e) c1 Else c2\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r (If(e) c1 Else c2)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>l\<bullet> While(e) c\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r (l\<bullet> While(e) c)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Jmp jump\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r (Jmp jump)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Throw e\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r (Throw e)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Try c1 Catch(C vn) c2\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r (Try c1 Catch(C vn) c2)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>c1 Finally c2\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r (c1 Finally c2)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Init C\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1r (Init C)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>NewC C\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (NewC C)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>New T[e]\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (New T[e])\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Cast T e\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (Cast T e)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>e InstOf T\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (e InstOf T)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Lit v\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (Lit v)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>UnOp unop e\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (UnOp unop e)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>BinOp binop e1 e2\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (BinOp binop e1 e2)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Super\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (Super)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Acc v\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (Acc v)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>v := e\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (v := e)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>c ? e1 : e2\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (c ? e1 : e2)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>{accC,statT,mode}e\<cdot>mn({pTs}args)\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l ({accC,statT,mode}e\<cdot>mn({pTs}args))\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Methd C sig\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (Methd C sig)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>Body D c\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In1l (Body D c)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>LVar vn\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In2 (LVar vn)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>{accC,statDeclC,stat}e..fn\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In2 ({accC,statDeclC,stat}e..fn)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>e1.[e2]\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In2 (e1.[e2])\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>[]::expr list\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In3 ([]::expr list)\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>\<langle>e#es\<rangle>\<guillemotright> A"
"Env\<turnstile> B \<guillemotright>In3 (e#es)\<guillemotright> A"
declare inj_term_sym_simps [simp del]
declare assigns_if.simps [simp]
declare split_paired_All [simp] split_paired_Ex [simp]
ML_setup {*
simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)
*}
(* To be able to eliminate both the versions with the overloaded brackets:
(B \<guillemotright>\<langle>Skip\<rangle>\<guillemotright> A) and with the explicit constructor (B \<guillemotright>In1r Skip\<guillemotright> A),
every rule appears in both versions
*)

lemma da_Skip: "A = \<lparr>nrm=B,brk=\<lambda> l. UNIV\<rparr> \<Longrightarrow> Env\<turnstile> B \<guillemotright>\<langle>Skip\<rangle>\<guillemotright> A"
by (auto intro: da.Skip)

lemma da_NewC: "A = \<lparr>nrm=B,brk=\<lambda> l. UNIV\<rparr> \<Longrightarrow> Env\<turnstile> B \<guillemotright>\<langle>NewC C\<rangle>\<guillemotright> A"
by (auto intro: da.NewC)

lemma da_Lit:  "A = \<lparr>nrm=B,brk=\<lambda> l. UNIV\<rparr> \<Longrightarrow> Env\<turnstile> B \<guillemotright>\<langle>Lit v\<rangle>\<guillemotright> A"
by (auto intro: da.Lit)

lemma da_Super: "\<lbrakk>This \<in> B;A = \<lparr>nrm=B,brk=\<lambda> l. UNIV\<rparr>\<rbrakk> \<Longrightarrow> Env\<turnstile> B \<guillemotright>\<langle>Super\<rangle>\<guillemotright> A"
by (auto intro: da.Super)

lemma da_Init: "A = \<lparr>nrm=B,brk=\<lambda> l. UNIV\<rparr> \<Longrightarrow> Env\<turnstile> B \<guillemotright>\<langle>Init C\<rangle>\<guillemotright> A"
by (auto intro: da.Init)

(*
For boolean expressions:

The following holds: "assignsE e \<subseteq> assigns_if True e \<inter> assigns_if False e"
but not vice versa:
"assigns_if True e \<inter> assigns_if False e \<subseteq> assignsE e"

Example:
e = ((x < 5) || (y = true)) && (y = true)

=  (   a    ||    b     ) &&    c

assigns_if True  a = {}
assigns_if False a = {}

assigns_if True  b = {y}
assigns_if False b = {y}

assigns_if True  c = {y}
assigns_if False c = {y}

assigns_if True (a || b) = assigns_if True a \<inter>
(assigns_if False a \<union> assigns_if True b)
= {} \<inter> ({} \<union> {y}) = {}
assigns_if False (a || b) = assigns_if False a \<union> assigns_if False b
= {} \<union> {y} = {y}

assigns_ifE True e =  assigns_if True (a || b) \<union> assigns_if True c
= {} \<union> {y} = {y}
assigns_ifE False e = assigns_if False (a || b) \<inter>
(assigns_if True (a || b) \<union> assigns_if False c)
= {y} \<inter> ({} \<union> {y}) = {y}

assignsE e = {}
*)

lemma assignsE_subseteq_assigns_ifs:
assumes boolEx: "E\<turnstile>e\<Colon>-PrimT Boolean" (is "?Boolean e")
shows "assignsE e \<subseteq> assigns_if True e \<inter> assigns_if False e" (is "?Incl e")
proof -
have True and "?Boolean e \<Longrightarrow> ?Incl e" and True and True
proof (induct _ and e and _ and _ rule: var_expr_stmt.induct)
case (Cast T e)
have "E\<turnstile>e\<Colon>- (PrimT Boolean)"
proof -
have "E\<turnstile>(Cast T e)\<Colon>- (PrimT Boolean)" .
then obtain Te where "E\<turnstile>e\<Colon>-Te"
"prg E\<turnstile>Te\<preceq>? PrimT Boolean"
by cases simp
thus ?thesis
by - (drule cast_Boolean2,simp)
qed
with Cast.hyps
show ?case
by simp
next
case (Lit val)
thus ?case
by - (erule wt_elim_cases, cases "val", auto simp add: empty_dt_def)
next
case (UnOp unop e)
thus ?case
by - (erule wt_elim_cases,cases unop,
next
case (BinOp binop e1 e2)
from BinOp.prems obtain e1T e2T
where "E\<turnstile>e1\<Colon>-e1T" and "E\<turnstile>e2\<Colon>-e2T" and "wt_binop (prg E) binop e1T e2T"
and "(binop_type binop) = Boolean"
by (elim wt_elim_cases) simp
with BinOp.hyps
show ?case
by - (cases binop, auto simp add: assignsE_const_simp)
next
case (Cond c e1 e2)
have hyp_c: "?Boolean c \<Longrightarrow> ?Incl c" .
have hyp_e1: "?Boolean e1 \<Longrightarrow> ?Incl e1" .
have hyp_e2: "?Boolean e2 \<Longrightarrow> ?Incl e2" .
have wt: "E\<turnstile>(c ? e1 : e2)\<Colon>-PrimT Boolean" .
then obtain
boolean_c:  "E\<turnstile>c\<Colon>-PrimT Boolean" and
boolean_e1: "E\<turnstile>e1\<Colon>-PrimT Boolean" and
boolean_e2: "E\<turnstile>e2\<Colon>-PrimT Boolean"
by (elim wt_elim_cases) (auto dest: widen_Boolean2)
show ?case
proof (cases "constVal c")
case None
with boolean_e1 boolean_e2
show ?thesis
using hyp_e1 hyp_e2
by (auto)
next
case (Some bv)
show ?thesis
proof (cases "the_Bool bv")
case True
with Some show ?thesis using hyp_e1 boolean_e1 by auto
next
case False
with Some show ?thesis using hyp_e2 boolean_e2 by auto
qed
qed
qed simp_all
with boolEx
show ?thesis
by blast
qed

(* Trick:
If you have a rule with the abstract term injections:
e.g:  da.Skip "B \<guillemotright>\<langle>Skip\<rangle>\<guillemotright> A"
and the current goal state as an concrete injection:
e.g: "B \<guillemotright>In1r Skip\<guillemotright> A"
you can convert the rule by: da.Skip [simplified]
if inj_term_simps is in the simpset

*)

lemma rmlab_same_label [simp]: "(rmlab l A) l = UNIV"

lemma rmlab_same_label1 [simp]: "l=l' \<Longrightarrow> (rmlab l A) l' = UNIV"

lemma rmlab_other_label [simp]: "l\<noteq>l'\<Longrightarrow> (rmlab l A) l' = A l'"

lemma range_inter_ts_subseteq [intro]: "\<forall> k. A k  \<subseteq> B k \<Longrightarrow>  \<Rightarrow>\<Inter>A \<subseteq> \<Rightarrow>\<Inter>B"

lemma range_inter_ts_subseteq':
"\<lbrakk>\<forall> k. A k  \<subseteq> B k; x \<in> \<Rightarrow>\<Inter>A\<rbrakk> \<Longrightarrow> x \<in> \<Rightarrow>\<Inter>B"

lemma da_monotone:
assumes      da: "Env\<turnstile> B  \<guillemotright>t\<guillemotright> A"   and
subseteq_B_B': "B \<subseteq> B'"          and
da': "Env\<turnstile> B' \<guillemotright>t\<guillemotright> A'"
shows "(nrm A \<subseteq> nrm A') \<and> (\<forall> l. (brk A l \<subseteq> brk A' l))"
proof -
from da
show "\<And> B' A'. \<lbrakk>Env\<turnstile> B' \<guillemotright>t\<guillemotright> A'; B \<subseteq> B'\<rbrakk>
\<Longrightarrow> (nrm A \<subseteq> nrm A') \<and> (\<forall> l. (brk A l \<subseteq> brk A' l))"
(is "PROP ?Hyp Env B t A")
proof (induct)
case Skip
from Skip.prems Skip.hyps
show ?case by cases simp
next
case Expr
from Expr.prems Expr.hyps
show ?case by cases simp
next
case (Lab A B C Env c l B' A')
have A: "nrm A = nrm C \<inter> brk C l" "brk A = rmlab l (brk C)" .
have "PROP ?Hyp Env B \<langle>c\<rangle> C" .
moreover
have "B \<subseteq> B'" .
moreover
obtain C'
where "Env\<turnstile> B' \<guillemotright>\<langle>c\<rangle>\<guillemotright> C'"
and A': "nrm A' = nrm C' \<inter> brk C' l" "brk A' = rmlab l (brk C')"
using Lab.prems
by - (erule da_elim_cases,simp)
ultimately
have "nrm C \<subseteq> nrm C'" and hyp_brk: "(\<forall>l. brk C l \<subseteq> brk C' l)" by auto
then
have "nrm C \<inter> brk C l \<subseteq> nrm C' \<inter> brk C' l" by auto
moreover
{
fix l'
from hyp_brk
have "rmlab l (brk C) l'  \<subseteq> rmlab l (brk C') l'"
by  (cases "l=l'") simp_all
}
moreover note A A'
ultimately show ?case
by simp
next
case (Comp A B C1 C2 Env c1 c2 B' A')
have A: "nrm A = nrm C2" "brk A = brk C1 \<Rightarrow>\<inter>  brk C2" .
have "Env\<turnstile> B' \<guillemotright>\<langle>c1;; c2\<rangle>\<guillemotright> A'" .
then obtain  C1' C2'
where da_c1: "Env\<turnstile> B' \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1'" and
da_c2: "Env\<turnstile> nrm C1' \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2'"  and
A': "nrm A' = nrm C2'" "brk A' = brk C1' \<Rightarrow>\<inter>  brk C2'"
by (rule da_elim_cases) auto
have "PROP ?Hyp Env B \<langle>c1\<rangle> C1" .
moreover have "B \<subseteq> B'" .
moreover note da_c1
ultimately have C1': "nrm C1 \<subseteq> nrm C1'" "(\<forall>l. brk C1 l \<subseteq> brk C1' l)"
by (auto)
have "PROP ?Hyp Env (nrm C1) \<langle>c2\<rangle> C2" .
with da_c2 C1'
have C2': "nrm C2 \<subseteq> nrm C2'" "(\<forall>l. brk C2 l \<subseteq> brk C2' l)"
by (auto)
with A A' C1'
show ?case
by auto
next
case (If A B C1 C2 E Env c1 c2 e B' A')
have A: "nrm A = nrm C1 \<inter> nrm C2" "brk A = brk C1 \<Rightarrow>\<inter>  brk C2" .
have "Env\<turnstile> B' \<guillemotright>\<langle>If(e) c1 Else c2\<rangle>\<guillemotright> A'" .
then obtain C1' C2'
where da_c1: "Env\<turnstile> B' \<union> assigns_if True e \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1'" and
da_c2: "Env\<turnstile> B' \<union> assigns_if False e \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2'" and
A': "nrm A' = nrm C1' \<inter> nrm C2'" "brk A' = brk C1' \<Rightarrow>\<inter>  brk C2'"
by (rule da_elim_cases) auto
have "PROP ?Hyp Env (B \<union> assigns_if True e) \<langle>c1\<rangle> C1" .
moreover have B': "B \<subseteq> B'" .
moreover note da_c1
ultimately obtain C1': "nrm C1 \<subseteq> nrm C1'" "(\<forall>l. brk C1 l \<subseteq> brk C1' l)"
by blast
have "PROP ?Hyp Env (B \<union> assigns_if False e) \<langle>c2\<rangle> C2" .
with da_c2 B'
obtain C2': "nrm C2 \<subseteq> nrm C2'" "(\<forall>l. brk C2 l \<subseteq> brk C2' l)"
by blast
with A A' C1'
show ?case
by auto
next
case (Loop A B C E Env c e l B' A')
have A: "nrm A = nrm C \<inter> (B \<union> assigns_if False e)"
"brk A = brk C" .
have "Env\<turnstile> B' \<guillemotright>\<langle>l\<bullet> While(e) c\<rangle>\<guillemotright> A'" .
then obtain C'
where
da_c': "Env\<turnstile> B' \<union> assigns_if True e \<guillemotright>\<langle>c\<rangle>\<guillemotright> C'" and
A': "nrm A' = nrm C' \<inter> (B' \<union> assigns_if False e)"
"brk A' = brk C'"
by (rule da_elim_cases) auto
have "PROP ?Hyp Env (B \<union> assigns_if True e) \<langle>c\<rangle> C" .
moreover have B': "B \<subseteq> B'" .
moreover note da_c'
ultimately obtain C': "nrm C \<subseteq> nrm C'" "(\<forall>l. brk C l \<subseteq> brk C' l)"
by blast
with A A' B'
have "nrm A \<subseteq> nrm A'"
by blast
moreover
{ fix l'
have  "brk A l' \<subseteq> brk A' l'"
proof (cases "constVal e")
case None
with A A' C'
show ?thesis
by (cases "l=l'") auto
next
case (Some bv)
with A A' C'
show ?thesis
by (cases "the_Bool bv", cases "l=l'") auto
qed
}
ultimately show ?case
by auto
next
case (Jmp A B Env jump B' A')
thus ?case by (elim da_elim_cases) (auto split: jump.splits)
next
case Throw thus ?case by -  (erule da_elim_cases, auto)
next
case (Try A B C C1 C2 Env c1 c2 vn B' A')
have A: "nrm A = nrm C1 \<inter> nrm C2"
"brk A = brk C1 \<Rightarrow>\<inter>  brk C2" .
have "Env\<turnstile> B' \<guillemotright>\<langle>Try c1 Catch(C vn) c2\<rangle>\<guillemotright> A'" .
then obtain C1' C2'
where da_c1': "Env\<turnstile> B' \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1'" and
da_c2': "Env\<lparr>lcl := lcl Env(VName vn\<mapsto>Class C)\<rparr>\<turnstile> B' \<union> {VName vn}
\<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2'" and
A': "nrm A' = nrm C1' \<inter> nrm C2'"
"brk A' = brk C1' \<Rightarrow>\<inter>  brk C2'"
by (rule da_elim_cases) auto
have "PROP ?Hyp Env B \<langle>c1\<rangle> C1" .
moreover have B': "B \<subseteq> B'" .
moreover note da_c1'
ultimately obtain C1': "nrm C1 \<subseteq> nrm C1'" "(\<forall>l. brk C1 l \<subseteq> brk C1' l)"
by blast
have "PROP ?Hyp (Env\<lparr>lcl := lcl Env(VName vn\<mapsto>Class C)\<rparr>)
(B \<union> {VName vn}) \<langle>c2\<rangle> C2" .
with B' da_c2'
obtain "nrm C2 \<subseteq> nrm C2'" "(\<forall>l. brk C2 l \<subseteq> brk C2' l)"
by blast
with C1' A A'
show ?case
by auto
next
case (Fin A B C1 C2 Env c1 c2 B' A')
have A: "nrm A = nrm C1 \<union> nrm C2"
"brk A = (brk C1 \<Rightarrow>\<union>\<^sub>\<forall> nrm C2) \<Rightarrow>\<inter> (brk C2)" .
have "Env\<turnstile> B' \<guillemotright>\<langle>c1 Finally c2\<rangle>\<guillemotright> A'" .
then obtain C1' C2'
where  da_c1': "Env\<turnstile> B' \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1'" and
da_c2': "Env\<turnstile> B' \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2'" and
A':  "nrm A' = nrm C1' \<union> nrm C2'"
"brk A' = (brk C1' \<Rightarrow>\<union>\<^sub>\<forall> nrm C2') \<Rightarrow>\<inter> (brk C2')"
by (rule da_elim_cases) auto
have "PROP ?Hyp Env B \<langle>c1\<rangle> C1" .
moreover have B': "B \<subseteq> B'" .
moreover note da_c1'
ultimately obtain C1': "nrm C1 \<subseteq> nrm C1'" "(\<forall>l. brk C1 l \<subseteq> brk C1' l)"
by blast
have hyp_c2: "PROP ?Hyp Env B \<langle>c2\<rangle> C2" .
from da_c2' B'
obtain "nrm C2 \<subseteq> nrm C2'" "(\<forall>l. brk C2 l \<subseteq> brk C2' l)"
by - (drule hyp_c2,auto)
with A A' C1'
show ?case
by auto
next
case Init thus ?case by -  (erule da_elim_cases, auto)
next
case NewC thus ?case by -  (erule da_elim_cases, auto)
next
case NewA thus ?case by -  (erule da_elim_cases, auto)
next
case Cast thus ?case by -  (erule da_elim_cases, auto)
next
case Inst thus ?case by -  (erule da_elim_cases, auto)
next
case Lit thus ?case by -  (erule da_elim_cases, auto)
next
case UnOp thus ?case by -  (erule da_elim_cases, auto)
next
case (CondAnd A B E1 E2 Env e1 e2 B' A')
have A: "nrm A = B \<union>
assigns_if True (BinOp CondAnd e1 e2) \<inter>
assigns_if False (BinOp CondAnd e1 e2)"
"brk A = (\<lambda>l. UNIV)" .
have "Env\<turnstile> B' \<guillemotright>\<langle>BinOp CondAnd e1 e2\<rangle>\<guillemotright> A'" .
then obtain  A': "nrm A' = B' \<union>
assigns_if True (BinOp CondAnd e1 e2) \<inter>
assigns_if False (BinOp CondAnd e1 e2)"
"brk A' = (\<lambda>l. UNIV)"
by (rule da_elim_cases) auto
have B': "B \<subseteq> B'" .
with A A' show ?case
by auto
next
case CondOr thus ?case by - (erule da_elim_cases, auto)
next
case BinOp thus ?case by -  (erule da_elim_cases, auto)
next
case Super thus ?case by -  (erule da_elim_cases, auto)
next
case AccLVar thus ?case by -  (erule da_elim_cases, auto)
next
case Acc thus ?case by -  (erule da_elim_cases, auto)
next
case AssLVar thus ?case by - (erule da_elim_cases, auto)
next
case Ass thus ?case by -  (erule da_elim_cases, auto)
next
case (CondBool A B C E1 E2 Env c e1 e2 B' A')
have A: "nrm A = B \<union>
assigns_if True (c ? e1 : e2) \<inter>
assigns_if False (c ? e1 : e2)"
"brk A = (\<lambda>l. UNIV)" .
have "Env\<turnstile> (c ? e1 : e2)\<Colon>- (PrimT Boolean)" .
moreover
have "Env\<turnstile> B' \<guillemotright>\<langle>c ? e1 : e2\<rangle>\<guillemotright> A'" .
ultimately
obtain A': "nrm A' = B' \<union>
assigns_if True (c ? e1 : e2) \<inter>
assigns_if False (c ? e1 : e2)"
"brk A' = (\<lambda>l. UNIV)"
by - (erule da_elim_cases,auto simp add: inj_term_simps)
(* inj_term_simps needed to handle wt (defined without \<langle>\<rangle>) *)
have B': "B \<subseteq> B'" .
with A A' show ?case
by auto
next
case (Cond A B C E1 E2 Env c e1 e2 B' A')
have A: "nrm A = nrm E1 \<inter> nrm E2"
"brk A = (\<lambda>l. UNIV)" .
have not_bool: "\<not> Env\<turnstile> (c ? e1 : e2)\<Colon>- (PrimT Boolean)" .
have "Env\<turnstile> B' \<guillemotright>\<langle>c ? e1 : e2\<rangle>\<guillemotright> A'" .
then obtain E1' E2'
where da_e1': "Env\<turnstile> B' \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1'" and
da_e2': "Env\<turnstile> B' \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2'" and
A': "nrm A' = nrm E1' \<inter> nrm E2'"
"brk A' = (\<lambda>l. UNIV)"
using not_bool
by  - (erule da_elim_cases, auto simp add: inj_term_simps)
(* inj_term_simps needed to handle wt (defined without \<langle>\<rangle>) *)
have "PROP ?Hyp Env (B \<union> assigns_if True c) \<langle>e1\<rangle> E1" .
moreover have B': "B \<subseteq> B'" .
moreover note da_e1'
ultimately obtain E1': "nrm E1 \<subseteq> nrm E1'" "(\<forall>l. brk E1 l \<subseteq> brk E1' l)"
by blast
have "PROP ?Hyp Env (B \<union> assigns_if False c) \<langle>e2\<rangle> E2" .
with B' da_e2'
obtain "nrm E2 \<subseteq> nrm E2'" "(\<forall>l. brk E2 l \<subseteq> brk E2' l)"
by blast
with E1' A A'
show ?case
by auto
next
case Call
from Call.prems and Call.hyps
show ?case by cases auto
next
case Methd thus ?case by -  (erule da_elim_cases, auto)
next
case Body thus ?case by -  (erule da_elim_cases, auto)
next
case LVar thus ?case by -  (erule da_elim_cases, auto)
next
case FVar thus ?case by -  (erule da_elim_cases, auto)
next
case AVar thus ?case by -  (erule da_elim_cases, auto)
next
case Nil thus ?case by -  (erule da_elim_cases, auto)
next
case Cons thus ?case by -  (erule da_elim_cases, auto)
qed
qed

lemma da_weaken:
assumes            da: "Env\<turnstile> B \<guillemotright>t\<guillemotright> A" and
subseteq_B_B': "B \<subseteq> B'"
shows "\<exists> A'. Env \<turnstile> B' \<guillemotright>t\<guillemotright> A'"
proof -
note assigned.select_convs [CPure.intro]
from da
show "\<And> B'. B \<subseteq> B' \<Longrightarrow> \<exists> A'. Env\<turnstile> B' \<guillemotright>t\<guillemotright> A'" (is "PROP ?Hyp Env B t")
proof (induct)
case Skip thus ?case by (rules intro: da.Skip)
next
case Expr thus ?case by (rules intro: da.Expr)
next
case (Lab A B C Env c l B')
have "PROP ?Hyp Env B \<langle>c\<rangle>" .
moreover
have B': "B \<subseteq> B'" .
ultimately obtain C' where "Env\<turnstile> B' \<guillemotright>\<langle>c\<rangle>\<guillemotright> C'"
by rules
then obtain A' where "Env\<turnstile> B' \<guillemotright>\<langle>Break l\<bullet> c\<rangle>\<guillemotright> A'"
by (rules intro: da.Lab)
thus ?case ..
next
case (Comp A B C1 C2 Env c1 c2 B')
have da_c1: "Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1" .
have "PROP ?Hyp Env B \<langle>c1\<rangle>" .
moreover
have B': "B \<subseteq> B'" .
ultimately obtain C1' where da_c1': "Env\<turnstile> B' \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1'"
by rules
with da_c1 B'
have
"nrm C1 \<subseteq> nrm C1'"
by (rule da_monotone [elim_format]) simp
moreover
have "PROP ?Hyp Env (nrm C1) \<langle>c2\<rangle>" .
ultimately obtain C2' where "Env\<turnstile> nrm C1' \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2'"
by rules
with da_c1' obtain A' where "Env\<turnstile> B' \<guillemotright>\<langle>c1;; c2\<rangle>\<guillemotright> A'"
by (rules intro: da.Comp)
thus ?case ..
next
case (If A B C1 C2 E Env c1 c2 e B')
have B': "B \<subseteq> B'" .
obtain  E' where "Env\<turnstile> B' \<guillemotright>\<langle>e\<rangle>\<guillemotright> E'"
proof -
have "PROP ?Hyp Env B \<langle>e\<rangle>" by (rule If.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain C1' where "Env\<turnstile> (B' \<union> assigns_if True e) \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1'"
proof -
from B'
have "(B \<union> assigns_if True e) \<subseteq> (B' \<union> assigns_if True e)"
by blast
moreover
have "PROP ?Hyp Env (B \<union> assigns_if True e) \<langle>c1\<rangle>" by (rule If.hyps)
ultimately
show ?thesis using that by rules
qed
moreover
obtain C2' where "Env\<turnstile> (B' \<union> assigns_if False e) \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2'"
proof -
from B' have "(B \<union> assigns_if False e) \<subseteq> (B' \<union> assigns_if False e)"
by blast
moreover
have "PROP ?Hyp Env (B \<union> assigns_if False e) \<langle>c2\<rangle>" by (rule If.hyps)
ultimately
show ?thesis using that by rules
qed
ultimately
obtain A' where "Env\<turnstile> B' \<guillemotright>\<langle>If(e) c1 Else c2\<rangle>\<guillemotright> A'"
by (rules intro: da.If)
thus ?case ..
next
case (Loop A B C E Env c e l B')
have B': "B \<subseteq> B'" .
obtain  E' where "Env\<turnstile> B' \<guillemotright>\<langle>e\<rangle>\<guillemotright> E'"
proof -
have "PROP ?Hyp Env B \<langle>e\<rangle>" by (rule Loop.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain C' where "Env\<turnstile> (B' \<union> assigns_if True e) \<guillemotright>\<langle>c\<rangle>\<guillemotright> C'"
proof -
from B'
have "(B \<union> assigns_if True e) \<subseteq> (B' \<union> assigns_if True e)"
by blast
moreover
have "PROP ?Hyp Env (B \<union> assigns_if True e) \<langle>c\<rangle>" by (rule Loop.hyps)
ultimately
show ?thesis using that by rules
qed
ultimately
obtain A' where "Env\<turnstile> B' \<guillemotright>\<langle>l\<bullet> While(e) c\<rangle>\<guillemotright> A'"
by (rules intro: da.Loop )
thus ?case ..
next
case (Jmp A B Env jump B')
have B': "B \<subseteq> B'" .
with Jmp.hyps have "jump = Ret \<longrightarrow> Result \<in> B' "
by auto
moreover
obtain A'::assigned
where  "nrm A' = UNIV"
"brk A' = (case jump of
Break l \<Rightarrow> \<lambda>k. if k = l then B' else UNIV
| Cont l \<Rightarrow> \<lambda>k. UNIV
| Ret \<Rightarrow> \<lambda>k. UNIV)"

by  rules
ultimately have "Env\<turnstile> B' \<guillemotright>\<langle>Jmp jump\<rangle>\<guillemotright> A'"
by (rule da.Jmp)
thus ?case ..
next
case Throw thus ?case by (rules intro: da.Throw )
next
case (Try A B C C1 C2 Env c1 c2 vn B')
have B': "B \<subseteq> B'" .
obtain C1' where "Env\<turnstile> B' \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1'"
proof -
have "PROP ?Hyp Env B \<langle>c1\<rangle>" by (rule Try.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain C2' where
"Env\<lparr>lcl := lcl Env(VName vn\<mapsto>Class C)\<rparr>\<turnstile> B' \<union> {VName vn} \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2'"
proof -
from B' have "B \<union> {VName vn} \<subseteq> B' \<union> {VName vn}" by blast
moreover
have "PROP ?Hyp (Env\<lparr>lcl := lcl Env(VName vn\<mapsto>Class C)\<rparr>)
(B \<union> {VName vn}) \<langle>c2\<rangle>"
by (rule Try.hyps)
ultimately
show ?thesis using that by rules
qed
ultimately
obtain A' where "Env\<turnstile> B' \<guillemotright>\<langle>Try c1 Catch(C vn) c2\<rangle>\<guillemotright> A'"
by (rules intro: da.Try )
thus ?case ..
next
case (Fin A B C1 C2 Env c1 c2 B')
have B': "B \<subseteq> B'" .
obtain C1' where C1': "Env\<turnstile> B' \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1'"
proof -
have "PROP ?Hyp Env B \<langle>c1\<rangle>" by (rule Fin.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain C2' where "Env\<turnstile> B' \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2'"
proof -
have "PROP ?Hyp Env B \<langle>c2\<rangle>" by (rule Fin.hyps)
with B'
show ?thesis using that by rules
qed
ultimately
obtain A' where "Env\<turnstile> B' \<guillemotright>\<langle>c1 Finally c2\<rangle>\<guillemotright> A'"
by (rules intro: da.Fin )
thus ?case ..
next
case Init thus ?case by (rules intro: da.Init)
next
case NewC thus ?case by (rules intro: da.NewC)
next
case NewA thus ?case by (rules intro: da.NewA)
next
case Cast thus ?case by (rules intro: da.Cast)
next
case Inst thus ?case by (rules intro: da.Inst)
next
case Lit thus ?case by (rules intro: da.Lit)
next
case UnOp thus ?case by (rules intro: da.UnOp)
next
case (CondAnd A B E1 E2 Env e1 e2 B')
have B': "B \<subseteq> B'" .
obtain E1' where "Env\<turnstile> B' \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1'"
proof -
have "PROP ?Hyp Env B \<langle>e1\<rangle>" by (rule CondAnd.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain E2' where "Env\<turnstile> B' \<union>  assigns_if True e1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2'"
proof -
from B' have "B \<union> assigns_if True e1 \<subseteq> B' \<union>  assigns_if True e1"
by blast
moreover
have "PROP ?Hyp Env (B \<union> assigns_if True e1) \<langle>e2\<rangle>" by (rule CondAnd.hyps)
ultimately show ?thesis using that by rules
qed
ultimately
obtain A' where "Env\<turnstile> B' \<guillemotright>\<langle>BinOp CondAnd e1 e2\<rangle>\<guillemotright> A'"
by (rules intro: da.CondAnd)
thus ?case ..
next
case (CondOr A B E1 E2 Env e1 e2 B')
have B': "B \<subseteq> B'" .
obtain E1' where "Env\<turnstile> B' \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1'"
proof -
have "PROP ?Hyp Env B \<langle>e1\<rangle>" by (rule CondOr.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain E2' where "Env\<turnstile> B' \<union> assigns_if False e1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2'"
proof -
from B' have "B \<union> assigns_if False e1 \<subseteq> B' \<union>  assigns_if False e1"
by blast
moreover
have "PROP ?Hyp Env (B \<union> assigns_if False e1) \<langle>e2\<rangle>" by (rule CondOr.hyps)
ultimately show ?thesis using that by rules
qed
ultimately
obtain A' where "Env\<turnstile> B' \<guillemotright>\<langle>BinOp CondOr e1 e2\<rangle>\<guillemotright> A'"
by (rules intro: da.CondOr)
thus ?case ..
next
case (BinOp A B E1 Env binop e1 e2 B')
have B': "B \<subseteq> B'" .
obtain E1' where E1': "Env\<turnstile> B' \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1'"
proof -
have "PROP ?Hyp Env B \<langle>e1\<rangle>" by (rule BinOp.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain A' where "Env\<turnstile> nrm E1' \<guillemotright>\<langle>e2\<rangle>\<guillemotright> A'"
proof -
have "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1" by (rule BinOp.hyps)
from this B' E1'
have "nrm E1 \<subseteq> nrm E1'"
by (rule da_monotone [THEN conjE])
moreover
have "PROP ?Hyp Env (nrm E1) \<langle>e2\<rangle>" by (rule BinOp.hyps)
ultimately show ?thesis using that by rules
qed
ultimately
have "Env\<turnstile> B' \<guillemotright>\<langle>BinOp binop e1 e2\<rangle>\<guillemotright> A'"
using BinOp.hyps by (rules intro: da.BinOp)
thus ?case ..
next
case (Super B Env B')
have B': "B \<subseteq> B'" .
with Super.hyps have "This \<in> B' "
by auto
thus ?case by (rules intro: da.Super)
next
case (AccLVar A B Env vn B')
have "vn \<in> B" .
moreover
have "B \<subseteq> B'" .
ultimately have "vn \<in> B'" by auto
thus ?case by (rules intro: da.AccLVar)
next
case Acc thus ?case by (rules intro: da.Acc)
next
case (AssLVar A B E Env e vn B')
have B': "B \<subseteq> B'" .
then obtain E' where "Env\<turnstile> B' \<guillemotright>\<langle>e\<rangle>\<guillemotright> E'"
by (rule AssLVar.hyps [elim_format]) rules
then obtain A' where
"Env\<turnstile> B' \<guillemotright>\<langle>LVar vn:=e\<rangle>\<guillemotright> A'"
by (rules intro: da.AssLVar)
thus ?case ..
next
case (Ass A B Env V e v B')
have B': "B \<subseteq> B'" .
have "\<forall>vn. v \<noteq> LVar vn".
moreover
obtain V' where V': "Env\<turnstile> B' \<guillemotright>\<langle>v\<rangle>\<guillemotright> V'"
proof -
have "PROP ?Hyp Env B \<langle>v\<rangle>" by (rule Ass.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain A' where "Env\<turnstile> nrm V' \<guillemotright>\<langle>e\<rangle>\<guillemotright> A'"
proof -
have "Env\<turnstile> B \<guillemotright>\<langle>v\<rangle>\<guillemotright> V" by (rule Ass.hyps)
from this B' V'
have "nrm V \<subseteq> nrm V'"
by (rule da_monotone [THEN conjE])
moreover
have "PROP ?Hyp Env (nrm V) \<langle>e\<rangle>" by (rule Ass.hyps)
ultimately show ?thesis using that by rules
qed
ultimately
have "Env\<turnstile> B' \<guillemotright>\<langle>v := e\<rangle>\<guillemotright> A'"
by (rules intro: da.Ass)
thus ?case ..
next
case (CondBool A B C E1 E2 Env c e1 e2 B')
have B': "B \<subseteq> B'" .
have "Env\<turnstile>(c ? e1 : e2)\<Colon>-(PrimT Boolean)" .
moreover obtain C' where C': "Env\<turnstile> B' \<guillemotright>\<langle>c\<rangle>\<guillemotright> C'"
proof -
have "PROP ?Hyp Env B \<langle>c\<rangle>" by (rule CondBool.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain E1' where "Env\<turnstile> B' \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1'"
proof -
from B'
have "(B \<union> assigns_if True c) \<subseteq> (B' \<union> assigns_if True c)"
by blast
moreover
have "PROP ?Hyp Env (B \<union> assigns_if True c) \<langle>e1\<rangle>" by (rule CondBool.hyps)
ultimately
show ?thesis using that by rules
qed
moreover
obtain E2' where "Env\<turnstile> B' \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2'"
proof -
from B'
have "(B \<union> assigns_if False c) \<subseteq> (B' \<union> assigns_if False c)"
by blast
moreover
have "PROP ?Hyp Env (B \<union> assigns_if False c) \<langle>e2\<rangle>" by(rule CondBool.hyps)
ultimately
show ?thesis using that by rules
qed
ultimately
obtain A' where "Env\<turnstile> B' \<guillemotright>\<langle>c ? e1 : e2\<rangle>\<guillemotright> A'"
by (rules intro: da.CondBool)
thus ?case ..
next
case (Cond A B C E1 E2 Env c e1 e2 B')
have B': "B \<subseteq> B'" .
have "\<not> Env\<turnstile>(c ? e1 : e2)\<Colon>-(PrimT Boolean)" .
moreover obtain C' where C': "Env\<turnstile> B' \<guillemotright>\<langle>c\<rangle>\<guillemotright> C'"
proof -
have "PROP ?Hyp Env B \<langle>c\<rangle>" by (rule Cond.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain E1' where "Env\<turnstile> B' \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1'"
proof -
from B'
have "(B \<union> assigns_if True c) \<subseteq> (B' \<union> assigns_if True c)"
by blast
moreover
have "PROP ?Hyp Env (B \<union> assigns_if True c) \<langle>e1\<rangle>" by (rule Cond.hyps)
ultimately
show ?thesis using that by rules
qed
moreover
obtain E2' where "Env\<turnstile> B' \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2'"
proof -
from B'
have "(B \<union> assigns_if False c) \<subseteq> (B' \<union> assigns_if False c)"
by blast
moreover
have "PROP ?Hyp Env (B \<union> assigns_if False c) \<langle>e2\<rangle>" by (rule Cond.hyps)
ultimately
show ?thesis using that by rules
qed
ultimately
obtain A' where "Env\<turnstile> B' \<guillemotright>\<langle>c ? e1 : e2\<rangle>\<guillemotright> A'"
by (rules intro: da.Cond)
thus ?case ..
next
case (Call A B E Env accC args e mn mode pTs statT B')
have B': "B \<subseteq> B'" .
obtain E' where E': "Env\<turnstile> B' \<guillemotright>\<langle>e\<rangle>\<guillemotright> E'"
proof -
have "PROP ?Hyp Env B \<langle>e\<rangle>" by (rule Call.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain A' where "Env\<turnstile> nrm E' \<guillemotright>\<langle>args\<rangle>\<guillemotright> A'"
proof -
have "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" by (rule Call.hyps)
from this B' E'
have "nrm E \<subseteq> nrm E'"
by (rule da_monotone [THEN conjE])
moreover
have "PROP ?Hyp Env (nrm E) \<langle>args\<rangle>" by (rule Call.hyps)
ultimately show ?thesis using that by rules
qed
ultimately
have "Env\<turnstile> B' \<guillemotright>\<langle>{accC,statT,mode}e\<cdot>mn( {pTs}args)\<rangle>\<guillemotright> A'"
by (rules intro: da.Call)
thus ?case ..
next
case Methd thus ?case by (rules intro: da.Methd)
next
case (Body A B C D Env c B')
have B': "B \<subseteq> B'" .
obtain C' where C': "Env\<turnstile> B' \<guillemotright>\<langle>c\<rangle>\<guillemotright> C'" and nrm_C': "nrm C \<subseteq> nrm C'"
proof -
have "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C" by (rule Body.hyps)
moreover note B'
moreover
from B' obtain C' where da_c: "Env\<turnstile> B' \<guillemotright>\<langle>c\<rangle>\<guillemotright> C'"
by (rule Body.hyps [elim_format]) blast
ultimately
have "nrm C \<subseteq> nrm C'"
by (rule da_monotone [THEN conjE])
with da_c that show ?thesis by rules
qed
moreover
have "Result \<in> nrm C" .
with nrm_C' have "Result \<in> nrm C'"
by blast
moreover have "jumpNestingOkS {Ret} c" .
ultimately obtain A' where
"Env\<turnstile> B' \<guillemotright>\<langle>Body D c\<rangle>\<guillemotright> A'"
by (rules intro: da.Body)
thus ?case ..
next
case LVar thus ?case by (rules intro: da.LVar)
next
case FVar thus ?case by (rules intro: da.FVar)
next
case (AVar A B E1 Env e1 e2 B')
have B': "B \<subseteq> B'" .
obtain E1' where E1': "Env\<turnstile> B' \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1'"
proof -
have "PROP ?Hyp Env B \<langle>e1\<rangle>" by (rule AVar.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain A' where "Env\<turnstile> nrm E1' \<guillemotright>\<langle>e2\<rangle>\<guillemotright> A'"
proof -
have "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1" by (rule AVar.hyps)
from this B' E1'
have "nrm E1 \<subseteq> nrm E1'"
by (rule da_monotone [THEN conjE])
moreover
have "PROP ?Hyp Env (nrm E1) \<langle>e2\<rangle>" by (rule AVar.hyps)
ultimately show ?thesis using that by rules
qed
ultimately
have "Env\<turnstile> B' \<guillemotright>\<langle>e1.[e2]\<rangle>\<guillemotright> A'"
by (rules intro: da.AVar)
thus ?case ..
next
case Nil thus ?case by (rules intro: da.Nil)
next
case (Cons A B E Env e es B')
have B': "B \<subseteq> B'" .
obtain E' where E': "Env\<turnstile> B' \<guillemotright>\<langle>e\<rangle>\<guillemotright> E'"
proof -
have "PROP ?Hyp Env B \<langle>e\<rangle>" by (rule Cons.hyps)
with B'
show ?thesis using that by rules
qed
moreover
obtain A' where "Env\<turnstile> nrm E' \<guillemotright>\<langle>es\<rangle>\<guillemotright> A'"
proof -
have "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" by (rule Cons.hyps)
from this B' E'
have "nrm E \<subseteq> nrm E'"
by (rule da_monotone [THEN conjE])
moreover
have "PROP ?Hyp Env (nrm E) \<langle>es\<rangle>" by (rule Cons.hyps)
ultimately show ?thesis using that by rules
qed
ultimately
have "Env\<turnstile> B' \<guillemotright>\<langle>e # es\<rangle>\<guillemotright> A'"
by (rules intro: da.Cons)
thus ?case ..
qed
qed

(* Remarks about the proof style:

"by (rule <Case>.hyps)" vs "."
--------------------------

with <Case>.hyps you state more precise were the rule comes from

. takes all assumptions into account, but looks more "light"
and is more resistent for cut and paste proof in different
cases.

"intro: da.intros" vs "da.<Case>"
---------------------------------
The first ist more convinient for cut and paste between cases,
*)

corollary da_weakenE [consumes 2]:
assumes          da: "Env\<turnstile> B  \<guillemotright>t\<guillemotright> A"   and
B': "B \<subseteq> B'"          and
ex_mono: "\<And> A'.  \<lbrakk>Env\<turnstile> B' \<guillemotright>t\<guillemotright> A'; nrm A \<subseteq> nrm A';
\<And> l. brk A l \<subseteq> brk A' l\<rbrakk> \<Longrightarrow> P"
shows "P"
proof -
from da B'
obtain A' where A': "Env\<turnstile> B' \<guillemotright>t\<guillemotright> A'"
by (rule da_weaken [elim_format]) rules
with da B'
have "nrm A \<subseteq> nrm A' \<and> (\<forall> l. brk A l \<subseteq> brk A' l)"
by (rule da_monotone)
with A' ex_mono
show ?thesis
by rules
qed

end