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theory DefiniteAssignmentheader {* Definite Assignment *} theory DefiniteAssignment imports WellType begin 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 => stmt => bool" primrec "jumpNestingOkS jmps (Skip) = True" "jumpNestingOkS jmps (Expr e) = True" "jumpNestingOkS jmps (j• s) = jumpNestingOkS ({j} ∪ jmps) s" "jumpNestingOkS jmps (c1;;c2) = (jumpNestingOkS jmps c1 ∧ jumpNestingOkS jmps c2)" "jumpNestingOkS jmps (If(e) c1 Else c2) = (jumpNestingOkS jmps c1 ∧ jumpNestingOkS jmps c2)" "jumpNestingOkS jmps (l• While(e) c) = jumpNestingOkS ({Cont l} ∪ jmps) c" --{* The label of the while loop only handles continue jumps. Breaks are only handled by @{term Lab} *} "jumpNestingOkS jmps (Jmp j) = (j ∈ jmps)" "jumpNestingOkS jmps (Throw e) = True" "jumpNestingOkS jmps (Try c1 Catch(C vn) c2) = (jumpNestingOkS jmps c1 ∧ jumpNestingOkS jmps c2)" "jumpNestingOkS jmps (c1 Finally c2) = (jumpNestingOkS jmps c1 ∧ 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 => term => bool" "jumpNestingOk jmps t ≡ (case t of In1 se => (case se of Inl e => True | Inr s => jumpNestingOkS jmps s) | In2 v => True | In3 es => True)" lemma jumpNestingOk_expr_simp [simp]: "jumpNestingOk jmps (In1l e) = True" by (simp add: jumpNestingOk_def) lemma jumpNestingOk_expr_simp1 [simp]: "jumpNestingOk jmps 〈e::expr〉 = True" by (simp add: inj_term_simps) lemma jumpNestingOk_stmt_simp [simp]: "jumpNestingOk jmps (In1r s) = jumpNestingOkS jmps s" by (simp add: jumpNestingOk_def) lemma jumpNestingOk_stmt_simp1 [simp]: "jumpNestingOk jmps 〈s::stmt〉 = jumpNestingOkS jmps s" by (simp add: inj_term_simps) lemma jumpNestingOk_var_simp [simp]: "jumpNestingOk jmps (In2 v) = True" by (simp add: jumpNestingOk_def) lemma jumpNestingOk_var_simp1 [simp]: "jumpNestingOk jmps 〈v::var〉 = True" by (simp add: inj_term_simps) lemma jumpNestingOk_expr_list_simp [simp]: "jumpNestingOk jmps (In3 es) = True" by (simp add: jumpNestingOk_def) lemma jumpNestingOk_expr_list_simp1 [simp]: "jumpNestingOk jmps 〈es::expr list〉 = True" by (simp add: inj_term_simps) section {* Calculation of assigned variables for boolean expressions*} subsection {* Very restricted calculation fallback calculation *} consts the_LVar_name:: "var => lname" primrec "the_LVar_name (LVar n) = n" consts assignsE :: "expr => lname set" assignsV :: "var => lname set" assignsEs:: "expr list => 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 ∨ binop=CondOr then (assignsE e1) else (assignsE e1) ∪ (assignsE e2))" "assignsE (Super) = {}" "assignsE (Acc v) = assignsV v" "assignsE (v:=e) = (assignsV v) ∪ (assignsE e) ∪ (if ∃ n. v=(LVar n) then {the_LVar_name v} else {})" "assignsE (b? e1 : e2) = (assignsE b) ∪ ((assignsE e1) ∩ (assignsE e2))" "assignsE ({accC,statT,mode}objRef·mn({pTs}args)) = (assignsE objRef) ∪ (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 ∪ assignsE e2" "assignsEs [] = {}" "assignsEs (e#es) = assignsE e ∪ assignsEs es" constdefs assigns:: "term => lname set" "assigns t ≡ (case t of In1 se => (case se of Inl e => assignsE e | Inr s => {}) | In2 v => assignsV v | In3 es => assignsEs es)" lemma assigns_expr_simp [simp]: "assigns (In1l e) = assignsE e" by (simp add: assigns_def) lemma assigns_expr_simp1 [simp]: "assigns (〈e〉) = assignsE e" by (simp add: inj_term_simps) lemma assigns_stmt_simp [simp]: "assigns (In1r s) = {}" by (simp add: assigns_def) lemma assigns_stmt_simp1 [simp]: "assigns (〈s::stmt〉) = {}" by (simp add: inj_term_simps) lemma assigns_var_simp [simp]: "assigns (In2 v) = assignsV v" by (simp add: assigns_def) lemma assigns_var_simp1 [simp]: "assigns (〈v〉) = assignsV v" by (simp add: inj_term_simps) lemma assigns_expr_list_simp [simp]: "assigns (In3 es) = assignsEs es" by (simp add: assigns_def) lemma assigns_expr_list_simp1 [simp]: "assigns (〈es〉) = assignsEs es" by (simp add: inj_term_simps) subsection "Analysis of constant expressions" consts constVal :: "expr => 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 => None | Some v => Some (eval_unop unop v))" "constVal (BinOp binop e1 e2) = (case (constVal e1) of None => None | Some v1 => (case (constVal e2) of None => None | Some v2 => 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 => None | Some bv=> (case the_Bool bv of True => (case (constVal e2) of None => None | Some v => constVal e1) | False=> (case (constVal e1) of None => None | Some v => 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: "!! v. P (Lit v)" and hyp_UnOp: "!! unop e'. P e' ==> P (UnOp unop e')" and hyp_BinOp: "!! binop e1 e2. [|P e1; P e2|] ==> P (BinOp binop e1 e2)" and hyp_CondL: "!! b bv e1 e2. [|constVal b = Some bv; the_Bool bv; P b; P e1|] ==> P (b? e1 : e2)" and hyp_CondR: "!! b bv e1 e2. [|constVal b = Some bv; ¬the_Bool bv; P b; P e2|] ==> P (b? e1 : e2)" shows "P e" proof - have "True" and "!! v. constVal e = Some v ==> 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 ==> 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 => expr => 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 => (if unop = UNot then assigns_if (¬b) e else UNIV) | Some v => (if v=Bool b then {} else UNIV))" "assigns_if b (BinOp binop e1 e2) = (case constVal (BinOp binop e1 e2) of None => (if binop=CondAnd then (case b of True => assigns_if True e1 ∪ assigns_if True e2 | False => assigns_if False e1 ∩ (assigns_if True e1 ∪ assigns_if False e2)) else (if binop=CondOr then (case b of True => assigns_if True e1 ∩ (assigns_if False e1 ∪ assigns_if True e2) | False => assigns_if False e1 ∪ assigns_if False e2) else assignsE e1 ∪ assignsE e2)) | Some v => (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) ∪ (case (constVal c) of None => (assigns_if b e1) ∩ (assigns_if b e2) | Some bv => (case the_Bool bv of True => assigns_if b e1 | False => assigns_if b e2))" "assigns_if b ({accC,statT,mode}objRef·mn({pTs}args)) = assignsE ({accC,statT,mode}objRef·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 "!! b. ?Const b e ==> ?Ass b e" and "True" and "True" proof (induct _ and e and _ and _ rule: var_expr_stmt.inducts) 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) note hyp_c = `!! b. ?Const b c ==> ?Ass b c` note hyp_e1 = `!! b. ?Const b e1 ==> ?Ass b e1` note hyp_e2 = `!! b. ?Const b e2 ==> ?Ass b e2` note 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 (¬b) e = UNIV" (is "?Ass b e") proof - have True and "!! b. ?Const b e ==> ?Ass b e" and True and True proof (induct _ and e and _ and _ rule: var_expr_stmt.inducts) 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) note hyp_c = `!! b. ?Const b c ==> ?Ass b c` note hyp_e1 = `!! b. ?Const b e1 ==> ?Ass b e1` note hyp_e2 = `!! b. ?Const b e2 ==> ?Ass b e2` note 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 => ('a,'b) tables => ('a,'b) tables" ("_ =>∪ _" [67,67] 65) "A =>∪ B ≡ λ k. A k ∪ B k" constdefs intersect_ts:: "('a,'b) tables => ('a,'b) tables => ('a,'b) tables" ("_ =>∩ _" [72,72] 71) "A =>∩ B ≡ λ k. A k ∩ B k" constdefs all_union_ts:: "('a,'b) tables => 'b set => ('a,'b) tables" (infixl "=>∪∀" 40) "A =>∪∀ B ≡ λ k. A k ∪ B" subsubsection {* Binary union of tables *} lemma union_ts_iff [simp]: "(c ∈ (A =>∪ B) k) = (c ∈ A k ∨ c ∈ B k)" by (unfold union_ts_def) blast lemma union_tsI1 [elim?]: "c ∈ A k ==> c ∈ (A =>∪ B) k" by simp lemma union_tsI2 [elim?]: "c ∈ B k ==> c ∈ (A =>∪ B) k" by simp lemma union_tsCI [intro!]: "(c ∉ B k ==> c ∈ A k) ==> c ∈ (A =>∪ B) k" by auto lemma union_tsE [elim!]: "[|c ∈ (A =>∪ B) k; (c ∈ A k ==> P); (c ∈ B k ==> P)|] ==> P" by (unfold union_ts_def) blast subsubsection {* Binary intersection of tables *} lemma intersect_ts_iff [simp]: "c ∈ (A =>∩ B) k = (c ∈ A k ∧ c ∈ B k)" by (unfold intersect_ts_def) blast lemma intersect_tsI [intro!]: "[|c ∈ A k; c ∈ B k|] ==> c ∈ (A =>∩ B) k" by simp lemma intersect_tsD1: "c ∈ (A =>∩ B) k ==> c ∈ A k" by simp lemma intersect_tsD2: "c ∈ (A =>∩ B) k ==> c ∈ B k" by simp lemma intersect_tsE [elim!]: "[|c ∈ (A =>∩ B) k; [|c ∈ A k; c ∈ B k|] ==> P|] ==> P" by simp subsubsection {* All-Union of tables and set *} lemma all_union_ts_iff [simp]: "(c ∈ (A =>∪∀ B) k) = (c ∈ A k ∨ c ∈ B)" by (unfold all_union_ts_def) blast lemma all_union_tsI1 [elim?]: "c ∈ A k ==> c ∈ (A =>∪∀ B) k" by simp lemma all_union_tsI2 [elim?]: "c ∈ B ==> c ∈ (A =>∪∀ B) k" by simp lemma all_union_tsCI [intro!]: "(c ∉ B ==> c ∈ A k) ==> c ∈ (A =>∪∀ B) k" by auto lemma all_union_tsE [elim!]: "[|c ∈ (A =>∪∀ B) k; (c ∈ A k ==> P); (c ∈ B ==> P)|] ==> 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 abrupt completion with a break *} constdefs rmlab :: "'a => ('a,'b) tables => ('a,'b) tables" "rmlab k A ≡ λ x. if x=k then UNIV else A x" (* constdefs setbrk :: "breakass => assigned => breakass set" "setbrk b A ≡ {b} ∪ {a| a. a∈ brk A ∧ lab a ≠ lab b}" *) constdefs range_inter_ts :: "('a,'b) tables => 'b set" ("=>\<Inter>_" 80) "=>\<Inter>A ≡ {x |x. ∀ k. x ∈ A k}" text {* In @{text "E\<turnstile> B »t» A"}, @{text B} denotes the ''assigned'' variables before evaluating term @{text t}, whereas @{text A} denotes the ''assigned'' variables after evaluating term @{text t}. The environment @{term E} is only needed for the conditional @{text "_ ? _ : _"}. The definite assignment rules refer to the typing rules here to distinguish boolean and other expressions. *} inductive da :: "env => lname set => term => assigned => bool" ("_\<turnstile> _ »_» _" [65,65,65,65] 71) where Skip: "Env\<turnstile> B »〈Skip〉» (|nrm=B,brk=λ l. UNIV|))," | Expr: "Env\<turnstile> B »〈e〉» A ==> Env\<turnstile> B »〈Expr e〉» A" | Lab: "[|Env\<turnstile> B »〈c〉» C; nrm A = nrm C ∩ (brk C) l; brk A = rmlab l (brk C)|] ==> Env\<turnstile> B »〈Break l• c〉» A" | Comp: "[|Env\<turnstile> B »〈c1〉» C1; Env\<turnstile> nrm C1 »〈c2〉» C2; nrm A = nrm C2; brk A = (brk C1) =>∩ (brk C2)|] ==> Env\<turnstile> B »〈c1;; c2〉» A" | If: "[|Env\<turnstile> B »〈e〉» E; Env\<turnstile> (B ∪ assigns_if True e) »〈c1〉» C1; Env\<turnstile> (B ∪ assigns_if False e) »〈c2〉» C2; nrm A = nrm C1 ∩ nrm C2; brk A = brk C1 =>∩ brk C2 |] ==> Env\<turnstile> B »〈If(e) c1 Else c2〉» 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 »〈e〉» E"} is just used to ensure 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: "[|Env\<turnstile> B »〈e〉» E; Env\<turnstile> (B ∪ assigns_if True e) »〈c〉» C; nrm A = nrm C ∩ (B ∪ assigns_if False e); brk A = brk C|] ==> Env\<turnstile> B »〈l• While(e) c〉» A" --{* The @{text Loop} rule resembles some of the ideas of the @{text If} rule. For the @{term "nrm A"} the set @{term "B ∪ 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: "[|jump=Ret --> Result ∈ B; nrm A = UNIV; brk A = (case jump of Break l => λ k. if k=l then B else UNIV | Cont l => λ k. UNIV | Ret => λ k. UNIV)|] ==> Env\<turnstile> B »〈Jmp jump〉» 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: "[|Env\<turnstile> B »〈e〉» E; nrm A = UNIV; brk A = (λ l. UNIV)|] ==> Env\<turnstile> B »〈Throw e〉» A" | Try: "[|Env\<turnstile> B »〈c1〉» C1; Env(|lcl := lcl Env(VName vn\<mapsto>Class C)|)),\<turnstile> (B ∪ {VName vn}) »〈c2〉» C2; nrm A = nrm C1 ∩ nrm C2; brk A = brk C1 =>∩ brk C2|] ==> Env\<turnstile> B »〈Try c1 Catch(C vn) c2〉» A" | Fin: "[|Env\<turnstile> B »〈c1〉» C1; Env\<turnstile> B »〈c2〉» C2; nrm A = nrm C1 ∪ nrm C2; brk A = ((brk C1) =>∪∀ (nrm C2)) =>∩ (brk C2)|] ==> Env\<turnstile> B »〈c1 Finally c2〉» 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 abruptly 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 "=>∪∀"}). 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 "λ 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 »〈Init C〉» (|nrm=B,brk=λ l. UNIV|))," --{* 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 »〈NewC C〉» (|nrm=B,brk=λ l. UNIV|))," | NewA: "Env\<turnstile> B »〈e〉» A ==> Env\<turnstile> B »〈New T[e]〉» A" | Cast: "Env\<turnstile> B »〈e〉» A ==> Env\<turnstile> B »〈Cast T e〉» A" | Inst: "Env\<turnstile> B »〈e〉» A ==> Env\<turnstile> B »〈e InstOf T〉» A" | Lit: "Env\<turnstile> B »〈Lit v〉» (|nrm=B,brk=λ l. UNIV|))," | UnOp: "Env\<turnstile> B »〈e〉» A ==> Env\<turnstile> B »〈UnOp unop e〉» A" | CondAnd: "[|Env\<turnstile> B »〈e1〉» E1; Env\<turnstile> (B ∪ assigns_if True e1) »〈e2〉» E2; nrm A = B ∪ (assigns_if True (BinOp CondAnd e1 e2) ∩ assigns_if False (BinOp CondAnd e1 e2)); brk A = (λ l. UNIV) |] ==> Env\<turnstile> B »〈BinOp CondAnd e1 e2〉» A" | CondOr: "[|Env\<turnstile> B »〈e1〉» E1; Env\<turnstile> (B ∪ assigns_if False e1) »〈e2〉» E2; nrm A = B ∪ (assigns_if True (BinOp CondOr e1 e2) ∩ assigns_if False (BinOp CondOr e1 e2)); brk A = (λ l. UNIV) |] ==> Env\<turnstile> B »〈BinOp CondOr e1 e2〉» A" | BinOp: "[|Env\<turnstile> B »〈e1〉» E1; Env\<turnstile> nrm E1 »〈e2〉» A; binop ≠ CondAnd; binop ≠ CondOr|] ==> Env\<turnstile> B »〈BinOp binop e1 e2〉» A" | Super: "This ∈ B ==> Env\<turnstile> B »〈Super〉» (|nrm=B,brk=λ l. UNIV|))," | AccLVar: "[|vn ∈ B; nrm A = B; brk A = (λ k. UNIV)|] ==> Env\<turnstile> B »〈Acc (LVar vn)〉» 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: "[|∀ vn. v ≠ LVar vn; Env\<turnstile> B »〈v〉» A|] ==> Env\<turnstile> B »〈Acc v〉» A" | AssLVar: "[|Env\<turnstile> B »〈e〉» E; nrm A = nrm E ∪ {vn}; brk A = brk E|] ==> Env\<turnstile> B »〈(LVar vn) := e〉» A" | Ass: "[|∀ vn. v ≠ LVar vn; Env\<turnstile> B »〈v〉» V; Env\<turnstile> nrm V »〈e〉» A|] ==> Env\<turnstile> B »〈v := e〉» A" | CondBool: "[|Env\<turnstile>(c ? e1 : e2)::-(PrimT Boolean); Env\<turnstile> B »〈c〉» C; Env\<turnstile> (B ∪ assigns_if True c) »〈e1〉» E1; Env\<turnstile> (B ∪ assigns_if False c) »〈e2〉» E2; nrm A = B ∪ (assigns_if True (c ? e1 : e2) ∩ assigns_if False (c ? e1 : e2)); brk A = (λ l. UNIV)|] ==> Env\<turnstile> B »〈c ? e1 : e2〉» A" | Cond: "[|¬ Env\<turnstile>(c ? e1 : e2)::-(PrimT Boolean); Env\<turnstile> B »〈c〉» C; Env\<turnstile> (B ∪ assigns_if True c) »〈e1〉» E1; Env\<turnstile> (B ∪ assigns_if False c) »〈e2〉» E2; nrm A = nrm E1 ∩ nrm E2; brk A = (λ l. UNIV)|] ==> Env\<turnstile> B »〈c ? e1 : e2〉» A" | Call: "[|Env\<turnstile> B »〈e〉» E; Env\<turnstile> nrm E »〈args〉» A|] ==> Env\<turnstile> B »〈{accC,statT,mode}e·mn({pTs}args)〉» 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: "[|methd (prg Env) D sig = Some m; Env\<turnstile> B »〈Body (declclass m) (stmt (mbody (mthd m)))〉» A |] ==> Env\<turnstile> B »〈Methd D sig〉» A" | Body: "[|Env\<turnstile> B »〈c〉» C; jumpNestingOkS {Ret} c; Result ∈ nrm C; nrm A = B; brk A = (λ l. UNIV)|] ==> Env\<turnstile> B »〈Body D c〉» 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 »〈LVar vn〉» (|nrm=B, brk=λ l. UNIV|))," | FVar: "Env\<turnstile> B »〈e〉» A ==> Env\<turnstile> B »〈{accC,statDeclC,stat}e..fn〉» A" | AVar: "[|Env\<turnstile> B »〈e1〉» E1; Env\<turnstile> nrm E1 »〈e2〉» A|] ==> Env\<turnstile> B »〈e1.[e2]〉» A" | Nil: "Env\<turnstile> B »〈[]::expr list〉» (|nrm=B, brk=λ l. UNIV|))," | Cons: "[|Env\<turnstile> B »〈e::expr〉» E; Env\<turnstile> nrm E »〈es〉» A|] ==> Env\<turnstile> B »〈e#es〉» A" declare inj_term_sym_simps [simp] declare assigns_if.simps [simp del] declare split_paired_All [simp del] split_paired_Ex [simp del] declaration {* K (Simplifier.map_ss (fn ss => ss delloop "split_all_tac")) *} inductive_cases da_elim_cases [cases set]: "Env\<turnstile> B »〈Skip〉» A" "Env\<turnstile> B »In1r Skip» A" "Env\<turnstile> B »〈Expr e〉» A" "Env\<turnstile> B »In1r (Expr e)» A" "Env\<turnstile> B »〈l• c〉» A" "Env\<turnstile> B »In1r (l• c)» A" "Env\<turnstile> B »〈c1;; c2〉» A" "Env\<turnstile> B »In1r (c1;; c2)» A" "Env\<turnstile> B »〈If(e) c1 Else c2〉» A" "Env\<turnstile> B »In1r (If(e) c1 Else c2)» A" "Env\<turnstile> B »〈l• While(e) c〉» A" "Env\<turnstile> B »In1r (l• While(e) c)» A" "Env\<turnstile> B »〈Jmp jump〉» A" "Env\<turnstile> B »In1r (Jmp jump)» A" "Env\<turnstile> B »〈Throw e〉» A" "Env\<turnstile> B »In1r (Throw e)» A" "Env\<turnstile> B »〈Try c1 Catch(C vn) c2〉» A" "Env\<turnstile> B »In1r (Try c1 Catch(C vn) c2)» A" "Env\<turnstile> B »〈c1 Finally c2〉» A" "Env\<turnstile> B »In1r (c1 Finally c2)» A" "Env\<turnstile> B »〈Init C〉» A" "Env\<turnstile> B »In1r (Init C)» A" "Env\<turnstile> B »〈NewC C〉» A" "Env\<turnstile> B »In1l (NewC C)» A" "Env\<turnstile> B »〈New T[e]〉» A" "Env\<turnstile> B »In1l (New T[e])» A" "Env\<turnstile> B »〈Cast T e〉» A" "Env\<turnstile> B »In1l (Cast T e)» A" "Env\<turnstile> B »〈e InstOf T〉» A" "Env\<turnstile> B »In1l (e InstOf T)» A" "Env\<turnstile> B »〈Lit v〉» A" "Env\<turnstile> B »In1l (Lit v)» A" "Env\<turnstile> B »〈UnOp unop e〉» A" "Env\<turnstile> B »In1l (UnOp unop e)» A" "Env\<turnstile> B »〈BinOp binop e1 e2〉» A" "Env\<turnstile> B »In1l (BinOp binop e1 e2)» A" "Env\<turnstile> B »〈Super〉» A" "Env\<turnstile> B »In1l (Super)» A" "Env\<turnstile> B »〈Acc v〉» A" "Env\<turnstile> B »In1l (Acc v)» A" "Env\<turnstile> B »〈v := e〉» A" "Env\<turnstile> B »In1l (v := e)» A" "Env\<turnstile> B »〈c ? e1 : e2〉» A" "Env\<turnstile> B »In1l (c ? e1 : e2)» A" "Env\<turnstile> B »〈{accC,statT,mode}e·mn({pTs}args)〉» A" "Env\<turnstile> B »In1l ({accC,statT,mode}e·mn({pTs}args))» A" "Env\<turnstile> B »〈Methd C sig〉» A" "Env\<turnstile> B »In1l (Methd C sig)» A" "Env\<turnstile> B »〈Body D c〉» A" "Env\<turnstile> B »In1l (Body D c)» A" "Env\<turnstile> B »〈LVar vn〉» A" "Env\<turnstile> B »In2 (LVar vn)» A" "Env\<turnstile> B »〈{accC,statDeclC,stat}e..fn〉» A" "Env\<turnstile> B »In2 ({accC,statDeclC,stat}e..fn)» A" "Env\<turnstile> B »〈e1.[e2]〉» A" "Env\<turnstile> B »In2 (e1.[e2])» A" "Env\<turnstile> B »〈[]::expr list〉» A" "Env\<turnstile> B »In3 ([]::expr list)» A" "Env\<turnstile> B »〈e#es〉» A" "Env\<turnstile> B »In3 (e#es)» A" declare inj_term_sym_simps [simp del] declare assigns_if.simps [simp] declare split_paired_All [simp] split_paired_Ex [simp] declaration {* K (Simplifier.map_ss (fn ss => ss addloop ("split_all_tac", split_all_tac))) *} (* To be able to eliminate both the versions with the overloaded brackets: (B »〈Skip〉» A) and with the explicit constructor (B »In1r Skip» A), every rule appears in both versions *) lemma da_Skip: "A = (|nrm=B,brk=λ l. UNIV|)), ==> Env\<turnstile> B »〈Skip〉» A" by (auto intro: da.Skip) lemma da_NewC: "A = (|nrm=B,brk=λ l. UNIV|)), ==> Env\<turnstile> B »〈NewC C〉» A" by (auto intro: da.NewC) lemma da_Lit: "A = (|nrm=B,brk=λ l. UNIV|)), ==> Env\<turnstile> B »〈Lit v〉» A" by (auto intro: da.Lit) lemma da_Super: "[|This ∈ B;A = (|nrm=B,brk=λ l. UNIV|)),|] ==> Env\<turnstile> B »〈Super〉» A" by (auto intro: da.Super) lemma da_Init: "A = (|nrm=B,brk=λ l. UNIV|)), ==> Env\<turnstile> B »〈Init C〉» A" by (auto intro: da.Init) (* For boolean expressions: The following holds: "assignsE e ⊆ assigns_if True e ∩ assigns_if False e" but not vice versa: "assigns_if True e ∩ assigns_if False e ⊆ 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 ∩ (assigns_if False a ∪ assigns_if True b) = {} ∩ ({} ∪ {y}) = {} assigns_if False (a || b) = assigns_if False a ∪ assigns_if False b = {} ∪ {y} = {y} assigns_ifE True e = assigns_if True (a || b) ∪ assigns_if True c = {} ∪ {y} = {y} assigns_ifE False e = assigns_if False (a || b) ∩ (assigns_if True (a || b) ∪ assigns_if False c) = {y} ∩ ({} ∪ {y}) = {y} assignsE e = {} *) lemma assignsE_subseteq_assigns_ifs: assumes boolEx: "E\<turnstile>e::-PrimT Boolean" (is "?Boolean e") shows "assignsE e ⊆ assigns_if True e ∩ assigns_if False e" (is "?Incl e") proof - have True and "?Boolean e ==> ?Incl e" and True and True proof (induct _ and e and _ and _ rule: var_expr_stmt.inducts) case (Cast T e) have "E\<turnstile>e::- (PrimT Boolean)" proof - from `E\<turnstile>(Cast T e)::- (PrimT Boolean)` obtain Te where "E\<turnstile>e::-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, (fastsimp simp add: assignsE_const_simp)+) next case (BinOp binop e1 e2) from BinOp.prems obtain e1T e2T where "E\<turnstile>e1::-e1T" and "E\<turnstile>e2::-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) note hyp_c = `?Boolean c ==> ?Incl c` note hyp_e1 = `?Boolean e1 ==> ?Incl e1` note hyp_e2 = `?Boolean e2 ==> ?Incl e2` note wt = `E\<turnstile>(c ? e1 : e2)::-PrimT Boolean` then obtain boolean_c: "E\<turnstile>c::-PrimT Boolean" and boolean_e1: "E\<turnstile>e1::-PrimT Boolean" and boolean_e2: "E\<turnstile>e2::-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 »〈Skip〉» A" and the current goal state as an concrete injection: e.g: "B »In1r Skip» 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" by (simp add: rmlab_def) lemma rmlab_same_label1 [simp]: "l=l' ==> (rmlab l A) l' = UNIV" by (simp add: rmlab_def) lemma rmlab_other_label [simp]: "l≠l'==> (rmlab l A) l' = A l'" by (auto simp add: rmlab_def) lemma range_inter_ts_subseteq [intro]: "∀ k. A k ⊆ B k ==> =>\<Inter>A ⊆ =>\<Inter>B" by (auto simp add: range_inter_ts_def) lemma range_inter_ts_subseteq': "[|∀ k. A k ⊆ B k; x ∈ =>\<Inter>A|] ==> x ∈ =>\<Inter>B" by (auto simp add: range_inter_ts_def) lemma da_monotone: assumes da: "Env\<turnstile> B »t» A" and "B ⊆ B'" and da': "Env\<turnstile> B' »t» A'" shows "(nrm A ⊆ nrm A') ∧ (∀ l. (brk A l ⊆ brk A' l))" proof - from da show "!! B' A'. [|Env\<turnstile> B' »t» A'; B ⊆ B'|] ==> (nrm A ⊆ nrm A') ∧ (∀ l. (brk A l ⊆ 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 Env B c C A l B' A') note A = `nrm A = nrm C ∩ brk C l` `brk A = rmlab l (brk C)` note `PROP ?Hyp Env B 〈c〉 C` moreover note `B ⊆ B'` moreover obtain C' where "Env\<turnstile> B' »〈c〉» C'" and A': "nrm A' = nrm C' ∩ brk C' l" "brk A' = rmlab l (brk C')" using Lab.prems by - (erule da_elim_cases,simp) ultimately have "nrm C ⊆ nrm C'" and hyp_brk: "(∀l. brk C l ⊆ brk C' l)" by auto then have "nrm C ∩ brk C l ⊆ nrm C' ∩ brk C' l" by auto moreover { fix l' from hyp_brk have "rmlab l (brk C) l' ⊆ rmlab l (brk C') l'" by (cases "l=l'") simp_all } moreover note A A' ultimately show ?case by simp next case (Comp Env B c1 C1 c2 C2 A B' A') note A = `nrm A = nrm C2` `brk A = brk C1 =>∩ brk C2` from `Env\<turnstile> B' »〈c1;; c2〉» A'` obtain C1' C2' where da_c1: "Env\<turnstile> B' »〈c1〉» C1'" and da_c2: "Env\<turnstile> nrm C1' »〈c2〉» C2'" and A': "nrm A' = nrm C2'" "brk A' = brk C1' =>∩ brk C2'" by (rule da_elim_cases) auto note `PROP ?Hyp Env B 〈c1〉 C1` moreover note `B ⊆ B'` moreover note da_c1 ultimately have C1': "nrm C1 ⊆ nrm C1'" "(∀l. brk C1 l ⊆ brk C1' l)" by auto note `PROP ?Hyp Env (nrm C1) 〈c2〉 C2` with da_c2 C1' have C2': "nrm C2 ⊆ nrm C2'" "(∀l. brk C2 l ⊆ brk C2' l)" by auto with A A' C1' show ?case by auto next case (If Env B e E c1 C1 c2 C2 A B' A') note A = `nrm A = nrm C1 ∩ nrm C2` `brk A = brk C1 =>∩ brk C2` from `Env\<turnstile> B' »〈If(e) c1 Else c2〉» A'` obtain C1' C2' where da_c1: "Env\<turnstile> B' ∪ assigns_if True e »〈c1〉» C1'" and da_c2: "Env\<turnstile> B' ∪ assigns_if False e »〈c2〉» C2'" and A': "nrm A' = nrm C1' ∩ nrm C2'" "brk A' = brk C1' =>∩ brk C2'" by (rule da_elim_cases) auto note `PROP ?Hyp Env (B ∪ assigns_if True e) 〈c1〉 C1` moreover note B' = `B ⊆ B'` moreover note da_c1 ultimately obtain C1': "nrm C1 ⊆ nrm C1'" "(∀l. brk C1 l ⊆ brk C1' l)" by blast note `PROP ?Hyp Env (B ∪ assigns_if False e) 〈c2〉 C2` with da_c2 B' obtain C2': "nrm C2 ⊆ nrm C2'" "(∀l. brk C2 l ⊆ brk C2' l)" by blast with A A' C1' show ?case by auto next case (Loop Env B e E c C A l B' A') note A = `nrm A = nrm C ∩ (B ∪ assigns_if False e)` `brk A = brk C` from `Env\<turnstile> B' »〈l• While(e) c〉» A'` obtain C' where da_c': "Env\<turnstile> B' ∪ assigns_if True e »〈c〉» C'" and A': "nrm A' = nrm C' ∩ (B' ∪ assigns_if False e)" "brk A' = brk C'" by (rule da_elim_cases) auto note `PROP ?Hyp Env (B ∪ assigns_if True e) 〈c〉 C` moreover note B' = `B ⊆ B'` moreover note da_c' ultimately obtain C': "nrm C ⊆ nrm C'" "(∀l. brk C l ⊆ brk C' l)" by blast with A A' B' have "nrm A ⊆ nrm A'" by blast moreover { fix l' have "brk A l' ⊆ 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 jump B A Env 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 Env B c1 C1 vn C c2 C2 A B' A') note A = `nrm A = nrm C1 ∩ nrm C2` `brk A = brk C1 =>∩ brk C2` from `Env\<turnstile> B' »〈Try c1 Catch(C vn) c2〉» A'` obtain C1' C2' where da_c1': "Env\<turnstile> B' »〈c1〉» C1'" and da_c2': "Env(|lcl := lcl Env(VName vn\<mapsto>Class C)|)),\<turnstile> B' ∪ {VName vn} »〈c2〉» C2'" and A': "nrm A' = nrm C1' ∩ nrm C2'" "brk A' = brk C1' =>∩ brk C2'" by (rule da_elim_cases) auto note `PROP ?Hyp Env B 〈c1〉 C1` moreover note B' = `B ⊆ B'` moreover note da_c1' ultimately obtain C1': "nrm C1 ⊆ nrm C1'" "(∀l. brk C1 l ⊆ brk C1' l)" by blast note `PROP ?Hyp (Env(|lcl := lcl Env(VName vn\<mapsto>Class C)|)),) (B ∪ {VName vn}) 〈c2〉 C2` with B' da_c2' obtain "nrm C2 ⊆ nrm C2'" "(∀l. brk C2 l ⊆ brk C2' l)" by blast with C1' A A' show ?case by auto next case (Fin Env B c1 C1 c2 C2 A B' A') note A = `nrm A = nrm C1 ∪ nrm C2` `brk A = (brk C1 =>∪∀ nrm C2) =>∩ (brk C2)` from `Env\<turnstile> B' »〈c1 Finally c2〉» A'` obtain C1' C2' where da_c1': "Env\<turnstile> B' »〈c1〉» C1'" and da_c2': "Env\<turnstile> B' »〈c2〉» C2'" and A': "nrm A' = nrm C1' ∪ nrm C2'" "brk A' = (brk C1' =>∪∀ nrm C2') =>∩ (brk C2')" by (rule da_elim_cases) auto note `PROP ?Hyp Env B 〈c1〉 C1` moreover note B' = `B ⊆ B'` moreover note da_c1' ultimately obtain C1': "nrm C1 ⊆ nrm C1'" "(∀l. brk C1 l ⊆ brk C1' l)" by blast note hyp_c2 = `PROP ?Hyp Env B 〈c2〉 C2` from da_c2' B' obtain "nrm C2 ⊆ nrm C2'" "(∀l. brk C2 l ⊆ 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 Env B e1 E1 e2 E2 A B' A') note A = `nrm A = B ∪ assigns_if True (BinOp CondAnd e1 e2) ∩ assigns_if False (BinOp CondAnd e1 e2)` `brk A = (λl. UNIV)` from `Env\<turnstile> B' »〈BinOp CondAnd e1 e2〉» A'` obtain A': "nrm A' = B' ∪ assigns_if True (BinOp CondAnd e1 e2) ∩ assigns_if False (BinOp CondAnd e1 e2)" "brk A' = (λl. UNIV)" by (rule da_elim_cases) auto note B' = `B ⊆ 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 Env c e1 e2 B C E1 E2 A B' A') note A = `nrm A = B ∪ assigns_if True (c ? e1 : e2) ∩ assigns_if False (c ? e1 : e2)` `brk A = (λl. UNIV)` note `Env\<turnstile> (c ? e1 : e2)::- (PrimT Boolean)` moreover note `Env\<turnstile> B' »〈c ? e1 : e2〉» A'` ultimately obtain A': "nrm A' = B' ∪ assigns_if True (c ? e1 : e2) ∩ assigns_if False (c ? e1 : e2)" "brk A' = (λl. UNIV)" by - (erule da_elim_cases,auto simp add: inj_term_simps) (* inj_term_simps needed to handle wt (defined without 〈〉) *) note B' = `B ⊆ B'` with A A' show ?case by auto next case (Cond Env c e1 e2 B C E1 E2 A B' A') note A = `nrm A = nrm E1 ∩ nrm E2` `brk A = (λl. UNIV)` note not_bool = `¬ Env\<turnstile> (c ? e1 : e2)::- (PrimT Boolean)` from `Env\<turnstile> B' »〈c ? e1 : e2〉» A'` obtain E1' E2' where da_e1': "Env\<turnstile> B' ∪ assigns_if True c »〈e1〉» E1'" and da_e2': "Env\<turnstile> B' ∪ assigns_if False c »〈e2〉» E2'" and A': "nrm A' = nrm E1' ∩ nrm E2'" "brk A' = (λl. UNIV)" using not_bool by - (erule da_elim_cases, auto simp add: inj_term_simps) (* inj_term_simps needed to handle wt (defined without 〈〉) *) note `PROP ?Hyp Env (B ∪ assigns_if True c) 〈e1〉 E1` moreover note B' = `B ⊆ B'` moreover note da_e1' ultimately obtain E1': "nrm E1 ⊆ nrm E1'" "(∀l. brk E1 l ⊆ brk E1' l)" by blast note `PROP ?Hyp Env (B ∪ assigns_if False c) 〈e2〉 E2` with B' da_e2' obtain "nrm E2 ⊆ nrm E2'" "(∀l. brk E2 l ⊆ 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 (rule da', rule `B ⊆ B'`) lemma da_weaken: assumes da: "Env\<turnstile> B »t» A" and "B ⊆ B'" shows "∃ A'. Env \<turnstile> B' »t» A'" proof - note assigned.select_convs [Pure.intro] from da show "!! B'. B ⊆ B' ==> ∃ A'. Env\<turnstile> B' »t» A'" (is "PROP ?Hyp Env B t") proof (induct) case Skip thus ?case by (iprover intro: da.Skip) next case Expr thus ?case by (iprover intro: da.Expr) next case (Lab Env B c C A l B') note `PROP ?Hyp Env B 〈c〉` moreover note B' = `B ⊆ B'` ultimately obtain C' where "Env\<turnstile> B' »〈c〉» C'" by iprover then obtain A' where "Env\<turnstile> B' »〈Break l• c〉» A'" by (iprover intro: da.Lab) thus ?case .. next case (Comp Env B c1 C1 c2 C2 A B') note da_c1 = `Env\<turnstile> B »〈c1〉» C1` note `PROP ?Hyp Env B 〈c1〉` moreover note B' = `B ⊆ B'` ultimately obtain C1' where da_c1': "Env\<turnstile> B' »〈c1〉» C1'" by iprover with da_c1 B' have "nrm C1 ⊆ nrm C1'" by (rule da_monotone [elim_format]) simp moreover note `PROP ?Hyp Env (nrm C1) 〈c2〉` ultimately obtain C2' where "Env\<turnstile> nrm C1' »〈c2〉» C2'" by iprover with da_c1' obtain A' where "Env\<turnstile> B' »〈c1;; c2〉» A'" by (iprover intro: da.Comp) thus ?case .. next case (If Env B e E c1 C1 c2 C2 A B') note B' = `B ⊆ B'` obtain E' where "Env\<turnstile> B' »〈e〉» E'" proof - have "PROP ?Hyp Env B 〈e〉" by (rule If.hyps) with B' show ?thesis using that by iprover qed moreover obtain C1' where "Env\<turnstile> (B' ∪ assigns_if True e) »〈c1〉» C1'" proof - from B' have "(B ∪ assigns_if True e) ⊆ (B' ∪ assigns_if True e)" by blast moreover have "PROP ?Hyp Env (B ∪ assigns_if True e) 〈c1〉" by (rule If.hyps) ultimately show ?thesis using that by iprover qed moreover obtain C2' where "Env\<turnstile> (B' ∪ assigns_if False e) »〈c2〉» C2'" proof - from B' have "(B ∪ assigns_if False e) ⊆ (B' ∪ assigns_if False e)" by blast moreover have "PROP ?Hyp Env (B ∪ assigns_if False e) 〈c2〉" by (rule If.hyps) ultimately show ?thesis using that by iprover qed ultimately obtain A' where "Env\<turnstile> B' »〈If(e) c1 Else c2〉» A'" by (iprover intro: da.If) thus ?case .. next case (Loop Env B e E c C A l B') note B' = `B ⊆ B'` obtain E' where "Env\<turnstile> B' »〈e〉» E'" proof - have "PROP ?Hyp Env B 〈e〉" by (rule Loop.hyps) with B' show ?thesis using that by iprover qed moreover obtain C' where "Env\<turnstile> (B' ∪ assigns_if True e) »〈c〉» C'" proof - from B' have "(B ∪ assigns_if True e) ⊆ (B' ∪ assigns_if True e)" by blast moreover have "PROP ?Hyp Env (B ∪ assigns_if True e) 〈c〉" by (rule Loop.hyps) ultimately show ?thesis using that by iprover qed ultimately obtain A' where "Env\<turnstile> B' »〈l• While(e) c〉» A'" by (iprover intro: da.Loop ) thus ?case .. next case (Jmp jump B A Env B') note B' = `B ⊆ B'` with Jmp.hyps have "jump = Ret --> Result ∈ B' " by auto moreover obtain A'::assigned where "nrm A' = UNIV" "brk A' = (case jump of Break l => λk. if k = l then B' else UNIV | Cont l => λk. UNIV | Ret => λk. UNIV)" by iprover ultimately have "Env\<turnstile> B' »〈Jmp jump〉» A'" by (rule da.Jmp) thus ?case .. next case Throw thus ?case by (iprover intro: da.Throw ) next case (Try Env B c1 C1 vn C c2 C2 A B') note B' = `B ⊆ B'` obtain C1' where "Env\<turnstile> B' »〈c1〉» C1'" proof - have "PROP ?Hyp Env B 〈c1〉" by (rule Try.hyps) with B' show ?thesis using that by iprover qed moreover obtain C2' where "Env(|lcl := lcl Env(VName vn\<mapsto>Class C)|)),\<turnstile> B' ∪ {VName vn} »〈c2〉» C2'" proof - from B' have "B ∪ {VName vn} ⊆ B' ∪ {VName vn}" by blast moreover have "PROP ?Hyp (Env(|lcl := lcl Env(VName vn\<mapsto>Class C)|)),) (B ∪ {VName vn}) 〈c2〉" by (rule Try.hyps) ultimately show ?thesis using that by iprover qed ultimately obtain A' where "Env\<turnstile> B' »〈Try c1 Catch(C vn) c2〉» A'" by (iprover intro: da.Try ) thus ?case .. next case (Fin Env B c1 C1 c2 C2 A B') note B' = `B ⊆ B'` obtain C1' where C1': "Env\<turnstile> B' »〈c1〉» C1'" proof - have "PROP ?Hyp Env B 〈c1〉" by (rule Fin.hyps) with B' show ?thesis using that by iprover qed moreover obtain C2' where "Env\<turnstile> B' »〈c2〉» C2'" proof - have "PROP ?Hyp Env B 〈c2〉" by (rule Fin.hyps) with B' show ?thesis using that by iprover qed ultimately obtain A' where "Env\<turnstile> B' »〈c1 Finally c2〉» A'" by (iprover intro: da.Fin ) thus ?case .. next case Init thus ?case by (iprover intro: da.Init) next case NewC thus ?case by (iprover intro: da.NewC) next case NewA thus ?case by (iprover intro: da.NewA) next case Cast thus ?case by (iprover intro: da.Cast) next case Inst thus ?case by (iprover intro: da.Inst) next case Lit thus ?case by (iprover intro: da.Lit) next case UnOp thus ?case by (iprover intro: da.UnOp) next case (CondAnd Env B e1 E1 e2 E2 A B') note B' = `B ⊆ B'` obtain E1' where "Env\<turnstile> B' »〈e1〉» E1'" proof - have "PROP ?Hyp Env B 〈e1〉" by (rule CondAnd.hyps) with B' show ?thesis using that by iprover qed moreover obtain E2' where "Env\<turnstile> B' ∪ assigns_if True e1 »〈e2〉» E2'" proof - from B' have "B ∪ assigns_if True e1 ⊆ B' ∪ assigns_if True e1" by blast moreover have "PROP ?Hyp Env (B ∪ assigns_if True e1) 〈e2〉" by (rule CondAnd.hyps) ultimately show ?thesis using that by iprover qed ultimately obtain A' where "Env\<turnstile> B' »〈BinOp CondAnd e1 e2〉» A'" by (iprover intro: da.CondAnd) thus ?case .. next case (CondOr Env B e1 E1 e2 E2 A B') note B' = `B ⊆ B'` obtain E1' where "Env\<turnstile> B' »〈e1〉» E1'" proof - have "PROP ?Hyp Env B 〈e1〉" by (rule CondOr.hyps) with B' show ?thesis using that by iprover qed moreover obtain E2' where "Env\<turnstile> B' ∪ assigns_if False e1 »〈e2〉» E2'" proof - from B' have "B ∪ assigns_if False e1 ⊆ B' ∪ assigns_if False e1" by blast moreover have "PROP ?Hyp Env (B ∪ assigns_if False e1) 〈e2〉" by (rule CondOr.hyps) ultimately show ?thesis using that by iprover qed ultimately obtain A' where "Env\<turnstile> B' »〈BinOp CondOr e1 e2〉» A'" by (iprover intro: da.CondOr) thus ?case .. next case (BinOp Env B e1 E1 e2 A binop B') note B' = `B ⊆ B'` obtain E1' where E1': "Env\<turnstile> B' »〈e1〉» E1'" proof - have "PROP ?Hyp Env B 〈e1〉" by (rule BinOp.hyps) with B' show ?thesis using that by iprover qed moreover obtain A' where "Env\<turnstile> nrm E1' »〈e2〉» A'" proof - have "Env\<turnstile> B »〈e1〉» E1" by (rule BinOp.hyps) from this B' E1' have "nrm E1 ⊆ nrm E1'" by (rule da_monotone [THEN conjE]) moreover have "PROP ?Hyp Env (nrm E1) 〈e2〉" by (rule BinOp.hyps) ultimately show ?thesis using that by iprover qed ultimately have "Env\<turnstile> B' »〈BinOp binop e1 e2〉» A'" using BinOp.hyps by (iprover intro: da.BinOp) thus ?case .. next case (Super B Env B') note B' = `B ⊆ B'` with Super.hyps have "This ∈ B'" by auto thus ?case by (iprover intro: da.Super) next case (AccLVar vn B A Env B') note `vn ∈ B` moreover note `B ⊆ B'` ultimately have "vn ∈ B'" by auto thus ?case by (iprover intro: da.AccLVar) next case Acc thus ?case by (iprover intro: da.Acc) next case (AssLVar Env B e E A vn B') note B' = `B ⊆ B'` then obtain E' where "Env\<turnstile> B' »〈e〉» E'" by (rule AssLVar.hyps [elim_format]) iprover then obtain A' where "Env\<turnstile> B' »〈LVar vn:=e〉» A'" by (iprover intro: da.AssLVar) thus ?case .. next case (Ass v Env B V e A B') note B' = `B ⊆ B'` note `∀vn. v ≠ LVar vn` moreover obtain V' where V': "Env\<turnstile> B' »〈v〉» V'" proof - have "PROP ?Hyp Env B 〈v〉" by (rule Ass.hyps) with B' show ?thesis using that by iprover qed moreover obtain A' where "Env\<turnstile> nrm V' »〈e〉» A'" proof - have "Env\<turnstile> B »〈v〉» V" by (rule Ass.hyps) from this B' V' have "nrm V ⊆ nrm V'" by (rule da_monotone [THEN conjE]) moreover have "PROP ?Hyp Env (nrm V) 〈e〉" by (rule Ass.hyps) ultimately show ?thesis using that by iprover qed ultimately have "Env\<turnstile> B' »〈v := e〉» A'" by (iprover intro: da.Ass) thus ?case .. next case (CondBool Env c e1 e2 B C E1 E2 A B') note B' = `B ⊆ B'` note `Env\<turnstile>(c ? e1 : e2)::-(PrimT Boolean)` moreover obtain C' where C': "Env\<turnstile> B' »〈c〉» C'" proof - have "PROP ?Hyp Env B 〈c〉" by (rule CondBool.hyps) with B' show ?thesis using that by iprover qed moreover obtain E1' where "Env\<turnstile> B' ∪ assigns_if True c »〈e1〉» E1'" proof - from B' have "(B ∪ assigns_if True c) ⊆ (B' ∪ assigns_if True c)" by blast moreover have "PROP ?Hyp Env (B ∪ assigns_if True c) 〈e1〉" by (rule CondBool.hyps) ultimately show ?thesis using that by iprover qed moreover obtain E2' where "Env\<turnstile> B' ∪ assigns_if False c »〈e2〉» E2'" proof - from B' have "(B ∪ assigns_if False c) ⊆ (B' ∪ assigns_if False c)" by blast moreover have "PROP ?Hyp Env (B ∪ assigns_if False c) 〈e2〉" by(rule CondBool.hyps) ultimately show ?thesis using that by iprover qed ultimately obtain A' where "Env\<turnstile> B' »〈c ? e1 : e2〉» A'" by (iprover intro: da.CondBool) thus ?case .. next case (Cond Env c e1 e2 B C E1 E2 A B') note B' = `B ⊆ B'` note `¬ Env\<turnstile>(c ? e1 : e2)::-(PrimT Boolean)` moreover obtain C' where C': "Env\<turnstile> B' »〈c〉» C'" proof - have "PROP ?Hyp Env B 〈c〉" by (rule Cond.hyps) with B' show ?thesis using that by iprover qed moreover obtain E1' where "Env\<turnstile> B' ∪ assigns_if True c »〈e1〉» E1'" proof - from B' have "(B ∪ assigns_if True c) ⊆ (B' ∪ assigns_if True c)" by blast moreover have "PROP ?Hyp Env (B ∪ assigns_if True c) 〈e1〉" by (rule Cond.hyps) ultimately show ?thesis using that by iprover qed moreover obtain E2' where "Env\<turnstile> B' ∪ assigns_if False c »〈e2〉» E2'" proof - from B' have "(B ∪ assigns_if False c) ⊆ (B' ∪ assigns_if False c)" by blast moreover have "PROP ?Hyp Env (B ∪ assigns_if False c) 〈e2〉" by (rule Cond.hyps) ultimately show ?thesis using that by iprover qed ultimately obtain A' where "Env\<turnstile> B' »〈c ? e1 : e2〉» A'" by (iprover intro: da.Cond) thus ?case .. next case (Call Env B e E args A accC statT mode mn pTs B') note B' = `B ⊆ B'` obtain E' where E': "Env\<turnstile> B' »〈e〉» E'" proof - have "PROP ?Hyp Env B 〈e〉" by (rule Call.hyps) with B' show ?thesis using that by iprover qed moreover obtain A' where "Env\<turnstile> nrm E' »〈args〉» A'" proof - have "Env\<turnstile> B »〈e〉» E" by (rule Call.hyps) from this B' E' have "nrm E ⊆ nrm E'" by (rule da_monotone [THEN conjE]) moreover have "PROP ?Hyp Env (nrm E) 〈args〉" by (rule Call.hyps) ultimately show ?thesis using that by iprover qed ultimately have "Env\<turnstile> B' »〈{accC,statT,mode}e·mn( {pTs}args)〉» A'" by (iprover intro: da.Call) thus ?case .. next case Methd thus ?case by (iprover intro: da.Methd) next case (Body Env B c C A D B') note B' = `B ⊆ B'` obtain C' where C': "Env\<turnstile> B' »〈c〉» C'" and nrm_C': "nrm C ⊆ nrm C'" proof - have "Env\<turnstile> B »〈c〉» C" by (rule Body.hyps) moreover note B' moreover from B' obtain C' where da_c: "Env\<turnstile> B' »〈c〉» C'" by (rule Body.hyps [elim_format]) blast ultimately have "nrm C ⊆ nrm C'" by (rule da_monotone [THEN conjE]) with da_c that show ?thesis by iprover qed moreover note `Result ∈ nrm C` with nrm_C' have "Result ∈ nrm C'" by blast moreover note `jumpNestingOkS {Ret} c` ultimately obtain A' where "Env\<turnstile> B' »〈Body D c〉» A'" by (iprover intro: da.Body) thus ?case .. next case LVar thus ?case by (iprover intro: da.LVar) next case FVar thus ?case by (iprover intro: da.FVar) next case (AVar Env B e1 E1 e2 A B') note B' = `B ⊆ B'` obtain E1' where E1': "Env\<turnstile> B' »〈e1〉» E1'" proof - have "PROP ?Hyp Env B 〈e1〉" by (rule AVar.hyps) with B' show ?thesis using that by iprover qed moreover obtain A' where "Env\<turnstile> nrm E1' »〈e2〉» A'" proof - have "Env\<turnstile> B »〈e1〉» E1" by (rule AVar.hyps) from this B' E1' have "nrm E1 ⊆ nrm E1'" by (rule da_monotone [THEN conjE]) moreover have "PROP ?Hyp Env (nrm E1) 〈e2〉" by (rule AVar.hyps) ultimately show ?thesis using that by iprover qed ultimately have "Env\<turnstile> B' »〈e1.[e2]〉» A'" by (iprover intro: da.AVar) thus ?case .. next case Nil thus ?case by (iprover intro: da.Nil) next case (Cons Env B e E es A B') note B' = `B ⊆ B'` obtain E' where E': "Env\<turnstile> B' »〈e〉» E'" proof - have "PROP ?Hyp Env B 〈e〉" by (rule Cons.hyps) with B' show ?thesis using that by iprover qed moreover obtain A' where "Env\<turnstile> nrm E' »〈es〉» A'" proof - have "Env\<turnstile> B »〈e〉» E" by (rule Cons.hyps) from this B' E' have "nrm E ⊆ nrm E'" by (rule da_monotone [THEN conjE]) moreover have "PROP ?Hyp Env (nrm E) 〈es〉" by (rule Cons.hyps) ultimately show ?thesis using that by iprover qed ultimately have "Env\<turnstile> B' »〈e # es〉» A'" by (iprover intro: da.Cons) thus ?case .. qed qed (rule `B ⊆ B'`) (* 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, the second is more informativ for the reader *) corollary da_weakenE [consumes 2]: assumes da: "Env\<turnstile> B »t» A" and B': "B ⊆ B'" and ex_mono: "!! A'. [|Env\<turnstile> B' »t» A'; nrm A ⊆ nrm A'; !! l. brk A l ⊆ brk A' l|] ==> P" shows "P" proof - from da B' obtain A' where A': "Env\<turnstile> B' »t» A'" by (rule da_weaken [elim_format]) iprover with da B' have "nrm A ⊆ nrm A' ∧ (∀ l. brk A l ⊆ brk A' l)" by (rule da_monotone) with A' ex_mono show ?thesis by iprover qed end
lemma jumpNestingOk_expr_simp:
jumpNestingOk jmps (In1l e) = True
lemma jumpNestingOk_expr_simp1:
jumpNestingOk jmps 〈e〉 = True
lemma jumpNestingOk_stmt_simp:
jumpNestingOk jmps (In1r s) = jumpNestingOkS jmps s
lemma jumpNestingOk_stmt_simp1:
jumpNestingOk jmps 〈s〉 = jumpNestingOkS jmps s
lemma jumpNestingOk_var_simp:
jumpNestingOk jmps (In2 v) = True
lemma jumpNestingOk_var_simp1:
jumpNestingOk jmps 〈v〉 = True
lemma jumpNestingOk_expr_list_simp:
jumpNestingOk jmps (In3 es) = True
lemma jumpNestingOk_expr_list_simp1:
jumpNestingOk jmps 〈es〉 = True
lemma assigns_expr_simp:
assigns (In1l e) = assignsE e
lemma assigns_expr_simp1:
assigns 〈e〉 = assignsE e
lemma assigns_stmt_simp:
assigns (In1r s) = {}
lemma assigns_stmt_simp1:
assigns 〈s〉 = {}
lemma assigns_var_simp:
assigns (In2 v) = assignsV v
lemma assigns_var_simp1:
assigns 〈v〉 = assignsV v
lemma assigns_expr_list_simp:
assigns (In3 es) = assignsEs es
lemma assigns_expr_list_simp1:
assigns 〈es〉 = assignsEs es
lemma constVal_Some_induct:
[| constVal e = Some v; !!v. P (Lit v); !!unop e'. P e' ==> P (UnOp unop e');
!!binop e1 e2. [| P e1; P e2 |] ==> P (BinOp binop e1 e2);
!!b bv e1 e2.
[| constVal b = Some bv; the_Bool bv; P b; P e1 |] ==> P (b ? e1 : e2);
!!b bv e1 e2.
[| constVal b = Some bv; ¬ the_Bool bv; P b; P e2 |]
==> P (b ? e1 : e2) |]
==> P e
lemma assignsE_const_simp:
constVal e = Some v ==> assignsE e = {}
lemma assigns_if_const_b_simp:
constVal e = Some (Bool b) ==> assigns_if b e = {}
lemma assigns_if_const_not_b_simp:
constVal e = Some (Bool b) ==> assigns_if (¬ b) e = UNIV
lemma union_ts_iff:
(c ∈ (A =>∪ B) k) = (c ∈ A k ∨ c ∈ B k)
lemma union_tsI1:
c ∈ A k ==> c ∈ (A =>∪ B) k
lemma union_tsI2:
c ∈ B k ==> c ∈ (A =>∪ B) k
lemma union_tsCI:
(c ∉ B k ==> c ∈ A k) ==> c ∈ (A =>∪ B) k
lemma union_tsE:
[| c ∈ (A =>∪ B) k; c ∈ A k ==> P; c ∈ B k ==> P |] ==> P
lemma intersect_ts_iff:
(c ∈ (A =>∩ B) k) = (c ∈ A k ∧ c ∈ B k)
lemma intersect_tsI:
[| c ∈ A k; c ∈ B k |] ==> c ∈ (A =>∩ B) k
lemma intersect_tsD1:
c ∈ (A =>∩ B) k ==> c ∈ A k
lemma intersect_tsD2:
c ∈ (A =>∩ B) k ==> c ∈ B k
lemma intersect_tsE:
[| c ∈ (A =>∩ B) k; [| c ∈ A k; c ∈ B k |] ==> P |] ==> P
lemma all_union_ts_iff:
(c ∈ (A =>∪∀ B) k) = (c ∈ A k ∨ c ∈ B)
lemma all_union_tsI1:
c ∈ A k ==> c ∈ (A =>∪∀ B) k
lemma all_union_tsI2:
c ∈ B ==> c ∈ (A =>∪∀ B) k
lemma all_union_tsCI:
(c ∉ B ==> c ∈ A k) ==> c ∈ (A =>∪∀ B) k
lemma all_union_tsE:
[| c ∈ (A =>∪∀ B) k; c ∈ A k ==> P; c ∈ B ==> P |] ==> P
lemma da_Skip:
A = (| nrm = B, brk = λl. UNIV |) ==> Env\<turnstile> B »〈Skip〉» A
lemma da_NewC:
A = (| nrm = B, brk = λl. UNIV |) ==> Env\<turnstile> B »〈NewC C〉» A
lemma da_Lit:
A = (| nrm = B, brk = λl. UNIV |) ==> Env\<turnstile> B »〈Lit v〉» A
lemma da_Super:
[| This ∈ B; A = (| nrm = B, brk = λl. UNIV |) |]
==> Env\<turnstile> B »〈Super〉» A
lemma da_Init:
A = (| nrm = B, brk = λl. UNIV |) ==> Env\<turnstile> B »〈Init C〉» A
lemma assignsE_subseteq_assigns_ifs:
E,empty_dt\<Turnstile>e::-PrimT Boolean
==> assignsE e ⊆ assigns_if True e ∩ assigns_if False e
lemma rmlab_same_label:
rmlab l A l = UNIV
lemma rmlab_same_label1:
l = l' ==> rmlab l A l' = UNIV
lemma rmlab_other_label:
l ≠ l' ==> rmlab l A l' = A l'
lemma range_inter_ts_subseteq:
∀k. A k ⊆ B k ==> =>\<Inter>A ⊆ =>\<Inter>B
lemma range_inter_ts_subseteq':
[| ∀k. A k ⊆ B k; x ∈ =>\<Inter>A |] ==> x ∈ =>\<Inter>B
lemma da_monotone:
[| Env\<turnstile> B »t» A; B ⊆ B'; Env\<turnstile> B' »t» A' |]
==> nrm A ⊆ nrm A' ∧ (∀l. brk A l ⊆ brk A' l)
lemma da_weaken:
[| Env\<turnstile> B »t» A; B ⊆ B' |] ==> ∃A'. Env\<turnstile> B' »t» A'
corollary da_weakenE:
[| Env\<turnstile> B »t» A; B ⊆ B';
!!A'. [| Env\<turnstile> B' »t» A'; nrm A ⊆ nrm A';
!!l. brk A l ⊆ brk A' l |]
==> P |]
==> P