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theory Effect(* Title: HOL/MicroJava/BV/Effect.thy ID: $Id: Effect.thy,v 1.11 2005/08/17 09:44:10 nipkow Exp $ Author: Gerwin Klein Copyright 2000 Technische Universitaet Muenchen *) header {* \isaheader{Effect of Instructions on the State Type} *} theory Effect imports JVMType "../JVM/JVMExceptions" begin types succ_type = "(p_count × state_type option) list" text {* Program counter of successor instructions: *} consts succs :: "instr => p_count => p_count list" primrec "succs (Load idx) pc = [pc+1]" "succs (Store idx) pc = [pc+1]" "succs (LitPush v) pc = [pc+1]" "succs (Getfield F C) pc = [pc+1]" "succs (Putfield F C) pc = [pc+1]" "succs (New C) pc = [pc+1]" "succs (Checkcast C) pc = [pc+1]" "succs Pop pc = [pc+1]" "succs Dup pc = [pc+1]" "succs Dup_x1 pc = [pc+1]" "succs Dup_x2 pc = [pc+1]" "succs Swap pc = [pc+1]" "succs IAdd pc = [pc+1]" "succs (Ifcmpeq b) pc = [pc+1, nat (int pc + b)]" "succs (Goto b) pc = [nat (int pc + b)]" "succs Return pc = [pc]" "succs (Invoke C mn fpTs) pc = [pc+1]" "succs Throw pc = [pc]" text "Effect of instruction on the state type:" consts eff' :: "instr × jvm_prog × state_type => state_type" recdef eff' "{}" "eff' (Load idx, G, (ST, LT)) = (ok_val (LT ! idx) # ST, LT)" "eff' (Store idx, G, (ts#ST, LT)) = (ST, LT[idx:= OK ts])" "eff' (LitPush v, G, (ST, LT)) = (the (typeof (λv. None) v) # ST, LT)" "eff' (Getfield F C, G, (oT#ST, LT)) = (snd (the (field (G,C) F)) # ST, LT)" "eff' (Putfield F C, G, (vT#oT#ST, LT)) = (ST,LT)" "eff' (New C, G, (ST,LT)) = (Class C # ST, LT)" "eff' (Checkcast C, G, (RefT rt#ST,LT)) = (Class C # ST,LT)" "eff' (Pop, G, (ts#ST,LT)) = (ST,LT)" "eff' (Dup, G, (ts#ST,LT)) = (ts#ts#ST,LT)" "eff' (Dup_x1, G, (ts1#ts2#ST,LT)) = (ts1#ts2#ts1#ST,LT)" "eff' (Dup_x2, G, (ts1#ts2#ts3#ST,LT)) = (ts1#ts2#ts3#ts1#ST,LT)" "eff' (Swap, G, (ts1#ts2#ST,LT)) = (ts2#ts1#ST,LT)" "eff' (IAdd, G, (PrimT Integer#PrimT Integer#ST,LT)) = (PrimT Integer#ST,LT)" "eff' (Ifcmpeq b, G, (ts1#ts2#ST,LT)) = (ST,LT)" "eff' (Goto b, G, s) = s" -- "Return has no successor instruction in the same method" "eff' (Return, G, s) = s" -- "Throw always terminates abruptly" "eff' (Throw, G, s) = s" "eff' (Invoke C mn fpTs, G, (ST,LT)) = (let ST' = drop (length fpTs) ST in (fst (snd (the (method (G,C) (mn,fpTs))))#(tl ST'),LT))" consts match_any :: "jvm_prog => p_count => exception_table => cname list" primrec "match_any G pc [] = []" "match_any G pc (e#es) = (let (start_pc, end_pc, handler_pc, catch_type) = e; es' = match_any G pc es in if start_pc <= pc ∧ pc < end_pc then catch_type#es' else es')" consts match :: "jvm_prog => xcpt => p_count => exception_table => cname list" primrec "match G X pc [] = []" "match G X pc (e#es) = (if match_exception_entry G (Xcpt X) pc e then [Xcpt X] else match G X pc es)" lemma match_some_entry: "match G X pc et = (if ∃e ∈ set et. match_exception_entry G (Xcpt X) pc e then [Xcpt X] else [])" by (induct et) auto consts xcpt_names :: "instr × jvm_prog × p_count × exception_table => cname list" recdef xcpt_names "{}" "xcpt_names (Getfield F C, G, pc, et) = match G NullPointer pc et" "xcpt_names (Putfield F C, G, pc, et) = match G NullPointer pc et" "xcpt_names (New C, G, pc, et) = match G OutOfMemory pc et" "xcpt_names (Checkcast C, G, pc, et) = match G ClassCast pc et" "xcpt_names (Throw, G, pc, et) = match_any G pc et" "xcpt_names (Invoke C m p, G, pc, et) = match_any G pc et" "xcpt_names (i, G, pc, et) = []" constdefs xcpt_eff :: "instr => jvm_prog => p_count => state_type option => exception_table => succ_type" "xcpt_eff i G pc s et == map (λC. (the (match_exception_table G C pc et), case s of None => None | Some s' => Some ([Class C], snd s'))) (xcpt_names (i,G,pc,et))" norm_eff :: "instr => jvm_prog => state_type option => state_type option" "norm_eff i G == option_map (λs. eff' (i,G,s))" eff :: "instr => jvm_prog => p_count => exception_table => state_type option => succ_type" "eff i G pc et s == (map (λpc'. (pc',norm_eff i G s)) (succs i pc)) @ (xcpt_eff i G pc s et)" constdefs isPrimT :: "ty => bool" "isPrimT T == case T of PrimT T' => True | RefT T' => False" isRefT :: "ty => bool" "isRefT T == case T of PrimT T' => False | RefT T' => True" lemma isPrimT [simp]: "isPrimT T = (∃T'. T = PrimT T')" by (simp add: isPrimT_def split: ty.splits) lemma isRefT [simp]: "isRefT T = (∃T'. T = RefT T')" by (simp add: isRefT_def split: ty.splits) lemma "list_all2 P a b ==> ∀(x,y) ∈ set (zip a b). P x y" by (simp add: list_all2_def) text "Conditions under which eff is applicable:" consts app' :: "instr × jvm_prog × p_count × nat × ty × state_type => bool" recdef app' "{}" "app' (Load idx, G, pc, maxs, rT, s) = (idx < length (snd s) ∧ (snd s) ! idx ≠ Err ∧ length (fst s) < maxs)" "app' (Store idx, G, pc, maxs, rT, (ts#ST, LT)) = (idx < length LT)" "app' (LitPush v, G, pc, maxs, rT, s) = (length (fst s) < maxs ∧ typeof (λt. None) v ≠ None)" "app' (Getfield F C, G, pc, maxs, rT, (oT#ST, LT)) = (is_class G C ∧ field (G,C) F ≠ None ∧ fst (the (field (G,C) F)) = C ∧ G \<turnstile> oT \<preceq> (Class C))" "app' (Putfield F C, G, pc, maxs, rT, (vT#oT#ST, LT)) = (is_class G C ∧ field (G,C) F ≠ None ∧ fst (the (field (G,C) F)) = C ∧ G \<turnstile> oT \<preceq> (Class C) ∧ G \<turnstile> vT \<preceq> (snd (the (field (G,C) F))))" "app' (New C, G, pc, maxs, rT, s) = (is_class G C ∧ length (fst s) < maxs)" "app' (Checkcast C, G, pc, maxs, rT, (RefT rt#ST,LT)) = (is_class G C)" "app' (Pop, G, pc, maxs, rT, (ts#ST,LT)) = True" "app' (Dup, G, pc, maxs, rT, (ts#ST,LT)) = (1+length ST < maxs)" "app' (Dup_x1, G, pc, maxs, rT, (ts1#ts2#ST,LT)) = (2+length ST < maxs)" "app' (Dup_x2, G, pc, maxs, rT, (ts1#ts2#ts3#ST,LT)) = (3+length ST < maxs)" "app' (Swap, G, pc, maxs, rT, (ts1#ts2#ST,LT)) = True" "app' (IAdd, G, pc, maxs, rT, (PrimT Integer#PrimT Integer#ST,LT)) = True" "app' (Ifcmpeq b, G, pc, maxs, rT, (ts#ts'#ST,LT)) = (0 ≤ int pc + b ∧ (isPrimT ts ∧ ts' = ts ∨ isRefT ts ∧ isRefT ts'))" "app' (Goto b, G, pc, maxs, rT, s) = (0 ≤ int pc + b)" "app' (Return, G, pc, maxs, rT, (T#ST,LT)) = (G \<turnstile> T \<preceq> rT)" "app' (Throw, G, pc, maxs, rT, (T#ST,LT)) = isRefT T" "app' (Invoke C mn fpTs, G, pc, maxs, rT, s) = (length fpTs < length (fst s) ∧ (let apTs = rev (take (length fpTs) (fst s)); X = hd (drop (length fpTs) (fst s)) in G \<turnstile> X \<preceq> Class C ∧ is_class G C ∧ method (G,C) (mn,fpTs) ≠ None ∧ list_all2 (λx y. G \<turnstile> x \<preceq> y) apTs fpTs))" "app' (i,G, pc,maxs,rT,s) = False" constdefs xcpt_app :: "instr => jvm_prog => nat => exception_table => bool" "xcpt_app i G pc et ≡ ∀C∈set(xcpt_names (i,G,pc,et)). is_class G C" app :: "instr => jvm_prog => nat => ty => nat => exception_table => state_type option => bool" "app i G maxs rT pc et s == case s of None => True | Some t => app' (i,G,pc,maxs,rT,t) ∧ xcpt_app i G pc et" lemma match_any_match_table: "C ∈ set (match_any G pc et) ==> match_exception_table G C pc et ≠ None" apply (induct et) apply simp apply simp apply clarify apply (simp split: split_if_asm) apply (auto simp add: match_exception_entry_def) done lemma match_X_match_table: "C ∈ set (match G X pc et) ==> match_exception_table G C pc et ≠ None" apply (induct et) apply simp apply (simp split: split_if_asm) done lemma xcpt_names_in_et: "C ∈ set (xcpt_names (i,G,pc,et)) ==> ∃e ∈ set et. the (match_exception_table G C pc et) = fst (snd (snd e))" apply (cases i) apply (auto dest!: match_any_match_table match_X_match_table dest: match_exception_table_in_et) done lemma 1: "2 < length a ==> (∃l l' l'' ls. a = l#l'#l''#ls)" proof (cases a) fix x xs assume "a = x#xs" "2 < length a" thus ?thesis by - (cases xs, simp, cases "tl xs", auto) qed auto lemma 2: "¬(2 < length a) ==> a = [] ∨ (∃ l. a = [l]) ∨ (∃ l l'. a = [l,l'])" proof -; assume "¬(2 < length a)" hence "length a < (Suc (Suc (Suc 0)))" by simp hence * : "length a = 0 ∨ length a = Suc 0 ∨ length a = Suc (Suc 0)" by (auto simp add: less_Suc_eq) { fix x assume "length x = Suc 0" hence "∃ l. x = [l]" by - (cases x, auto) } note 0 = this have "length a = Suc (Suc 0) ==> ∃l l'. a = [l,l']" by (cases a, auto dest: 0) with * show ?thesis by (auto dest: 0) qed lemmas [simp] = app_def xcpt_app_def text {* \medskip simp rules for @{term app} *} lemma appNone[simp]: "app i G maxs rT pc et None = True" by simp lemma appLoad[simp]: "(app (Load idx) G maxs rT pc et (Some s)) = (∃ST LT. s = (ST,LT) ∧ idx < length LT ∧ LT!idx ≠ Err ∧ length ST < maxs)" by (cases s, simp) lemma appStore[simp]: "(app (Store idx) G maxs rT pc et (Some s)) = (∃ts ST LT. s = (ts#ST,LT) ∧ idx < length LT)" by (cases s, cases "2 < length (fst s)", auto dest: 1 2) lemma appLitPush[simp]: "(app (LitPush v) G maxs rT pc et (Some s)) = (∃ST LT. s = (ST,LT) ∧ length ST < maxs ∧ typeof (λv. None) v ≠ None)" by (cases s, simp) lemma appGetField[simp]: "(app (Getfield F C) G maxs rT pc et (Some s)) = (∃ oT vT ST LT. s = (oT#ST, LT) ∧ is_class G C ∧ field (G,C) F = Some (C,vT) ∧ G \<turnstile> oT \<preceq> (Class C) ∧ (∀x ∈ set (match G NullPointer pc et). is_class G x))" by (cases s, cases "2 <length (fst s)", auto dest!: 1 2) lemma appPutField[simp]: "(app (Putfield F C) G maxs rT pc et (Some s)) = (∃ vT vT' oT ST LT. s = (vT#oT#ST, LT) ∧ is_class G C ∧ field (G,C) F = Some (C, vT') ∧ G \<turnstile> oT \<preceq> (Class C) ∧ G \<turnstile> vT \<preceq> vT' ∧ (∀x ∈ set (match G NullPointer pc et). is_class G x))" by (cases s, cases "2 <length (fst s)", auto dest!: 1 2) lemma appNew[simp]: "(app (New C) G maxs rT pc et (Some s)) = (∃ST LT. s=(ST,LT) ∧ is_class G C ∧ length ST < maxs ∧ (∀x ∈ set (match G OutOfMemory pc et). is_class G x))" by (cases s, simp) lemma appCheckcast[simp]: "(app (Checkcast C) G maxs rT pc et (Some s)) = (∃rT ST LT. s = (RefT rT#ST,LT) ∧ is_class G C ∧ (∀x ∈ set (match G ClassCast pc et). is_class G x))" by (cases s, cases "fst s", simp add: app_def) (cases "hd (fst s)", auto) lemma appPop[simp]: "(app Pop G maxs rT pc et (Some s)) = (∃ts ST LT. s = (ts#ST,LT))" by (cases s, cases "2 <length (fst s)", auto dest: 1 2) lemma appDup[simp]: "(app Dup G maxs rT pc et (Some s)) = (∃ts ST LT. s = (ts#ST,LT) ∧ 1+length ST < maxs)" by (cases s, cases "2 <length (fst s)", auto dest: 1 2) lemma appDup_x1[simp]: "(app Dup_x1 G maxs rT pc et (Some s)) = (∃ts1 ts2 ST LT. s = (ts1#ts2#ST,LT) ∧ 2+length ST < maxs)" by (cases s, cases "2 <length (fst s)", auto dest: 1 2) lemma appDup_x2[simp]: "(app Dup_x2 G maxs rT pc et (Some s)) = (∃ts1 ts2 ts3 ST LT. s = (ts1#ts2#ts3#ST,LT) ∧ 3+length ST < maxs)" by (cases s, cases "2 <length (fst s)", auto dest: 1 2) lemma appSwap[simp]: "app Swap G maxs rT pc et (Some s) = (∃ts1 ts2 ST LT. s = (ts1#ts2#ST,LT))" by (cases s, cases "2 <length (fst s)", auto dest: 1 2) lemma appIAdd[simp]: "app IAdd G maxs rT pc et (Some s) = (∃ ST LT. s = (PrimT Integer#PrimT Integer#ST,LT))" (is "?app s = ?P s") proof (cases (open) s) case Pair have "?app (a,b) = ?P (a,b)" proof (cases "a") fix t ts assume a: "a = t#ts" show ?thesis proof (cases t) fix p assume p: "t = PrimT p" show ?thesis proof (cases p) assume ip: "p = Integer" show ?thesis proof (cases ts) fix t' ts' assume t': "ts = t' # ts'" show ?thesis proof (cases t') fix p' assume "t' = PrimT p'" with t' ip p a show ?thesis by - (cases p', auto) qed (auto simp add: a p ip t') qed (auto simp add: a p ip) qed (auto simp add: a p) qed (auto simp add: a) qed auto with Pair show ?thesis by simp qed lemma appIfcmpeq[simp]: "app (Ifcmpeq b) G maxs rT pc et (Some s) = (∃ts1 ts2 ST LT. s = (ts1#ts2#ST,LT) ∧ 0 ≤ int pc + b ∧ ((∃ p. ts1 = PrimT p ∧ ts2 = PrimT p) ∨ (∃r r'. ts1 = RefT r ∧ ts2 = RefT r')))" by (cases s, cases "2 <length (fst s)", auto dest!: 1 2) lemma appReturn[simp]: "app Return G maxs rT pc et (Some s) = (∃T ST LT. s = (T#ST,LT) ∧ (G \<turnstile> T \<preceq> rT))" by (cases s, cases "2 <length (fst s)", auto dest: 1 2) lemma appGoto[simp]: "app (Goto b) G maxs rT pc et (Some s) = (0 ≤ int pc + b)" by simp lemma appThrow[simp]: "app Throw G maxs rT pc et (Some s) = (∃T ST LT r. s=(T#ST,LT) ∧ T = RefT r ∧ (∀C ∈ set (match_any G pc et). is_class G C))" by (cases s, cases "2 < length (fst s)", auto dest: 1 2) lemma appInvoke[simp]: "app (Invoke C mn fpTs) G maxs rT pc et (Some s) = (∃apTs X ST LT mD' rT' b'. s = ((rev apTs) @ (X # ST), LT) ∧ length apTs = length fpTs ∧ is_class G C ∧ G \<turnstile> X \<preceq> Class C ∧ (∀(aT,fT)∈set(zip apTs fpTs). G \<turnstile> aT \<preceq> fT) ∧ method (G,C) (mn,fpTs) = Some (mD', rT', b') ∧ (∀C ∈ set (match_any G pc et). is_class G C))" (is "?app s = ?P s") proof (cases (open) s) note list_all2_def [simp] case Pair have "?app (a,b) ==> ?P (a,b)" proof - assume app: "?app (a,b)" hence "a = (rev (rev (take (length fpTs) a))) @ (drop (length fpTs) a) ∧ length fpTs < length a" (is "?a ∧ ?l") by (auto simp add: app_def) hence "?a ∧ 0 < length (drop (length fpTs) a)" (is "?a ∧ ?l") by auto hence "?a ∧ ?l ∧ length (rev (take (length fpTs) a)) = length fpTs" by (auto simp add: min_def) hence "∃apTs ST. a = rev apTs @ ST ∧ length apTs = length fpTs ∧ 0 < length ST" by blast hence "∃apTs ST. a = rev apTs @ ST ∧ length apTs = length fpTs ∧ ST ≠ []" by blast hence "∃apTs ST. a = rev apTs @ ST ∧ length apTs = length fpTs ∧ (∃X ST'. ST = X#ST')" by (simp add: neq_Nil_conv) hence "∃apTs X ST. a = rev apTs @ X # ST ∧ length apTs = length fpTs" by blast with app show ?thesis by (unfold app_def, clarsimp) blast qed with Pair have "?app s ==> ?P s" by (simp only:) moreover have "?P s ==> ?app s" by (unfold app_def) (clarsimp simp add: min_def) ultimately show ?thesis by (rule iffI) qed lemma effNone: "(pc', s') ∈ set (eff i G pc et None) ==> s' = None" by (auto simp add: eff_def xcpt_eff_def norm_eff_def) lemma xcpt_app_lemma [code]: "xcpt_app i G pc et = list_all (is_class G) (xcpt_names (i, G, pc, et))" by (simp add: list_all_iff) lemmas [simp del] = app_def xcpt_app_def end
lemma match_some_entry:
match G X pc et = (if ∃e∈set et. match_exception_entry G (Xcpt X) pc e then [Xcpt X] else [])
lemma isPrimT:
isPrimT T = (∃T'. T = PrimT T')
lemma isRefT:
isRefT T = (∃T'. T = RefT T')
lemma
list_all2 P a b ==> ∀(x, y)∈set (zip a b). P x y
lemma match_any_match_table:
C ∈ set (match_any G pc et) ==> match_exception_table G C pc et ≠ None
lemma match_X_match_table:
C ∈ set (match G X pc et) ==> match_exception_table G C pc et ≠ None
lemma xcpt_names_in_et:
C ∈ set (xcpt_names (i, G, pc, et)) ==> ∃e∈set et. the (match_exception_table G C pc et) = fst (snd (snd e))
lemma 1:
2 < length a ==> ∃l l' l'' ls. a = l # l' # l'' # ls
lemma 2:
¬ 2 < length a ==> a = [] ∨ (∃l. a = [l]) ∨ (∃l l'. a = [l, l'])
lemmas
app i G maxs rT pc et s == case s of None => True | Some t => app' (i, G, pc, maxs, rT, t) ∧ xcpt_app i G pc et
xcpt_app i G pc et == ∀C∈set (xcpt_names (i, G, pc, et)). is_class G C
lemmas
app i G maxs rT pc et s == case s of None => True | Some t => app' (i, G, pc, maxs, rT, t) ∧ xcpt_app i G pc et
xcpt_app i G pc et == ∀C∈set (xcpt_names (i, G, pc, et)). is_class G C
lemma appNone:
app i G maxs rT pc et None = True
lemma appLoad:
app (Load idx) G maxs rT pc et (Some s) = (∃ST LT. s = (ST, LT) ∧ idx < length LT ∧ LT ! idx ≠ Err ∧ length ST < maxs)
lemma appStore:
app (Store idx) G maxs rT pc et (Some s) = (∃ts ST LT. s = (ts # ST, LT) ∧ idx < length LT)
lemma appLitPush:
app (LitPush v) G maxs rT pc et (Some s) = (∃ST LT. s = (ST, LT) ∧ length ST < maxs ∧ typeof empty v ≠ None)
lemma appGetField:
app (Getfield F C) G maxs rT pc et (Some s) = (∃oT vT ST LT. s = (oT # ST, LT) ∧ is_class G C ∧ field (G, C) F = Some (C, vT) ∧ G |- oT <= Class C ∧ (∀x∈set (match G NullPointer pc et). is_class G x))
lemma appPutField:
app (Putfield F C) G maxs rT pc et (Some s) = (∃vT vT' oT ST LT. s = (vT # oT # ST, LT) ∧ is_class G C ∧ field (G, C) F = Some (C, vT') ∧ G |- oT <= Class C ∧ G |- vT <= vT' ∧ (∀x∈set (match G NullPointer pc et). is_class G x))
lemma appNew:
app (New C) G maxs rT pc et (Some s) = (∃ST LT. s = (ST, LT) ∧ is_class G C ∧ length ST < maxs ∧ (∀x∈set (match G OutOfMemory pc et). is_class G x))
lemma appCheckcast:
app (Checkcast C) G maxs rT pc et (Some s) = (∃rT ST LT. s = (RefT rT # ST, LT) ∧ is_class G C ∧ (∀x∈set (match G ClassCast pc et). is_class G x))
lemma appPop:
app Pop G maxs rT pc et (Some s) = (∃ts ST LT. s = (ts # ST, LT))
lemma appDup:
app Dup G maxs rT pc et (Some s) = (∃ts ST LT. s = (ts # ST, LT) ∧ 1 + length ST < maxs)
lemma appDup_x1:
app Dup_x1 G maxs rT pc et (Some s) = (∃ts1 ts2 ST LT. s = (ts1 # ts2 # ST, LT) ∧ 2 + length ST < maxs)
lemma appDup_x2:
app Dup_x2 G maxs rT pc et (Some s) = (∃ts1 ts2 ts3 ST LT. s = (ts1 # ts2 # ts3 # ST, LT) ∧ 3 + length ST < maxs)
lemma appSwap:
app Swap G maxs rT pc et (Some s) = (∃ts1 ts2 ST LT. s = (ts1 # ts2 # ST, LT))
lemma appIAdd:
app IAdd G maxs rT pc et (Some s) = (∃ST LT. s = (PrimT Integer # PrimT Integer # ST, LT))
lemma appIfcmpeq:
app (Ifcmpeq b) G maxs rT pc et (Some s) = (∃ts1 ts2 ST LT. s = (ts1 # ts2 # ST, LT) ∧ 0 ≤ int pc + b ∧ ((∃p. ts1 = PrimT p ∧ ts2 = PrimT p) ∨ (∃r r'. ts1 = RefT r ∧ ts2 = RefT r')))
lemma appReturn:
app Return G maxs rT pc et (Some s) = (∃T ST LT. s = (T # ST, LT) ∧ G |- T <= rT)
lemma appGoto:
app (Goto b) G maxs rT pc et (Some s) = (0 ≤ int pc + b)
lemma appThrow:
app Throw G maxs rT pc et (Some s) = (∃T ST LT r. s = (T # ST, LT) ∧ T = RefT r ∧ (∀C∈set (match_any G pc et). is_class G C))
lemma appInvoke:
app (Invoke C mn fpTs) G maxs rT pc et (Some s) = (∃apTs X ST LT mD' rT' b'. s = (rev apTs @ X # ST, LT) ∧ length apTs = length fpTs ∧ is_class G C ∧ G |- X <= Class C ∧ (∀(aT, fT)∈set (zip apTs fpTs). G |- aT <= fT) ∧ method (G, C) (mn, fpTs) = Some (mD', rT', b') ∧ (∀C∈set (match_any G pc et). is_class G C))
lemma effNone:
(pc', s') ∈ set (eff i G pc et None) ==> s' = None
lemma xcpt_app_lemma:
xcpt_app i G pc et = list_all (is_class G) (xcpt_names (i, G, pc, et))
lemmas
app i G maxs rT pc et s == case s of None => True | Some t => app' (i, G, pc, maxs, rT, t) ∧ xcpt_app i G pc et
xcpt_app i G pc et == ∀C∈set (xcpt_names (i, G, pc, et)). is_class G C
lemmas
app i G maxs rT pc et s == case s of None => True | Some t => app' (i, G, pc, maxs, rT, t) ∧ xcpt_app i G pc et
xcpt_app i G pc et == ∀C∈set (xcpt_names (i, G, pc, et)). is_class G C