--- a/src/HOL/MicroJava/BV/Step.thy Fri Aug 11 14:52:39 2000 +0200
+++ b/src/HOL/MicroJava/BV/Step.thy Fri Aug 11 14:52:52 2000 +0200
@@ -12,7 +12,7 @@
text "Effect of instruction on the state type"
consts
-step :: "instr \\<times> jvm_prog \\<times> state_type \\<Rightarrow> state_type option"
+step :: "instr \<times> jvm_prog \<times> state_type \<Rightarrow> state_type option"
recdef step "{}"
"step (Load idx, G, (ST, LT)) = Some (the (LT ! idx) # ST, LT)"
@@ -41,22 +41,22 @@
text "Conditions under which step is applicable"
consts
-app :: "instr \\<times> jvm_prog \\<times> ty \\<times> state_type \\<Rightarrow> bool"
+app :: "instr \<times> jvm_prog \<times> ty \<times> state_type \<Rightarrow> bool"
recdef app "{}"
-"app (Load idx, G, rT, s) = (idx < length (snd s) \\<and> (snd s) ! idx \\<noteq> None)"
+"app (Load idx, G, rT, s) = (idx < length (snd s) \<and> (snd s) ! idx \<noteq> None)"
"app (Store idx, G, rT, (ts#ST, LT)) = (idx < length LT)"
"app (Bipush i, G, rT, s) = True"
"app (Aconst_null, G, rT, s) = True"
-"app (Getfield F C, G, rT, (oT#ST, LT)) = (is_class G C \\<and>
- field (G,C) F \\<noteq> None \\<and>
- fst (the (field (G,C) F)) = C \\<and>
- G \\<turnstile> oT \\<preceq> (Class C))"
-"app (Putfield F C, G, rT, (vT#oT#ST, LT)) = (is_class G C \\<and>
- field (G,C) F \\<noteq> None \\<and>
- fst (the (field (G,C) F)) = C \\<and>
- G \\<turnstile> oT \\<preceq> (Class C) \\<and>
- G \\<turnstile> vT \\<preceq> (snd (the (field (G,C) F))))"
+"app (Getfield F C, G, rT, (oT#ST, LT)) = (is_class G C \<and>
+ field (G,C) F \<noteq> None \<and>
+ fst (the (field (G,C) F)) = C \<and>
+ G \<turnstile> oT \<preceq> (Class C))"
+"app (Putfield F C, G, rT, (vT#oT#ST, LT)) = (is_class G C \<and>
+ field (G,C) F \<noteq> None \<and>
+ fst (the (field (G,C) F)) = C \<and>
+ G \<turnstile> oT \<preceq> (Class C) \<and>
+ G \<turnstile> vT \<preceq> (snd (the (field (G,C) F))))"
"app (New C, G, rT, s) = (is_class G C)"
"app (Checkcast C, G, rT, (RefT rt#ST,LT)) = (is_class G C)"
"app (Pop, G, rT, (ts#ST,LT)) = True"
@@ -66,19 +66,19 @@
"app (Swap, G, rT, (ts1#ts2#ST,LT)) = True"
"app (IAdd, G, rT, (PrimT Integer#PrimT Integer#ST,LT))
= True"
-"app (Ifcmpeq b, G, rT, (ts1#ts2#ST,LT)) = ((\\<exists> p. ts1 = PrimT p \\<and> ts1 = PrimT p) \\<or>
- (\\<exists>r r'. ts1 = RefT r \\<and> ts2 = RefT r'))"
+"app (Ifcmpeq b, G, rT, (ts1#ts2#ST,LT)) = ((\<exists> p. ts1 = PrimT p \<and> ts1 = PrimT p) \<or>
+ (\<exists>r r'. ts1 = RefT r \<and> ts2 = RefT r'))"
"app (Goto b, G, rT, s) = True"
-"app (Return, G, rT, (T#ST,LT)) = (G \\<turnstile> T \\<preceq> rT)"
+"app (Return, G, rT, (T#ST,LT)) = (G \<turnstile> T \<preceq> rT)"
app_invoke:
-"app (Invoke C mn fpTs, G, rT, s) = (length fpTs < length (fst s) \\<and>
+"app (Invoke C mn fpTs, G, rT, s) = (length fpTs < length (fst s) \<and>
(let
apTs = rev (take (length fpTs) (fst s));
X = hd (drop (length fpTs) (fst s))
in
- G \\<turnstile> X \\<preceq> Class C \\<and>
- (\\<forall>(aT,fT)\\<in>set(zip apTs fpTs). G \\<turnstile> aT \\<preceq> fT) \\<and>
- method (G,C) (mn,fpTs) \\<noteq> None
+ G \<turnstile> X \<preceq> Class C \<and>
+ (\<forall>(aT,fT)\<in>set(zip apTs fpTs). G \<turnstile> aT \<preceq> fT) \<and>
+ method (G,C) (mn,fpTs) \<noteq> None
))"
"app (i,G,rT,s) = False"
@@ -87,7 +87,7 @@
text {* \isa{p_count} of successor instructions *}
consts
-succs :: "instr \\<Rightarrow> p_count \\<Rightarrow> p_count set"
+succs :: "instr \<Rightarrow> p_count \<Rightarrow> p_count set"
primrec
"succs (Load idx) pc = {pc+1}"
@@ -110,25 +110,25 @@
"succs (Invoke C mn fpTs) pc = {pc+1}"
-lemma 1: "2 < length a \\<Longrightarrow> (\\<exists>l l' l'' ls. a = l#l'#l''#ls)"
+lemma 1: "2 < length a \<Longrightarrow> (\<exists>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: "\\<not>(2 < length a) \\<Longrightarrow> a = [] \\<or> (\\<exists> l. a = [l]) \\<or> (\\<exists> l l'. a = [l,l'])"
+lemma 2: "\<not>(2 < length a) \<Longrightarrow> a = [] \<or> (\<exists> l. a = [l]) \<or> (\<exists> l l'. a = [l,l'])"
proof -;
- assume "\\<not>(2 < length a)"
+ assume "\<not>(2 < length a)"
hence "length a < (Suc 2)" by simp
- hence * : "length a = 0 \\<or> length a = 1 \\<or> length a = 2" by (auto simp add: less_Suc_eq)
+ hence * : "length a = 0 \<or> length a = 1 \<or> length a = 2" by (auto simp add: less_Suc_eq)
{
fix x
assume "length x = 1"
- hence "\\<exists> l. x = [l]" by - (cases x, auto)
+ hence "\<exists> l. x = [l]" by - (cases x, auto)
} note 0 = this
- have "length a = 2 \\<Longrightarrow> \\<exists>l l'. a = [l,l']" by (cases a, auto dest: 0)
+ have "length a = 2 \<Longrightarrow> \<exists>l l'. a = [l,l']" by (cases a, auto dest: 0)
with * show ?thesis by (auto dest: 0)
qed
@@ -139,56 +139,56 @@
*}
lemma appStore[simp]:
-"app (Store idx, G, rT, s) = (\\<exists> ts ST LT. s = (ts#ST,LT) \\<and> idx < length LT)"
+"app (Store idx, G, rT, s) = (\<exists> ts ST LT. s = (ts#ST,LT) \<and> idx < length LT)"
by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
lemma appGetField[simp]:
-"app (Getfield F C, G, rT, s) = (\\<exists> oT ST LT. s = (oT#ST, LT) \\<and> is_class G C \\<and>
- fst (the (field (G,C) F)) = C \\<and>
- field (G,C) F \\<noteq> None \\<and> G \\<turnstile> oT \\<preceq> (Class C))"
+"app (Getfield F C, G, rT, s) = (\<exists> oT ST LT. s = (oT#ST, LT) \<and> is_class G C \<and>
+ fst (the (field (G,C) F)) = C \<and>
+ field (G,C) F \<noteq> None \<and> G \<turnstile> oT \<preceq> (Class C))"
by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
lemma appPutField[simp]:
-"app (Putfield F C, G, rT, s) = (\\<exists> vT oT ST LT. s = (vT#oT#ST, LT) \\<and> is_class G C \\<and>
- field (G,C) F \\<noteq> None \\<and> fst (the (field (G,C) F)) = C \\<and>
- G \\<turnstile> oT \\<preceq> (Class C) \\<and>
- G \\<turnstile> vT \\<preceq> (snd (the (field (G,C) F))))"
+"app (Putfield F C, G, rT, s) = (\<exists> vT oT ST LT. s = (vT#oT#ST, LT) \<and> is_class G C \<and>
+ field (G,C) F \<noteq> None \<and> fst (the (field (G,C) F)) = C \<and>
+ G \<turnstile> oT \<preceq> (Class C) \<and>
+ G \<turnstile> vT \<preceq> (snd (the (field (G,C) F))))"
by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
lemma appCheckcast[simp]:
-"app (Checkcast C, G, rT, s) = (\\<exists>rT ST LT. s = (RefT rT#ST,LT) \\<and> is_class G C)"
+"app (Checkcast C, G, rT, s) = (\<exists>rT ST LT. s = (RefT rT#ST,LT) \<and> is_class G C)"
by (cases s, cases "fst s", simp, cases "hd (fst s)", auto)
lemma appPop[simp]:
-"app (Pop, G, rT, s) = (\\<exists>ts ST LT. s = (ts#ST,LT))"
+"app (Pop, G, rT, s) = (\<exists>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, rT, s) = (\\<exists>ts ST LT. s = (ts#ST,LT))"
+"app (Dup, G, rT, s) = (\<exists>ts ST LT. s = (ts#ST,LT))"
by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
lemma appDup_x1[simp]:
-"app (Dup_x1, G, rT, s) = (\\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT))"
+"app (Dup_x1, G, rT, s) = (\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT))"
by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
lemma appDup_x2[simp]:
-"app (Dup_x2, G, rT, s) = (\\<exists>ts1 ts2 ts3 ST LT. s = (ts1#ts2#ts3#ST,LT))"
+"app (Dup_x2, G, rT, s) = (\<exists>ts1 ts2 ts3 ST LT. s = (ts1#ts2#ts3#ST,LT))"
by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
lemma appSwap[simp]:
-"app (Swap, G, rT, s) = (\\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT))"
+"app (Swap, G, rT, s) = (\<exists>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, rT, s) = (\\<exists> ST LT. s = (PrimT Integer#PrimT Integer#ST,LT))" (is "?app s = ?P s")
+"app (IAdd, G, rT, s) = (\<exists> ST LT. s = (PrimT Integer#PrimT Integer#ST,LT))" (is "?app s = ?P s")
proof (cases s)
case Pair
have "?app (a,b) = ?P (a,b)"
@@ -218,44 +218,49 @@
lemma appIfcmpeq[simp]:
-"app (Ifcmpeq b, G, rT, s) = (\\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT) \\<and>
- ((\\<exists> p. ts1 = PrimT p \\<and> ts1 = PrimT p) \\<or>
- (\\<exists>r r'. ts1 = RefT r \\<and> ts2 = RefT r')))"
+"app (Ifcmpeq b, G, rT, s) = (\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT) \<and>
+ ((\<exists> p. ts1 = PrimT p \<and> ts1 = PrimT p) \<or>
+ (\<exists>r r'. ts1 = RefT r \<and> ts2 = RefT r')))"
by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
lemma appReturn[simp]:
-"app (Return, G, rT, s) = (\\<exists>T ST LT. s = (T#ST,LT) \\<and> (G \\<turnstile> T \\<preceq> rT))"
+"app (Return, G, rT, s) = (\<exists>T ST LT. s = (T#ST,LT) \<and> (G \<turnstile> T \<preceq> rT))"
by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
lemma appInvoke[simp]:
-"app (Invoke C mn fpTs, G, rT, s) = (\\<exists>apTs X ST LT.
- s = ((rev apTs) @ (X # ST), LT) \\<and>
- length apTs = length fpTs \\<and>
- G \\<turnstile> X \\<preceq> Class C \\<and>
- (\\<forall>(aT,fT)\\<in>set(zip apTs fpTs). G \\<turnstile> aT \\<preceq> fT) \\<and>
- method (G,C) (mn,fpTs) \\<noteq> None)" (is "?app s = ?P s")
+"app (Invoke C mn fpTs, G, rT, s) = (\<exists>apTs X ST LT.
+ s = ((rev apTs) @ (X # ST), LT) \<and>
+ length apTs = length fpTs \<and>
+ G \<turnstile> X \<preceq> Class C \<and>
+ (\<forall>(aT,fT)\<in>set(zip apTs fpTs). G \<turnstile> aT \<preceq> fT) \<and>
+ method (G,C) (mn,fpTs) \<noteq> None)" (is "?app s = ?P s")
proof (cases s)
case Pair
- have "?app (a,b) \\<Longrightarrow> ?P (a,b)"
+ have "?app (a,b) \<Longrightarrow> ?P (a,b)"
proof -
assume app: "?app (a,b)"
- hence "a = (rev (rev (take (length fpTs) a))) @ (drop (length fpTs) a) \\<and> length fpTs < length a"
- (is "?a \\<and> ?l") by auto
- hence "?a \\<and> 0 < length (drop (length fpTs) a)" (is "?a \\<and> ?l") by auto
- hence "?a \\<and> ?l \\<and> length (rev (take (length fpTs) a)) = length fpTs" by (auto simp add: min_def)
- hence "\\<exists>apTs ST. a = rev apTs @ ST \\<and> length apTs = length fpTs \\<and> 0 < length ST" by blast
- hence "\\<exists>apTs ST. a = rev apTs @ ST \\<and> length apTs = length fpTs \\<and> ST \\<noteq> []" by blast
- hence "\\<exists>apTs ST. a = rev apTs @ ST \\<and> length apTs = length fpTs \\<and> (\\<exists>X ST'. ST = X#ST')" by (simp add: neq_Nil_conv)
- hence "\\<exists>apTs X ST. a = rev apTs @ X # ST \\<and> length apTs = length fpTs" by blast
+ hence "a = (rev (rev (take (length fpTs) a))) @ (drop (length fpTs) a) \<and> length fpTs < length a"
+ (is "?a \<and> ?l") by auto
+ hence "?a \<and> 0 < length (drop (length fpTs) a)" (is "?a \<and> ?l") by auto
+ hence "?a \<and> ?l \<and> length (rev (take (length fpTs) a)) = length fpTs" by (auto simp add: min_def)
+ hence "\<exists>apTs ST. a = rev apTs @ ST \<and> length apTs = length fpTs \<and> 0 < length ST" by blast
+ hence "\<exists>apTs ST. a = rev apTs @ ST \<and> length apTs = length fpTs \<and> ST \<noteq> []" by blast
+ hence "\<exists>apTs ST. a = rev apTs @ ST \<and> length apTs = length fpTs \<and> (\<exists>X ST'. ST = X#ST')" by (simp add: neq_Nil_conv)
+ hence "\<exists>apTs X ST. a = rev apTs @ X # ST \<and> length apTs = length fpTs" by blast
with app
show ?thesis by auto blast
qed
- with Pair have "?app s \\<Longrightarrow> ?P s" by simp
+ with Pair have "?app s \<Longrightarrow> ?P s" by simp
thus ?thesis by auto
qed
lemmas [simp del] = app_invoke
+
+lemma app_step_some:
+ "\<lbrakk>app (i,G,rT,s); succs i pc \<noteq> {}\<rbrakk> \<Longrightarrow> step (i,G,s) \<noteq> None";
+ by (cases s, cases i, auto)
+
end