--- a/src/HOL/MicroJava/BV/StepMono.thy Fri Aug 11 14:52:39 2000 +0200
+++ b/src/HOL/MicroJava/BV/StepMono.thy Fri Aug 11 14:52:52 2000 +0200
@@ -1,4 +1,4 @@
-(* Title: HOL/MicroJava/BV/Step.thy
+(* Title: HOL/MicroJava/BV/StepMono.thy
ID: $Id$
Author: Gerwin Klein
Copyright 2000 Technische Universitaet Muenchen
@@ -11,39 +11,31 @@
lemmas [trans] = sup_state_trans sup_loc_trans widen_trans
-ML_setup {* bind_thm ("widen_RefT", widen_RefT) *}
-ML_setup {* bind_thm ("widen_RefT2", widen_RefT2) *}
-
-
-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)
-
lemma sup_state_length:
-"G \\<turnstile> s2 <=s s1 \\<Longrightarrow> length (fst s2) = length (fst s1) \\<and> length (snd s2) = length (snd s1)"
+"G \<turnstile> s2 <=s s1 \<Longrightarrow> length (fst s2) = length (fst s1) \<and> length (snd s2) = length (snd s1)"
by (cases s1, cases s2, simp add: sup_state_length_fst sup_state_length_snd)
-lemma PrimT_PrimT: "(G \\<turnstile> xb \\<preceq> PrimT p) = (xb = PrimT p)"
+lemma PrimT_PrimT: "(G \<turnstile> xb \<preceq> PrimT p) = (xb = PrimT p)"
proof
- show "xb = PrimT p \\<Longrightarrow> G \\<turnstile> xb \\<preceq> PrimT p" by blast
+ show "xb = PrimT p \<Longrightarrow> G \<turnstile> xb \<preceq> PrimT p" by blast
- show "G\\<turnstile> xb \\<preceq> PrimT p \\<Longrightarrow> xb = PrimT p"
+ show "G\<turnstile> xb \<preceq> PrimT p \<Longrightarrow> xb = PrimT p"
proof (cases xb)
fix prim
- assume "xb = PrimT prim" "G\\<turnstile>xb\\<preceq>PrimT p"
+ assume "xb = PrimT prim" "G\<turnstile>xb\<preceq>PrimT p"
thus ?thesis by simp
next
fix ref
- assume "G\\<turnstile>xb\\<preceq>PrimT p" "xb = RefT ref"
+ assume "G\<turnstile>xb\<preceq>PrimT p" "xb = RefT ref"
thus ?thesis by simp (rule widen_RefT [elimify], auto)
qed
qed
lemma sup_loc_some [rulify]:
-"\\<forall> y n. (G \\<turnstile> b <=l y) \\<longrightarrow> n < length y \\<longrightarrow> y!n = Some t \\<longrightarrow> (\\<exists>t. b!n = Some t \\<and> (G \\<turnstile> (b!n) <=o (y!n)))" (is "?P b")
+"\<forall> y n. (G \<turnstile> b <=l y) \<longrightarrow> n < length y \<longrightarrow> y!n = Some t \<longrightarrow> (\<exists>t. b!n = Some t \<and> (G \<turnstile> (b!n) <=o (y!n)))" (is "?P b")
proof (induct "?P" b)
show "?P []" by simp
@@ -52,10 +44,10 @@
proof (clarsimp simp add: list_all2_Cons1 sup_loc_def)
fix z zs n
assume * :
- "G \\<turnstile> a <=o z" "list_all2 (sup_ty_opt G) list zs"
+ "G \<turnstile> a <=o z" "list_all2 (sup_ty_opt G) list zs"
"n < Suc (length zs)" "(z # zs) ! n = Some t"
- show "(\\<exists>t. (a # list) ! n = Some t) \\<and> G \\<turnstile>(a # list) ! n <=o Some t"
+ show "(\<exists>t. (a # list) ! n = Some t) \<and> G \<turnstile>(a # list) ! n <=o Some t"
proof (cases n)
case 0
with * show ?thesis by (simp add: sup_ty_opt_some)
@@ -69,8 +61,8 @@
lemma all_widen_is_sup_loc:
-"\\<forall>b. length a = length b \\<longrightarrow> (\\<forall>x\\<in>set (zip a b). x \\<in> widen G) = (G \\<turnstile> (map Some a) <=l (map Some b))"
- (is "\\<forall>b. length a = length b \\<longrightarrow> ?Q a b" is "?P a")
+"\<forall>b. length a = length b \<longrightarrow> (\<forall>x\<in>set (zip a b). x \<in> widen G) = (G \<turnstile> (map Some a) <=l (map Some b))"
+ (is "\<forall>b. length a = length b \<longrightarrow> ?Q a b" is "?P a")
proof (induct "a")
show "?P []" by simp
@@ -87,7 +79,7 @@
lemma append_length_n [rulify]:
-"\\<forall>n. n \\<le> length x \\<longrightarrow> (\\<exists>a b. x = a@b \\<and> length a = n)" (is "?P x")
+"\<forall>n. n \<le> length x \<longrightarrow> (\<exists>a b. x = a@b \<and> length a = n)" (is "?P x")
proof (induct "?P" "x")
show "?P []" by simp
@@ -96,22 +88,22 @@
show "?P (l#ls)"
proof (intro allI impI)
fix n
- assume l: "n \\<le> length (l # ls)"
+ assume l: "n \<le> length (l # ls)"
- show "\\<exists>a b. l # ls = a @ b \\<and> length a = n"
+ show "\<exists>a b. l # ls = a @ b \<and> length a = n"
proof (cases n)
assume "n=0" thus ?thesis by simp
next
fix "n'" assume s: "n = Suc n'"
with l
- have "n' \\<le> length ls" by simp
- hence "\\<exists>a b. ls = a @ b \\<and> length a = n'" by (rule Cons [rulify])
+ have "n' \<le> length ls" by simp
+ hence "\<exists>a b. ls = a @ b \<and> length a = n'" by (rule Cons [rulify])
thus ?thesis
proof elim
fix a b
assume "ls = a @ b" "length a = n'"
with s
- have "l # ls = (l#a) @ b \\<and> length (l#a) = n" by simp
+ have "l # ls = (l#a) @ b \<and> length (l#a) = n" by simp
thus ?thesis by blast
qed
qed
@@ -121,23 +113,23 @@
lemma rev_append_cons:
-"\\<lbrakk>n < length x\\<rbrakk> \\<Longrightarrow> \\<exists>a b c. x = (rev a) @ b # c \\<and> length a = n"
+"\<lbrakk>n < length x\<rbrakk> \<Longrightarrow> \<exists>a b c. x = (rev a) @ b # c \<and> length a = n"
proof -
assume n: "n < length x"
- hence "n \\<le> length x" by simp
- hence "\\<exists>a b. x = a @ b \\<and> length a = n" by (rule append_length_n)
+ hence "n \<le> length x" by simp
+ hence "\<exists>a b. x = a @ b \<and> length a = n" by (rule append_length_n)
thus ?thesis
proof elim
fix r d assume x: "x = r@d" "length r = n"
with n
- have "\\<exists>b c. d = b#c" by (simp add: neq_Nil_conv)
+ have "\<exists>b c. d = b#c" by (simp add: neq_Nil_conv)
thus ?thesis
proof elim
fix b c
assume "d = b#c"
with x
- have "x = (rev (rev r)) @ b # c \\<and> length (rev r) = n" by simp
+ have "x = (rev (rev r)) @ b # c \<and> length (rev r) = n" by simp
thus ?thesis by blast
qed
qed
@@ -145,9 +137,9 @@
lemma app_mono:
-"\\<lbrakk>G \\<turnstile> s2 <=s s1; app (i, G, rT, s1)\\<rbrakk> \\<Longrightarrow> app (i, G, rT, s2)";
+"\<lbrakk>G \<turnstile> s2 <=s s1; app (i, G, rT, s1)\<rbrakk> \<Longrightarrow> app (i, G, rT, s2)";
proof -
- assume G: "G \\<turnstile> s2 <=s s1"
+ assume G: "G \<turnstile> s2 <=s s1"
assume app: "app (i, G, rT, s1)"
show ?thesis
@@ -158,7 +150,7 @@
have l: "length (snd s1) = length (snd s2)" by (simp add: sup_state_length)
from G Load app
- have "G \\<turnstile> snd s2 <=l snd s1" by (auto simp add: sup_state_def)
+ have "G \<turnstile> snd s2 <=l snd s1" by (auto simp add: sup_state_def)
with G Load app l
show ?thesis by clarsimp (drule sup_loc_some, simp+)
@@ -191,22 +183,22 @@
where s1: "s1 = (vT # oT # ST, LT)" and
"field (G, cname) vname = Some (cname, b)"
"is_class G cname" and
- oT: "G\\<turnstile> oT\\<preceq> (Class cname)" and
- vT: "G\\<turnstile> vT\\<preceq> b"
+ oT: "G\<turnstile> oT\<preceq> (Class cname)" and
+ vT: "G\<turnstile> vT\<preceq> b"
by simp (elim exE conjE, simp, rule that)
moreover
from s1 G
obtain vT' oT' ST' LT'
where s2: "s2 = (vT' # oT' # ST', LT')" and
- oT': "G\\<turnstile> oT' \\<preceq> oT" and
- vT': "G\\<turnstile> vT' \\<preceq> vT"
+ oT': "G\<turnstile> oT' \<preceq> oT" and
+ vT': "G\<turnstile> vT' \<preceq> vT"
by - (cases s2, simp add: sup_state_Cons2, elim exE conjE, simp, rule that)
moreover
from vT' vT
- have "G \\<turnstile> vT' \\<preceq> b" by (rule widen_trans)
+ have "G \<turnstile> vT' \<preceq> b" by (rule widen_trans)
moreover
from oT' oT
- have "G\\<turnstile> oT' \\<preceq> (Class cname)" by (rule widen_trans)
+ have "G\<turnstile> oT' \<preceq> (Class cname)" by (rule widen_trans)
ultimately
show ?thesis
by (auto simp add: Putfield)
@@ -214,12 +206,12 @@
case Checkcast
with app G
show ?thesis
- by - (cases s2, auto intro: widen_RefT2 simp add: sup_state_Cons2)
+ by - (cases s2, auto intro!: widen_RefT2 simp add: sup_state_Cons2)
next
case Return
with app G
show ?thesis
- by - (cases s2, clarsimp simp add: sup_state_Cons2, rule widen_trans)
+ by - (cases s2, auto simp add: sup_state_Cons2, rule widen_trans)
next
case Pop
with app G
@@ -266,9 +258,9 @@
obtain apTs X ST LT where
s1: "s1 = (rev apTs @ X # ST, LT)" and
l: "length apTs = length list" and
- C: "G \\<turnstile> X \\<preceq> Class cname" and
- w: "\\<forall>x \\<in> set (zip apTs list). x \\<in> widen G" and
- m: "method (G, cname) (mname, list) \\<noteq> None"
+ C: "G \<turnstile> X \<preceq> Class cname" and
+ w: "\<forall>x \<in> set (zip apTs list). x \<in> widen G" and
+ m: "method (G, cname) (mname, list) \<noteq> None"
by (simp del: not_None_eq, elim exE conjE) (rule that)
obtain apTs' X' ST' LT' where
@@ -278,7 +270,7 @@
from l s1 G
have "length list < length (fst s2)"
by (simp add: sup_state_length)
- hence "\\<exists>a b c. (fst s2) = rev a @ b # c \\<and> length a = length list"
+ hence "\<exists>a b c. (fst s2) = rev a @ b # c \<and> length a = length list"
by (rule rev_append_cons [rulify])
thus ?thesis
by - (cases s2, elim exE conjE, simp, rule that)
@@ -289,26 +281,26 @@
from this s1 s2 G
obtain
- G': "G \\<turnstile> (apTs',LT') <=s (apTs,LT)" and
- X : "G \\<turnstile> X' \\<preceq> X" and "G \\<turnstile> (ST',LT') <=s (ST,LT)"
+ G': "G \<turnstile> (apTs',LT') <=s (apTs,LT)" and
+ X : "G \<turnstile> X' \<preceq> X" and "G \<turnstile> (ST',LT') <=s (ST,LT)"
by (simp add: sup_state_rev_fst sup_state_append_fst sup_state_Cons1);
with C
- have C': "G \\<turnstile> X' \\<preceq> Class cname"
+ have C': "G \<turnstile> X' \<preceq> Class cname"
by - (rule widen_trans, auto)
from G'
- have "G \\<turnstile> map Some apTs' <=l map Some apTs"
+ have "G \<turnstile> map Some apTs' <=l map Some apTs"
by (simp add: sup_state_def)
also
from l w
- have "G \\<turnstile> map Some apTs <=l map Some list"
+ have "G \<turnstile> map Some apTs <=l map Some list"
by (simp add: all_widen_is_sup_loc)
finally
- have "G \\<turnstile> map Some apTs' <=l map Some list" .
+ have "G \<turnstile> map Some apTs' <=l map Some list" .
with l'
- have w': "\\<forall>x \\<in> set (zip apTs' list). x \\<in> widen G"
+ have w': "\<forall>x \<in> set (zip apTs' list). x \<in> widen G"
by (simp add: all_widen_is_sup_loc)
from Invoke s2 l' w' C' m
@@ -319,14 +311,14 @@
lemma step_mono:
-"\\<lbrakk>succs i pc \\<noteq> {}; app (i,G,rT,s2); G \\<turnstile> s1 <=s s2\\<rbrakk> \\<Longrightarrow>
- G \\<turnstile> the (step (i,G,s1)) <=s the (step (i,G,s2))"
+"\<lbrakk>succs i pc \<noteq> {}; app (i,G,rT,s2); G \<turnstile> s1 <=s s2\<rbrakk> \<Longrightarrow>
+ G \<turnstile> the (step (i,G,s1)) <=s the (step (i,G,s2))"
proof (cases s1, cases s2)
fix a1 b1 a2 b2
assume s: "s1 = (a1,b1)" "s2 = (a2,b2)"
- assume succs: "succs i pc \\<noteq> {}"
+ assume succs: "succs i pc \<noteq> {}"
assume app2: "app (i,G,rT,s2)"
- assume G: "G \\<turnstile> s1 <=s s2"
+ assume G: "G \<turnstile> s1 <=s s2"
from G app2
have app1: "app (i,G,rT,s1)" by (rule app_mono)
@@ -334,9 +326,9 @@
from app1 app2 succs
obtain a1' b1' a2' b2'
where step: "step (i,G,s1) = Some (a1',b1')" "step (i,G,s2) = Some (a2',b2')";
- by (auto dest: app_step_some);
+ by (auto dest!: app_step_some);
- have "G \\<turnstile> (a1',b1') <=s (a2',b2')"
+ have "G \<turnstile> (a1',b1') <=s (a2',b2')"
proof (cases i)
case Load
@@ -349,10 +341,10 @@
y': "nat < length b2" "b2 ! nat = Some y'" by clarsimp
from G s
- have "G \\<turnstile> b1 <=l b2" by (simp add: sup_state_def)
+ have "G \<turnstile> b1 <=l b2" by (simp add: sup_state_def)
with y y'
- have "G \\<turnstile> y \\<preceq> y'"
+ have "G \<turnstile> y \<preceq> y'"
by - (drule sup_loc_some, simp+)
with Load G y y' s step app1 app2
@@ -411,17 +403,17 @@
have lr: "length a = length a'" by simp
from lr G s s1 s2
- have "G \\<turnstile> (ST, b1) <=s (ST', b2)"
+ have "G \<turnstile> (ST, b1) <=s (ST', b2)"
by (simp add: sup_state_append_fst sup_state_Cons1)
moreover
from Invoke G s step app1 app2
- have "b1 = b1' \\<and> b2 = b2'" by simp
+ have "b1 = b1' \<and> b2 = b2'" by simp
ultimately
- have "G \\<turnstile> (ST, b1') <=s (ST', b2')" by simp
+ have "G \<turnstile> (ST, b1') <=s (ST', b2')" by simp
with Invoke G s step app1 app2 s1 s2 l l'
show ?thesis