--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MicroJava/BV/EffectMono.thy Sun Dec 16 00:17:44 2001 +0100
@@ -0,0 +1,442 @@
+(* Title: HOL/MicroJava/BV/EffMono.thy
+ ID: $Id$
+ Author: Gerwin Klein
+ Copyright 2000 Technische Universitaet Muenchen
+*)
+
+header {* Monotonicity of eff and app *}
+
+theory EffectMono = Effect:
+
+
+lemma PrimT_PrimT: "(G \<turnstile> xb \<preceq> PrimT p) = (xb = PrimT p)"
+ by (auto elim: widen.elims)
+
+
+lemma sup_loc_some [rule_format]:
+"\<forall>y n. (G \<turnstile> b <=l y) --> n < length y --> y!n = OK t -->
+ (\<exists>t. b!n = OK t \<and> (G \<turnstile> (b!n) <=o (y!n)))" (is "?P b")
+proof (induct (open) ?P b)
+ show "?P []" by simp
+
+ case Cons
+ show "?P (a#list)"
+ proof (clarsimp simp add: list_all2_Cons1 sup_loc_def Listn.le_def lesub_def)
+ fix z zs n
+ assume * :
+ "G \<turnstile> a <=o z" "list_all2 (sup_ty_opt G) list zs"
+ "n < Suc (length list)" "(z # zs) ! n = OK t"
+
+ show "(\<exists>t. (a # list) ! n = OK t) \<and> G \<turnstile>(a # list) ! n <=o OK t"
+ proof (cases n)
+ case 0
+ with * show ?thesis by (simp add: sup_ty_opt_OK)
+ next
+ case Suc
+ with Cons *
+ show ?thesis by (simp add: sup_loc_def Listn.le_def lesub_def)
+ qed
+ qed
+qed
+
+
+lemma all_widen_is_sup_loc:
+"\<forall>b. length a = length b -->
+ (\<forall>x\<in>set (zip a b). x \<in> widen G) = (G \<turnstile> (map OK a) <=l (map OK b))"
+ (is "\<forall>b. length a = length b --> ?Q a b" is "?P a")
+proof (induct "a")
+ show "?P []" by simp
+
+ fix l ls assume Cons: "?P ls"
+
+ show "?P (l#ls)"
+ proof (intro allI impI)
+ fix b
+ assume "length (l # ls) = length (b::ty list)"
+ with Cons
+ show "?Q (l # ls) b" by - (cases b, auto)
+ qed
+qed
+
+
+lemma append_length_n [rule_format]:
+"\<forall>n. n \<le> length x --> (\<exists>a b. x = a@b \<and> length a = n)" (is "?P x")
+proof (induct (open) ?P x)
+ show "?P []" by simp
+
+ fix l ls assume Cons: "?P ls"
+
+ show "?P (l#ls)"
+ proof (intro allI impI)
+ fix n
+ assume l: "n \<le> length (l # ls)"
+
+ 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 [rule_format])
+ then obtain a b where "ls = a @ b" "length a = n'" by rules
+ with s have "l # ls = (l#a) @ b \<and> length (l#a) = n" by simp
+ thus ?thesis by blast
+ qed
+ qed
+qed
+
+lemma rev_append_cons:
+"n < length x ==> \<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)
+ then obtain r d where x: "x = r@d" "length r = n" by rules
+ with n have "\<exists>b c. d = b#c" by (simp add: neq_Nil_conv)
+ then obtain b c where "d = b#c" by rules
+ with x have "x = (rev (rev r)) @ b # c \<and> length (rev r) = n" by simp
+ thus ?thesis by blast
+qed
+
+lemma sup_loc_length_map:
+ "G \<turnstile> map f a <=l map g b \<Longrightarrow> length a = length b"
+proof -
+ assume "G \<turnstile> map f a <=l map g b"
+ hence "length (map f a) = length (map g b)" by (rule sup_loc_length)
+ thus ?thesis by simp
+qed
+
+lemmas [iff] = not_Err_eq
+
+lemma app_mono:
+"[|G \<turnstile> s <=' s'; app i G m rT pc et s'|] ==> app i G m rT pc et s"
+proof -
+
+ { fix s1 s2
+ assume G: "G \<turnstile> s2 <=s s1"
+ assume app: "app i G m rT pc et (Some s1)"
+
+ note [simp] = sup_loc_length sup_loc_length_map
+
+ have "app i G m rT pc et (Some s2)"
+ proof (cases (open) i)
+ case Load
+
+ from G Load app
+ have "G \<turnstile> snd s2 <=l snd s1" by (auto simp add: sup_state_conv)
+
+ with G Load app show ?thesis
+ by (cases s2) (auto simp add: sup_state_conv dest: sup_loc_some)
+ next
+ case Store
+ with G app show ?thesis
+ by (cases s2, auto simp add: map_eq_Cons sup_loc_Cons2 sup_state_conv)
+ next
+ case LitPush
+ with G app show ?thesis by (cases s2, auto simp add: sup_state_conv)
+ next
+ case New
+ with G app show ?thesis by (cases s2, auto simp add: sup_state_conv)
+ next
+ case Getfield
+ with app G show ?thesis
+ by (cases s2) (clarsimp simp add: sup_state_Cons2, rule widen_trans)
+ next
+ case Putfield
+
+ with app
+ obtain vT oT ST LT b
+ 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" and
+ xc: "is_class G (Xcpt NullPointer)"
+ by force
+ 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"
+ 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)
+ moreover
+ from oT' oT
+ have "G\<turnstile> oT' \<preceq> (Class cname)" by (rule widen_trans)
+ ultimately
+ show ?thesis by (auto simp add: Putfield xc)
+ next
+ case Checkcast
+ with app G show ?thesis
+ by (cases s2, auto intro!: widen_RefT2 simp add: sup_state_Cons2)
+ next
+ case Return
+ with app G show ?thesis
+ by (cases s2) (auto simp add: sup_state_Cons2, rule widen_trans)
+ next
+ case Pop
+ with app G show ?thesis
+ by (cases s2, clarsimp simp add: sup_state_Cons2)
+ next
+ case Dup
+ with app G show ?thesis
+ by (cases s2, clarsimp simp add: sup_state_Cons2,
+ auto dest: sup_state_length)
+ next
+ case Dup_x1
+ with app G show ?thesis
+ by (cases s2, clarsimp simp add: sup_state_Cons2,
+ auto dest: sup_state_length)
+ next
+ case Dup_x2
+ with app G show ?thesis
+ by (cases s2, clarsimp simp add: sup_state_Cons2,
+ auto dest: sup_state_length)
+ next
+ case Swap
+ with app G show ?thesis
+ by (cases s2, clarsimp simp add: sup_state_Cons2)
+ next
+ case IAdd
+ with app G show ?thesis
+ by (cases s2, auto simp add: sup_state_Cons2 PrimT_PrimT)
+ next
+ case Goto
+ with app show ?thesis by simp
+ next
+ case Ifcmpeq
+ with app G show ?thesis
+ by (cases s2, auto simp add: sup_state_Cons2 PrimT_PrimT widen_RefT2)
+ next
+ case Invoke
+
+ with app
+ obtain apTs X ST LT mD' rT' b' where
+ s1: "s1 = (rev apTs @ X # ST, LT)" and
+ l: "length apTs = length list" and
+ c: "is_class G cname" 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) = Some (mD', rT', b')" and
+ x: "\<forall>C \<in> set (match_any G pc et). is_class G C"
+ by (simp del: not_None_eq, elim exE conjE) (rule that)
+
+ obtain apTs' X' ST' LT' where
+ s2: "s2 = (rev apTs' @ X' # ST', LT')" and
+ l': "length apTs' = length list"
+ proof -
+ from l s1 G
+ have "length list < length (fst s2)"
+ by simp
+ hence "\<exists>a b c. (fst s2) = rev a @ b # c \<and> length a = length list"
+ by (rule rev_append_cons [rule_format])
+ thus ?thesis
+ by - (cases s2, elim exE conjE, simp, rule that)
+ qed
+
+ from l l'
+ have "length (rev apTs') = length (rev apTs)" by simp
+
+ 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)"
+ by (simp add: sup_state_rev_fst sup_state_append_fst sup_state_Cons1)
+
+ with C
+ have C': "G \<turnstile> X' \<preceq> Class cname"
+ by - (rule widen_trans, auto)
+
+ from G'
+ have "G \<turnstile> map OK apTs' <=l map OK apTs"
+ by (simp add: sup_state_conv)
+ also
+ from l w
+ have "G \<turnstile> map OK apTs <=l map OK list"
+ by (simp add: all_widen_is_sup_loc)
+ finally
+ have "G \<turnstile> map OK apTs' <=l map OK list" .
+
+ with l'
+ 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 c x
+ show ?thesis
+ by (simp del: split_paired_Ex) blast
+ next
+ case Throw
+ with app G show ?thesis
+ by (cases s2, clarsimp simp add: sup_state_Cons2 widen_RefT2)
+ qed
+ } note this [simp]
+
+ assume "G \<turnstile> s <=' s'" "app i G m rT pc et s'"
+ thus ?thesis by (cases s, cases s', auto)
+qed
+
+lemmas [simp del] = split_paired_Ex
+
+lemma eff'_mono:
+"[| app i G m rT pc et (Some s2); G \<turnstile> s1 <=s s2 |] ==>
+ G \<turnstile> eff' (i,G,s1) <=s eff' (i,G,s2)"
+proof (cases s1, cases s2)
+ fix a1 b1 a2 b2
+ assume s: "s1 = (a1,b1)" "s2 = (a2,b2)"
+ assume app2: "app i G m rT pc et (Some s2)"
+ assume G: "G \<turnstile> s1 <=s s2"
+
+ note [simp] = eff_def
+
+ hence "G \<turnstile> (Some s1) <=' (Some s2)" by simp
+ from this app2
+ have app1: "app i G m rT pc et (Some s1)" by (rule app_mono)
+
+ show ?thesis
+ proof (cases (open) i)
+ case Load
+
+ with s app1
+ obtain y where
+ y: "nat < length b1" "b1 ! nat = OK y" by clarsimp
+
+ from Load s app2
+ obtain y' where
+ y': "nat < length b2" "b2 ! nat = OK y'" by clarsimp
+
+ from G s
+ have "G \<turnstile> b1 <=l b2" by (simp add: sup_state_conv)
+
+ with y y'
+ have "G \<turnstile> y \<preceq> y'"
+ by - (drule sup_loc_some, simp+)
+
+ with Load G y y' s app1 app2
+ show ?thesis by (clarsimp simp add: sup_state_conv)
+ next
+ case Store
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_conv sup_loc_update)
+ next
+ case LitPush
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case New
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case Getfield
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case Putfield
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case Checkcast
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case Invoke
+
+ with s app1
+ obtain a X ST where
+ s1: "s1 = (a @ X # ST, b1)" and
+ l: "length a = length list"
+ by (simp, elim exE conjE, simp)
+
+ from Invoke s app2
+ obtain a' X' ST' where
+ s2: "s2 = (a' @ X' # ST', b2)" and
+ l': "length a' = length list"
+ by (simp, elim exE conjE, simp)
+
+ from l l'
+ have lr: "length a = length a'" by simp
+
+ from lr G s s1 s2
+ have "G \<turnstile> (ST, b1) <=s (ST', b2)"
+ by (simp add: sup_state_append_fst sup_state_Cons1)
+
+ moreover
+
+ obtain b1' b2' where eff':
+ "b1' = snd (eff' (i,G,s1))"
+ "b2' = snd (eff' (i,G,s2))" by simp
+
+ from Invoke G s eff' app1 app2
+ obtain "b1 = b1'" "b2 = b2'" by simp
+
+ ultimately
+
+ have "G \<turnstile> (ST, b1') <=s (ST', b2')" by simp
+
+ with Invoke G s app1 app2 eff' s1 s2 l l'
+ show ?thesis
+ by (clarsimp simp add: sup_state_conv)
+ next
+ case Return
+ with G
+ show ?thesis
+ by simp
+ next
+ case Pop
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case Dup
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case Dup_x1
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case Dup_x2
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case Swap
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case IAdd
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case Goto
+ with G s app1 app2
+ show ?thesis by simp
+ next
+ case Ifcmpeq
+ with G s app1 app2
+ show ?thesis
+ by (clarsimp simp add: sup_state_Cons1)
+ next
+ case Throw
+ with G
+ show ?thesis
+ by simp
+ qed
+qed
+
+lemmas [iff del] = not_Err_eq
+
+end
+