(* Title: HOL/MicroJava/BV/StepMono.thy
ID: $Id$
Author: Gerwin Klein
Copyright 2000 Technische Universitaet Muenchen
*)
header {* Monotonicity of \isa{step} and \isa{app} *}
theory StepMono = Step :
lemmas [trans] = sup_state_trans sup_loc_trans widen_trans
lemma sup_state_length:
"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)"
proof
show "xb = PrimT p \<Longrightarrow> G \<turnstile> xb \<preceq> PrimT p" by blast
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"
thus ?thesis by simp
next
fix 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")
proof (induct "?P" b)
show "?P []" by simp
case Cons
show "?P (a#list)"
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"
"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"
proof (cases n)
case 0
with * show ?thesis by (simp add: sup_ty_opt_some)
next
case Suc
with Cons *
show ?thesis by (simp add: sup_loc_def)
qed
qed
qed
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")
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 [rulify]:
"\<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
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 [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
thus ?thesis by blast
qed
qed
qed
qed
lemma rev_append_cons:
"\<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)
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)
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
thus ?thesis by blast
qed
qed
qed
lemma app_mono:
"\<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 app: "app (i, G, rT, s1)"
show ?thesis
proof (cases i)
case Load
from G
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)
with G Load app l
show ?thesis by clarsimp (drule sup_loc_some, simp+)
next
case Store
with G app
show ?thesis
by - (cases s2,
auto dest: map_hd_tl simp add: sup_loc_Cons2 sup_loc_length sup_state_def)
next
case Bipush
thus ?thesis by simp
next
case Aconst_null
thus ?thesis by simp
next
case New
with app
show ?thesis by simp
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"
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"
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)
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)
next
case Dup_x1
with app G
show ?thesis
by - (cases s2, clarsimp simp add: sup_state_Cons2)
next
case Dup_x2
with app G
show ?thesis
by - (cases s2, clarsimp simp add: sup_state_Cons2)
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 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"
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 add: sup_state_length)
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)
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 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"
by (simp add: all_widen_is_sup_loc)
finally
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"
by (simp add: all_widen_is_sup_loc)
from Invoke s2 l' w' C' m
show ?thesis
by simp blast
qed
qed
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))"
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 app2: "app (i,G,rT,s2)"
assume G: "G \<turnstile> s1 <=s s2"
from G app2
have app1: "app (i,G,rT,s1)" by (rule app_mono)
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);
have "G \<turnstile> (a1',b1') <=s (a2',b2')"
proof (cases i)
case Load
with s app1
obtain y where
y: "nat < length b1" "b1 ! nat = Some y" by clarsimp
from Load s app2
obtain y' where
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)
with y y'
have "G \<turnstile> y \<preceq> y'"
by - (drule sup_loc_some, simp+)
with Load G y y' s step app1 app2
show ?thesis by (clarsimp simp add: sup_state_def)
next
case Store
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_def sup_loc_update)
next
case Bipush
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case New
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case Aconst_null
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case Getfield
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case Putfield
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case Checkcast
with G s step 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
from Invoke G s step app1 app2
have "b1 = b1' \<and> b2 = b2'" by simp
ultimately
have "G \<turnstile> (ST, b1') <=s (ST', b2')" by simp
with Invoke G s step app1 app2 s1 s2 l l'
show ?thesis
by (clarsimp simp add: sup_state_def)
next
case Return
with succs have "False" by simp
thus ?thesis by blast
next
case Pop
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case Dup
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case Dup_x1
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case Dup_x2
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case Swap
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case IAdd
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
next
case Goto
with G s step app1 app2
show ?thesis by simp
next
case Ifcmpeq
with G s step app1 app2
show ?thesis
by (clarsimp simp add: sup_state_Cons1)
qed
with step
show ?thesis by auto
qed
end