Theory JVM

theory JVM
imports Typing_Framework_JVM
(*  Title:      HOL/MicroJava/BV/JVM.thy
Author: Tobias Nipkow, Gerwin Klein
Copyright 2000 TUM
*)


header {* \isaheader{Kildall for the JVM}\label{sec:JVM} *}

theory JVM
imports Typing_Framework_JVM
begin

definition kiljvm :: "jvm_prog => nat => nat => ty => exception_table =>
instr list => JVMType.state list => JVMType.state list"
where
"kiljvm G maxs maxr rT et bs ==
kildall (JVMType.le G maxs maxr) (JVMType.sup G maxs maxr) (exec G maxs rT et bs)"


definition wt_kil :: "jvm_prog => cname => ty list => ty => nat => nat =>
exception_table => instr list => bool"
where
"wt_kil G C pTs rT mxs mxl et ins ==
check_bounded ins et ∧ 0 < size ins ∧
(let first = Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err));
start = OK first#(replicate (size ins - 1) (OK None));
result = kiljvm G mxs (1+size pTs+mxl) rT et ins start
in ∀n < size ins. result!n ≠ Err)"


definition wt_jvm_prog_kildall :: "jvm_prog => bool" where
"wt_jvm_prog_kildall G ==
wf_prog (λG C (sig,rT,(maxs,maxl,b,et)). wt_kil G C (snd sig) rT maxs maxl et b) G"


theorem is_bcv_kiljvm:
"[| wf_prog wf_mb G; bounded (exec G maxs rT et bs) (size bs) |] ==>
is_bcv (JVMType.le G maxs maxr) Err (exec G maxs rT et bs)
(size bs) (states G maxs maxr) (kiljvm G maxs maxr rT et bs)"

apply (unfold kiljvm_def sl_triple_conv)
apply (rule is_bcv_kildall)
apply (simp (no_asm) add: sl_triple_conv [symmetric])
apply (force intro!: semilat_JVM_slI dest: wf_acyclic
simp add: symmetric sl_triple_conv)
apply (simp (no_asm) add: JVM_le_unfold)
apply (blast intro!: order_widen wf_converse_subcls1_impl_acc_subtype
dest: wf_subcls1 wf_acyclic wf_prog_ws_prog)
apply (simp add: JVM_le_unfold)
apply (erule exec_pres_type)
apply assumption
apply (drule wf_prog_ws_prog, erule exec_mono, assumption)
done

lemma subset_replicate: "set (replicate n x) ⊆ {x}"
by (induct n) auto

lemma in_set_replicate:
"x ∈ set (replicate n y) ==> x = y"
proof -
assume "x ∈ set (replicate n y)"
also have "set (replicate n y) ⊆ {y}" by (rule subset_replicate)
finally have "x ∈ {y}" .
thus ?thesis by simp
qed

theorem wt_kil_correct:
assumes wf: "wf_prog wf_mb G"
assumes C: "is_class G C"
assumes pTs: "set pTs ⊆ types G"

assumes wtk: "wt_kil G C pTs rT maxs mxl et bs"

shows "∃phi. wt_method G C pTs rT maxs mxl bs et phi"
proof -
let ?start = "OK (Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err)))
#(replicate (size bs - 1) (OK None))"


from wtk obtain maxr r where
bounded: "check_bounded bs et" and
result: "r = kiljvm G maxs maxr rT et bs ?start" and
success: "∀n < size bs. r!n ≠ Err" and
instrs: "0 < size bs" and
maxr: "maxr = Suc (length pTs + mxl)"
by (unfold wt_kil_def) simp

from bounded have "bounded (exec G maxs rT et bs) (size bs)"
by (unfold exec_def) (intro bounded_lift check_bounded_is_bounded)
with wf have bcv:
"is_bcv (JVMType.le G maxs maxr) Err (exec G maxs rT et bs)
(size bs) (states G maxs maxr) (kiljvm G maxs maxr rT et bs)"

by (rule is_bcv_kiljvm)

from C pTs instrs maxr
have "?start ∈ list (length bs) (states G maxs maxr)"
apply (unfold JVM_states_unfold)
apply (rule listI)
apply (auto intro: list_appendI dest!: in_set_replicate)
apply force
done

with bcv success result have
"∃ts∈list (length bs) (states G maxs maxr).
?start <=[JVMType.le G maxs maxr] ts ∧
wt_step (JVMType.le G maxs maxr) Err (exec G maxs rT et bs) ts"

by (unfold is_bcv_def) auto
then obtain phi' where
phi': "phi' ∈ list (length bs) (states G maxs maxr)" and
s: "?start <=[JVMType.le G maxs maxr] phi'" and
w: "wt_step (JVMType.le G maxs maxr) Err (exec G maxs rT et bs) phi'"
by blast
hence wt_err_step:
"wt_err_step (sup_state_opt G) (exec G maxs rT et bs) phi'"
by (simp add: wt_err_step_def exec_def JVM_le_Err_conv)

from s have le: "JVMType.le G maxs maxr (?start ! 0) (phi'!0)"
by (drule_tac p=0 in le_listD) (simp add: lesub_def)+

from phi' have l: "size phi' = size bs" by simp
with instrs w have "phi' ! 0 ≠ Err" by (unfold wt_step_def) simp
with instrs l have phi0: "OK (map ok_val phi' ! 0) = phi' ! 0"
by auto

from phi' have "check_types G maxs maxr phi'" by(simp add: check_types_def)
also from w have "phi' = map OK (map ok_val phi')"
by (auto simp add: wt_step_def intro!: nth_equalityI)
finally
have check_types:
"check_types G maxs maxr (map OK (map ok_val phi'))" .

from l bounded
have "bounded (λpc. eff (bs!pc) G pc et) (length phi')"
by (simp add: exec_def check_bounded_is_bounded)
hence bounded': "bounded (exec G maxs rT et bs) (length bs)"
by (auto intro: bounded_lift simp add: exec_def l)
with wt_err_step
have "wt_app_eff (sup_state_opt G) (λpc. app (bs!pc) G maxs rT pc et)
(λpc. eff (bs!pc) G pc et) (map ok_val phi')"

by (auto intro: wt_err_imp_wt_app_eff simp add: l exec_def)
with instrs l le bounded bounded' check_types maxr
have "wt_method G C pTs rT maxs mxl bs et (map ok_val phi')"
apply (unfold wt_method_def wt_app_eff_def)
apply simp
apply (rule conjI)
apply (unfold wt_start_def)
apply (rule JVM_le_convert [THEN iffD1])
apply (simp (no_asm) add: phi0)
apply clarify
apply (erule allE, erule impE, assumption)
apply (elim conjE)
apply (clarsimp simp add: lesub_def wt_instr_def)
apply (simp add: exec_def)
apply (drule bounded_err_stepD, assumption+)
apply blast
done

thus ?thesis by blast
qed


theorem wt_kil_complete:
assumes wf: "wf_prog wf_mb G"
assumes C: "is_class G C"
assumes pTs: "set pTs ⊆ types G"

assumes wtm: "wt_method G C pTs rT maxs mxl bs et phi"

shows "wt_kil G C pTs rT maxs mxl et bs"
proof -
let ?mxr = "1+size pTs+mxl"

from wtm obtain
instrs: "0 < length bs" and
len: "length phi = length bs" and
bounded: "check_bounded bs et" and
ck_types: "check_types G maxs ?mxr (map OK phi)" and
wt_start: "wt_start G C pTs mxl phi" and
wt_ins: "∀pc. pc < length bs -->
wt_instr (bs ! pc) G rT phi maxs (length bs) et pc"

by (unfold wt_method_def) simp

from ck_types len
have istype_phi:
"map OK phi ∈ list (length bs) (states G maxs (1+size pTs+mxl))"
by (auto simp add: check_types_def intro!: listI)

let ?eff = "λpc. eff (bs!pc) G pc et"
let ?app = "λpc. app (bs!pc) G maxs rT pc et"

from bounded
have bounded_exec: "bounded (exec G maxs rT et bs) (size bs)"
by (unfold exec_def) (intro bounded_lift check_bounded_is_bounded)

from wt_ins
have "wt_app_eff (sup_state_opt G) ?app ?eff phi"
apply (unfold wt_app_eff_def wt_instr_def lesub_def)
apply (simp (no_asm) only: len)
apply blast
done
with bounded_exec
have "wt_err_step (sup_state_opt G) (err_step (size phi) ?app ?eff) (map OK phi)"
by - (erule wt_app_eff_imp_wt_err,simp add: exec_def len)
hence wt_err:
"wt_err_step (sup_state_opt G) (exec G maxs rT et bs) (map OK phi)"
by (unfold exec_def) (simp add: len)

from wf bounded_exec
have is_bcv:
"is_bcv (JVMType.le G maxs ?mxr) Err (exec G maxs rT et bs)
(size bs) (states G maxs ?mxr) (kiljvm G maxs ?mxr rT et bs)"

by (rule is_bcv_kiljvm)

let ?start = "OK (Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err)))
#(replicate (size bs - 1) (OK None))"


from C pTs instrs
have start: "?start ∈ list (length bs) (states G maxs ?mxr)"
apply (unfold JVM_states_unfold)
apply (rule listI)
apply (auto intro!: list_appendI dest!: in_set_replicate)
apply force
done

let ?phi = "map OK phi"
have less_phi: "?start <=[JVMType.le G maxs ?mxr] ?phi"
proof -
from len instrs
have "length ?start = length (map OK phi)" by simp
moreover
{ fix n
from wt_start
have "G \<turnstile> ok_val (?start!0) <=' phi!0"
by (simp add: wt_start_def)
moreover
from instrs len
have "0 < length phi" by simp
ultimately
have "JVMType.le G maxs ?mxr (?start!0) (?phi!0)"
by (simp add: JVM_le_Err_conv Err.le_def lesub_def)
moreover
{ fix n'
have "JVMType.le G maxs ?mxr (OK None) (?phi!n)"
by (auto simp add: JVM_le_Err_conv Err.le_def lesub_def
split: err.splits)
hence "[| n = Suc n'; n < length ?start |]
==> JVMType.le G maxs ?mxr (?start!n) (?phi!n)"

by simp
}
ultimately
have "n < length ?start ==> (?start!n) <=_(JVMType.le G maxs ?mxr) (?phi!n)"
by (unfold lesub_def) (cases n, blast+)
}
ultimately show ?thesis by (rule le_listI)
qed

from wt_err
have "wt_step (JVMType.le G maxs ?mxr) Err (exec G maxs rT et bs) ?phi"
by (simp add: wt_err_step_def JVM_le_Err_conv)
with start istype_phi less_phi is_bcv
have "∀p. p < length bs --> kiljvm G maxs ?mxr rT et bs ?start ! p ≠ Err"
by (unfold is_bcv_def) auto
with bounded instrs
show "wt_kil G C pTs rT maxs mxl et bs" by (unfold wt_kil_def) simp
qed


theorem jvm_kildall_sound_complete:
"wt_jvm_prog_kildall G = (∃Phi. wt_jvm_prog G Phi)"
proof
let ?Phi = "λC sig. let (C,rT,(maxs,maxl,ins,et)) = the (method (G,C) sig) in
SOME phi. wt_method G C (snd sig) rT maxs maxl ins et phi"


assume "wt_jvm_prog_kildall G"
hence "wt_jvm_prog G ?Phi"
apply (unfold wt_jvm_prog_def wt_jvm_prog_kildall_def)
apply (erule jvm_prog_lift)
apply (auto dest!: wt_kil_correct intro: someI)
done
thus "∃Phi. wt_jvm_prog G Phi" by fast
next
assume "∃Phi. wt_jvm_prog G Phi"
thus "wt_jvm_prog_kildall G"
apply (clarify)
apply (unfold wt_jvm_prog_def wt_jvm_prog_kildall_def)
apply (erule jvm_prog_lift)
apply (auto intro: wt_kil_complete)
done
qed

end