Theory Conform

theory Conform
imports WellType Exceptions
(*  Title:      HOL/MicroJava/J/Conform.thy
Author: David von Oheimb
Copyright 1999 Technische Universitaet Muenchen
*)


header {* \isaheader{Conformity Relations for Type Soundness Proof} *}

theory Conform imports State WellType Exceptions begin

type_synonym 'c env' = "'c prog × (vname \<rightharpoonup> ty)" -- "same as @{text env} of @{text WellType.thy}"

definition hext :: "aheap => aheap => bool" ("_ <=| _" [51,51] 50) where
"h<=|h' == ∀a C fs. h a = Some(C,fs) --> (∃fs'. h' a = Some(C,fs'))"

definition conf :: "'c prog => aheap => val => ty => bool"
("_,_ |- _ ::<= _" [51,51,51,51] 50) where
"G,h|-v::<=T == ∃T'. typeof (Option.map obj_ty o h) v = Some T' ∧ G\<turnstile>T'\<preceq>T"

definition lconf :: "'c prog => aheap => ('a \<rightharpoonup> val) => ('a \<rightharpoonup> ty) => bool"
("_,_ |- _ [::<=] _" [51,51,51,51] 50) where
"G,h|-vs[::<=]Ts == ∀n T. Ts n = Some T --> (∃v. vs n = Some v ∧ G,h|-v::<=T)"

definition oconf :: "'c prog => aheap => obj => bool" ("_,_ |- _ [ok]" [51,51,51] 50) where
"G,h|-obj [ok] == G,h|-snd obj[::<=]map_of (fields (G,fst obj))"

definition hconf :: "'c prog => aheap => bool" ("_ |-h _ [ok]" [51,51] 50) where
"G|-h h [ok] == ∀a obj. h a = Some obj --> G,h|-obj [ok]"

definition xconf :: "aheap => val option => bool" where
"xconf hp vo == preallocated hp ∧ (∀ v. (vo = Some v) --> (∃ xc. v = (Addr (XcptRef xc))))"

definition conforms :: "xstate => java_mb env' => bool" ("_ ::<= _" [51,51] 50) where
"s::<=E == prg E|-h heap (store s) [ok] ∧
prg E,heap (store s)|-locals (store s)[::<=]localT E ∧
xconf (heap (store s)) (abrupt s)"



notation (xsymbols)
hext ("_ ≤| _" [51,51] 50) and
conf ("_,_ \<turnstile> _ ::\<preceq> _" [51,51,51,51] 50) and
lconf ("_,_ \<turnstile> _ [::\<preceq>] _" [51,51,51,51] 50) and
oconf ("_,_ \<turnstile> _ \<surd>" [51,51,51] 50) and
hconf ("_ \<turnstile>h _ \<surd>" [51,51] 50) and
conforms ("_ ::\<preceq> _" [51,51] 50)


section "hext"

lemma hextI:
" ∀a C fs . h a = Some (C,fs) -->
(∃fs'. h' a = Some (C,fs')) ==> h≤|h'"

apply (unfold hext_def)
apply auto
done

lemma hext_objD: "[|h≤|h'; h a = Some (C,fs) |] ==> ∃fs'. h' a = Some (C,fs')"
apply (unfold hext_def)
apply (force)
done

lemma hext_refl [simp]: "h≤|h"
apply (rule hextI)
apply (fast)
done

lemma hext_new [simp]: "h a = None ==> h≤|h(a\<mapsto>x)"
apply (rule hextI)
apply auto
done

lemma hext_trans: "[|h≤|h'; h'≤|h''|] ==> h≤|h''"
apply (rule hextI)
apply (fast dest: hext_objD)
done

lemma hext_upd_obj: "h a = Some (C,fs) ==> h≤|h(a\<mapsto>(C,fs'))"
apply (rule hextI)
apply auto
done


section "conf"

lemma conf_Null [simp]: "G,h\<turnstile>Null::\<preceq>T = G\<turnstile>RefT NullT\<preceq>T"
apply (unfold conf_def)
apply (simp (no_asm))
done

lemma conf_litval [rule_format (no_asm), simp]:
"typeof (λv. None) v = Some T --> G,h\<turnstile>v::\<preceq>T"
apply (unfold conf_def)
apply (rule val.induct)
apply auto
done

lemma conf_AddrI: "[|h a = Some obj; G\<turnstile>obj_ty obj\<preceq>T|] ==> G,h\<turnstile>Addr a::\<preceq>T"
apply (unfold conf_def)
apply (simp)
done

lemma conf_obj_AddrI: "[|h a = Some (C,fs); G\<turnstile>C\<preceq>C D|] ==> G,h\<turnstile>Addr a::\<preceq> Class D"
apply (unfold conf_def)
apply (simp)
done

lemma defval_conf [rule_format (no_asm)]:
"is_type G T --> G,h\<turnstile>default_val T::\<preceq>T"
apply (unfold conf_def)
apply (rule_tac y = "T" in ty.exhaust)
apply (erule ssubst)
apply (rule_tac y = "prim_ty" in prim_ty.exhaust)
apply (auto simp add: widen.null)
done

lemma conf_upd_obj:
"h a = Some (C,fs) ==> (G,h(a\<mapsto>(C,fs'))\<turnstile>x::\<preceq>T) = (G,h\<turnstile>x::\<preceq>T)"
apply (unfold conf_def)
apply (rule val.induct)
apply auto
done

lemma conf_widen [rule_format (no_asm)]:
"wf_prog wf_mb G ==> G,h\<turnstile>x::\<preceq>T --> G\<turnstile>T\<preceq>T' --> G,h\<turnstile>x::\<preceq>T'"
apply (unfold conf_def)
apply (rule val.induct)
apply (auto intro: widen_trans)
done

lemma conf_hext [rule_format (no_asm)]: "h≤|h' ==> G,h\<turnstile>v::\<preceq>T --> G,h'\<turnstile>v::\<preceq>T"
apply (unfold conf_def)
apply (rule val.induct)
apply (auto dest: hext_objD)
done

lemma new_locD: "[|h a = None; G,h\<turnstile>Addr t::\<preceq>T|] ==> t≠a"
apply (unfold conf_def)
apply auto
done

lemma conf_RefTD [rule_format]:
"G,h\<turnstile>a'::\<preceq>RefT T ==> a' = Null ∨
(∃a obj T'. a' = Addr a ∧ h a = Some obj ∧ obj_ty obj = T' ∧ G\<turnstile>T'\<preceq>RefT T)"

unfolding conf_def by (induct a') auto

lemma conf_NullTD: "G,h\<turnstile>a'::\<preceq>RefT NullT ==> a' = Null"
apply (drule conf_RefTD)
apply auto
done

lemma non_npD: "[|a' ≠ Null; G,h\<turnstile>a'::\<preceq>RefT t|] ==>
∃a C fs. a' = Addr a ∧ h a = Some (C,fs) ∧ G\<turnstile>Class C\<preceq>RefT t"

apply (drule conf_RefTD)
apply auto
done

lemma non_np_objD: "!!G. [|a' ≠ Null; G,h\<turnstile>a'::\<preceq> Class C|] ==>
(∃a C' fs. a' = Addr a ∧ h a = Some (C',fs) ∧ G\<turnstile>C'\<preceq>C C)"

apply (fast dest: non_npD)
done

lemma non_np_objD' [rule_format (no_asm)]:
"a' ≠ Null ==> wf_prog wf_mb G ==> G,h\<turnstile>a'::\<preceq>RefT t -->
(∃a C fs. a' = Addr a ∧ h a = Some (C,fs) ∧ G\<turnstile>Class C\<preceq>RefT t)"

apply(rule_tac y = "t" in ref_ty.exhaust)
apply (fast dest: conf_NullTD)
apply (fast dest: non_np_objD)
done

lemma conf_list_gext_widen [rule_format (no_asm)]:
"wf_prog wf_mb G ==> ∀Ts Ts'. list_all2 (conf G h) vs Ts -->
list_all2 (λT T'. G\<turnstile>T\<preceq>T') Ts Ts' --> list_all2 (conf G h) vs Ts'"

apply(induct_tac "vs")
apply(clarsimp)
apply(clarsimp)
apply(frule list_all2_lengthD [THEN sym])
apply(simp (no_asm_use) add: length_Suc_conv)
apply(safe)
apply(frule list_all2_lengthD [THEN sym])
apply(simp (no_asm_use) add: length_Suc_conv)
apply(clarify)
apply(fast elim: conf_widen)
done


section "lconf"

lemma lconfD: "[| G,h\<turnstile>vs[::\<preceq>]Ts; Ts n = Some T |] ==> G,h\<turnstile>(the (vs n))::\<preceq>T"
apply (unfold lconf_def)
apply (force)
done

lemma lconf_hext [elim]: "[| G,h\<turnstile>l[::\<preceq>]L; h≤|h' |] ==> G,h'\<turnstile>l[::\<preceq>]L"
apply (unfold lconf_def)
apply (fast elim: conf_hext)
done

lemma lconf_upd: "!!X. [| G,h\<turnstile>l[::\<preceq>]lT;
G,h\<turnstile>v::\<preceq>T; lT va = Some T |] ==> G,h\<turnstile>l(va\<mapsto>v)[::\<preceq>]lT"

apply (unfold lconf_def)
apply auto
done

lemma lconf_init_vars_lemma [rule_format (no_asm)]:
"∀x. P x --> R (dv x) x ==> (∀x. map_of fs f = Some x --> P x) -->
(∀T. map_of fs f = Some T -->
(∃v. map_of (map (λ(f,ft). (f, dv ft)) fs) f = Some v ∧ R v T))"

apply( induct_tac "fs")
apply auto
done

lemma lconf_init_vars [intro!]:
"∀n. ∀T. map_of fs n = Some T --> is_type G T ==> G,h\<turnstile>init_vars fs[::\<preceq>]map_of fs"
apply (unfold lconf_def init_vars_def)
apply auto
apply( rule lconf_init_vars_lemma)
apply( erule_tac [3] asm_rl)
apply( intro strip)
apply( erule defval_conf)
apply auto
done

lemma lconf_ext: "[|G,s\<turnstile>l[::\<preceq>]L; G,s\<turnstile>v::\<preceq>T|] ==> G,s\<turnstile>l(vn\<mapsto>v)[::\<preceq>]L(vn\<mapsto>T)"
apply (unfold lconf_def)
apply auto
done

lemma lconf_ext_list [rule_format (no_asm)]:
"G,h\<turnstile>l[::\<preceq>]L ==> ∀vs Ts. distinct vns --> length Ts = length vns -->
list_all2 (λv T. G,h\<turnstile>v::\<preceq>T) vs Ts --> G,h\<turnstile>l(vns[\<mapsto>]vs)[::\<preceq>]L(vns[\<mapsto>]Ts)"

apply (unfold lconf_def)
apply( induct_tac "vns")
apply( clarsimp)
apply( clarsimp)
apply( frule list_all2_lengthD)
apply( auto simp add: length_Suc_conv)
done

lemma lconf_restr: "[|lT vn = None; G, h \<turnstile> l [::\<preceq>] lT(vn\<mapsto>T)|] ==> G, h \<turnstile> l [::\<preceq>] lT"
apply (unfold lconf_def)
apply (intro strip)
apply (case_tac "n = vn")
apply auto
done

section "oconf"

lemma oconf_hext: "G,h\<turnstile>obj\<surd> ==> h≤|h' ==> G,h'\<turnstile>obj\<surd>"
apply (unfold oconf_def)
apply (fast)
done

lemma oconf_obj: "G,h\<turnstile>(C,fs)\<surd> =
(∀T f. map_of(fields (G,C)) f = Some T --> (∃v. fs f = Some v ∧ G,h\<turnstile>v::\<preceq>T))"

apply (unfold oconf_def lconf_def)
apply auto
done

lemmas oconf_objD = oconf_obj [THEN iffD1, THEN spec, THEN spec, THEN mp]


section "hconf"

lemma hconfD: "[|G\<turnstile>h h\<surd>; h a = Some obj|] ==> G,h\<turnstile>obj\<surd>"
apply (unfold hconf_def)
apply (fast)
done

lemma hconfI: "∀a obj. h a=Some obj --> G,h\<turnstile>obj\<surd> ==> G\<turnstile>h h\<surd>"
apply (unfold hconf_def)
apply (fast)
done


section "xconf"

lemma xconf_raise_if: "xconf h x ==> xconf h (raise_if b xcn x)"
by (simp add: xconf_def raise_if_def)



section "conforms"

lemma conforms_heapD: "(x, (h, l))::\<preceq>(G, lT) ==> G\<turnstile>h h\<surd>"
apply (unfold conforms_def)
apply (simp)
done

lemma conforms_localD: "(x, (h, l))::\<preceq>(G, lT) ==> G,h\<turnstile>l[::\<preceq>]lT"
apply (unfold conforms_def)
apply (simp)
done

lemma conforms_xcptD: "(x, (h, l))::\<preceq>(G, lT) ==> xconf h x"
apply (unfold conforms_def)
apply (simp)
done

lemma conformsI: "[|G\<turnstile>h h\<surd>; G,h\<turnstile>l[::\<preceq>]lT; xconf h x|] ==> (x, (h, l))::\<preceq>(G, lT)"
apply (unfold conforms_def)
apply auto
done

lemma conforms_restr: "[|lT vn = None; s ::\<preceq> (G, lT(vn\<mapsto>T)) |] ==> s ::\<preceq> (G, lT)"
by (simp add: conforms_def, fast intro: lconf_restr)

lemma conforms_xcpt_change: "[| (x, (h,l))::\<preceq> (G, lT); xconf h x --> xconf h x' |] ==> (x', (h,l))::\<preceq> (G, lT)"
by (simp add: conforms_def)


lemma preallocated_hext: "[| preallocated h; h≤|h'|] ==> preallocated h'"
by (simp add: preallocated_def hext_def)

lemma xconf_hext: "[| xconf h vo; h≤|h'|] ==> xconf h' vo"
by (simp add: xconf_def preallocated_def hext_def)

lemma conforms_hext: "[|(x,(h,l))::\<preceq>(G,lT); h≤|h'; G\<turnstile>h h'\<surd> |]
==> (x,(h',l))::\<preceq>(G,lT)"

by (fast dest: conforms_localD conforms_xcptD elim!: conformsI xconf_hext)


lemma conforms_upd_obj:
"[|(x,(h,l))::\<preceq>(G, lT); G,h(a\<mapsto>obj)\<turnstile>obj\<surd>; h≤|h(a\<mapsto>obj)|]
==> (x,(h(a\<mapsto>obj),l))::\<preceq>(G, lT)"

apply(rule conforms_hext)
apply auto
apply(rule hconfI)
apply(drule conforms_heapD)
apply(auto elim: oconf_hext dest: hconfD)
done

lemma conforms_upd_local:
"[|(x,(h, l))::\<preceq>(G, lT); G,h\<turnstile>v::\<preceq>T; lT va = Some T|]
==> (x,(h, l(va\<mapsto>v)))::\<preceq>(G, lT)"

apply (unfold conforms_def)
apply( auto elim: lconf_upd)
done

end