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loop_to_wordProofScript.sml
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(*
Correctness proof for loop_to_word
*)
open preamble
loopSemTheory loopPropsTheory
wordLangTheory wordSemTheory wordPropsTheory
pan_commonTheory pan_commonPropsTheory
loop_to_wordTheory loop_removeProofTheory
wordConvsTheory;
val _ = new_theory "loop_to_wordProof";
val _ = temp_delsimps ["lift_disj_eq", "lift_imp_disj"];
val _ = temp_delsimps ["fromAList_def", "domain_union",
"domain_inter", "domain_difference",
"domain_map", "sptree.map_def", "sptree.lookup_rwts",
"sptree.insert_notEmpty", "sptree.isEmpty_union"];
Definition locals_rel_def:
locals_rel ctxt l1 l2 ⇔
INJ (find_var ctxt) (domain ctxt) UNIV ∧
(∀n m. lookup n ctxt = SOME m ==> m ≠ 0) ∧
∀n v. lookup n l1 = SOME v ⇒
∃m. lookup n ctxt = SOME m ∧ lookup m l2 = SOME v
End
Definition globals_rel_def:
globals_rel g1 g2 =
∀n v. FLOOKUP g1 n = SOME v ⇒ FLOOKUP g2 (Temp n) = SOME v
End
Definition code_rel_def:
code_rel s_code t_code =
∀name params body.
lookup name s_code = SOME (params,body) ⇒
lookup name t_code = SOME (LENGTH params+1, comp_func name params body) ∧
no_Loops body ∧ ALL_DISTINCT params
End
Definition state_rel_def:
state_rel s t <=>
t.memory = s.memory ∧
t.mdomain = s.mdomain ∧
t.sh_mdomain = s.sh_mdomain ∧
t.clock = s.clock ∧
t.be = s.be ∧
t.ffi = s.ffi ∧
ALOOKUP (fmap_to_alist t.store) CurrHeap = SOME (Word s.base_addr) ∧
globals_rel s.globals t.store ∧
code_rel s.code t.code
End
val goal =
``λ(prog:α loopLang$prog, s). ∀res s1 t ctxt retv l.
evaluate (prog,s) = (res,s1) ∧ res ≠ SOME Error ∧
state_rel s t ∧ locals_rel ctxt s.locals t.locals ∧
ALOOKUP (fmap_to_alist t.store) CurrHeap = SOME (Word s.base_addr) ∧
lookup 0 t.locals = SOME retv ∧ no_Loops prog ∧
good_dimindex(:'a) ∧
~(isWord retv) ∧
domain (acc_vars prog LN) ⊆ domain ctxt ⇒
∃t1 res1.
evaluate (FST (comp ctxt prog l),t) = (res1,t1) ∧
state_rel s1 t1 ∧
case res of
| NONE => res1 = NONE ∧ lookup 0 t1.locals = SOME retv ∧
(* always return to the label stored in Var 0 for wordLang's prog *)
locals_rel ctxt s1.locals t1.locals ∧
t1.stack = t.stack ∧ t1.handler = t.handler
| SOME (Result v) => res1 = SOME (Result retv v) ∧
t1.stack = t.stack ∧ t1.handler = t.handler
| SOME TimeOut => res1 = SOME TimeOut
| SOME (FinalFFI f) => res1 = SOME (FinalFFI f)
| SOME (Exception v) =>
(res1 ≠ SOME Error ⇒ ∃u1 u2. res1 = SOME (Exception u1 u2)) ∧
∀r l1 l2. jump_exc (t1 with <| stack := t.stack;
handler := t.handler |>) = SOME (r,l1,l2) ⇒
res1 = SOME (Exception (Loc l1 l2) v) ∧ r = t1
| _ => F``
local
val ind_thm = loopSemTheory.evaluate_ind
|> ISPEC goal
|> CONV_RULE (DEPTH_CONV PairRules.PBETA_CONV) |> REWRITE_RULE [];
fun list_dest_conj tm = if not (is_conj tm) then [tm] else let
val (c1,c2) = dest_conj tm in list_dest_conj c1 @ list_dest_conj c2 end
val ind_goals = ind_thm |> concl |> dest_imp |> fst |> list_dest_conj
in
fun get_goal s = first (can (find_term (can (match_term (Term [QUOTE s]))))) ind_goals
fun compile_correct_tm () = ind_thm |> concl |> rand
fun the_ind_thm () = ind_thm
end
Theorem locals_rel_intro:
locals_rel ctxt l1 l2 ==>
INJ (find_var ctxt) (domain ctxt) UNIV ∧
(∀n m. lookup n ctxt = SOME m ==> m ≠ 0) ∧
∀n v. lookup n l1 = SOME v ⇒
∃m. lookup n ctxt = SOME m ∧ lookup m l2 = SOME v
Proof
rw [locals_rel_def]
QED
Theorem globals_rel_intro:
globals_rel g1 g2 ==>
∀n v. FLOOKUP g1 n = SOME v ⇒ FLOOKUP g2 (Temp n) = SOME v
Proof
rw [globals_rel_def]
QED
Theorem code_rel_intro:
code_rel s_code t_code ==>
∀name params body.
lookup name s_code = SOME (params,body) ⇒
lookup name t_code = SOME (LENGTH params+1, comp_func name params body) ∧
no_Loops body ∧ ALL_DISTINCT params
Proof
rw [code_rel_def] >> metis_tac []
QED
Theorem state_rel_intro:
state_rel s t ==>
t.memory = s.memory ∧
t.mdomain = s.mdomain ∧
t.clock = s.clock ∧
t.be = s.be ∧
t.ffi = s.ffi ∧
ALOOKUP (fmap_to_alist t.store) CurrHeap = SOME (Word s.base_addr) ∧
globals_rel s.globals t.store ∧
code_rel s.code t.code
Proof
rw [state_rel_def]
QED
Theorem find_var_neq_0:
v ∈ domain ctxt ∧ locals_rel ctxt lcl lcl' ⇒
find_var ctxt v ≠ 0
Proof
fs [locals_rel_def, find_var_def] >> rw [] >>
Cases_on ‘lookup var_name ctxt’ >> fs [] >>
fs [domain_lookup] >> res_tac >> metis_tac []
QED
Theorem locals_rel_insert:
locals_rel ctxt lcl lcl' ∧ v IN domain ctxt ⇒
locals_rel ctxt (insert v w lcl)
(insert (find_var ctxt v) w lcl')
Proof
fs [locals_rel_def,lookup_insert] >> rw [] >>
fs [CaseEq"bool"] >> rveq >> fs [] >>
fs [domain_lookup,find_var_def] >>
res_tac >> fs [] >>
disj2_tac >> CCONTR_TAC >> fs [] >> rveq >> fs [] >>
fs [INJ_DEF,domain_lookup] >>
first_x_assum (qspecl_then [‘v’,‘n’] mp_tac) >>
fs [] >> fs [find_var_def]
QED
Theorem locals_rel_get_var:
locals_rel ctxt l t.locals ∧ lookup n l = SOME w ⇒
wordSem$get_var (find_var ctxt n) t = SOME w
Proof
fs [locals_rel_def] >> rw[] >> res_tac >>
fs [find_var_def, get_var_def]
QED
Theorem locals_rel_get_vars:
∀argvars argvals.
locals_rel ctxt s.locals t.locals ∧
loopSem$get_vars argvars s = SOME argvals ⇒
wordSem$get_vars (MAP (find_var ctxt) argvars) t = SOME argvals ∧
LENGTH argvals = LENGTH argvars
Proof
Induct >> fs [loopSemTheory.get_vars_def,get_vars_def,CaseEq"option"]
>> rw [] >> imp_res_tac locals_rel_get_var >> fs []
QED
Triviality state_rel_IMP:
state_rel s t ⇒ t.clock = s.clock
Proof
fs [state_rel_def]
QED
Theorem set_fromNumSet:
set (fromNumSet t) = domain t
Proof
fs [fromNumSet_def,EXTENSION,MEM_MAP,EXISTS_PROD,MEM_toAList,domain_lookup]
QED
Theorem domain_toNumSet:
domain (toNumSet ps) = set ps
Proof
Induct_on ‘ps’ >> fs [toNumSet_def]
QED
Theorem domain_make_ctxt:
∀n ps l. domain (make_ctxt n ps l) = domain l UNION set ps
Proof
Induct_on ‘ps’ >> fs [make_ctxt_def] >> fs [EXTENSION] >> metis_tac []
QED
Theorem make_ctxt_inj:
∀xs l n. (∀x y v. lookup x l = SOME v ∧ lookup y l = SOME v ⇒ x = y ∧ v < n) ⇒
(∀x y v. lookup x (make_ctxt n xs l) = SOME v ∧
lookup y (make_ctxt n xs l) = SOME v ⇒ x = y)
Proof
Induct_on ‘xs’ >> fs [make_ctxt_def] >> rw []
>> first_x_assum (qspecl_then [‘insert h n l’,‘n+2’] mp_tac)
>> impl_tac >-
(fs [lookup_insert] >> rw []
>> CCONTR_TAC >> fs [] >> res_tac >> fs [])
>> metis_tac []
QED
Triviality make_ctxt_APPEND:
∀xs ys n l.
make_ctxt n (xs ++ ys) l =
make_ctxt (n + 2 * LENGTH xs) ys (make_ctxt n xs l)
Proof
Induct >> fs [make_ctxt_def,MULT_CLAUSES]
QED
Triviality make_ctxt_NOT_MEM:
∀xs n l x. ~MEM x xs ⇒ lookup x (make_ctxt n xs l) = lookup x l
Proof
Induct >> fs [make_ctxt_def,lookup_insert]
QED
Theorem lookup_EL_make_ctxt:
∀params k n l.
k < LENGTH params ∧ ALL_DISTINCT params ⇒
lookup (EL k params) (make_ctxt n params l) = SOME (2 * k + n)
Proof
Induct >> Cases_on ‘k’ >> fs [] >> fs [make_ctxt_def,make_ctxt_NOT_MEM]
QED
Theorem lookup_make_ctxt_range:
∀xs m l n y.
lookup n (make_ctxt m xs l) = SOME y ⇒
lookup n l = SOME y ∨ m ≤ y
Proof
Induct >> fs [make_ctxt_def] >> rw []
>> first_x_assum drule
>> fs [lookup_insert] >> rw [] >> fs []
QED
Theorem locals_rel_make_ctxt:
ALL_DISTINCT params ∧ DISJOINT (set params) (set xs) ∧
LENGTH params = LENGTH l ⇒
locals_rel (make_ctxt 2 (params ++ xs) LN)
(fromAList (ZIP (params,l))) (fromList2 (retv::l))
Proof
fs [locals_rel_def] >> rw []
>-
(fs [INJ_DEF,find_var_def,domain_lookup] >> rw [] >> rfs []
>> rveq >> fs []
>> imp_res_tac (MP_CANON make_ctxt_inj) >> fs [lookup_def])
>-
(Cases_on ‘lookup n (make_ctxt 2 (params ++ xs) LN)’ >> simp []
>> drule lookup_make_ctxt_range >> fs [lookup_def])
>> fs [lookup_fromAList]
>> imp_res_tac ALOOKUP_MEM
>> rfs [MEM_ZIP] >> rveq >> fs [make_ctxt_APPEND]
>> rename [‘k < LENGTH _’]
>> ‘k < LENGTH params’ by fs []
>> drule EL_MEM >> strip_tac
>> ‘~MEM (EL k params) xs’ by (fs [IN_DISJOINT] >> metis_tac [])
>> fs [make_ctxt_NOT_MEM]
>> fs [lookup_EL_make_ctxt]
>> fs [lookup_fromList2,EVEN_ADD,EVEN_DOUBLE]
>> ‘2 * k + 2 = (SUC k) * 2’ by fs []
>> asm_rewrite_tac [MATCH_MP MULT_DIV (DECIDE “0 < 2:num”)]
>> fs [lookup_fromList]
>> rewrite_tac [GSYM ADD1,EL,TL]
QED
Theorem domain_mk_new_cutset_not_empty:
domain (mk_new_cutset ctxt x1) ≠ ∅
Proof
fs [mk_new_cutset_def]
QED
Theorem cut_env_mk_new_cutset:
locals_rel ctxt l1 l2 ∧ domain x1 ⊆ domain l1 ∧ lookup 0 l2 = SOME y ⇒
∃env1. cut_env (mk_new_cutset ctxt x1) l2 = SOME env1 ∧
locals_rel ctxt (inter l1 x1) env1
Proof
fs [locals_rel_def,SUBSET_DEF,cut_env_def] >> fs [lookup_inter_alt]
>> fs [mk_new_cutset_def,domain_toNumSet,MEM_MAP,set_fromNumSet,PULL_EXISTS]
>> fs [DISJ_IMP_THM,PULL_EXISTS]
>> strip_tac >> fs [domain_lookup]
>> rw [] >> res_tac >> fs [] >> rveq >> fs [find_var_def]
>> rw [] >> res_tac >> fs [] >> rveq >> fs [find_var_def]
>> disj2_tac >> qexists_tac ‘n’ >> fs []
QED
Theorem env_to_list_IMP:
env_to_list env1 t.permute = (l,permute) ⇒
domain (fromAList l) = domain env1 ∧
∀x. lookup x (fromAList l) = lookup x env1
Proof
strip_tac >> drule env_to_list_lookup_equiv
>> fs [EXTENSION,domain_lookup,lookup_fromAList]
QED
Theorem cut_env_mk_new_cutset_IMP:
cut_env (mk_new_cutset ctxt x1) l1 = SOME l2 ⇒
lookup 0 l2 = lookup 0 l1
Proof
fs [cut_env_def] >> rw []
>> fs [SUBSET_DEF,mk_new_cutset_def]
>> fs [lookup_inter_alt]
QED
Triviality LASTN_ADD_CONS:
~(LENGTH xs <= n) ⇒ LASTN (n + 1) (x::xs) = LASTN (n + 1) xs
Proof
fs [LASTN_CONS]
QED
Theorem comp_exp_preserves_eval:
!s e v t ctxt.
eval s e = SOME v ∧
state_rel s t /\ locals_rel ctxt s.locals t.locals ==>
word_exp t (comp_exp ctxt e) = SOME v
Proof
ho_match_mp_tac eval_ind >>
rw [] >>
fs [eval_def, comp_exp_def, word_exp_def]
>- (
fs [find_var_def, locals_rel_def] >>
TOP_CASE_TAC >> fs [] >>
first_x_assum drule >>
strip_tac >> fs [] >> rveq >> fs [])
>- fs [state_rel_def, globals_rel_def]
>- (
cases_on ‘eval s e’ >> fs [] >>
cases_on ‘x’ >> fs [] >>
first_x_assum drule_all >> fs [] >>
strip_tac >>
fs [state_rel_def, mem_load_def,
loopSemTheory.mem_load_def])
>- (
fs [CaseEq "option"] >>
qsuff_tac
‘the_words (MAP (λa. word_exp t a)
(MAP (λa. comp_exp ctxt a) wexps)) = SOME ws’
>- fs [] >>
ntac 2 (pop_assum mp_tac) >>
ntac 2 (pop_assum kall_tac) >>
rpt (pop_assum mp_tac) >>
qid_spec_tac ‘ws’ >>
qid_spec_tac ‘wexps’ >>
Induct >> rw [] >>
last_assum mp_tac >>
impl_tac >- metis_tac [] >>
fs [the_words_def, CaseEq"option", CaseEq"word_loc"] >>
rveq >> fs [])
>- (fs [CaseEq"option", CaseEq"word_loc"] >> rveq >> fs []) >>
fs[state_rel_def] >>
Cases_on ‘FLOOKUP t.store CurrHeap’ >> fs[] >>
rename1 ‘x’ >>
Cases_on ‘x’ >> fs[theWord_def, isWord_def]
QED
Theorem compile_Skip:
^(get_goal "comp _ loopLang$Skip") ∧
^(get_goal "comp _ loopLang$Fail") ∧
^(get_goal "comp _ loopLang$Tick")
Proof
rpt strip_tac >>
fs [loopSemTheory.evaluate_def, comp_def,
evaluate_def] >>
rveq >> fs [] >>
TOP_CASE_TAC >>
fs [flush_state_def, state_rel_def,
loopSemTheory.dec_clock_def, dec_clock_def] >> rveq >>
fs []
QED
Theorem compile_Loop:
^(get_goal "comp _ loopLang$Continue") ∧
^(get_goal "comp _ loopLang$Break") ∧
^(get_goal "comp _ (loopLang$Loop _ _ _)")
Proof
rpt strip_tac >>
fs [no_Loops_def, every_prog_def] >>
fs [no_Loop_def, every_prog_def]
QED
Theorem compile_Mark:
^(get_goal "comp _ (Mark _)")
Proof
rpt strip_tac >>
fs [loopSemTheory.evaluate_def, comp_def,
evaluate_def, no_Loops_def,
loopLangTheory.acc_vars_def, no_Loop_def, every_prog_def]
QED
Theorem compile_Return:
^(get_goal "loopLang$Return")
Proof
rpt strip_tac >>
fs [loopSemTheory.evaluate_def, comp_def, evaluate_def] >>
cases_on ‘lookup n s.locals’ >>
fs [] >> rveq >>
TOP_CASE_TAC >>
fs [find_var_def, locals_rel_def, get_var_def] >>
res_tac >> rveq >>
TOP_CASE_TAC >> fs [isWord_def] >>
fs [flush_state_def, state_rel_def,
loopSemTheory.call_env_def]
QED
Theorem compile_Raise:
^(get_goal "loopLang$Raise")
Proof
fs [comp_def,loopSemTheory.evaluate_def,CaseEq"option"] >>
rw [] >> fs [evaluate_def] >>
imp_res_tac locals_rel_get_var >> fs [] >>
Cases_on ‘jump_exc t’ >> fs []
>- fs [jump_exc_def, state_rel_def, loopSemTheory.call_env_def] >>
fs [CaseEq"prod",PULL_EXISTS] >>
PairCases_on ‘x’ >> fs [loopSemTheory.call_env_def] >>
pop_assum mp_tac >>
fs [CaseEq"option",CaseEq"prod", jump_exc_def,
CaseEq"stack_frame", CaseEq"list"] >>
strip_tac >> fs [] >> rveq >> fs [] >>
fs [state_rel_def]
QED
Theorem compile_Seq:
^(get_goal "comp _ (loopLang$Seq _ _)")
Proof
rpt strip_tac >>
fs [loopSemTheory.evaluate_def] >>
pairarg_tac >> fs [comp_def] >>
rpt (pairarg_tac >> fs []) >>
fs [evaluate_def] >>
rpt (pairarg_tac >> fs []) >>
first_x_assum (qspecl_then [‘t’,‘ctxt’,‘retv’,‘l’] mp_tac) >>
impl_tac
>- (
fs [] >>
conj_tac >- (CCONTR_TAC >> fs []) >>
fs [no_Loops_def, no_Loop_def, every_prog_def] >>
qpat_x_assum ‘_ ⊆ domain ctxt’ mp_tac >>
fs [loopLangTheory.acc_vars_def] >>
once_rewrite_tac [acc_vars_acc] >> fs []) >>
fs [] >> strip_tac >>
reverse (Cases_on ‘res'’) >> fs [] >> rveq >> fs []
>- (
Cases_on ‘x’ >> fs [] >>
IF_CASES_TAC >> fs []) >>
rename [‘state_rel s2 t2’] >>
first_x_assum drule >>
fs[state_rel_def]>>
rpt (disch_then drule) >>
disch_then (qspec_then ‘l'’ mp_tac) >>
impl_tac
>- (
qpat_x_assum ‘_ ⊆ domain ctxt’ mp_tac >>
fs [no_Loops_def, no_Loop_def, every_prog_def] >>
fs [loopLangTheory.acc_vars_def] >>
once_rewrite_tac [acc_vars_acc] >> fs []
) >>
fs [] >> strip_tac >> fs [] >>
Cases_on ‘res’ >> fs [] >>
Cases_on ‘x’ >> fs []
QED
Theorem compile_Assign:
^(get_goal "loopLang$Assign") ∧
^(get_goal "loopLang$LocValue")
Proof
rpt strip_tac >>
fs [loopSemTheory.evaluate_def,
comp_def, evaluate_def]
>- (
cases_on ‘eval s exp’ >> fs [] >>
rveq >> fs [] >>
imp_res_tac comp_exp_preserves_eval >>
fs [loopSemTheory.set_var_def, set_var_def] >>
conj_tac >- fs [state_rel_def] >>
conj_tac
>- (
fs [lookup_insert, CaseEq "bool", loopLangTheory.acc_vars_def] >>
imp_res_tac find_var_neq_0 >> fs []) >>
match_mp_tac locals_rel_insert >>
fs [loopLangTheory.acc_vars_def]) >>
fs [CaseEq "bool"] >> rveq >> fs [] >>
fs [loopSemTheory.set_var_def, set_var_def] >>
conj_tac
>- (
fs [state_rel_def,
code_rel_def,domain_lookup,EXISTS_PROD] >>
metis_tac []) >>
conj_tac >- fs [state_rel_def] >>
conj_tac
>- (
fs [lookup_insert, CaseEq "bool", loopLangTheory.acc_vars_def] >>
imp_res_tac find_var_neq_0 >> fs []) >>
match_mp_tac locals_rel_insert >>
fs [loopLangTheory.acc_vars_def]
QED
Theorem compile_Store:
^(get_goal "loopLang$Store") ∧
^(get_goal "loopLang$StoreByte")
Proof
rpt strip_tac >>
fs [loopSemTheory.evaluate_def,
comp_def, evaluate_def]
>- (
fs [CaseEq "option", CaseEq "word_loc"] >> rveq >>
imp_res_tac comp_exp_preserves_eval >>
fs [] >>
drule_all locals_rel_get_var >>
strip_tac >> fs [] >>
fs [loopSemTheory.mem_store_def, mem_store_def] >>
rveq >> fs [state_rel_def]) >>
fs [CaseEq "option", CaseEq "word_loc"] >> rveq >>
fs [inst_def, word_exp_def] >>
drule locals_rel_intro >>
strip_tac >>
res_tac >> fs [] >>
fs [find_var_def, the_words_def, word_op_def] >>
fs [get_var_def] >>
fs [state_rel_def]
QED
Theorem compile_LoadByte:
^(get_goal "loopLang$LoadByte")
Proof
rpt strip_tac >>
fs [loopSemTheory.evaluate_def,
comp_def, evaluate_def] >>
fs [CaseEq "option", CaseEq "word_loc"] >> rveq >>
fs [inst_def, word_exp_def] >>
drule locals_rel_intro >>
strip_tac >>
res_tac >> fs [] >>
fs [find_var_def, the_words_def, word_op_def] >>
drule state_rel_intro >>
strip_tac >> fs [] >>
fs [loopSemTheory.set_var_def, set_var_def] >>
conj_tac >- fs [state_rel_def] >>
fs [loopLangTheory.acc_vars_def] >>
imp_res_tac find_var_neq_0 >>
fs [domain_lookup, lookup_insert, CaseEq "bool"] >>
conj_tac
>- (CCONTR_TAC >> res_tac >> fs []) >>
drule locals_rel_insert >>
disch_then (qspecl_then [‘Word (w2w b)’, ‘v’] mp_tac) >>
fs [domain_lookup, find_var_def]
QED
Theorem compile_SetGlobal:
^(get_goal "loopLang$SetGlobal")
Proof
rpt strip_tac >>
fs [loopSemTheory.evaluate_def,
comp_def, evaluate_def] >>
fs [CaseEq "option"] >>
rveq >> fs [] >>
imp_res_tac comp_exp_preserves_eval >>
fs [] >>
fs [state_rel_def, set_store_def,
loopSemTheory.set_globals_def, globals_rel_def] >>
rw [FLOOKUP_UPDATE]
QED
Theorem acc_vars_acc'[local] =
acc_vars_acc |> CONV_RULE SWAP_FORALL_CONV |> SPEC “acc_vars q LN”;
Theorem compile_If:
^(get_goal "loopLang$If")
Proof
rpt strip_tac >>
‘no_Loops c1 ∧ no_Loops c2’ by (gvs[no_Loops_def,no_Loop_def,every_prog_def]) >>
fs [loopSemTheory.evaluate_def, comp_def] >>
rpt(pairarg_tac >> simp[]) >>
fs [find_var_def, get_var_def] >>
imp_res_tac locals_rel_intro >> fs [] >>
gvs[AllCaseEqs(),
DefnBase.one_line_ify NONE loopSemTheory.cut_res_def,
DefnBase.one_line_ify NONE cut_state_def,
loopLangTheory.acc_vars_def,
acc_vars_acc'
] >>
first_assum $ drule_then strip_assume_tac >>
gvs[AllCaseEqs(),
DefnBase.one_line_ify NONE loopSemTheory.get_var_imm_def] >>
first_assum $ drule_then strip_assume_tac >>
simp[evaluate_def,get_var_def,find_reg_imm_def,get_var_imm_def,find_var_def] >>
rw[] >> fs[] >>
last_x_assum drule >> rpt(disch_then drule) >>
rename1 ‘comp ctxt cv’ >>
rename1 ‘comp ctxt cv the_l’ >>
disch_then $ qspec_then ‘the_l’ strip_assume_tac >>
gvs[] >>
(* the every_case_tac replaces a deeply nested TRY block, so hopefully it's
a forgivable sin *)
every_case_tac >> gvs[] >>
gvs[state_rel_def,flush_state_def,
AllCaseEqs(),
DefnBase.one_line_ify NONE loopSemTheory.cut_state_def,
loopSemTheory.dec_clock_def,wordSemTheory.dec_clock_def
] >>
gvs[locals_rel_def,lookup_inter_alt]
QED
Theorem compile_Call:
^(get_goal "comp _ (loopLang$Call _ _ _ _)")
Proof
rw [] >> qpat_x_assum ‘evaluate _ = (res,_)’ mp_tac >> simp [loopSemTheory.evaluate_def]
>> simp [CaseEq"option"]
>> strip_tac >> fs []
>> rename [‘find_code _ _ _ = SOME x’]
>> PairCases_on ‘x’ >> fs []
>> rename [‘find_code _ _ _ = SOME (new_env,new_code)’]
>> ‘~bad_dest_args dest (MAP (find_var ctxt) argvars)’ by
(pop_assum kall_tac >> Cases_on ‘dest’ >> fs [bad_dest_args_def]
>> fs [loopSemTheory.find_code_def]
>> imp_res_tac locals_rel_get_vars >> CCONTR_TAC >> fs [])
>> Cases_on ‘ret’ >> fs []
>-
(fs [comp_def,evaluate_def]
>> imp_res_tac locals_rel_get_vars >> fs [add_ret_loc_def]
>> fs [get_vars_def,get_var_def]
>> simp [bad_dest_args_def,call_env_def,dec_clock_def]
>> ‘∃args1 prog1 ss1 name1 ctxt1 l1.
find_code dest (retv::argvals) t.code t.stack_size = SOME (args1,prog1,ss1) ∧
FST (comp ctxt1 new_code l1) = prog1 ∧
lookup 0 (fromList2 args1) = SOME retv ∧
locals_rel ctxt1 new_env (fromList2 args1) ∧ no_Loops new_code ∧
domain (acc_vars new_code LN) ⊆ domain ctxt1’ by
(qpat_x_assum ‘_ = (res,_)’ kall_tac
>> Cases_on ‘dest’ >> fs [loopSemTheory.find_code_def]
>-
(fs [CaseEq"word_loc",CaseEq"num",CaseEq"option",CaseEq"prod",CaseEq"bool"]
>> rveq >> fs [code_rel_def,state_rel_def]
>> first_x_assum drule >> strip_tac >> fs []
>> fs [find_code_def]
>> ‘∃x l. argvals = SNOC x l’ by metis_tac [SNOC_CASES]
>> qpat_x_assum ‘_ = Loc loc 0’ mp_tac
>> rveq >> rewrite_tac [GSYM SNOC,LAST_SNOC,FRONT_SNOC] >> fs []
>> strip_tac >> rveq >> fs []
>> simp [comp_func_def]
>> qmatch_goalsub_abbrev_tac ‘comp ctxt2 _ ll2’
>> qexists_tac ‘ctxt2’ >> qexists_tac ‘ll2’ >> fs []
>> conj_tac >- fs [lookup_fromList2,lookup_fromList]
>> simp [Abbr‘ctxt2’,domain_make_ctxt,set_fromNumSet,
domain_difference,domain_toNumSet, SUBSET_DEF]
>> match_mp_tac locals_rel_make_ctxt
>> fs [IN_DISJOINT,set_fromNumSet,domain_difference,
domain_toNumSet,GSYM IMP_DISJ_THM])
>> fs [CaseEq"word_loc",CaseEq"num",CaseEq"option",CaseEq"prod",CaseEq"bool"]
>> rveq >> fs [code_rel_def,state_rel_def]
>> first_x_assum drule >> strip_tac >> fs []
>> fs [find_code_def]
>> simp [comp_func_def]
>> qmatch_goalsub_abbrev_tac ‘comp ctxt2 _ ll2’
>> qexists_tac ‘ctxt2’ >> qexists_tac ‘ll2’ >> fs []
>> conj_tac >- fs [lookup_fromList2,lookup_fromList]
>> simp [Abbr‘ctxt2’,domain_make_ctxt,set_fromNumSet,
domain_difference,domain_toNumSet, SUBSET_DEF]
>> match_mp_tac locals_rel_make_ctxt
>> fs [IN_DISJOINT,set_fromNumSet,domain_difference,
domain_toNumSet,GSYM IMP_DISJ_THM])
>> fs [] >> imp_res_tac state_rel_IMP
>> fs [] >> IF_CASES_TAC >> fs []
>-
(fs [CaseEq"bool"] >> rveq >> fs []
>> fs [state_rel_def,flush_state_def])
>> Cases_on ‘handler = NONE’ >> fs [] >> rveq
>> Cases_on ‘evaluate (new_code,dec_clock s with locals := new_env)’ >> fs []
>> Cases_on ‘q’ >> fs []
>> Cases_on ‘x = Error’ >> rveq >> fs []
>> qmatch_goalsub_abbrev_tac ‘wordSem$evaluate (_,tt)’
>> first_x_assum (qspecl_then [‘tt’,‘ctxt1’,‘retv’,‘l1’] mp_tac)
>> impl_tac
>- (fs [Abbr‘tt’] >>
fs [state_rel_def,loopSemTheory.dec_clock_def])
>> strip_tac >> fs []
>> Cases_on ‘x’ >> fs [] >> rveq >> fs []
>- fs [Abbr‘tt’]
>> qexists_tac ‘t1’ >> fs []
>> qexists_tac ‘res1’ >> fs []
>> conj_tac >- (Cases_on ‘res1’ >> simp [CaseEq"option"] >> fs [])
>> rpt gen_tac >> strip_tac >> pop_assum mp_tac
>> qunabbrev_tac ‘tt’ >> fs [])
>> fs [comp_def,evaluate_def]
>> imp_res_tac locals_rel_get_vars >> fs [add_ret_loc_def]
>> fs [get_vars_def,get_var_def]
>> simp [bad_dest_args_def,call_env_def,dec_clock_def]
>> PairCases_on ‘x’ >> PairCases_on ‘l’
>> fs [] >> imp_res_tac state_rel_IMP
>> ‘∃args1 prog1 ss1 name1 ctxt1 l2.
find_code dest (Loc l0 l1::argvals) t.code t.stack_size = SOME (args1,prog1,ss1) ∧
FST (comp ctxt1 new_code l2) = prog1 ∧
lookup 0 (fromList2 args1) = SOME (Loc l0 l1) ∧
locals_rel ctxt1 new_env (fromList2 args1) ∧ no_Loops new_code ∧
domain (acc_vars new_code LN) ⊆ domain ctxt1’ by
(qpat_x_assum ‘_ = (res,_)’ kall_tac
>> rpt (qpat_x_assum ‘∀x. _’ kall_tac)
>> Cases_on ‘dest’ >> fs [loopSemTheory.find_code_def]
>-
(fs [CaseEq"word_loc",CaseEq"num",CaseEq"option",CaseEq"prod",CaseEq"bool"]
>> rveq >> fs [code_rel_def,state_rel_def]
>> first_x_assum drule >> strip_tac >> fs []
>> fs [find_code_def]
>> ‘∃x l. argvals = SNOC x l’ by metis_tac [SNOC_CASES]
>> qpat_x_assum ‘_ = Loc loc 0’ mp_tac
>> rveq >> rewrite_tac [GSYM SNOC,LAST_SNOC,FRONT_SNOC] >> fs []
>> strip_tac >> rveq >> fs []
>> simp [comp_func_def]
>> qmatch_goalsub_abbrev_tac ‘comp ctxt2 _ ll2’
>> qexists_tac ‘ctxt2’ >> qexists_tac ‘ll2’ >> fs []
>> conj_tac >- fs [lookup_fromList2,lookup_fromList]
>> simp [Abbr‘ctxt2’,domain_make_ctxt,set_fromNumSet,
domain_difference,domain_toNumSet, SUBSET_DEF]
>> match_mp_tac locals_rel_make_ctxt
>> fs [IN_DISJOINT,set_fromNumSet,domain_difference,
domain_toNumSet,GSYM IMP_DISJ_THM])
>> fs [CaseEq"word_loc",CaseEq"num",CaseEq"option",CaseEq"prod",CaseEq"bool"]
>> rveq >> fs [code_rel_def,state_rel_def]
>> first_x_assum drule >> strip_tac >> fs []
>> fs [find_code_def]
>> simp [comp_func_def]
>> qmatch_goalsub_abbrev_tac ‘comp ctxt2 _ ll2’
>> qexists_tac ‘ctxt2’ >> qexists_tac ‘ll2’ >> fs []
>> conj_tac >- fs [lookup_fromList2,lookup_fromList]
>> simp [Abbr‘ctxt2’,domain_make_ctxt,set_fromNumSet,
domain_difference,domain_toNumSet, SUBSET_DEF]
>> match_mp_tac locals_rel_make_ctxt
>> fs [IN_DISJOINT,set_fromNumSet,domain_difference,
domain_toNumSet,GSYM IMP_DISJ_THM])
>> Cases_on ‘handler’ >> fs []
>-
(fs [evaluate_def,add_ret_loc_def,domain_mk_new_cutset_not_empty,cut_res_def]
>> fs [loopSemTheory.cut_state_def]
>> Cases_on ‘domain x1 ⊆ domain s.locals’ >> fs []
>> qpat_x_assum ‘locals_rel _ s.locals _’ assume_tac
>> drule cut_env_mk_new_cutset
>> rpt (disch_then drule) >> strip_tac >> fs []
>> (IF_CASES_TAC >> fs [] >- (rveq >> fs [flush_state_def,state_rel_def]))
>> fs [CaseEq"prod",CaseEq"option"] >> fs [] >> rveq >> fs []
>> rename [‘_ = (SOME res2,st)’]
>> qmatch_goalsub_abbrev_tac ‘wordSem$evaluate (_,tt)’
>> fs [PULL_EXISTS]
>> Cases_on ‘res2 = Error’ >> fs []
>> first_x_assum (qspecl_then [‘tt’,‘ctxt1’,‘Loc l0 l1’,‘l2’] mp_tac)
>> (impl_tac >-
(fs [Abbr‘tt’,call_env_def,push_env_def,isWord_def]
>> pairarg_tac >> fs [dec_clock_def,loopSemTheory.dec_clock_def,state_rel_def]))
>> strip_tac >> fs []
>> Cases_on ‘res2’ >> fs [] >> rveq >> fs []
>-
(fs [Abbr‘tt’,call_env_def,push_env_def,dec_clock_def]
>> pairarg_tac >> fs [pop_env_def,set_var_def]
>> imp_res_tac env_to_list_IMP
>> fs [loopSemTheory.set_var_def,loopSemTheory.dec_clock_def]
>> fs [state_rel_def]
>> rename [‘find_var ctxt var_name’]
>> ‘var_name IN domain ctxt’ by fs [loopLangTheory.acc_vars_def]
>> simp [lookup_insert]
>> imp_res_tac find_var_neq_0 >> fs []
>> imp_res_tac cut_env_mk_new_cutset_IMP >> fs []
>> match_mp_tac locals_rel_insert >> fs []
>> fs [locals_rel_def])
>> qunabbrev_tac ‘tt’
>> pop_assum mp_tac
>> Cases_on ‘res1’ >- fs []
>> disch_then (fn th => assume_tac (REWRITE_RULE [IMP_DISJ_THM] th))
>> fs [] >> Cases_on ‘x’ >> fs []
>> fs [state_rel_def]
>> fs [call_env_def,push_env_def] >> pairarg_tac >> fs [dec_clock_def]
>> fs [jump_exc_def,NOT_LESS]
>> Cases_on ‘LENGTH t.stack <= t.handler’ >> fs [LASTN_ADD_CONS]
>> simp [CaseEq"option",CaseEq"prod",CaseEq"bool",set_var_def,CaseEq"list",
CaseEq"stack_frame"] >> rw [] >> fs [])
>> PairCases_on ‘x’ >> fs []
>> rpt (pairarg_tac >> fs [])
>> fs [evaluate_def,add_ret_loc_def,domain_mk_new_cutset_not_empty,cut_res_def]
>> fs [loopSemTheory.cut_state_def]
>> Cases_on ‘domain x1 ⊆ domain s.locals’ >> fs []
>> qpat_x_assum ‘locals_rel _ s.locals _’ assume_tac
>> drule cut_env_mk_new_cutset
>> rpt (disch_then drule) >> strip_tac >> fs []
>> (IF_CASES_TAC >> fs [] >- (rveq >> fs [flush_state_def,state_rel_def]))
>> fs [CaseEq"prod",CaseEq"option"] >> fs [] >> rveq >> fs []
>> rename [‘_ = (SOME res2,st)’]
>> qmatch_goalsub_abbrev_tac ‘wordSem$evaluate (_,tt)’
>> fs [PULL_EXISTS]
>> Cases_on ‘res2 = Error’ >> fs []
>> first_x_assum (qspecl_then [‘tt’,‘ctxt1’,‘Loc l0 l1’,‘l2’] mp_tac)
>> (impl_tac >-
(fs [Abbr‘tt’] >> rename [‘SOME (find_var _ _,p1,l8)’]
>> PairCases_on ‘l8’ >> fs [call_env_def,push_env_def,isWord_def]
>> pairarg_tac >> fs [dec_clock_def,loopSemTheory.dec_clock_def,state_rel_def]))
>> strip_tac >> fs []
>> Cases_on ‘res2’ >> fs [] >> rveq >> fs []
>> fs [loopSemTheory.dec_clock_def]
>-
(rename [‘loopSem$set_var hvar w _’]
>> Cases_on ‘evaluate (x2,set_var hvar w (st with locals := inter s.locals x1))’
>> fs []
>> Cases_on ‘q = SOME Error’ >- fs [cut_res_def] >> fs []
>> fs [pop_env_def,Abbr‘tt’] >> fs [call_env_def,push_env_def]
>> rename [‘SOME (find_var _ _,p1,l8)’]
>> PairCases_on ‘l8’ >> fs [call_env_def,push_env_def]
>> pairarg_tac >> fs [dec_clock_def,loopSemTheory.dec_clock_def]
>> pop_assum mp_tac
>> pairarg_tac >> fs [dec_clock_def,loopSemTheory.dec_clock_def]
>> reverse IF_CASES_TAC >- (imp_res_tac env_to_list_IMP >> fs [])
>> strip_tac >> fs [] >> pop_assum mp_tac >> fs [set_var_def]
>> fs [cut_res_def]
>> qmatch_goalsub_abbrev_tac ‘wordSem$evaluate (_,tt)’ >> strip_tac
>> first_x_assum (qspecl_then [‘tt’,‘ctxt’,‘retv’,‘l1'’] mp_tac)
>> impl_tac >-
(fs [loopSemTheory.set_var_def,state_rel_def,Abbr‘tt’]
>> qpat_x_assum ‘_ SUBSET domain ctxt’ mp_tac
>> simp [loopLangTheory.acc_vars_def]
>> once_rewrite_tac [acc_vars_acc]
>> once_rewrite_tac [acc_vars_acc] >> fs [] >> strip_tac
>> qpat_x_assum ‘no_Loops (Call _ _ _ _)’ mp_tac
>> simp [no_Loops_def,every_prog_def,no_Loop_def] >> strip_tac
>> imp_res_tac env_to_list_IMP >> fs []
>> fs [lookup_insert]
>> imp_res_tac find_var_neq_0 >> fs []
>> imp_res_tac cut_env_mk_new_cutset_IMP >> fs []
>> match_mp_tac locals_rel_insert >> fs [locals_rel_def])
>> fs [] >> strip_tac
>> Cases_on ‘q’ >> fs [] >> rveq >> fs []
>-
(rename [‘cut_state names s9’]
>> fs [loopSemTheory.cut_state_def]
>> Cases_on ‘domain names ⊆ domain s9.locals’ >> fs []
>> imp_res_tac state_rel_IMP >> fs []
>> IF_CASES_TAC
>> fs [flush_state_def] >> rveq >> fs [] >> fs [state_rel_def,dec_clock_def]
>> fs [loopSemTheory.dec_clock_def,Abbr‘tt’]
>> fs [locals_rel_def,lookup_inter_alt])
>> Cases_on ‘x’ >> fs []
>- fs [Abbr‘tt’]
>> pop_assum mp_tac >> rewrite_tac [IMP_DISJ_THM]
>> IF_CASES_TAC >> fs []
>> fs [Abbr‘tt’] >> metis_tac [])
>> qpat_x_assum ‘∀x. _’ (assume_tac o REWRITE_RULE [IMP_DISJ_THM])
>> rename [‘loopSem$set_var hvar w _’]
>> Cases_on ‘evaluate (x1',set_var hvar w (st with locals := inter s.locals x1))’
>> fs []
>> Cases_on ‘q = SOME Error’ >- fs [cut_res_def] >> fs []
>> fs [pop_env_def,Abbr‘tt’] >> fs [call_env_def,push_env_def]
>> rename [‘SOME (find_var _ _,p1,l8)’]
>> PairCases_on ‘l8’ >> fs [call_env_def,push_env_def]
>> pairarg_tac >> fs [dec_clock_def,loopSemTheory.dec_clock_def]
>> pop_assum mp_tac
>> pairarg_tac >> fs [dec_clock_def,loopSemTheory.dec_clock_def]
>> Cases_on ‘res1’ >> fs [] >> rveq >> fs []
>> qpat_x_assum ‘∀x. _’ mp_tac
>> simp [jump_exc_def]
>> qmatch_goalsub_abbrev_tac ‘LASTN n1 xs1’
>> ‘LASTN n1 xs1 = xs1’ by
(qsuff_tac ‘n1 = LENGTH xs1’ >> fs [LASTN_LENGTH_ID]
>> unabbrev_all_tac >> fs [])
>> fs [] >> fs [Abbr‘n1’,Abbr‘xs1’] >> strip_tac >> rveq >> fs []
>> ‘t1.locals = fromAList l ∧
t1.stack = t.stack ∧ t1.handler = t.handler’ by fs [state_component_equality]
>> reverse IF_CASES_TAC >- (imp_res_tac env_to_list_IMP >> fs [] >> rfs [])
>> strip_tac >> fs []
>> pop_assum mp_tac >> fs [set_var_def]
>> fs [cut_res_def]
>> qmatch_goalsub_abbrev_tac ‘wordSem$evaluate (_,tt)’ >> strip_tac
>> first_x_assum (qspecl_then [‘tt’,‘ctxt’,‘retv’,‘(l0,l1 + 1)’] mp_tac)
>> impl_tac >-
(fs [loopSemTheory.set_var_def,state_rel_def,Abbr‘tt’]
>> qpat_x_assum ‘_ SUBSET domain ctxt’ mp_tac
>> simp [loopLangTheory.acc_vars_def]
>> once_rewrite_tac [acc_vars_acc]
>> once_rewrite_tac [acc_vars_acc] >> fs [] >> strip_tac
>> qpat_x_assum ‘no_Loops (Call _ _ _ _)’ mp_tac
>> simp [no_Loops_def,every_prog_def,no_Loop_def] >> strip_tac
>> imp_res_tac env_to_list_IMP >> fs []
>> fs [lookup_insert]
>> imp_res_tac find_var_neq_0 >> fs []
>> imp_res_tac cut_env_mk_new_cutset_IMP >> fs []
>> match_mp_tac locals_rel_insert >> fs [locals_rel_def])
>> fs [] >> strip_tac
>> Cases_on ‘q’ >> fs [] >> rveq >> fs []
>-
(rename [‘cut_state names s9’]
>> fs [loopSemTheory.cut_state_def]
>> Cases_on ‘domain names ⊆ domain s9.locals’ >> fs []
>> imp_res_tac state_rel_IMP >> fs []
>> IF_CASES_TAC
>> fs [flush_state_def] >> rveq >> fs [] >> fs [state_rel_def,dec_clock_def]
>> fs [loopSemTheory.dec_clock_def,Abbr‘tt’]
>> fs [locals_rel_def,lookup_inter_alt])
>> pop_assum (assume_tac o REWRITE_RULE [IMP_DISJ_THM])
>> Cases_on ‘x’ >> fs []
>- fs [Abbr‘tt’]
>> rveq >> fs []
>> pop_assum mp_tac
>> fs [Abbr‘tt’,jump_exc_def]
>> metis_tac []
QED
Theorem compile_FFI:
^(get_goal "loopLang$FFI")
Proof
rpt strip_tac >>
fs [loopSemTheory.evaluate_def,
comp_def, evaluate_def] >>
fs [CaseEq "option", CaseEq "word_loc"] >>
rveq >> fs [] >>
fs [find_var_def, get_var_def] >>
imp_res_tac state_rel_intro >>
imp_res_tac locals_rel_intro >>
res_tac >> fs [] >>
fs [loopSemTheory.cut_state_def] >> rveq >>
drule_all cut_env_mk_new_cutset >>
strip_tac >> fs [] >>
TOP_CASE_TAC >> fs [] >> rveq >> fs [] >>
fs [state_rel_def, flush_state_def,
loopSemTheory.call_env_def] >>
fs [cut_env_def] >>
rveq >> fs [] >>
fs [lookup_inter] >>
TOP_CASE_TAC >>
fs [mk_new_cutset_def]
QED
Theorem compile_Arith:
^(get_goal "loopLang$Arith")
Proof
rpt strip_tac >>
gvs [loopSemTheory.evaluate_def,
comp_def, evaluate_def,DefnBase.one_line_ify NONE loop_arith_def,
AllCaseEqs(),inst_def,PULL_EXISTS,get_vars_def,find_var_def,get_var_def,
loopSemTheory.set_var_def,wordSemTheory.set_var_def,
state_rel_def,SUBSET_DEF,loopLangTheory.acc_vars_def,
SF DNF_ss
] >>
imp_res_tac locals_rel_intro >>
gvs[lookup_insert,lookup_insert,domain_lookup] >>
rw[] >>
gvs[locals_rel_def,lookup_insert] >>
rw[] >>
res_tac >> gvs[] >> rw[] >>
gvs[INJ_DEF,domain_lookup,PULL_EXISTS,find_var_def]
QED
Theorem TAKE_1_word_to_bytes:
good_dimindex(:'a) ⇒ TAKE 1 (word_to_bytes (w:'a word) F) = [get_byte 0w w F]
Proof
rw[word_to_bytes_def,good_dimindex_def] >> rw[word_to_bytes_aux_compute]
QED
Theorem compile_ShMem:
^(get_goal "loopLang$ShMem")
Proof
rpt strip_tac >>
gvs [loopSemTheory.evaluate_def,comp_def,evaluate_def,
DefnBase.one_line_ify NONE loopSemTheory.sh_mem_op_def,
AllCaseEqs(),PULL_EXISTS,
loopSemTheory.sh_mem_load_def,
loopSemTheory.sh_mem_store_def,
loopLangTheory.acc_vars_def,
domain_lookup
] >>
imp_res_tac comp_exp_preserves_eval >>
gvs[DefnBase.one_line_ify NONE wordSemTheory.share_inst_def,
PULL_EXISTS,
sh_mem_load_def,
sh_mem_load_byte_def,
sh_mem_load32_def,
sh_mem_store_def,
sh_mem_store_byte_def,
sh_mem_store32_def,
sh_mem_set_var_def,
find_var_def,
state_rel_def,
set_var_def,
loopSemTheory.set_var_def,
locals_rel_def,
lookup_insert,
get_var_def
] >>
rw[] >>
gvs[flush_state_def,loopSemTheory.call_env_def] >>
res_tac >>