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milExecutableIOScript.sml
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milExecutableIOScript.sml
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open HolKernel boolLib Parse bossLib wordsLib optionTheory wordsTheory finite_mapTheory pred_setTheory listTheory rich_listTheory sortingTheory ottTheory milUtilityTheory milTheory milSemanticsUtilityTheory milTracesTheory milMetaTheory milInitializationTheory milExecutableUtilityTheory milExecutableTransitionTheory milExecutableCompletenessTheory;
(* ======================================== *)
(* IO execution of bounded number of steps *)
(* ======================================== *)
val _ = new_theory "milExecutableIO";
(* --------------------------------- *)
(* Function and property definitions *)
(* --------------------------------- *)
(* TODO: easy optimization to put transitions at head of exec, reverse the final list *)
Definition IO_bounded_execution_acc:
IO_bounded_execution_acc
(f_tran: v -> t -> i list)
(f_sem: e -> (t |-> v) -> v option)
(State_st_list l s cs fs) (pos:num) (n:num)
(exec:(State_list # ll # State_list) list) :
((State_list # ll # State_list) list) option =
case n of
| 0 => SOME exec
| SUC n' =>
(case DROP pos l of
| [] => SOME exec
| i_assign t c mop::il =>
(case FLOOKUP s t of
| NONE =>
(case f_sem c s of
| NONE => NONE
| SOME v =>
if v <> val_false then
(case sem_instr_exe f_sem (i_assign t c mop) (State_st_list l s cs fs) of
| NONE => NONE
| SOME (v',obs) =>
let (ll,stl) = OoO_Exe_list_instr_not_stored_guard_true_sem_instr
(State_st_list l s cs fs) (i_assign t c mop) v' obs in
case mop of
| o_store res_MEM ta tv =>
(case OoO_Cmt_list_stored f_sem stl t ta tv of
| NONE => NONE
| SOME (ll',stl') =>
IO_bounded_execution_acc f_tran f_sem stl' (SUC pos) n'
(exec++[(State_st_list l s cs fs,ll,stl);(stl,ll',stl')]))
| o_store res_PC ta tv =>
(case OoO_Ftc_list_stored f_tran f_sem stl t v' of
| NONE => NONE
| SOME (ll',stl') =>
IO_bounded_execution_acc f_tran f_sem stl' (SUC pos) n'
(exec++[(State_st_list l s cs fs,ll,stl);(stl,ll',stl')]))
| _ => (* instruction is completed, move on *)
IO_bounded_execution_acc f_tran f_sem stl (SUC pos) n'
(exec++[(State_st_list l s cs fs,ll,stl)]))
else (* instruction is completed, move on *)
IO_bounded_execution_acc f_tran f_sem (State_st_list l s cs fs) (SUC pos) n' exec)
| SOME v =>
(case mop of
| o_store res_MEM ta tv =>
if MEM t cs then (* instruction is completed, move on *)
IO_bounded_execution_acc f_tran f_sem (State_st_list l s cs fs) (SUC pos) n' exec
else
(case OoO_Cmt_list_stored_incomplete f_sem (State_st_list l s cs fs) t ta tv of
| NONE => NONE
| SOME (ll,stl) => (* instruction is completed, move on *)
IO_bounded_execution_acc f_tran f_sem stl (SUC pos) n'
(exec++[(State_st_list l s cs fs,ll,stl)]))
| o_store res_PC ta tv =>
if MEM t fs then (* instruction is completed, move on *)
IO_bounded_execution_acc f_tran f_sem (State_st_list l s cs fs) (SUC pos) n' exec
else
(case OoO_Ftc_list_stored_incomplete f_tran f_sem (State_st_list l s cs fs) t v of
| NONE => NONE
| SOME (ll,stl) => (* instruction is completed, move on *)
IO_bounded_execution_acc f_tran f_sem stl (SUC pos) n'
(exec++[(State_st_list l s cs fs,ll,stl)]))
| _ => (* instruction is completed, move on *)
IO_bounded_execution_acc f_tran f_sem (State_st_list l s cs fs) (SUC pos) n' exec)))
End
(* FIXME: double check how PRE affects all theorem statements *)
Definition IO_bounded_execution:
IO_bounded_execution
(f_tran : v -> t -> i list)
(f_sem: e -> (t |-> v) -> v option)
(stl:State_list) (pos:num) (n:num) :
((State_list # ll # State_list) list) option =
IO_bounded_execution_acc f_tran f_sem stl (PRE pos) n []
End
Definition step_execution_list_OoO_HD:
step_execution_list_OoO_HD stl pi n =
(FST (HD pi) = stl /\
step_execution out_of_order_step_list pi /\
LENGTH pi <= 2*n)
End
Definition step_execution_list_IO_HD:
step_execution_list_IO_HD stl pi pos n =
(FST (HD pi) = stl /\
step_execution in_order_step_list pi /\
Completed_list_up_to sem_expr (SND (SND (LAST pi))) (pos + n) /\
LENGTH pi <= 2*n)
End
(* ----------------------- *)
(* Execution OoO soundness *)
(* ----------------------- *)
Theorem NO_DUPLICATES_FIND_instr:
!l t i.
NO_DUPLICATES l ==>
MEM i l ==>
bound_name_instr i = t ==>
FIND_instr t l = SOME i
Proof
rw [] >>
Cases_on `FIND_instr (bound_name_instr i) l` >-
METIS_TAC [FIND_instr_eq_NONE,MEM_MAP,NO_DUPLICATES_bound_name_instr] >>
METIS_TAC [FIND_instr_eq_SOME,NO_DUPLICATES_bound_name_instr]
QED
(* FIXME: extract lemma on one-step cases *)
Theorem IO_bounded_execution_acc_OoO_execution_sound:
!n stl pos pi pi'.
State_st_list_well_formed_ok stl ==>
step_execution out_of_order_step_list pi ==>
SND (SND (LAST pi)) = stl ==>
IO_bounded_execution_acc translate_val_list sem_expr stl pos n pi = SOME pi' ==>
?pi''. pi' = pi ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2*n
Proof
Induct_on `n` >>
Cases_on `stl` >> rename1 `State_st_list l s cs fs` >-
fs [IO_bounded_execution_acc] >>
once_rewrite_tac [IO_bounded_execution_acc] >>
fs [] >> rw [] >>
Cases_on `DROP pos l` >> fs [] >>
Cases_on `h` >>
rename1 `i_assign t' c mop::l'` >> rename1 `i_assign t c mop` >>
fs [] >>
Cases_on `FLOOKUP s t` >> fs [] >-
(Cases_on `sem_expr c s` >> fs [] >>
rename1 `sem_expr c s = SOME v` >>
Cases_on `v = val_false` >> fs [] >-
(`?pi''. pi' = pi ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
Q.EXISTS_TAC `pi''` >> rw []) >>
Cases_on `sem_instr_exe sem_expr (i_assign t c mop) (State_st_list l s cs fs)` >> fs [] >>
Cases_on `x` >> rename1 `(v',obs)` >> fs [] >>
Cases_on `OoO_Exe_list_instr_not_stored_guard_true_sem_instr
(State_st_list l s cs fs) (i_assign t c mop) v' obs` >>
fs [] >> rename1 `(ll,stl)` >>
`MEM (i_assign t c mop) l` by METIS_TAC [DROP_HEAD_MEM] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [State_st_list_well_formed_ok_OoO_Exe_list_instr_not_stored_guard_true_sem_instr] >>
`State_st_list_ok (State_st_list l s cs fs)` by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`FIND_instr t l = SOME (i_assign t c mop)`
by METIS_TAC [State_st_list_ok,NO_DUPLICATES_FIND_instr,bound_name_instr] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
Cases_on `mop` >> fs [] >| [
Q.ABBREV_TAC `pi0 = pi ++ [(State_st_list l s cs fs,ll,stl)]` >>
`State_st_list_well_formed_ok (SND (SND (LAST pi0)))` by fs [Abbr `pi0`] >>
sg `step_execution out_of_order_step_list pi0` >-
(fs [Abbr `pi0`,step_list_to_step] >>
`pi <> []` by METIS_TAC [step_execution_not_empty_list] >>
`pi = [] \/ ?e pi1. pi = SNOC e pi1` by fs [SNOC_CASES] >> fs [SNOC_APPEND] >>
Cases_on `e'` >> fs [] >> Cases_on `r` >> rename1 `(stl1,ll1,stl2)` >>
fs [step_list_to_step] >> rw [] >>
`pi1 ++ [(stl1,ll1,State_st_list l s cs fs)] ++ [(State_st_list l s cs fs,ll,stl)] =
pi1 ++ [(stl1,ll1,State_st_list l s cs fs);(State_st_list l s cs fs,ll,stl)]`
by fs [] >>
rw [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl` by fs [Abbr `pi0`] >>
`?pi''. pi' = pi0 ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
fs [Abbr `pi0`],
Q.ABBREV_TAC `pi0 = pi ++ [(State_st_list l s cs fs,ll,stl)]` >>
`State_st_list_well_formed_ok (SND (SND (LAST pi0)))` by fs [Abbr `pi0`] >>
sg `step_execution out_of_order_step_list pi0` >-
(fs [Abbr `pi0`,step_list_to_step] >>
`pi <> []` by METIS_TAC [step_execution_not_empty_list] >>
`pi = [] \/ ?e pi1. pi = SNOC e pi1` by fs [SNOC_CASES] >> fs [SNOC_APPEND] >>
Cases_on `e` >> fs [] >> Cases_on `r'` >> rename1 `(stl1,ll1,stl2)` >>
fs [step_list_to_step] >> rw [] >>
`pi1 ++ [(stl1,ll1,State_st_list l s cs fs)] ++ [(State_st_list l s cs fs,ll,stl)] =
pi1 ++ [(stl1,ll1,State_st_list l s cs fs);(State_st_list l s cs fs,ll,stl)]`
by fs [] >>
rw [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl` by fs [Abbr `pi0`] >>
`?pi''. pi' = pi0 ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
fs [Abbr `pi0`],
Cases_on `r` >> fs [] >| [
Cases_on `OoO_Ftc_list_stored translate_val_list sem_expr stl t v'` >> fs [] >>
Cases_on `x` >> fs [] >> rename1 `(ll',stl')` >>
rename1 `o_store res_PC ta tv` >>
`stl = State_st_list l (s |+ (t,v')) cs fs`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_result_form] >>
rw [] >>
Q.ABBREV_TAC `pi0 = pi ++ [(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs);
(State_st_list l (s |+ (t,v')) cs fs,ll',stl')]` >>
`FLOOKUP (s |+ (t,v')) t = SOME v'` by fs [FLOOKUP_UPDATE] >>
`State_st_list_well_formed_ok stl'`
by METIS_TAC [State_st_list_well_formed_ok_OoO_Ftc_list_stored] >>
`out_of_order_step_list (State_st_list l (s |+ (t,v')) cs fs) ll' stl'`
by METIS_TAC [OoO_Ftc_list_stored_SOME_out_of_order_step_list] >>
`State_st_list_well_formed_ok (SND (SND (LAST pi0)))` by fs [Abbr `pi0`] >>
sg `step_execution out_of_order_step_list pi0` >-
(fs [Abbr `pi0`,step_list_to_step] >>
`step_execution out_of_order_step_list
(pi ++ [(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs)])`
suffices_by METIS_TAC [step_execution_rules] >>
`pi <> []` by METIS_TAC [step_execution_not_empty_list] >>
`pi = [] \/ ?e pi1. pi = SNOC e pi1` by fs [SNOC_CASES] >> fs [SNOC_APPEND] >>
Cases_on `e` >> fs [] >> Cases_on `r` >> rename1 `(stl1,ll1,stl2)` >>
fs [step_list_to_step] >> rw [] >>
`pi1 ++ [(stl1,ll1,State_st_list l s cs fs)] ++
[(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs)] =
pi1 ++ [(stl1,ll1,State_st_list l s cs fs);
(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs)]`
by fs [] >>
rw [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl'` by fs [Abbr `pi0`] >>
`?pi''. pi' = pi0 ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
fs [Abbr `pi0`],
Q.ABBREV_TAC `pi0 = pi ++ [(State_st_list l s cs fs,ll,stl)]` >>
`State_st_list_well_formed_ok (SND (SND (LAST pi0)))` by fs [Abbr `pi0`] >>
sg `step_execution out_of_order_step_list pi0` >-
(fs [Abbr `pi0`,step_list_to_step] >>
`pi <> []` by METIS_TAC [step_execution_not_empty_list] >>
`pi = [] \/ ?e pi1. pi = SNOC e pi1` by fs [SNOC_CASES] >> fs [SNOC_APPEND] >>
Cases_on `e` >> fs [] >> Cases_on `r` >> rename1 `(stl1,ll1,stl2)` >>
fs [step_list_to_step] >> rw [] >>
`pi1 ++ [(stl1,ll1,State_st_list l s cs fs)] ++
[(State_st_list l s cs fs,ll,stl)] =
pi1 ++ [(stl1,ll1,State_st_list l s cs fs);
(State_st_list l s cs fs,ll,stl)]`
by fs [] >>
rw [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl` by fs [Abbr `pi0`] >>
`?pi''. pi' = pi0 ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
fs [Abbr `pi0`],
rename1 `o_store res_MEM ta tv` >>
Cases_on `OoO_Cmt_list_stored sem_expr stl t ta tv` >> fs [] >>
Cases_on `x` >> fs [] >> rename1 `(ll',stl')` >>
`stl = State_st_list l (s |+ (t,v')) cs fs`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_result_form] >>
rw [] >>
Q.ABBREV_TAC `pi0 = pi ++ [(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs);
(State_st_list l (s |+ (t,v')) cs fs,ll',stl')]` >>
`FLOOKUP (s |+ (t,v')) t = SOME v'` by fs [FLOOKUP_UPDATE] >>
`FLOOKUP (s |+ (t,v')) t <> NONE` by fs [] >>
`State_st_list_well_formed_ok stl'`
by METIS_TAC [State_st_list_well_formed_ok_OoO_Cmt_list_stored] >>
`out_of_order_step_list (State_st_list l (s |+ (t,v')) cs fs) ll' stl'`
by METIS_TAC [OoO_Cmt_list_stored_SOME_out_of_order_step_list] >>
`State_st_list_well_formed_ok (SND (SND (LAST pi0)))` by fs [Abbr `pi0`] >>
sg `step_execution out_of_order_step_list pi0` >-
(fs [Abbr `pi0`,step_list_to_step] >>
`step_execution out_of_order_step_list
(pi ++ [(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs)])`
suffices_by METIS_TAC [step_execution_rules] >>
`pi <> []` by METIS_TAC [step_execution_not_empty_list] >>
`pi = [] \/ ?e pi1. pi = SNOC e pi1` by fs [SNOC_CASES] >> fs [SNOC_APPEND] >>
Cases_on `e` >> fs [] >> Cases_on `r` >> rename1 `(stl1,ll1,stl2)` >>
fs [step_list_to_step] >> rw [] >>
`pi1 ++ [(stl1,ll1,State_st_list l s cs fs)] ++
[(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs)] =
pi1 ++ [(stl1,ll1,State_st_list l s cs fs);
(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs)]`
by fs [] >>
rw [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl'` by fs [Abbr `pi0`] >>
`?pi''. pi' = pi0 ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
fs [Abbr `pi0`]
]
]) >>
`MEM (i_assign t c mop) l` by METIS_TAC [DROP_HEAD_MEM] >>
`FIND_instr t l = SOME (i_assign t c mop)`
by METIS_TAC [State_st_list_well_formed_ok,State_st_list_ok,NO_DUPLICATES_FIND_instr,bound_name_instr] >>
Cases_on `mop` >> fs [] >| [
`?pi''. pi' = pi ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
Q.EXISTS_TAC `pi''` >> rw [],
`?pi''. pi' = pi ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
Q.EXISTS_TAC `pi''` >> rw [],
Cases_on `r` >> fs [] >| [
Cases_on `MEM t fs` >> fs [] >-
(`?pi''. pi' = pi ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
Q.EXISTS_TAC `pi''` >> rw []) >>
Cases_on `OoO_Ftc_list_stored_incomplete translate_val_list sem_expr (State_st_list l s cs fs) t x` >> fs [] >>
Cases_on `x'` >> fs [] >> rename1 `(ll,stl)` >> rename1 `o_store res_PC ta tv` >>
rename1 `FLOOKUP s t = SOME v` >>
`State_st_list_ok (State_st_list l s cs fs)`
by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Ftc_list_stored_incomplete_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
Q.ABBREV_TAC `pi0 = pi ++ [(State_st_list l s cs fs,ll,stl)]` >>
`State_st_list_well_formed_ok (SND (SND (LAST pi0)))` by fs [Abbr `pi0`] >>
sg `step_execution out_of_order_step_list pi0` >-
(fs [Abbr `pi0`,step_list_to_step] >>
`pi <> []` by METIS_TAC [step_execution_not_empty_list] >>
`pi = [] \/ ?e pi1. pi = SNOC e pi1` by fs [SNOC_CASES] >> fs [SNOC_APPEND] >>
Cases_on `e` >> fs [] >> Cases_on `r` >> rename1 `(stl1,ll1,stl2)` >>
fs [step_list_to_step] >> rw [] >>
`pi1 ++ [(stl1,ll1,State_st_list l s cs fs)] ++
[(State_st_list l s cs fs,ll,stl)] =
pi1 ++ [(stl1,ll1,State_st_list l s cs fs);
(State_st_list l s cs fs,ll,stl)]`
by fs [] >>
rw [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl` by fs [Abbr `pi0`] >>
`?pi''. pi' = pi0 ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
fs [Abbr `pi0`],
`?pi''. pi' = pi ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
Q.EXISTS_TAC `pi''` >> rw [],
Cases_on `MEM t cs` >> fs [] >-
(`?pi''. pi' = pi ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
Q.EXISTS_TAC `pi''` >> rw []) >>
rename1 `o_store res_MEM ta tv` >>
Cases_on `OoO_Cmt_list_stored_incomplete sem_expr (State_st_list l s cs fs) t ta tv` >> fs [] >>
Cases_on `x'` >> fs [] >> rename1 `(ll,stl)` >>
rename1 `FLOOKUP s t = SOME v` >>
`FLOOKUP s t <> NONE` by rw [] >>
`State_st_list_ok (State_st_list l s cs fs)`
by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Cmt_list_stored_incomplete_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
Q.ABBREV_TAC `pi0 = pi ++ [(State_st_list l s cs fs,ll,stl)]` >>
`State_st_list_well_formed_ok (SND (SND (LAST pi0)))` by fs [Abbr `pi0`] >>
sg `step_execution out_of_order_step_list pi0` >-
(fs [Abbr `pi0`,step_list_to_step] >>
`pi <> []` by METIS_TAC [step_execution_not_empty_list] >>
`pi = [] \/ ?e pi1. pi = SNOC e pi1` by fs [SNOC_CASES] >> fs [SNOC_APPEND] >>
Cases_on `e` >> fs [] >> Cases_on `r` >> rename1 `(stl1,ll1,stl2)` >>
fs [step_list_to_step] >> rw [] >>
`pi1 ++ [(stl1,ll1,State_st_list l s cs fs)] ++
[(State_st_list l s cs fs,ll,stl)] =
pi1 ++ [(stl1,ll1,State_st_list l s cs fs);
(State_st_list l s cs fs,ll,stl)]`
by fs [] >>
rw [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl` by fs [Abbr `pi0`] >>
`?pi''. pi' = pi0 ++ pi'' /\
step_execution out_of_order_step_list pi' /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
fs [Abbr `pi0`]
]
]
QED
Theorem IO_bounded_execution_acc_OoO_empty_sound:
!l s cs fs pos n i il pi.
State_st_list_well_formed_ok (State_st_list l s cs fs) ==>
DROP pos l = i::il ==>
~(Completed_list sem_expr (State_st_list l s cs fs) i) ==>
IO_bounded_execution_acc translate_val_list sem_expr (State_st_list l s cs fs) pos (SUC n) [] = SOME pi ==>
step_execution_list_OoO_HD (State_st_list l s cs fs) pi (SUC n)
Proof
once_rewrite_tac [IO_bounded_execution_acc] >>
fs [] >> rw [] >>
Cases_on `i` >> rename1 `i_assign t c mop` >> fs [] >>
`MEM (i_assign t c mop) l` by METIS_TAC [DROP_HEAD_MEM] >>
`FIND_instr t l = SOME (i_assign t c mop)`
by METIS_TAC [State_st_list_well_formed_ok,State_st_list_ok,NO_DUPLICATES_FIND_instr,bound_name_instr] >>
Cases_on `FLOOKUP s t` >> fs [] >> Cases_on `mop` >> fs [Completed_list] >| [
Cases_on `sem_expr c s` >> fs [] >>
rename1 `v <> val_false` >>
Cases_on `v = val_false` >> fs [] >>
Cases_on `sem_instr_exe sem_expr (i_assign t c (o_internal e)) (State_st_list l s cs fs)` >> fs [] >>
Cases_on `x` >> fs [] >> rename1 `(v',obs)` >>
Cases_on `OoO_Exe_list_instr_not_stored_guard_true_sem_instr
(State_st_list l s cs fs) (i_assign t c (o_internal e)) v' obs` >>
fs [] >> rename1 `(ll,stl)` >>
`State_st_list_ok (State_st_list l s cs fs)` by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
sg `step_execution out_of_order_step_list [(State_st_list l s cs fs,ll,stl)]` >-
(fs [step_list_to_step] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST [(State_st_list l s cs fs,ll,stl)])) = stl` by fs [] >>
`?pi'. pi = [(State_st_list l s cs fs,ll,stl)] ++ pi' /\
step_execution out_of_order_step_list pi /\
LENGTH pi' <= 2 * n`
by METIS_TAC [IO_bounded_execution_acc_OoO_execution_sound] >>
`FST (HD ([(State_st_list l s cs fs,ll,stl)] ++ pi')) = State_st_list l s cs fs` by fs [] >>
`LENGTH pi <= 2 * (SUC n)` by rw [] >>
METIS_TAC [step_execution_list_OoO_HD],
Cases_on `sem_expr c s` >> fs [] >>
rename1 `v <> val_false` >>
Cases_on `v = val_false` >> fs [] >>
Cases_on `sem_instr_exe sem_expr (i_assign t c (o_load r n')) (State_st_list l s cs fs)` >> fs [] >>
Cases_on `x` >> fs [] >> rename1 `(v',obs)` >>
Cases_on `OoO_Exe_list_instr_not_stored_guard_true_sem_instr (State_st_list l s cs fs) (i_assign t c (o_load r n')) v' obs` >>
fs [] >> rename1 `(ll,stl)` >>
`State_st_list_ok (State_st_list l s cs fs)` by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
sg `step_execution out_of_order_step_list [(State_st_list l s cs fs,ll,stl)]` >-
(fs [step_list_to_step] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST [(State_st_list l s cs fs,ll,stl)])) = stl` by fs [] >>
`?pi'. pi = [(State_st_list l s cs fs,ll,stl)] ++ pi' /\
step_execution out_of_order_step_list pi /\
LENGTH pi' <= 2 * n`
by METIS_TAC [IO_bounded_execution_acc_OoO_execution_sound] >>
`FST (HD ([(State_st_list l s cs fs,ll,stl)] ++ pi')) = State_st_list l s cs fs` by fs [] >>
`LENGTH pi <= 2 * (SUC n)` by rw [] >>
METIS_TAC [step_execution_list_OoO_HD],
Cases_on `sem_expr c s` >> fs [] >>
rename1 `v <> val_false` >>
Cases_on `v = val_false` >> Cases_on `r` >> fs [Completed_list] >| [
rename1 `o_store res_PC ta tv` >>
Cases_on `sem_instr_exe sem_expr (i_assign t c (o_store res_PC ta tv)) (State_st_list l s cs fs)` >> fs [] >>
Cases_on `x` >> fs [] >> rename1 `(v',obs)` >>
Cases_on `OoO_Exe_list_instr_not_stored_guard_true_sem_instr (State_st_list l s cs fs) (i_assign t c (o_store res_PC ta tv)) v' obs` >> fs [] >>
rename1 `(ll,stl)` >>
Cases_on `OoO_Ftc_list_stored translate_val_list sem_expr stl t v'` >> fs [] >>
Cases_on `x` >> fs [] >> rename1 `(ll',stl')` >>
`State_st_list_ok (State_st_list l s cs fs)` by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
`stl = State_st_list l (s |+ (t,v')) cs fs`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_result_form] >>
rw [] >>
`FLOOKUP (s |+ (t,v')) t = SOME v'` by fs [FLOOKUP_UPDATE] >>
`FLOOKUP (s |+ (t,v')) t <> NONE` by fs [] >>
`out_of_order_step_list (State_st_list l (s |+ (t,v')) cs fs) ll' stl'`
by METIS_TAC [OoO_Ftc_list_stored_SOME_out_of_order_step_list] >>
`State_st_list_ok stl'` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl'`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
Q.ABBREV_TAC `pi0 = [(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs);
(State_st_list l (s |+ (t,v')) cs fs,ll',stl')]` >>
sg `step_execution out_of_order_step_list pi0` >-
(rw [Abbr `pi0`,step_list_to_step] >>
`step_execution out_of_order_step_list ([] ++
[(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs);
(State_st_list l (s |+ (t,v')) cs fs,ll',stl')])`
suffices_by fs [] >>
`step_execution out_of_order_step_list ([] ++
[(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs)])`
suffices_by METIS_TAC [step_execution_rules] >>
fs [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl'` by fs [Abbr `pi0`] >>
`?pi'. pi = pi0 ++ pi' /\
step_execution out_of_order_step_list pi /\
LENGTH pi' <= 2 * n`
by METIS_TAC [IO_bounded_execution_acc_OoO_execution_sound] >>
`FST (HD (pi0 ++ pi')) = State_st_list l s cs fs` by fs [Abbr `pi0`] >>
`LENGTH pi <= 2 * (SUC n)` by rw [Abbr `pi0`] >>
METIS_TAC [step_execution_list_OoO_HD],
rename1 `o_store res_REG ta tv` >>
Cases_on `sem_instr_exe sem_expr (i_assign t c (o_store res_REG ta tv)) (State_st_list l s cs fs)` >> fs [] >>
Cases_on `x` >> rename1 `(v',obs)` >> fs [] >>
Cases_on `OoO_Exe_list_instr_not_stored_guard_true_sem_instr (State_st_list l s cs fs)
(i_assign t c (o_store res_REG ta tv)) v' obs` >>
fs [] >> rename1 `(ll,stl)` >>
`State_st_list_ok (State_st_list l s cs fs)` by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
sg `step_execution out_of_order_step_list [(State_st_list l s cs fs,ll,stl)]` >-
(fs [step_list_to_step] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST [(State_st_list l s cs fs,ll,stl)])) = stl` by fs [] >>
`?pi'. pi = [(State_st_list l s cs fs,ll,stl)] ++ pi' /\
step_execution out_of_order_step_list pi /\
LENGTH pi' <= 2 * n`
by METIS_TAC [IO_bounded_execution_acc_OoO_execution_sound] >>
`FST (HD ([(State_st_list l s cs fs,ll,stl)] ++ pi')) = State_st_list l s cs fs` by fs [] >>
`LENGTH pi <= 2 * (SUC n)` by rw [] >>
METIS_TAC [step_execution_list_OoO_HD],
rename1 `o_store res_MEM ta tv` >>
Cases_on `sem_instr_exe sem_expr (i_assign t c (o_store res_MEM ta tv)) (State_st_list l s cs fs)` >> fs [] >>
Cases_on `x` >> fs [] >> rename1 `(v',obs)` >>
Cases_on `OoO_Exe_list_instr_not_stored_guard_true_sem_instr (State_st_list l s cs fs) (i_assign t c (o_store res_MEM ta tv)) v' obs` >> fs [] >>
rename1 `(ll,stl)` >>
Cases_on `OoO_Cmt_list_stored sem_expr stl t ta tv` >> fs [] >>
Cases_on `x` >> fs [] >> rename1 `(ll',stl')` >>
`State_st_list_ok (State_st_list l s cs fs)` by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
`stl = State_st_list l (s |+ (t,v')) cs fs`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_result_form] >>
rw [] >>
`FLOOKUP (s |+ (t,v')) t = SOME v'` by fs [FLOOKUP_UPDATE] >>
`FLOOKUP (s |+ (t,v')) t <> NONE` by fs [] >>
`out_of_order_step_list (State_st_list l (s |+ (t,v')) cs fs) ll' stl'`
by METIS_TAC [OoO_Cmt_list_stored_SOME_out_of_order_step_list] >>
`State_st_list_ok stl'` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl'`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
Q.ABBREV_TAC `pi0 = [(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs);
(State_st_list l (s |+ (t,v')) cs fs,ll',stl')]` >>
sg `step_execution out_of_order_step_list pi0` >-
(rw [Abbr `pi0`,step_list_to_step] >>
`step_execution out_of_order_step_list ([] ++
[(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs);
(State_st_list l (s |+ (t,v')) cs fs,ll',stl')])`
suffices_by fs [] >>
`step_execution out_of_order_step_list ([] ++
[(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs)])`
suffices_by METIS_TAC [step_execution_rules] >>
fs [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl'` by fs [Abbr `pi0`] >>
`?pi'. pi = pi0 ++ pi' /\
step_execution out_of_order_step_list pi /\
LENGTH pi' <= 2 * n`
by METIS_TAC [IO_bounded_execution_acc_OoO_execution_sound] >>
`FST (HD (pi0 ++ pi')) = State_st_list l s cs fs` by fs [Abbr `pi0`] >>
`LENGTH pi <= 2 * (SUC n)` by rw [Abbr `pi0`] >>
METIS_TAC [step_execution_list_OoO_HD]
],
rename1 `o_store r ta tv` >>
rename1 `FLOOKUP s t = SOME v` >>
Cases_on `r` >> fs [Completed_list] >-
(Cases_on `MEM t fs` >> fs [] >>
Cases_on `OoO_Ftc_list_stored_incomplete translate_val_list sem_expr (State_st_list l s cs fs) t v` >> fs [] >>
Cases_on `x` >> fs [] >>
rename1 `(ll,stl)` >>
`State_st_list_ok (State_st_list l s cs fs)` by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Ftc_list_stored_incomplete_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
sg `step_execution out_of_order_step_list [(State_st_list l s cs fs,ll,stl)]` >-
(fs [step_list_to_step] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST [(State_st_list l s cs fs,ll,stl)])) = stl` by fs [] >>
`?pi'. pi = [(State_st_list l s cs fs,ll,stl)] ++ pi' /\
step_execution out_of_order_step_list pi /\
LENGTH pi' <= 2 * n`
by METIS_TAC [IO_bounded_execution_acc_OoO_execution_sound] >>
`FST (HD ([(State_st_list l s cs fs,ll,stl)] ++ pi')) = State_st_list l s cs fs` by fs [] >>
`LENGTH pi <= 2 * (SUC n)` by rw [] >>
METIS_TAC [step_execution_list_OoO_HD]) >>
Cases_on `MEM t cs` >> fs [] >>
Cases_on `OoO_Cmt_list_stored_incomplete sem_expr (State_st_list l s cs fs) t ta tv` >> fs [] >>
Cases_on `x` >> fs [] >>
rename1 `(ll,stl)` >>
`FLOOKUP s t <> NONE` by fs [] >>
`State_st_list_ok (State_st_list l s cs fs)` by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Cmt_list_stored_incomplete_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
sg `step_execution out_of_order_step_list [(State_st_list l s cs fs,ll,stl)]` >-
(fs [step_list_to_step] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST [(State_st_list l s cs fs,ll,stl)])) = stl` by fs [] >>
`?pi'. pi = [(State_st_list l s cs fs,ll,stl)] ++ pi' /\
step_execution out_of_order_step_list pi /\
LENGTH pi' <= 2 * n`
by METIS_TAC [IO_bounded_execution_acc_OoO_execution_sound] >>
`FST (HD ([(State_st_list l s cs fs,ll,stl)] ++ pi')) = State_st_list l s cs fs` by fs [] >>
`LENGTH pi <= 2 * (SUC n)` by rw [] >>
METIS_TAC [step_execution_list_OoO_HD]
]
QED
(* FIXME: replace by NTH version just below? *)
Theorem IO_bounded_execution_out_of_order_step_list_sound:
!l s cs fs pos n i il pi.
State_st_list_well_formed_ok (State_st_list l s cs fs) ==>
DROP (PRE pos) l = i::il ==>
~(Completed_list sem_expr (State_st_list l s cs fs) i) ==>
IO_bounded_execution translate_val_list sem_expr (State_st_list l s cs fs) pos (SUC n) = SOME pi ==>
FST (HD pi) = State_st_list l s cs fs /\
step_execution out_of_order_step_list pi /\
LENGTH pi <= 2 * n + 2
Proof
rw [IO_bounded_execution] >>
`2 * SUC n = 2 * n + 2` by DECIDE_TAC >>
METIS_TAC [IO_bounded_execution_acc_OoO_empty_sound,step_execution_list_OoO_HD]
QED
Theorem IO_bounded_execution_out_of_order_step_list_sound_NTH:
!l s cs fs pos n i pi.
State_st_list_well_formed_ok (State_st_list l s cs fs) ==>
NTH (PRE pos) l = SOME i ==>
~(Completed_list sem_expr (State_st_list l s cs fs) i) ==>
IO_bounded_execution translate_val_list sem_expr (State_st_list l s cs fs) pos (SUC n) = SOME pi ==>
FST (HD pi) = State_st_list l s cs fs /\
step_execution out_of_order_step_list pi /\
LENGTH pi <= 2 * n + 2
Proof
METIS_TAC [
NTH_SOME_DROP,
IO_bounded_execution_out_of_order_step_list_sound
]
QED
(* FIXME: replace by NTH version just below? *)
Theorem IO_bounded_execution_out_of_order_step_sound:
!l s cs fs pos n i il pi.
State_st_list_well_formed_ok (State_st_list l s cs fs) ==>
DROP (PRE pos) l = i::il ==>
~(Completed_list sem_expr (State_st_list l s cs fs) i) ==>
IO_bounded_execution translate_val_list sem_expr (State_st_list l s cs fs) pos (SUC n) = SOME pi ==>
FST (HD pi) = State_st_list l s cs fs /\
step_execution out_of_order_step (MAP step_list_to_step pi) /\
LENGTH pi <= 2 * n + 2
Proof
METIS_TAC [
step_execution_out_of_order_step_list_step,
IO_bounded_execution_out_of_order_step_list_sound
]
QED
Theorem IO_bounded_execution_out_of_order_step_sound_NTH:
!stl pos n i pi.
State_st_list_well_formed_ok stl ==>
NTH (PRE pos) (state_program_list stl) = SOME i ==>
~(Completed_list sem_expr stl i) ==>
IO_bounded_execution translate_val_list sem_expr stl pos (SUC n) = SOME pi ==>
FST (HD pi) = stl /\
step_execution out_of_order_step (MAP step_list_to_step pi) /\
LENGTH pi <= 2 * n + 2
Proof
Cases_on `stl` >>
rw [state_program_list] >>
METIS_TAC [
NTH_SOME_DROP,
IO_bounded_execution_out_of_order_step_sound
]
QED
(* ---------------------- *)
(* Execution IO soundness *)
(* ---------------------- *)
Theorem IO_bounded_execution_acc_IO_execution_sound:
translate_val_list_SORTED ==>
!n l s cs fs pos pi pi'.
State_st_list_well_formed_ok (State_st_list l s cs fs) ==>
SORTED bound_name_instr_le l ==>
Completed_list_up_to sem_expr (State_st_list l s cs fs) pos ==>
step_execution in_order_step_list pi ==>
SND (SND (LAST pi)) = State_st_list l s cs fs ==>
IO_bounded_execution_acc translate_val_list sem_expr (State_st_list l s cs fs) pos n pi = SOME pi' ==>
?pi''. pi' = pi ++ pi'' /\
step_execution in_order_step_list pi' /\
Completed_list_up_to sem_expr (SND (SND (LAST pi'))) (pos + n) /\
LENGTH pi'' <= 2 * n
Proof
once_rewrite_tac [translate_val_list_SORTED] >>
strip_tac >>
Induct_on `n` >- fs [IO_bounded_execution_acc] >>
once_rewrite_tac [IO_bounded_execution_acc] >>
fs [] >> rw [] >>
Cases_on `DROP pos l` >> fs [] >-
(rw [] >> fs [Completed_list_up_to] >> rw [] >>
`TAKE (pos + SUC n) l = TAKE pos l` suffices_by METIS_TAC [] \\
rw [LENGTH_TAKE_PLUS]) >>
Cases_on `h` >>
rename1 `i_assign t' c mop::l'` >> rename1 `i_assign t c mop` >>
fs [] >>
Cases_on `FLOOKUP s t` >> fs [] >-
(Cases_on `sem_expr c s` >> fs [] >>
rename1 `sem_expr c s = SOME v` >>
Cases_on `v = val_false` >> fs [] >-
(`Completed_list sem_expr (State_st_list l s cs fs) (i_assign t c mop)`
by (Cases_on `mop` >> rw [Completed_list] >> Cases_on `r` >> rw [Completed_list]) >>
sg `Completed_list_up_to sem_expr (State_st_list l s cs fs) (SUC pos)` >-
(rw [Completed_list_up_to] >>
`TAKE (SUC pos) l = SNOC (i_assign t c mop) (TAKE pos l)`
by rw [DROP_TAKE_SNOC] >>
fs [Completed_list_up_to]) >>
Q.PAT_X_ASSUM `!l s cs. P`
(STRIP_ASSUME_TAC o (Q.SPECL [`l`,`s`,`cs`,`fs`,`SUC pos`, `pi`,`pi'`])) >>
`?pi''. pi' = pi ++ pi'' /\
step_execution in_order_step_list pi' /\
Completed_list_up_to sem_expr (SND (SND (LAST pi'))) (n + (SUC pos)) /\
LENGTH pi'' ≤ 2 * n` by METIS_TAC [] >>
Q.EXISTS_TAC `pi''` >> rw [] >>
`pos + SUC n = n + SUC pos` by rw [] >>
rw []) >>
Cases_on `sem_instr_exe sem_expr (i_assign t c mop) (State_st_list l s cs fs)` >> fs [] >>
Cases_on `x` >> rename1 `(v',obs)` >> fs [] >>
Cases_on `OoO_Exe_list_instr_not_stored_guard_true_sem_instr
(State_st_list l s cs fs) (i_assign t c mop) v' obs` >>
fs [] >> rename1 `(ll,stl)` >>
`MEM (i_assign t c mop) l` by METIS_TAC [DROP_HEAD_MEM] >>
`FIND_instr t l = SOME (i_assign t c mop)`
by METIS_TAC [State_st_list_well_formed_ok,State_st_list_ok,NO_DUPLICATES_FIND_instr,bound_name_instr] >>
`State_st_list_ok (State_st_list l s cs fs)` by fs [State_st_list_ok,State_st_list_well_formed_ok] >>
`out_of_order_step_list (State_st_list l s cs fs) ll stl`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_SOME_out_of_order_step_list] >>
`State_st_list_ok stl` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
sg `in_order_step_list (State_st_list l s cs fs) ll stl` >-
(`ll = ll_lb obs act_exe_list t /\ stl = State_st_list l (s |+ (t,v')) cs fs`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_result_form] >>
rw [in_order_step_list,IO_step_list] >>
fs [out_of_order_step_list] >>
rw [EVERY_MEM] >>
fs [MEM_FILTER] >>
`bound_name_instr i < t` by DECIDE_TAC >>
`MEM i (TAKE pos l)` suffices_by fs [Completed_list_up_to] >>
`MEM i (TAKE pos l) \/ MEM i (DROP pos l)`
by METIS_TAC [TAKE_DROP,MEM_APPEND] >>
`i <> i_assign t c mop` by (Cases_on `i` >> fs [bound_name_instr]) >>
`MEM i (i_assign t c mop::l')` by METIS_TAC [] >>
fs [] >> rw [] >>
sg `SORTED bound_name_instr_le ((TAKE pos l ++ [i_assign t c mop]) ++ l')` >-
(`(TAKE pos l ++ [i_assign t c mop]) ++ l' = l` suffices_by rw [] >>
`l = TAKE pos l ++ (i_assign t c mop::l')` by METIS_TAC [TAKE_DROP] >>
once_rewrite_tac [GSYM APPEND_ASSOC] >>
`[i_assign t c mop] ++ l' = i_assign t c mop::l'` by rw [] >>
METIS_TAC []) >>
`!x y. MEM x (TAKE pos l ++ [i_assign t c mop]) ==> MEM y l' ==> bound_name_instr_le x y`
by METIS_TAC [SORTED_APPEND,transitive_bound_name_instr_le] >>
`MEM (i_assign t c mop) (TAKE pos l ++ [i_assign t c mop])` by rw [] >>
`bound_name_instr_le (i_assign t c mop) i` by METIS_TAC [] >>
fs [bound_name_instr_le,name_le,bound_name_instr]) >>
Cases_on `mop` >> fs [] >| [
Q.ABBREV_TAC `pi0 = pi ++ [(State_st_list l s cs fs,ll,stl)]` >>
`State_st_list_well_formed_ok (SND (SND (LAST pi0)))` by fs [Abbr `pi0`] >>
sg `step_execution in_order_step_list pi0` >-
(fs [Abbr `pi0`,step_list_to_step] >>
`pi <> []` by METIS_TAC [step_execution_not_empty_list] >>
`pi = [] \/ ?e pi1. pi = SNOC e pi1` by fs [SNOC_CASES] >> fs [SNOC_APPEND] >>
Cases_on `e'` >> fs [] >> Cases_on `r` >> rename1 `(stl1,ll1,stl2)` >>
fs [step_list_to_step] >> rw [] >>
`pi1 ++ [(stl1,ll1,State_st_list l s cs fs)] ++ [(State_st_list l s cs fs,ll,stl)] =
pi1 ++ [(stl1,ll1,State_st_list l s cs fs);(State_st_list l s cs fs,ll,stl)]`
by fs [] >>
rw [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl` by fs [Abbr `pi0`] >>
Cases_on `stl` >>
rename1 `(ll,State_st_list l'' s'' cs'' fs'')` >>
`SORTED bound_name_instr_le l''` by fs [OoO_Exe_list_instr_not_stored_guard_true_sem_instr] >>
sg `Completed_list sem_expr (State_st_list l'' s'' cs'' fs'') (i_assign t c (o_internal e))` >-
(fs [Completed_list,OoO_Exe_list_instr_not_stored_guard_true_sem_instr] >>
rw [FLOOKUP_DEF]) >>
sg `Completed_list_up_to sem_expr (State_st_list l'' s'' cs'' fs'') (SUC pos)` >-
(rw [Completed_list_up_to] >>
`ll = ll_lb obs act_exe_list t /\
State_st_list l'' s'' cs'' fs'' = State_st_list l (s |+ (t,v')) cs fs`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_result_form] >>
rw [] >>
`TAKE (SUC pos) l = SNOC (i_assign t c (o_internal e)) (TAKE pos l)` by rw [DROP_TAKE_SNOC] >>
fs [] >>
`Completed_list sem_expr (State_st_list l s cs fs) i`
by fs [Completed_list_up_to] >>
`MEM i l` by METIS_TAC [MEM_TAKE] >>
(* TODO: adapt well_formed_state_OoO_completed_t_preserved for lists *)
`Completed (state_list_to_state (State_st_list l s cs fs)) i`
by METIS_TAC [Completed_list_correct] >>
`Completed (state_list_to_state (State_st_list l (s |+ (t,v')) cs fs)) i`
suffices_by METIS_TAC [Completed_list_correct] >>
`Completed_t (state_list_to_state (State_st_list l s cs fs)) (bound_name_instr i)`
by (fs [state_list_to_state] >> rw [Completed_t] >> METIS_TAC []) >>
`Completed_t (state_list_to_state (State_st_list l (s |+ (t,v')) cs fs)) (bound_name_instr i)`
by METIS_TAC [state_list_to_state,
out_of_order_step_list_sound,
well_formed_state_OoO_completed_t_preserved,
State_st_list_well_formed_ok] >>
fs [state_list_to_state,Completed_t,State_st_list_well_formed_ok] >>
METIS_TAC [wfs_unique_instr_names]) >>
Q.PAT_X_ASSUM `!l s cs. P`
(STRIP_ASSUME_TAC o (Q.SPECL [`l''`,`s''`,`cs''`,`fs''`,`SUC pos`, `pi0`,`pi'`])) >>
`?pi''. pi' = pi0 ++ pi'' /\
step_execution in_order_step_list pi' /\
Completed_list_up_to sem_expr (SND (SND (LAST pi'))) (n + SUC pos) /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
Q.EXISTS_TAC `(State_st_list l s cs fs,ll,State_st_list l'' s'' cs'' fs'')::pi''` >>
`pos + SUC n = n + SUC pos` by DECIDE_TAC >>
rw [Abbr `pi0`],
Q.ABBREV_TAC `pi0 = pi ++ [(State_st_list l s cs fs,ll,stl)]` >>
`State_st_list_well_formed_ok (SND (SND (LAST pi0)))` by fs [Abbr `pi0`] >>
sg `step_execution in_order_step_list pi0` >-
(fs [Abbr `pi0`,step_list_to_step] >>
`pi <> []` by METIS_TAC [step_execution_not_empty_list] >>
`pi = [] \/ ?e pi1. pi = SNOC e pi1` by fs [SNOC_CASES] >> fs [SNOC_APPEND] >>
Cases_on `e` >> fs [] >> Cases_on `r'` >> rename1 `(stl1,ll1,stl2)` >>
fs [step_list_to_step] >> rw [] >>
`pi1 ++ [(stl1,ll1,State_st_list l s cs fs)] ++ [(State_st_list l s cs fs,ll,stl)] =
pi1 ++ [(stl1,ll1,State_st_list l s cs fs);(State_st_list l s cs fs,ll,stl)]`
by fs [] >>
rw [] >> METIS_TAC [step_execution_rules]) >>
`SND (SND (LAST pi0)) = stl` by fs [Abbr `pi0`] >>
Cases_on `stl` >>
rename1 `(ll,State_st_list l'' s'' cs'' fs'')` >>
`SORTED bound_name_instr_le l''` by fs [OoO_Exe_list_instr_not_stored_guard_true_sem_instr] >>
sg `Completed_list sem_expr (State_st_list l'' s'' cs'' fs'') (i_assign t c (o_load r n'))` >-
(fs [Completed_list,OoO_Exe_list_instr_not_stored_guard_true_sem_instr] >>
rw [FLOOKUP_DEF]) >>
sg `Completed_list_up_to sem_expr (State_st_list l'' s'' cs'' fs'') (SUC pos)` >-
(rw [Completed_list_up_to] >>
`ll = ll_lb obs act_exe_list t /\
State_st_list l'' s'' cs'' fs'' = State_st_list l (s |+ (t,v')) cs fs`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_result_form] >>
rw [] >>
`TAKE (SUC pos) l = SNOC (i_assign t c (o_load r n')) (TAKE pos l)` by rw [DROP_TAKE_SNOC] >>
fs [] >>
`Completed_list sem_expr (State_st_list l s cs fs) i`
by fs [Completed_list_up_to] >>
`MEM i l` by METIS_TAC [MEM_TAKE] >>
(* TODO: adapt well_formed_state_OoO_completed_t_preserved for lists *)
`Completed (state_list_to_state (State_st_list l s cs fs)) i`
by METIS_TAC [Completed_list_correct] >>
`Completed (state_list_to_state (State_st_list l (s |+ (t,v')) cs fs)) i`
suffices_by METIS_TAC [Completed_list_correct] >>
`Completed_t (state_list_to_state (State_st_list l s cs fs)) (bound_name_instr i)`
by (fs [state_list_to_state] >> rw [Completed_t] >> METIS_TAC []) >>
`Completed_t (state_list_to_state (State_st_list l (s |+ (t,v')) cs fs)) (bound_name_instr i)`
by METIS_TAC [state_list_to_state,
out_of_order_step_list_sound,
well_formed_state_OoO_completed_t_preserved,
State_st_list_well_formed_ok] >>
fs [state_list_to_state,Completed_t,State_st_list_well_formed_ok] >>
METIS_TAC [wfs_unique_instr_names]) >>
Q.PAT_X_ASSUM `!l s cs. P`
(STRIP_ASSUME_TAC o (Q.SPECL [`l''`,`s''`,`cs''`,`fs''`,`SUC pos`, `pi0`,`pi'`])) >>
`?pi''. pi' = pi0 ++ pi'' /\
step_execution in_order_step_list pi' /\
Completed_list_up_to sem_expr (SND (SND (LAST pi'))) (n + SUC pos) /\
LENGTH pi'' <= 2 * n` by METIS_TAC [] >>
Q.EXISTS_TAC `(State_st_list l s cs fs,ll,State_st_list l'' s'' cs'' fs'')::pi''` >>
`pos + SUC n = n + SUC pos` by DECIDE_TAC >>
rw [Abbr `pi0`],
Cases_on `r` >> fs [] >| [
Cases_on `OoO_Ftc_list_stored translate_val_list sem_expr stl t v'` >> fs [] >>
Cases_on `x` >> fs [] >> rename1 `(ll',stl')` >>
rename1 `o_store res_PC ta tv` >>
`stl = State_st_list l (s |+ (t,v')) cs fs`
by METIS_TAC [OoO_Exe_list_instr_not_stored_guard_true_sem_instr_result_form] >>
rw [] >>
Q.ABBREV_TAC `pi0 = pi ++ [(State_st_list l s cs fs,ll,State_st_list l (s |+ (t,v')) cs fs);
(State_st_list l (s |+ (t,v')) cs fs,ll',stl')]` >>
`FLOOKUP (s |+ (t,v')) t = SOME v'` by fs [FLOOKUP_UPDATE] >>
`out_of_order_step_list (State_st_list l (s |+ (t,v')) cs fs) ll' stl'`
by METIS_TAC [OoO_Ftc_list_stored_SOME_out_of_order_step_list] >>
`State_st_list_ok stl'` by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_ok] >>
`State_st_list_well_formed_ok stl'`
by METIS_TAC [step_invariant,step_invariant_out_of_order_step_list_well_formed_ok] >>
`State_st_list_well_formed_ok (SND (SND (LAST pi0)))` by fs [Abbr `pi0`] >>
sg `Completed_list_up_to sem_expr (State_st_list l (s |+ (t,v')) cs fs) pos` >-
(rw [Completed_list_up_to] >>
`MEM i l` by METIS_TAC [MEM_TAKE] >>
`Completed_list sem_expr (State_st_list l s cs fs) i`
by fs [Completed_list_up_to] >>
`Completed (state_list_to_state (State_st_list l s cs fs)) i`
by METIS_TAC [Completed_list_correct] >>
`Completed (state_list_to_state (State_st_list l (s |+ (t,v')) cs fs)) i`
suffices_by METIS_TAC [Completed_list_correct] >>
`Completed_t (state_list_to_state (State_st_list l s cs fs)) (bound_name_instr i)`
by (fs [state_list_to_state] >> rw [Completed_t] >> METIS_TAC []) >>
`Completed_t (state_list_to_state (State_st_list l (s |+ (t,v')) cs fs)) (bound_name_instr i)`
by METIS_TAC [state_list_to_state,
out_of_order_step_list_sound,
well_formed_state_OoO_completed_t_preserved,
State_st_list_well_formed_ok] >>
fs [state_list_to_state,Completed_t,State_st_list_well_formed_ok] >>
METIS_TAC [wfs_unique_instr_names]) >>
sg `in_order_step_list (State_st_list l (s |+ (t,v')) cs fs) ll' stl'` >-
(`ll' = ll_lb (obs_il v') (act_ftc_list (translate_val_list v' (max_bound_name_list l))) t /\
stl' = State_st_list (l ++ translate_val_list v' (max_bound_name_list l)) (s |+ (t,v')) cs (t::fs)`
by METIS_TAC [OoO_Ftc_list_stored_result_form] >>
rw [in_order_step_list,IO_step_list] >>
fs [out_of_order_step_list] >>
rw [EVERY_MEM] >>
fs [MEM_FILTER] >>
`bound_name_instr i < t` by DECIDE_TAC >>
`MEM i (TAKE pos l)` suffices_by fs [Completed_list_up_to] >>
`MEM i (TAKE pos l) \/ MEM i (DROP pos l)`
by METIS_TAC [TAKE_DROP,MEM_APPEND] >>
`i <> i_assign t c (o_store res_PC ta tv)` by (Cases_on `i` >> fs [bound_name_instr]) >>
`MEM i (i_assign t c (o_store res_PC ta tv)::l')` by METIS_TAC [] >>
fs [] >> rw [] >>
sg `SORTED bound_name_instr_le ((TAKE pos l ++ [i_assign t c (o_store res_PC ta tv)]) ++ l')` >-
(`(TAKE pos l ++ [i_assign t c (o_store res_PC ta tv)]) ++ l' = l` suffices_by rw [] >>
`l = TAKE pos l ++ (i_assign t c (o_store res_PC ta tv)::l')` by METIS_TAC [TAKE_DROP] >>
once_rewrite_tac [GSYM APPEND_ASSOC] >>
`[i_assign t c (o_store res_PC ta tv)] ++ l' = i_assign t c (o_store res_PC ta tv)::l'` by rw [] >>
METIS_TAC []) >>
`!x y. MEM x (TAKE pos l ++ [i_assign t c (o_store res_PC ta tv)]) ==> MEM y l' ==> bound_name_instr_le x y`
by METIS_TAC [SORTED_APPEND,transitive_bound_name_instr_le] >>
`MEM (i_assign t c (o_store res_PC ta tv)) (TAKE pos l ++ [i_assign t c (o_store res_PC ta tv)])` by rw [] >>
`bound_name_instr_le (i_assign t c (o_store res_PC ta tv)) i` by METIS_TAC [] >>
fs [bound_name_instr_le,name_le,bound_name_instr]) >>
sg `SORTED bound_name_instr_le (l ++ translate_val_list v' (max_bound_name_list l))` >-
(`!x y. MEM x l ==> MEM y (translate_val_list v' (max_bound_name_list l)) ==> bound_name_instr_le x y`
suffices_by METIS_TAC [SORTED_APPEND,transitive_bound_name_instr_le] >>
rw [bound_name_instr_le,name_le] >>
`bound_name_instr x < bound_name_instr y` suffices_by rw [] >>
`y IN (translate_val v' (MAX_SET (bound_names_program (set l))))`
by METIS_TAC [translate_val_list_correct,max_bound_name_list_correct] >>
`FINITE (set l)` by rw [] >>
METIS_TAC [translate_val_max_name_lt]) >>
`Completed_list sem_expr (State_st_list (l ++ translate_val_list v' (max_bound_name_list l))
(s |+ (t,v')) cs (t::fs)) (i_assign t c (o_store res_PC ta tv))` by rw [Completed_list] >>
sg `Completed_list_up_to sem_expr stl' (SUC pos)` >-
(`stl' = State_st_list (l ++ translate_val_list v' (max_bound_name_list l)) (s |+ (t,v')) cs (t::fs)`
by METIS_TAC [OoO_Ftc_list_stored_result_form] >>
rw [Completed_list_up_to] >>
`DROP pos (l ++ translate_val_list v' (max_bound_name_list l)) =
i_assign t c (o_store res_PC ta tv)::l' ++ translate_val_list v' (max_bound_name_list l)`
by rw [DROP_CONS_APPEND] >>
`TAKE (SUC pos) (l ++ translate_val_list v' (max_bound_name_list l)) =
SNOC (i_assign t c (o_store res_PC ta tv)) (TAKE pos (l ++ translate_val_list v' (max_bound_name_list l)))`
by rw [DROP_TAKE_SNOC] >>
fs [] >>
`MEM i (TAKE pos l)` by METIS_TAC [DROP_CONS_TAKE_APPEND] >>
`MEM i l` by METIS_TAC [MEM_TAKE] >>
`Completed_list sem_expr (State_st_list l (s |+ (t,v')) cs fs) i`
by fs [Completed_list_up_to] >>
`Completed (state_list_to_state (State_st_list l (s |+ (t,v')) cs fs)) i`
by fs [Completed_list_correct] >>
`Completed_t (state_list_to_state (State_st_list l (s |+ (t,v')) cs fs)) (bound_name_instr i)`
by (fs [state_list_to_state] >> rw [Completed_t] >> METIS_TAC []) >>
`Completed_t (state_list_to_state (State_st_list (l ++ translate_val_list v' (max_bound_name_list l))
(s |+ (t,v')) cs (t::fs))) (bound_name_instr i)`
by METIS_TAC [state_list_to_state,
out_of_order_step_list_sound,
well_formed_state_OoO_completed_t_preserved,
State_st_list_well_formed_ok] >>
`Completed (state_list_to_state (State_st_list (l ++ translate_val_list v' (max_bound_name_list l))
(s |+ (t,v')) cs (t::fs))) i` suffices_by METIS_TAC [Completed_list_correct] >>