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milMaxExeTraceExampleSpectreOOOScript.sml
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open HolKernel boolLib Parse bossLib wordsLib wordsTheory finite_mapTheory pred_setTheory relationTheory listTheory rich_listTheory pairTheory milUtilityTheory milTheory milSemanticsUtilityTheory milMetaTheory milTracesTheory milMetaIOTheory milExecutableUtilityTheory milExecutableTransitionTheory milExecutableIOTheory milExecutableIOTraceTheory milExecutableIOCompletenessTheory milExecutableExamplesTheory milNoninterferenceTheory milMaxExeTraceUtilityTheory;
(* ================================================== *)
(* Maximal execution and trace of example spectreOoO *)
(* ================================================== *)
val _ = new_theory "milMaxExeTraceExampleSpectreOOO";
(* example Spectre OoO after initialization *)
Definition example_spectre_OoO_exe_init:
example_spectre_OoO_exe_init a b c d r1 r2 z b1 b2 t =
[i_assign (t+1) (e_val 1w) (o_internal (e_val b1));
i_assign (t+2) (e_val 1w) (o_internal (e_val a));
i_assign (t+3) (e_val 1w) (o_store res_MEM (t+1) (t+2));
i_assign (t+4) (e_val 1w) (o_internal (e_val z));
i_assign (t+5) (e_val 1w) (o_internal (e_val b));
i_assign (t+6) (e_val 1w) (o_store res_REG (t+4) (t+5));
i_assign (t+7) (e_val 1w) (o_internal (e_val r2));
i_assign (t+8) (e_val 1w) (o_internal (e_val c));
i_assign (t+9) (e_val 1w) (o_store res_REG (t+7) (t+8));
i_assign (t+10) (e_val 1w) (o_internal (e_val 0w));
i_assign (t+11) (e_val 1w) (o_internal (e_val d));
i_assign (t+12) (e_val 1w) (o_store res_PC (t+10) (t+11));
i_assign (t+13) (e_val 1w) (o_internal (e_val 0w));
i_assign (t+14) (e_val 1w) (o_internal (e_val r1));
i_assign (t+15) (e_val 1w) (o_internal (e_val r2));
i_assign (t+16) (e_val 1w) (o_internal (e_val z));
i_assign (t+17) (e_val 1w) (o_internal (e_val b1));
i_assign (t+18) (e_val 1w) (o_internal (e_val b2));
i_assign (t+19) (e_val 1w) (o_load res_MEM (t+17));
i_assign (t+20) (e_val 1w) (o_store res_REG (t+14) (t+19));
i_assign (t+21) (e_val 1w) (o_load res_REG (t+16));
i_assign (t+22) (e_eq (e_name (t+21)) (e_val 1w)) (o_load res_REG (t+14));
i_assign (t+23) (e_eq (e_name (t+21)) (e_val 1w)) (o_store res_REG (t+15) (t+22));
i_assign (t+24) (e_val 1w) (o_load res_REG (t+15));
i_assign (t+25) (e_val 1w) (o_store res_MEM (t+18) (t+24));
i_assign (t+26) (e_val 1w) (o_load res_PC (t+13));
i_assign (t+27) (e_val 1w) (o_internal (e_add (e_name (t+26)) (e_val 4w)));
i_assign (t+28) (e_val 1w) (o_store res_PC (t+13) (t+27))]
End
Definition example_spectre_OoO_list:
example_spectre_OoO_list a b c d r1 r2 z b1 b2 = example_spectre_OoO_exe_init a b c d r1 r2 z b1 b2 0
End
Definition example_spectre_OoO_store:
example_spectre_OoO_store a b c d r2 z b1 : num |-> v =
(FEMPTY |+ (1,b1) |+ (2,a) |+ (3,a) |+ (4,z) |+ (5,b) |+ (6,b) |+
(7,r2) |+ (8,c) |+ (9,c) |+ (10,0w) |+ (11,d) |+ (12,d))
End
Definition example_spectre_OoO_state_st_list:
example_spectre_OoO_state_st_list a b c d r1 r2 z b1 b2 =
State_st_list (example_spectre_OoO_list a b c d r1 r2 z b1 b2)
(example_spectre_OoO_store a b c d r2 z b1) [3] [12]
End
Definition example_spectre_OoO_state_st_list_final:
example_spectre_OoO_state_st_list_final a b c d b1 b2 =
State_st_list
[i_assign 1 (e_val 1w) (o_internal (e_val b1));
i_assign 2 (e_val 1w) (o_internal (e_val a));
i_assign 3 (e_val 1w) (o_store res_MEM 1 2);
i_assign 4 (e_val 1w) (o_internal (e_val 3w));
i_assign 5 (e_val 1w) (o_internal (e_val b));
i_assign 6 (e_val 1w) (o_store res_REG 4 5);
i_assign 7 (e_val 1w) (o_internal (e_val 2w));
i_assign 8 (e_val 1w) (o_internal (e_val c));
i_assign 9 (e_val 1w) (o_store res_REG 7 8);
i_assign 10 (e_val 1w) (o_internal (e_val 0w));
i_assign 11 (e_val 1w) (o_internal (e_val d));
i_assign 12 (e_val 1w) (o_store res_PC 10 11);
i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]
(FEMPTY |+ (1,b1) |+ (2,a) |+ (3,a) |+ (4,3w) |+ (5,b) |+
(6,b) |+ (7,2w) |+ (8,c) |+ (9,c) |+ (10,0w) |+ (11,d) |+
(12,d) |+ (13,0w) |+ (14,1w) |+ (15,2w) |+ (16,3w) |+ (17,b1) |+
(18,b2) |+ (19,a) |+ (20,a) |+ (21,b) |+ (24,c) |+ (25,c) |+
(26,d) |+ (27,d + 4w) |+ (28,d + 4w)) [25; 3] [28; 12]
End
(* Use 1 - 5 for r1, r2, z, a, b to execute
* a, b, c and d are initial values for b1, z, r2, PC respectively.
*)
Definition example_spectre_OoO_maximal_execution:
example_spectre_OoO_maximal_execution a b c d b1 b2 =
IO_bounded_execution (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 13 16
End
Definition example_spectre_OoO_maximal_trace_aux:
example_spectre_OoO_maximal_trace_aux a b c d b1 b2 =
IO_bounded_trace_aux (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 13 16
End
Definition example_spectre_OoO_maximal_trace:
example_spectre_OoO_maximal_trace a b c d b1 b2 =
IO_bounded_trace (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 13 16
End
Theorem IO_bounded_trace_last_example_spectre_OoO[local]:
translate_val_list = (\v t. []) ==>
sem_expr = sem_expr_exe ==>
!f_tran stl' tr a b c d b1 b2.
IO_bounded_trace_acc (\v t. []) sem_expr_exe (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [] = SOME (stl',tr) ==>
SND (SND (LAST (THE (IO_bounded_execution_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [])))) = stl'
Proof
rw [example_spectre_OoO_state_st_list] >>
`DROP 12 (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2) =
[i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]`
by (fs [example_spectre_OoO_list,example_spectre_OoO_exe_init] >> EVAL_TAC) >>
Q.ABBREV_TAC `i = i_assign 13 (e_val 1w) (o_internal (e_val 0w))` >>
Q.ABBREV_TAC `il = [i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]` >>
`~(Completed_list sem_expr (State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) i)`
by (fs [example_spectre_OoO_list,example_spectre_OoO_exe_init,example_spectre_OoO_store,Abbr `i`] >> EVAL_TAC) >>
`?x. IO_bounded_execution_acc (\v t. []) sem_expr_exe
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) 12 16 [] = SOME x`
by METIS_TAC [IO_bounded_trace_execution_acc_SOME] >> fs [] >>
`IO_bounded_execution_acc translate_val_list sem_expr
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) 12 (SUC 15) [] = SOME x` by fs [] >>
`IO_bounded_trace_acc translate_val_list sem_expr
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) 12 (SUC 15) [] = SOME (stl',tr)` by fs [] >>
Q.ABBREV_TAC `l = example_spectre_OoO_list a b c d 1w 2w 3w b1 b2` >>
Q.ABBREV_TAC `s = example_spectre_OoO_store a b c d 2w 3w b1` >>
PROVE_TAC [IO_bounded_trace_acc_empty_LAST]
QED
Theorem example_spectre_OoO_IO_bounded_trace_acc_not_true[local]:
!a b c d b1 b2.
b <> 1w ==>
IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_dl b1; obs_ds b2; obs_il (d + 4w)])
Proof
rw [example_spectre_OoO_state_st_list,Once IO_bounded_trace_acc,example_spectre_OoO_list,
example_spectre_OoO_exe_init,example_spectre_OoO_store,
FLOOKUP_DEF,OoO_Ftc_list_stored_incomplete] >>
NTAC 6 (fs [Once IO_bounded_trace_acc,FLOOKUP_DEF,sem_expr_exe,val_false,sem_instr_exe,sem_expr_exe,
OoO_Exe_list_instr_not_stored_guard_true_sem_instr,obs_of_ll,obs_visible]) >>
fs [Once IO_bounded_trace_acc] >>
fs [Once FLOOKUP_DEF,sem_expr_exe,val_false,sem_instr_exe,sem_expr_exe] >>
qmatch_goalsub_abbrev_tac `State_st_list instr_list store _ _` >>
`str_act_list sem_expr_exe (State_st_list instr_list store [3] [12]) 19 =
[i_assign 3 (e_val 1w) (o_store res_MEM 1 2)]` by (fs [Abbr `instr_list`,Abbr `store`] >> EVAL_TAC) >>
fs [bound_names_program_list,bound_name_instr,FLOOKUP_DEF,
OoO_Exe_list_instr_not_stored_guard_true_sem_instr,obs_of_ll,obs_visible,Abbr `store`] >>
qmatch_goalsub_abbrev_tac `store ' 3` >>
`store ' 3 = a` by (fs [Abbr `store`] >> EVAL_TAC) >>
`store ' 17 = b1` by (fs [Abbr `store`] >> EVAL_TAC) >> fs [Abbr `instr_list`,Abbr `store`] >>
fs [Once IO_bounded_trace_acc] >>
fs [FLOOKUP_DEF,sem_expr_exe,val_false,sem_instr_exe,sem_expr_exe,
OoO_Exe_list_instr_not_stored_guard_true_sem_instr,obs_of_ll,obs_visible] >>
fs [Once IO_bounded_trace_acc] >>
fs [Once FLOOKUP_DEF,sem_expr_exe,val_false,sem_instr_exe,sem_expr_exe] >>
qmatch_goalsub_abbrev_tac `State_st_list instr_list store _ _` >>
`str_act_list sem_expr_exe (State_st_list instr_list store [3] [12]) 21 =
[i_assign 6 (e_val 1w) (o_store res_REG 4 5)]` by (fs [Abbr `instr_list`,Abbr `store`] >> EVAL_TAC) >>
fs [bound_names_program_list,bound_name_instr,Once FLOOKUP_DEF,Abbr `store`] >>
fs [Once FLOOKUP_DEF,OoO_Exe_list_instr_not_stored_guard_true_sem_instr,
obs_of_ll,obs_visible,Abbr `instr_list`] >>
fs [Once IO_bounded_trace_acc] >>
fs [FLOOKUP_DEF,sem_expr_exe,v_eq,val_false] >>
qmatch_goalsub_abbrev_tac `store ' 6` >>
`store ' 6 = b` by (fs [Abbr `store`] >> EVAL_TAC) >> fs [Abbr `store`] >>
NTAC 3 (rw [Once IO_bounded_trace_acc] >> fs [FLOOKUP_DEF,sem_expr_exe,v_eq,val_false]) >>
rw [Once sem_instr_exe] >>
fs [str_act_list,addr_of_list,FIND_instr,bound_name_instr,str_may_list] >>
qmatch_goalsub_abbrev_tac `State_st_list instr_list store _ _` >>
`str_may_list_find sem_expr_exe instr_list store 24 res_REG 15 = [i_assign 9 (e_val 1w) (o_store res_REG 7 8)]`
by (fs [Abbr `instr_list`,Abbr `store`] >>EVAL_TAC >> fs []) >>
fs [str_act_list_find,str_act_list_cond,sem_expr_exe,Abbr `instr_list`,Abbr `store`] >>
NTAC 3 (rw [Once FLOOKUP_DEF]) >>
fs [bound_names_program_list] >>
rw [Once FLOOKUP_DEF] >>
fs [bound_name_instr,FLOOKUP_DEF,OoO_Exe_list_instr_not_stored_guard_true_sem_instr,
obs_visible,obs_of_ll] >>
EVAL_TAC >>
Cases_on `b1 = b2` >> fs [] >> EVAL_TAC
QED
Theorem example_spectre_OoO_maximal_execution_last_state_all_completed[local]:
translate_val_list = (\v t. []) ==>
sem_expr = sem_expr_exe ==>
!a b c d b1 b2.
all_instructions_completed_list sem_expr_exe
(SND (SND (LAST (THE (example_spectre_OoO_maximal_execution a b c d b1 b2)))))
Proof
rw [] >>
Cases_on `b = 1w` >-
(rw [] >> EVAL_TAC >> Cases_on `b1 <> b2` >> fs [] >> EVAL_TAC >> rw [] >> EVAL_TAC) >>
rw [example_spectre_OoO_maximal_execution,IO_bounded_execution] >>
`IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_dl b1; obs_ds b2; obs_il (d + 4w)])`
by fs [example_spectre_OoO_IO_bounded_trace_acc_not_true] >>
`SND (SND (LAST (THE (IO_bounded_execution_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [])))) =
example_spectre_OoO_state_st_list_final a b c d b1 b2`
by METIS_TAC [IO_bounded_trace_last_example_spectre_OoO] >>
rw [example_spectre_OoO_state_st_list_final] >> EVAL_TAC >> rw [] >> EVAL_TAC >> fs []
QED
Theorem State_st_list_well_formed_ok_example_spectre_OoO_state_st_list:
sem_expr_exe = sem_expr ==>
!a b c d b1 b2.
State_st_list_well_formed_ok (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)
Proof
rw [State_st_list_well_formed_ok,example_spectre_OoO_state_st_list,
example_spectre_OoO_list,example_spectre_OoO_exe_init,example_spectre_OoO_store] >-
EVAL_TAC >>
rw [state_list_to_state] >>
SIMP_TAC std_ss [well_formed_state] >>
REPEAT STRIP_TAC >|[
EVAL_TAC,
EVAL_TAC,
rw [bound_names_program] >> METIS_TAC [bound_name_instr],
fs [bound_name_instr] >> rw [] >>
fs [free_names_instr,names_e,names_o],
fs [] >> rw [] >> fs [bound_name_instr],
fs [] >> rw [] >>
fs [free_names_instr,names_e,names_o] >> METIS_TAC [bound_name_instr],
fs [],
fs [],
fs [map_down,FLOOKUP_DEF],
fs [map_down,FLOOKUP_DEF] >> rw [] >> fs [] >> EVAL_TAC,
fs [map_down,FLOOKUP_DEF] >> rw [] >> fs [sem_expr_correct,val_false],
fs [val_true,val_zero],
fs [val_true] >> rw [] >> fs [names_e],
fs [] >> rw [] >> fs [names_o,names_e,sem_expr_correct,val_false],
fs [sem_instr,sem_expr_correct,FLOOKUP_DEF] >> rw [] >> fs [] >> EVAL_TAC,
fs [val_true,val_zero],
(* sem_expr = sem_expr_exe needed here *)
`NO_DUPLICATES (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)`
by (fs [example_spectre_OoO_state_st_list] >> EVAL_TAC) >>
`State_st_list_ok (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)`
by fs [example_spectre_OoO_state_st_list,State_st_list_ok] >>
`str_may_list sem_expr_exe (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 3 = []` by EVAL_TAC >>
`str_may (state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)) 3 =
LIST_TO_SET (str_may_list sem_expr (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 3)`
by METIS_TAC [str_may_list_correct] >>
`str_may (state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)) 3 = {}`
by METIS_TAC [LIST_TO_SET] >>
fs [state_list_to_state,example_spectre_OoO_state_st_list,
example_spectre_OoO_list,example_spectre_OoO_exe_init,
example_spectre_OoO_store,bound_names_program],
`NO_DUPLICATES (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)`
by (fs [example_spectre_OoO_state_st_list] >> EVAL_TAC) >>
`State_st_list_ok (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)`
by fs [example_spectre_OoO_state_st_list,State_st_list_ok] >>
`str_may_list sem_expr_exe (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 = []` by EVAL_TAC >>
`str_may (state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)) 12 =
LIST_TO_SET (str_may_list sem_expr (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12)`
by METIS_TAC [str_may_list_correct] >>
`str_may (state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)) 12 = {}`
by METIS_TAC [LIST_TO_SET] >>
fs [state_list_to_state,example_spectre_OoO_state_st_list,
example_spectre_OoO_list,example_spectre_OoO_exe_init,
example_spectre_OoO_store,bound_names_program]
]
QED
Theorem HD_MAP_step_list_to_step_and_state_list_to_state[local]:
!pi stl.
pi <> [] ==>
FST (HD pi) = stl ==>
FST (HD (MAP step_list_to_step pi)) = state_list_to_state stl
Proof
Cases_on `pi` >> rw [] >>
Cases_on `h` >> Cases_on `r` >> rw [step_list_to_step]
QED
Theorem LAST_MAP_step_list_to_step_and_state_list_to_state[local]:
!pi.
pi <> [] ==>
SND (SND (LAST (MAP step_list_to_step pi))) = state_list_to_state (SND (SND (LAST pi)))
Proof
Cases_on `pi` >> rw [LAST_MAP] >>
Cases_on `LAST (h::t)` >> Cases_on `r` >> rw [step_list_to_step]
QED
Theorem example_spectre_OoO_maximal_execution_sound[local]:
sem_expr = sem_expr_exe ==>
translate_val_list = (\v t. []) ==>
!a b c d b1 b2 pi.
example_spectre_OoO_maximal_execution a b c d b1 b2 = SOME pi ==>
State_st_list_well_formed_ok (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) ==>
step_execution in_order_step (MAP step_list_to_step pi) /\
FST (HD pi) = example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2
Proof
rw [] >>
`translate_val_list_SORTED` by fs [translate_val_list_SORTED] >>
fs [example_spectre_OoO_maximal_execution,example_spectre_OoO_state_st_list] >>
`SORTED bound_name_instr_le (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)` by (fs [] >> EVAL_TAC) >>
`Completed_list_up_to sem_expr (State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) (PRE 13)`
by (fs [example_spectre_OoO_list,example_spectre_OoO_exe_init,example_spectre_OoO_store] >> EVAL_TAC >> rw [] >> EVAL_TAC) >>
`DROP (PRE 13) (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2) =
[i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]`
by (fs [example_spectre_OoO_list,example_spectre_OoO_exe_init] >> EVAL_TAC) >>
Q.ABBREV_TAC `i = i_assign 13 (e_val 1w) (o_internal (e_val 0w))` >>
Q.ABBREV_TAC `il = [i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]` >>
`~(Completed_list sem_expr (State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) i)`
by (fs [example_spectre_OoO_list,example_spectre_OoO_exe_init,example_spectre_OoO_store,Abbr `i`] >> EVAL_TAC) >>
`IO_bounded_execution translate_val_list sem_expr
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) 13 (SUC 15) = SOME pi` by fs [] >>
PROVE_TAC [IO_bounded_execution_in_order_step_sound]
QED
Theorem example_spectre_OoO_maximal_execution_correct[local]:
sem_expr = sem_expr_exe ==>
translate_val_list = (\v t. []) ==>
!a b c d b1 b2 pi.
example_spectre_OoO_maximal_execution a b c d b1 b2 = SOME pi ==>
pi <> [] ==>
maximal_execution_form in_order_step (state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2))
(MAP step_list_to_step pi)
Proof
rw [maximal_execution_form] >>
`State_st_list_well_formed_ok (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)`
by fs [State_st_list_well_formed_ok_example_spectre_OoO_state_st_list] >-
METIS_TAC [example_spectre_OoO_maximal_execution_sound] >-
(`FST (HD pi) = example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2`
by METIS_TAC [example_spectre_OoO_maximal_execution_sound] >>
fs [HD_MAP_step_list_to_step_and_state_list_to_state]) >>
`SND (SND (LAST (MAP step_list_to_step (THE (example_spectre_OoO_maximal_execution a b c d b1 b2))))) =
state_list_to_state (SND (SND (LAST (THE (example_spectre_OoO_maximal_execution a b c d b1 b2)))))`
by fs [LAST_MAP_step_list_to_step_and_state_list_to_state] >>
`all_instructions_completed_list sem_expr_exe
(SND (SND (LAST (THE (example_spectre_OoO_maximal_execution a b c d b1 b2)))))`
by fs [example_spectre_OoO_maximal_execution_last_state_all_completed] >>
`THE (example_spectre_OoO_maximal_execution a b c d b1 b2) = pi` by fs [] >> fs [] >>
`all_instructions_completed_list sem_expr (SND (SND (LAST pi)))` by fs [] >>
`?stl'.(SND (SND (LAST pi))) = stl'` by fs [] >>
`well_formed_state (state_list_to_state stl')`
by (fs [example_spectre_OoO_maximal_execution,example_spectre_OoO_state_st_list] >>
`DROP (PRE 13) (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2) =
[i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]`
by (fs [example_spectre_OoO_list] >> EVAL_TAC) >>
Q.ABBREV_TAC `i = i_assign 13 (e_val 1w) (o_internal (e_val 0w))` >>
Q.ABBREV_TAC `il = [i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]` >>
`~(Completed_list sem_expr (State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) i)`
by (fs [example_spectre_OoO_list,example_spectre_OoO_store,Abbr `i`] >> EVAL_TAC) >>
`IO_bounded_execution translate_val_list sem_expr
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] [12]) 13 (SUC 15) = SOME pi` by fs [] >>
PROVE_TAC [State_st_list_well_formed_ok_IO_bounded_execution_last_state]) >>
`SND (SND (step_list_to_step (LAST pi))) = SND (SND (LAST (MAP step_list_to_step pi)))` by METIS_TAC [LAST_MAP] >>
fs [all_instructions_completed_list_implies_termination_IO]
QED
Theorem example_spectre_OoO_maximal_trace_result[local]:
!a b c d b1 b2.
example_spectre_OoO_maximal_trace a b c d b1 b2 =
SOME [obs_dl b1; obs_ds b2; obs_il (d + 4w)]
Proof
rw [] >> Cases_on `b = 1w` >-
(rw [] >> EVAL_TAC >> Cases_on `b1 = b2` >> fs [] >> EVAL_TAC) >>
rw [example_spectre_OoO_maximal_trace,IO_bounded_trace,IO_bounded_trace_aux] >>
`IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 12 16 [] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,[obs_dl b1; obs_ds b2; obs_il (d + 4w)])`
by fs [example_spectre_OoO_IO_bounded_trace_acc_not_true] >>
fs [IO_bounded_trace_aux]
QED
Theorem example_spectre_OoO_maximal_execution_exists_result[local]:
!a b c d b1 b2.
?pi. example_spectre_OoO_maximal_execution a b c d b1 b2 = SOME pi /\ pi <> []
Proof
rw [] >>
`example_spectre_OoO_maximal_trace a b c d b1 b2 = SOME [obs_dl b1; obs_ds b2; obs_il (d + 4w)]`
by rw [example_spectre_OoO_maximal_trace_result] >>
fs [example_spectre_OoO_maximal_trace,example_spectre_OoO_maximal_execution] >>
`?pi. IO_bounded_execution (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 13 16 = SOME pi /\
[obs_dl b1; obs_ds b2; obs_il (d + 4w)] = trace obs_of_ll obs_visible pi`
by fs [IO_bounded_execution_for_trace] >>
Q.EXISTS_TAC `pi` >> fs [trace] >>
Cases_on `pi` >> fs []
QED
Theorem example_spectre_OoO_same_obs[local]:
!a a' b b' c c' d d' b1 b1' b2 b2'.
example_spectre_OoO_maximal_trace a b c d b1 b2 =
example_spectre_OoO_maximal_trace a' b' c' d' b1' b2' ==>
d = d' /\ b1 = b1' /\ b2 = b2'
Proof
rw [] >> fs [example_spectre_OoO_maximal_trace_result]
QED
Theorem noninterference_example_spectre_OoO_trace:
sem_expr = sem_expr_exe ==>
translate_val_list = (\v t. []) ==>
!a a' b b' c c' d d' b1 b1' b2 b2' s1 s2.
state_list_to_state (example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) = s1 ==>
state_list_to_state (example_spectre_OoO_state_st_list a' b' c' d' 1w 2w 3w b1' b2') = s2 ==>
trace_indist_IO s1 s2 ==>
d = d' /\ b1 = b1' /\ b2 = b2'
Proof
REWRITE_TAC [trace_indist_IO,trace_indist] >>
NTAC 2 STRIP_TAC >>
REPEAT GEN_TAC >> NTAC 3 STRIP_TAC >> fs [] >>
`?p1 p2. example_spectre_OoO_maximal_execution a b c d b1 b2 = SOME p1 /\
example_spectre_OoO_maximal_execution a' b' c' d' b1' b2' = SOME p2 /\
p1 <> [] /\ p2 <> []` by METIS_TAC [example_spectre_OoO_maximal_execution_exists_result] >>
`maximal_execution_form in_order_step (state_list_to_state
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2)) (MAP step_list_to_step p1) /\
maximal_execution_form in_order_step (state_list_to_state
(example_spectre_OoO_state_st_list a' b' c' d' 1w 2w 3w b1' b2')) (MAP step_list_to_step p2)`
by fs [example_spectre_OoO_maximal_execution_correct] >>
Q.ABBREV_TAC `stl1 = example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2` >>
Q.ABBREV_TAC `stl2 = example_spectre_OoO_state_st_list a' b' c' d' 1w 2w 3w b1' b2'` >>
`State_st_list_well_formed_ok stl1 /\ State_st_list_well_formed_ok stl2`
by fs [Abbr `stl1`,Abbr `stl2`,State_st_list_well_formed_ok_example_spectre_OoO_state_st_list] >>
`well_formed_state (state_list_to_state stl1) /\ well_formed_state (state_list_to_state stl2)`
by (Cases_on `stl1` >> Cases_on `stl2` >> fs [State_st_list_well_formed_ok]) >>
`well_formed_state s1 /\ well_formed_state s2` by METIS_TAC [] >>
Q.ABBREV_TAC `max_org_pi1 = MAP step_list_to_step p1` >>
Q.ABBREV_TAC `max_org_pi2 = MAP step_list_to_step p2` >>
`maximal_execution_form in_order_step s1 max_org_pi1 /\
maximal_execution_form in_order_step s2 max_org_pi2` by METIS_TAC [] >>
`maximal_execution_form in_order_step' s1 (MAP update_lbl max_org_pi1) /\
maximal_execution_form in_order_step' s2 (MAP update_lbl max_org_pi2)`
by fs [max_execution_format_step_and_step'] >>
Q.ABBREV_TAC `max_pi1 = MAP update_lbl max_org_pi1` >>
Q.ABBREV_TAC `max_pi2 = MAP update_lbl max_org_pi2` >>
`trace obs_of_l obs_visible max_pi1 = trace obs_of_l obs_visible max_pi2`
by REWRITE_TAC [UNDISCH_ALL (SPEC_ALL noninterference_implies_maximal_execution_form_in_order_step')] >>
`trace obs_of_l obs_visible max_org_pi1 = trace obs_of_l obs_visible max_org_pi2` by METIS_TAC [MAP_update_lbl_same_obs,Abbr `max_pi1`,Abbr `max_pi2`] >>
`example_spectre_OoO_maximal_trace a b c d b1 b2 <> NONE /\
example_spectre_OoO_maximal_trace a' b' c' d' b1' b2' <> NONE`
by fs [example_spectre_OoO_maximal_trace_result] >>
`?tr1 tr2. example_spectre_OoO_maximal_trace a b c d b1 b2 = SOME tr1 /\
example_spectre_OoO_maximal_trace a' b' c' d' b1' b2' = SOME tr2`
by (Cases_on `example_spectre_OoO_maximal_trace a b c d b1 b2` >>
Cases_on `example_spectre_OoO_maximal_trace a' b' c' d' b1' b2'` >> fs []) >>
`(trace obs_of_l obs_visible max_org_pi1 = tr1) /\ (trace obs_of_l obs_visible max_org_pi2 = tr2)`
by METIS_TAC [example_spectre_OoO_maximal_execution,example_spectre_OoO_maximal_trace,
IO_bounded_execution_and_trace,Abbr `max_org_pi1`,Abbr `max_org_pi2`] >>
`example_spectre_OoO_maximal_trace a b c d b1 b2 = example_spectre_OoO_maximal_trace a' b' c' d' b1' b2'` by fs [] >>
METIS_TAC [example_spectre_OoO_same_obs]
QED
(* Unused definitions and theorems of example SpectreOoO
Definition example_spectre_OoO_state_st_list_mid:
example_spectre_OoO_state_st_list_mid a b c d b1 b2 =
State_st_list
[i_assign 1 (e_val 1w) (o_internal (e_val b1));
i_assign 2 (e_val 1w) (o_internal (e_val a));
i_assign 3 (e_val 1w) (o_store res_MEM 1 2);
i_assign 4 (e_val 1w) (o_internal (e_val 3w));
i_assign 5 (e_val 1w) (o_internal (e_val b));
i_assign 6 (e_val 1w) (o_store res_REG 4 5);
i_assign 7 (e_val 1w) (o_internal (e_val 2w));
i_assign 8 (e_val 1w) (o_internal (e_val c));
i_assign 9 (e_val 1w) (o_store res_REG 7 8);
i_assign 10 (e_val 1w) (o_internal (e_val 0w));
i_assign 11 (e_val 1w) (o_internal (e_val d));
i_assign 12 (e_val 1w) (o_store res_PC 10 11);
i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]
(FEMPTY |+ (1,b1) |+ (2,a) |+ (3,a) |+ (4,3w) |+ (5,b) |+ (6,b) |+
(7,2w) |+ (8,c) |+ (9,c) |+ (10,0w) |+ (11,d) |+ (12,d) |+ (13,0w) |+
(14,1w) |+ (15,2w) |+ (16,3w) |+ (17,b1) |+ (18,b2) |+ (19,a) |+ (20,a) |+ (21,b)) [3] [12]
End
Theorem IO_bounded_execution_acc_complete_steps[local]:
translate_val_list_SORTED ==>
!l s cs fs l' s' cs' fs' pi n pos.
State_st_list_well_formed_initialized_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 ==>
(!t. MEM t (bound_names_program_list l) /\ t > pos ==> t NOTIN FDOM s) ==>
FST (HD pi) = State_st_list l s cs fs ==>
step_execution in_order_step (MAP step_list_to_step pi) ==>
SND (SND (LAST pi)) = State_st_list l' s' cs' fs' ==>
Completed_list_up_to sem_expr (State_st_list l' s' cs' fs') (pos + n) ==>
(!t. MEM t (bound_names_program_list l') /\ t > (pos+n) ==> t NOTIN FDOM s') ==>
IO_bounded_execution_acc translate_val_list sem_expr (State_st_list l s cs fs) pos n [] = SOME pi
Proof
cheat
QED
Theorem APPEND_HD_FST[local]:
!l1 l2 l.
l = l1 ++ l2 ==> l1 <> [] ==> HD l = HD l1
Proof
rw [] >> Cases_on `l1` >> fs []
QED
Theorem IO_bounded_execution_acc_subresults[local]:
translate_val_list_SORTED ==>
!l l' l'' s s' s'' cs cs' cs'' fs fs' fs'' stl'' pos m n pi pi' i il.
State_st_list_well_formed_initialized_ok (State_st_list l s cs fs) ==>
State_st_list_well_formed_ok (State_st_list l' s' cs' fs') ==>
(!t. MEM t (bound_names_program_list l) /\ t > pos ==> t NOTIN FDOM s) ==>
(!t. MEM t (bound_names_program_list l'') /\ t > (pos+m+n) ==> t NOTIN FDOM s'') ==>
SORTED bound_name_instr_le l ==>
SORTED bound_name_instr_le l' ==>
Completed_list_up_to sem_expr (State_st_list l s cs fs) pos ==>
DROP pos l = i::il ==>
~Completed_list sem_expr (State_st_list l s cs fs) i ==>
pi <> [] ==>
IO_bounded_execution_acc translate_val_list sem_expr (State_st_list l s cs fs) pos m [] = SOME pi ==>
SND (SND (LAST pi)) = State_st_list l' s' cs' fs' ==>
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+m) n pi = SOME pi' ==>
IO_bounded_execution_acc translate_val_list sem_expr (State_st_list l s cs fs) pos (m+n) [] = SOME pi'
Proof
rw [] >>
Cases_on `m` >-
fs [Once IO_bounded_execution_acc] >>
`State_st_list_well_formed_ok (State_st_list l s cs fs)` by fs [State_st_list_well_formed_ok,State_st_list_well_formed_initialized_ok] >>
`step_execution_list_IO_HD (State_st_list l s cs fs) pi pos (SUC n')` by fs [IO_bounded_execution_acc_IO_empty_sound] >>
fs [step_execution_list_IO_HD] >>
`Completed_list_up_to sem_expr (State_st_list l' s' cs' fs') (pos + SUC n')` by METIS_TAC [] >>
`?pi''. pi' = pi ++ pi'' /\ step_execution in_order_step (MAP step_list_to_step pi') /\
Completed_list_up_to sem_expr (SND (SND (LAST pi'))) (pos + SUC n' + n) /\
LENGTH pi'' <= 2 * n` by fs [IO_bounded_execution_acc_IO_execution_sound] >>
`HD pi' = HD pi` by fs [APPEND_HD_FST] >>
`FST (HD pi') = State_st_list l s cs fs` by fs [] >>
`pos + SUC n' + n = pos + (n + SUC n') /\ n + (pos + SUC n') = pos + (n + SUC n')` by rw [] >>
`Completed_list_up_to sem_expr (SND (SND (LAST pi'))) (pos + (n + SUC n'))` by METIS_TAC [] >>
`(!t. MEM t (bound_names_program_list l'') /\ t > (pos+(n + SUC n')) ==> t NOTIN FDOM s'')` by METIS_TAC [] >>
METIS_TAC [IO_bounded_execution_acc_complete_steps]
QED
Theorem IO_bounded_trace_acc_subresults[local]:
!f_tran f_sem l s s' cs cs' fs fs' stl'' pos m n tr tr'.
IO_bounded_trace_acc f_tran f_sem (State_st_list l s cs fs) pos m [] = SOME (State_st_list l s' cs' fs',tr) ==>
IO_bounded_trace_acc f_tran f_sem (State_st_list l s' cs' fs') (pos+m) n tr = SOME (stl'',tr') ==>
IO_bounded_trace_acc f_tran f_sem (State_st_list l s cs fs) pos (m+n) [] = SOME (stl'',tr')
Proof
cheat
QED
Theorem example_spectre_OoO_IO_bounded_trace_acc_not_true_part1[local]:
!a b c d b1 b2.
b <> 1w ==>
IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 11 10 [] =
SOME (example_spectre_OoO_state_st_list_mid a b c d b1 b2, [obs_il d; obs_dl b1])
Proof
rw [example_spectre_OoO_state_st_list_mid] >> EVAL_TAC
QED
Theorem example_spectre_OoO_IO_bounded_trace_acc_not_true_part2[local]:
!a b c d b1 b2.
b <> 1w ==>
IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list_mid a b c d b1 b2) 21 7 [obs_il d; obs_dl b1] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_il d; obs_dl b1; obs_ds b2; obs_il (d + 4w)])
Proof
rw [example_spectre_OoO_state_st_list_mid,example_spectre_OoO_state_st_list_final] >>
NTAC 3 (rw [Once IO_bounded_trace_acc] >> fs [FLOOKUP_DEF,sem_expr_exe,v_eq,val_false]) >>
rw [Once sem_instr_exe] >>
fs [str_act_list,addr_of_list,FIND_instr,bound_name_instr,str_may_list] >>
Q.ABBREV_TAC `l1 = [i_assign 1 (e_val 1w) (o_internal (e_val b1));
i_assign 2 (e_val 1w) (o_internal (e_val a));
i_assign 3 (e_val 1w) (o_store res_MEM 1 2);
i_assign 4 (e_val 1w) (o_internal (e_val 3w));
i_assign 5 (e_val 1w) (o_internal (e_val b));
i_assign 6 (e_val 1w) (o_store res_REG 4 5);
i_assign 7 (e_val 1w) (o_internal (e_val 2w));
i_assign 8 (e_val 1w) (o_internal (e_val c));
i_assign 9 (e_val 1w) (o_store res_REG 7 8);
i_assign 10 (e_val 1w) (o_internal (e_val 0w));
i_assign 11 (e_val 1w) (o_internal (e_val d));
i_assign 12 (e_val 1w) (o_store res_PC 10 11);
i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]` >>
Q.ABBREV_TAC `(s1:num |-> v) = (FEMPTY |+ (1,b1) |+ (2,a) |+ (3,a) |+ (4,3w) |+
(5,b) |+ (6,b) |+ (7,2w) |+ (8,c) |+ (9,c) |+
(10,0w) |+ (11,d) |+ (12,d) |+ (13,0w) |+ (14,1w) |+
(15,2w) |+ (16,3w) |+ (17,b1) |+ (18,b2) |+ (19,a) |+
(20,a) |+ (21,b))` >>
`str_may_list_find sem_expr_exe l1 s1 24 res_REG 15 = [i_assign 9 (e_val 1w) (o_store res_REG 7 8)]`
by (fs [Abbr `l1`,Abbr `s1`] >> EVAL_TAC >> fs []) >>
fs [str_act_list_find,str_act_list_cond,sem_expr_exe,Abbr `s1`] >>
NTAC 3 (rw [Once FLOOKUP_DEF]) >>
fs [bound_names_program_list] >>
rw [Once FLOOKUP_DEF] >>
fs [bound_name_instr,FLOOKUP_DEF,OoO_Exe_list_instr_not_stored_guard_true_sem_instr,
obs_visible,obs_of_ll] >>
fs [Abbr `l1`] >> EVAL_TAC >>
Cases_on `b1 = b2` >> fs [] >> EVAL_TAC
QED
Theorem example_spectre_OoO_IO_bounded_trace_acc_not_true[local]:
!a b c d b1 b2.
b <> 1w ==>
IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 11 17 [] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_il d; obs_dl b1; obs_ds b2; obs_il (d + 4w)])
Proof
rw [] >>
`IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list a b c d 1w 2w 3w b1 b2) 11 10 [] =
SOME (example_spectre_OoO_state_st_list_mid a b c d b1 b2, [obs_il d; obs_dl b1])`
by fs [example_spectre_OoO_IO_bounded_trace_acc_not_true_part1] >>
`IO_bounded_trace_acc (\v t. []) sem_expr_exe
(example_spectre_OoO_state_st_list_mid a b c d b1 b2) (11+10) 7 [obs_il d; obs_dl b1] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_il d; obs_dl b1; obs_ds b2; obs_il (d + 4w)])`
by fs [example_spectre_OoO_IO_bounded_trace_acc_not_true_part2] >>
fs [example_spectre_OoO_state_st_list,example_spectre_OoO_state_st_list_mid] >>
`example_spectre_OoO_list a b c d 1w 2w 3w b1 b2 =
[i_assign 1 (e_val 1w) (o_internal (e_val b1));
i_assign 2 (e_val 1w) (o_internal (e_val a));
i_assign 3 (e_val 1w) (o_store res_MEM 1 2);
i_assign 4 (e_val 1w) (o_internal (e_val 3w));
i_assign 5 (e_val 1w) (o_internal (e_val b));
i_assign 6 (e_val 1w) (o_store res_REG 4 5);
i_assign 7 (e_val 1w) (o_internal (e_val 2w));
i_assign 8 (e_val 1w) (o_internal (e_val c));
i_assign 9 (e_val 1w) (o_store res_REG 7 8);
i_assign 10 (e_val 1w) (o_internal (e_val 0w));
i_assign 11 (e_val 1w) (o_internal (e_val d));
i_assign 12 (e_val 1w) (o_store res_PC 10 11);
i_assign 13 (e_val 1w) (o_internal (e_val 0w));
i_assign 14 (e_val 1w) (o_internal (e_val 1w));
i_assign 15 (e_val 1w) (o_internal (e_val 2w));
i_assign 16 (e_val 1w) (o_internal (e_val 3w));
i_assign 17 (e_val 1w) (o_internal (e_val b1));
i_assign 18 (e_val 1w) (o_internal (e_val b2));
i_assign 19 (e_val 1w) (o_load res_MEM 17);
i_assign 20 (e_val 1w) (o_store res_REG 14 19);
i_assign 21 (e_val 1w) (o_load res_REG 16);
i_assign 22 (e_eq (e_name 21) (e_val 1w)) (o_load res_REG 14);
i_assign 23 (e_eq (e_name 21) (e_val 1w)) (o_store res_REG 15 22);
i_assign 24 (e_val 1w) (o_load res_REG 15);
i_assign 25 (e_val 1w) (o_store res_MEM 18 24);
i_assign 26 (e_val 1w) (o_load res_PC 13);
i_assign 27 (e_val 1w) (o_internal (e_add (e_name 26) (e_val 4w)));
i_assign 28 (e_val 1w) (o_store res_PC 13 27)]` by fs [example_spectre_OoO_list] >>
`IO_bounded_trace_acc (\v t. []) sem_expr_exe
(State_st_list (example_spectre_OoO_list a b c d 1w 2w 3w b1 b2)
(example_spectre_OoO_store a b c d 2w 3w b1) [3] []) 11 (10+7) [] =
SOME (example_spectre_OoO_state_st_list_final a b c d b1 b2,
[obs_il d; obs_dl b1; obs_ds b2; obs_il (d + 4w)])` by fs [IO_bounded_trace_acc_subresults] >>
FULL_SIMP_TAC arith_ss []
QED
*)
val _ = export_theory ();