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p4_deterScript.sml
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p4_deterScript.sml
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open HolKernel boolLib liteLib simpLib Parse bossLib;
open arithmeticTheory stringTheory containerTheory pred_setTheory
listTheory finite_mapTheory;
open p4Lib;
open blastLib bitstringLib;
open p4Theory p4_auxTheory ottTheory;
open p4_exec_semTheory;
open bitstringTheory;
open wordsTheory;
open optionTheory;
open sumTheory;
open stringTheory;
open ottTheory;
open pairTheory;
open rich_listTheory;
val _ = new_theory "p4_deter";
(*Tactics*)
fun OPEN_EXP_RED_TAC exp_term =
(Q.PAT_X_ASSUM `e_red j scope scopest ^exp_term exp2 fr` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once e_red_cases] thm)))
fun OPEN_STMT_RED_TAC stm_term =
(Q.PAT_X_ASSUM `stmt_red ct (ab, gsl,[(fun,[^stm_term],gam)],st) stat` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once stmt_red_cases] thm)))
(*********** SAME SATE DEF *****************)
val same_state_def = Define `
same_state (st:'a state) st' =
(st = st')
`;
(******** STMT DETERMINISIM DEF ************)
val det_stmt_def = Define `
det_stmt (stm) (ty:'a itself) = ! (c: 'a ctx) ascope sl s' s'' g_scope_list f status .
(stmt_red c ( ascope, g_scope_list, ([(f,[stm],sl)]), status) (s'))
/\
(stmt_red c ( ascope, g_scope_list, ([(f,[stm],sl)]), status) (s''))
==>
( same_state s' s'')
`;
(********* SAME FRAME and EXP DEF *************)
val same_frame_exp_def = Define `
same_frame_exp (frame:frame_list) frame' (e:e) e' =
((frame = frame') /\ (e = e'))
`;
(********** EXPRESSION DETER DEF *************)
(* Based on the e_induction definition, this is P0
wich shows that each exression reduction is determ.*)
val det_exp_def = Define `
det_exp e (ty:'a itself) = ! (c: 'a ctx) scope scopest e' e'' frame frame'.
(e_red (c: 'a ctx) scope scopest e e' frame )
/\
(e_red (c: 'a ctx) scope scopest e e'' frame' )
==>
(same_frame_exp frame frame' e' e'')
`;
(********** EXPRESSION List P3 ***************)
(* Supposed to be a property that states the following:
every expression element of the list is deterministic
for clarification, check the definition of e_induction
in P4 theory, it is basically P3 *)
val det_exp_list_def = Define `
det_exp_list (l : e list) (ty:'a itself) = !(x : e). MEM x l ==> det_exp (x) (ty:'a itself)
`;
(********** STR EXPRESSION List P1 ***************)
(* Supposed to be a property that states the following:
every expression element of the list of tuples of expression
and strings is deterministic for clarification, check
the definition of e_induction in P4 theory, it is P1 *)
val det_strexp_list_def = Define `
det_strexp_list (l : (string#e) list) (ty:'a itself)
= ! (e:e) . MEM e (SND (UNZIP l)) ==> det_exp(e) (ty:'a itself)
`;
(********** STR EXPRESSION tuple P2 ***************)
(* Supposed to be a property that states the following:
A tuple of (expression, string) in the struct or header
is deterministic. Based on e_induction this is P2 *)
val det_strexp_tuple_def = Define `
det_strexp_tup (tup : (string#e)) (ty:'a itself)
= det_exp (SND(tup)) (ty:'a itself)
`;
(******** stmt list DETERMINISIM DEF *******)
val det_stmtl_def = Define `
det_stmtl stmtl (ty:'a itself) = ! (c:'a ctx) (s':'a state) s'' (g_scope_list:scope list) status (ss:scope list) (f:funn) ascope.
(stmt_red c ( ascope, g_scope_list , ([(f,stmtl,ss)]) , status) (s'))
/\
(stmt_red c ( ascope, g_scope_list , ([(f,stmtl,ss)]) , status) (s''))
==>
( same_state s' s'')
`;
(******** Single frame DETERMINISIM DEF *******)
val det_frame_def = Define `
det_frame frame (ty:'a itself) = ! (c:'a ctx) (s':'a state) s'' (g_scope_list:scope list) status ascope.
(frames_red c ( ascope, g_scope_list , [frame] , status) (s'))
/\
(frames_red c ( ascope, g_scope_list , [frame] , status) (s''))
==>
( same_state s' s'')
`;
(******** Frame list DETERMINISIM DEF *******)
val det_framel_def = Define `
det_framel framel (ty:'a itself) = ! (c:'a ctx) (s':'a state) s'' (g_scope_list:scope list) status ascope.
(frames_red c ( ascope, g_scope_list , framel, status) (s'))
/\
(frames_red c ( ascope, g_scope_list , framel, status) (s''))
==>
( same_state s' s'')
`;
(*lemmas and thms*)
(********* lookup_funn_sig_body eq ************)
val fun_body_map_EQ =
prove(`` ! funn func_map b_func_map ext_map stmt (sdl') (sdl: (string # d) list).
lookup_funn_sig_body funn func_map b_func_map ext_map = SOME (stmt, sdl ) /\
lookup_funn_sig funn func_map b_func_map ext_map = SOME (sdl') ==>
(sdl = sdl')
``,
REPEAT STRIP_TAC >>
rfs [lookup_funn_sig_def]
);
(*determinism proof of expressions*)
Theorem P4_exp_det:
! (ty:'a itself) .
(! e . det_exp e ty) /\
(! (l1: e list). det_exp_list l1 ty) /\
(! (l2: (string#e) list) . det_strexp_list l2 ty) /\
(! tup. det_strexp_tup tup ty)
Proof
STRIP_TAC >>
Induct >|[
(*****************************)
(* e_v case *)
(*****************************)
RW_TAC (srw_ss()) [same_state_def, det_exp_def, Once e_red_cases]
,
(*****************************)
(* e_var case *)
(*****************************)
RW_TAC (srw_ss()) [det_exp_def] >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def, Once e_red_cases, same_frame_exp_def] >>
METIS_TAC [option_case_def]
,
(*****************************)
(* e_list case *)
(*****************************)
RW_TAC (srw_ss()) [det_exp_def] >>
FULL_SIMP_TAC (srw_ss()) [Once e_red_cases, same_frame_exp_def]
,
(*****************************)
(* e_acc case *)
(*****************************)
NTAC 2 (SIMP_TAC (srw_ss()) [det_exp_def] >>
REPEAT STRIP_TAC >>
OPEN_EXP_RED_TAC ``e_acc e e'`` >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
RW_TAC (srw_ss()) []) >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def,lemma_v_red_forall, same_frame_exp_def] >>
RES_TAC >>
FULL_SIMP_TAC (srw_ss()) [same_frame_exp_def]
,
(*****************************)
(* UNOP case *)
(*****************************)
SIMP_TAC (srw_ss()) [det_exp_def] >>
REPEAT STRIP_TAC >>
OPEN_EXP_RED_TAC ``(e_unop u e)`` >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
RW_TAC (srw_ss()) [] >>
OPEN_EXP_RED_TAC ``(e_unop u e)`` >>
RW_TAC (srw_ss()) [] >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def] >>
RES_TAC >>
FULL_SIMP_TAC (srw_ss()) [same_frame_exp_def] >>
REPEAT (IMP_RES_TAC lemma_v_red_forall >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def] )
,
(*****************************)
(* cast case *)
(*****************************)
SIMP_TAC (srw_ss()) [det_exp_def] >>
REPEAT STRIP_TAC >>
OPEN_EXP_RED_TAC ``(e_cast c e)`` >>
OPEN_EXP_RED_TAC ``(e_cast c e)`` >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
RW_TAC (srw_ss()) [] >>
FULL_SIMP_TAC (srw_ss()) [same_frame_exp_def] >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def] >>
RES_TAC >>
gvs[same_frame_exp_def]>>
REPEAT (IMP_RES_TAC lemma_v_red_forall >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def] )
,
(*****************************)
(* binop case *)
(*****************************)
(
SIMP_TAC (srw_ss()) [det_exp_def] >>
REPEAT STRIP_TAC >>
OPEN_EXP_RED_TAC ``(e_binop e b e')`` >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
RW_TAC (srw_ss()) []
) >- ( (*e_binop e b e'*)
OPEN_EXP_RED_TAC ``(e_binop e b e')`` >>
RW_TAC (srw_ss()) [] >>
IMP_RES_TAC lemma_v_red_forall >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def,lemma_v_red_forall, same_state_def] >>
RES_TAC >>
FULL_SIMP_TAC (srw_ss()) [Once same_frame_exp_def]
) >- (
(*e_binop (e_v v) b e'*)
OPEN_EXP_RED_TAC ``(e_binop (e_v v) b e')`` >>
RW_TAC (srw_ss()) []>>
IMP_RES_TAC lemma_v_red_forall >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def] >>
RES_TAC >>
FULL_SIMP_TAC (srw_ss()) [Once same_frame_exp_def] >>
FULL_SIMP_TAC (srw_ss()) [is_short_circuitable_def]
) >>
(
(*e_mul case and after, use those as much as needed.*)
OPEN_EXP_RED_TAC ``(e_binop a b c)`` >>
RW_TAC (srw_ss()) [] >>
IMP_RES_TAC lemma_v_red_forall >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def,lemma_v_red_forall, same_frame_exp_def, option_case_def] >>
FULL_SIMP_TAC (srw_ss()) [is_short_circuitable_def] )
,
(*****************************)
(* e_concat *)
(*****************************)
NTAC 2 (SIMP_TAC (srw_ss()) [det_exp_def] >>
REPEAT STRIP_TAC >>
OPEN_EXP_RED_TAC ``e_concat e e'`` >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
RW_TAC (srw_ss()) []) >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def,lemma_v_red_forall, same_frame_exp_def] >>
RES_TAC >>
FULL_SIMP_TAC (srw_ss()) [same_frame_exp_def]
,
(*****************************)
(* e_slice *)
(*****************************)
NTAC 2 (SIMP_TAC (srw_ss()) [det_exp_def] >>
REPEAT STRIP_TAC >>
OPEN_EXP_RED_TAC ``e_slice e e' e''`` >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
RW_TAC (srw_ss()) []) >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def,lemma_v_red_forall, same_frame_exp_def] >>
RES_TAC >>
FULL_SIMP_TAC (srw_ss()) [same_frame_exp_def]
,
(*****************************)
(* e_func_call *)
(*****************************)
SIMP_TAC (srw_ss()) [det_exp_def] >>
REPEAT STRIP_TAC >>
OPEN_EXP_RED_TAC ``(e_call s l)`` >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
RW_TAC (srw_ss()) [] >| [
(*first subgoal*)
OPEN_EXP_RED_TAC ``(e_call s l)`` >>
RW_TAC (srw_ss()) [] >| [
` (stmt',MAP (λ(e_,x_,d_). (x_,d_)) e_x_d_list') =
(stmt,MAP (λ(e_,x_,d_). (x_,d_)) e_x_d_list)
` by METIS_TAC [SOME_EL,SOME_11] >>
` (MAP (λ(e_,x_,d_). (x_,d_)) e_x_d_list') =
(MAP (λ(e_,x_,d_). (x_,d_)) e_x_d_list)
` by METIS_TAC [CLOSED_PAIR_EQ] >>
IMP_RES_TAC lemma_MAP3>>
REV_FULL_SIMP_TAC (list_ss) [rich_listTheory.MAP_FST_funs, same_frame_exp_def, option_case_def] >>
REV_FULL_SIMP_TAC (std_ss++optionSimps.OPTION_ss) [option_case_def] >>
` SOME scope' = SOME scope''
` by METIS_TAC [SOME_EL,SOME_11] >>
REV_FULL_SIMP_TAC (std_ss++optionSimps.OPTION_ss) [option_case_def]
,
REV_FULL_SIMP_TAC (srw_ss()) [same_frame_exp_def] >>
ASSUME_TAC fun_body_map_EQ >>
Q.PAT_X_ASSUM `∀funn func_map ext_map stmt sdl' sdl. _`
( STRIP_ASSUME_TAC o (Q.SPECL [
`f`,
`func_map`,
`b_func_map`,
`ext_map` ,
`stmt` ,
`MAP (λ(e_,e'_,x_,d_). (x_,d_)) (e_e'_x_d_list:(e#e#string#d)list)`,
`MAP (λ(e_,x_,d_). (x_,d_)) (e_x_d_list:(e#string#d)list)` ])) >>
rfs [] >>
IMP_RES_TAC lemma_MAP1>>
RES_TAC >>
IMP_RES_TAC lemma_args >>
METIS_TAC[]
]
,
(*second subgoal*)
OPEN_EXP_RED_TAC ``(e_call s l)`` >>
RW_TAC (srw_ss()) [] >|[
REV_FULL_SIMP_TAC (srw_ss()) [same_frame_exp_def] >>
ASSUME_TAC fun_body_map_EQ >>
Q.PAT_X_ASSUM `∀funn func_map ext_map stmt sdl' sdl. _`
( STRIP_ASSUME_TAC o (Q.SPECL [
`f`,
`func_map`,
`b_func_map`,
`ext_map` ,
`stmt` ,
`MAP (λ(e_,e'_,x_,d_). (x_,d_)) (e_e'_x_d_list:(e#e#string#d)list)`,
`MAP (λ(e_,x_,d_). (x_,d_)) (e_x_d_list:(e#string#d)list)` ])) >>
rfs [] >>
IMP_RES_TAC lemma_MAP1>>
RES_TAC >>
IMP_RES_TAC lemma_args >>
METIS_TAC[]
,
(*start the forth sub poof here*)
`SOME (MAP (λ(e_,e'_,x_,d_). (x_,d_)) e_e'_x_d_list) =
SOME (MAP (λ(e_,e'_,x_,d_). (x_,d_)) e_e'_x_d_list') ` by METIS_TAC [SOME_EL,SOME_11] >> rw[] >>
(*REV_FULL_SIMP_TAC (srw_ss()) [same_frame_exp_def] >> *)
(**first show that the d is the same in both lists, thus the i = i'*)
REV_FULL_SIMP_TAC (srw_ss()) [] >>
IMP_RES_TAC lemma_MAP2 >>
`i = i'` by METIS_TAC [ option_case_def]>> rw[] >> rfs[] >>
(*Now try to show that the EL i l is deterministic*)
REV_FULL_SIMP_TAC (srw_ss()) [det_exp_list_def] >>
PAT_ASSUM `` ∀x._``
( STRIP_ASSUME_TAC o (Q.SPECL [`(EL i (MAP (λ(e_,e'_,x_,d_). e_) (e_e'_x_d_list:(e#e#string#d)list)))` ])) >>
rw[]>>
IMP_RES_TAC unred_arg_index_in_range >>
IMP_RES_TAC EL_MEM >> rfs[]>>
FULL_SIMP_TAC list_ss [det_exp_def] >>
RES_TAC >> rw[] >>
FULL_SIMP_TAC list_ss [same_frame_exp_def]>> rfs[] >>
rw[] >> rw[]
] ]
,
(*****************************)
(* e_select *)
(*****************************)
SIMP_TAC (srw_ss()) [det_exp_def] >>
REPEAT STRIP_TAC >>
OPEN_EXP_RED_TAC ``(e_select e l s)`` >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
(*first + second + beginning of third subgoal*)
(OPEN_EXP_RED_TAC ``(e_select e l s)`` >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
FULL_SIMP_TAC (srw_ss()) [same_frame_exp_def] >>
IMP_RES_TAC lemma_v_red_forall) >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def]>>
RES_TAC>>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def, Once same_frame_exp_def ]
,
(*****************************)
(* e_struct *)
(*****************************)
NTAC 2 (SIMP_TAC (srw_ss()) [det_exp_def] >>
REPEAT STRIP_TAC >>
OPEN_EXP_RED_TAC ``(e_struct l)`` >>
REV_FULL_SIMP_TAC (srw_ss()) []) >>
fs[] >> rw[] >| [
(*first subgoal *)
IMP_RES_TAC lemma_MAP5 >>
fs[SOME_EL,SOME_11] >>
rw[] >>
REV_FULL_SIMP_TAC (srw_ss()) [det_strexp_list_def] >>
PAT_ASSUM `` ∀e._``
( STRIP_ASSUME_TAC o (Q.SPECL [`EL i (MAP (λ(f_,e_,e'_). (e_:e)) (f_e_e'_list':(string # e # e) list))`])) >>
IMP_RES_TAC ured_mem_length >>
IMP_RES_TAC EL_MEM >>
IMP_RES_TAC mem_el_map2 >>
FULL_SIMP_TAC list_ss [] >>
IMP_RES_TAC det_exp_def >>
fs[same_frame_exp_def] >>
` MAP (λ(f_,e_,e'_). e'_) f_e_e'_list =
MAP (λ(f_,e_,e'_). e'_) f_e_e'_list'
` by METIS_TAC [SOME_EL,SOME_11] >>
fs[ lemma_MAP6]
,
(*second subgoal *)
IMP_RES_TAC unred_mem_not_const >>
IMP_RES_TAC lemma_MAP7 >>
fs[]
,
(*third subgoal *)
IMP_RES_TAC unred_mem_not_const >>
IMP_RES_TAC lemma_MAP7 >>
fs[]
,
(*fourth subgoal *)
IMP_RES_TAC lemma_MAP5 >>
rw[] >>
FULL_SIMP_TAC list_ss [] >>
` MAP (λ(f_,e_,v_). v_) f_e_v_list =
MAP (λ(f_,e_,v_). v_) f_e_v_list'
` by METIS_TAC [SOME_EL,SOME_11] >>
IMP_RES_TAC lemma_MAP6 >>
fs[same_frame_exp_def]
]
,
(*****************************)
(* e_header *)
(*****************************)
NTAC 2 (SIMP_TAC (srw_ss()) [det_exp_def] >>
REPEAT STRIP_TAC >>
OPEN_EXP_RED_TAC ``(e_header b l)`` >>
REV_FULL_SIMP_TAC (srw_ss()) []) >>
fs[] >> rw[] >| [
(*first subgoal *)
IMP_RES_TAC lemma_MAP5 >>
fs[SOME_EL,SOME_11] >>
rw[] >>
REV_FULL_SIMP_TAC (srw_ss()) [det_strexp_list_def] >>
PAT_ASSUM `` ∀e._``
( STRIP_ASSUME_TAC o (Q.SPECL [`EL i (MAP (λ(f_,e_,e'_). (e_:e)) (f_e_e'_list':(string # e # e) list))`])) >>
IMP_RES_TAC ured_mem_length >>
IMP_RES_TAC EL_MEM >>
IMP_RES_TAC mem_el_map2 >>
FULL_SIMP_TAC list_ss [] >>
IMP_RES_TAC det_exp_def >>
fs[same_frame_exp_def] >>
` MAP (λ(f_,e_,e'_). e'_) f_e_e'_list =
MAP (λ(f_,e_,e'_). e'_) f_e_e'_list'
` by METIS_TAC [SOME_EL,SOME_11] >>
fs[ lemma_MAP6]
,
IMP_RES_TAC unred_mem_not_const >>
IMP_RES_TAC lemma_MAP7 >>
fs[]
,
IMP_RES_TAC unred_mem_not_const >>
IMP_RES_TAC lemma_MAP7 >>
fs[]
,
IMP_RES_TAC lemma_MAP5 >>
rw[] >>
FULL_SIMP_TAC list_ss [] >>
` MAP (λ(f_,e_,v_). v_) f_e_v_list =
MAP (λ(f_,e_,v_). v_) f_e_v_list'
` by METIS_TAC [SOME_EL,SOME_11] >>
IMP_RES_TAC lemma_MAP6 >>
fs[same_frame_exp_def]
]
,
(*****************************)
(* strexcp [] *)
(*****************************)
FULL_SIMP_TAC list_ss [det_strexp_list_def]
,
(*****************************)
(* strexcp l *)
(*****************************)
Cases_on `tup` >>
FULL_SIMP_TAC list_ss [det_strexp_tuple_def ] >>
FULL_SIMP_TAC list_ss [det_strexp_list_def] >>
FULL_SIMP_TAC list_ss [det_exp_def ] >>
NTAC 2 STRIP_TAC >|[
(*first case*)
REPEAT STRIP_TAC >>
rw [] >>
PAT_ASSUM `` ∀c scope scopest e' e'' frame frame'.
e_red c scope scopest e e' frame ∧
e_red c scope scopest e e'' frame' ⇒
same_frame_exp frame frame' e' e''``
( STRIP_ASSUME_TAC o (Q.SPECL [`c`, `scope`, `scopest`, `e'`, `e''`, `frame`, `frame'`])) >>
FULL_SIMP_TAC list_ss [same_frame_exp_def]
,
(*second case*)
REPEAT STRIP_TAC >>
rw [] >>
PAT_ASSUM `` ∀e. MEM e (SND (UNZIP l2)) ⇒
∀c scope scopest e' e'' frame frame'.
e_red c scope scopest e e' frame ∧
e_red c scope scopest e e'' frame' ⇒
same_frame_exp frame frame' e' e''``
( STRIP_ASSUME_TAC o (Q.SPECL [`e`])) >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
PAT_ASSUM `` ∀c scope scopest e' e'' frame frame'.
e_red c scope scopest e e' frame ∧
e_red c scope scopest e e'' frame' ⇒
same_frame_exp frame frame' e' e''``
( STRIP_ASSUME_TAC o (Q.SPECL [`c`, `scope`, `scopest`, `e'`, `e''`, `frame`, `frame'`])) >>
FULL_SIMP_TAC list_ss [same_frame_exp_def]
]
,
(*****************************)
(* str_exp_tup *)
(*****************************)
FULL_SIMP_TAC list_ss [det_strexp_tuple_def ]
,
(*****************************)
(* P2 [] *)
(*****************************)
FULL_SIMP_TAC list_ss [det_exp_list_def]
,
(*****************************)
(* P2 l *)
(*****************************)
FULL_SIMP_TAC list_ss [det_exp_list_def] >>
REPEAT STRIP_TAC >>
rw []
]
QED
(**************************************************)
(**************************************************)
(*determinism proof single frame stmt *)
(**************************************************)
(**************************************************)
Theorem P4_stmt_det:
!stmt ty. det_stmt stmt ty
Proof
Induct >|[
(*****************************)
(* stmt_empty *)
(*****************************)
RW_TAC (srw_ss()) [same_state_def, det_stmt_def, Once e_red_cases] >>
OPEN_STMT_RED_TAC ``stmt_empty`` >>
fs []
,
(*****************************)
(* stmt_ass *)
(*****************************)
SIMP_TAC (srw_ss()) [det_stmt_def] >>
REPEAT STRIP_TAC >>
NTAC 2 (OPEN_STMT_RED_TAC ``(stmt_ass l e)``) >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
(*first + second + third subgoal*)
RW_TAC (srw_ss()) [assign_def, same_state_def]>>
IMP_RES_TAC lemma_v_red_forall >>
TRY (`SOME scopes_list' = SOME scopes_list''` by METIS_TAC [CLOSED_PAIR_EQ] >>
fs []) >>
rw[] >> rfs[] >>
(*fourth subgoal*)
rfs[] >>
Cases_on `assign (sl ⧺ g_scope_list) v l` >>
fs[] >>
Cases_on `separate x` >>
rw[] >>
ASSUME_TAC P4_exp_det >>
fs [det_exp_def] >>
RES_TAC >>
fs [same_frame_exp_def]
,
(*****************************)
(* stmt_cond *)
(*****************************)
SIMP_TAC (srw_ss()) [det_stmt_def] >>
REPEAT STRIP_TAC >>
NTAC 2 (OPEN_STMT_RED_TAC ``stmt_cond e stm stm'`` >>
REV_FULL_SIMP_TAC (srw_ss()) []) >>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def]>>
IMP_RES_TAC lemma_v_red_forall>>
ASSUME_TAC P4_exp_det >>
fs [det_exp_def] >>
RES_TAC >>
fs [same_frame_exp_def]
,
(*****************************)
(* stmt_block *)
(*****************************)
NTAC 2 (SIMP_TAC (srw_ss()) [det_stmt_def] >>
REPEAT STRIP_TAC >>
OPEN_STMT_RED_TAC ``stmt_block l stm`` >>
REV_FULL_SIMP_TAC (srw_ss()) []) >>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def]
,
(*****************************)
(* stmt_ret *)
(*****************************)
(NTAC 2 ( SIMP_TAC (srw_ss()) [det_stmt_def] >>
REPEAT STRIP_TAC >>
OPEN_STMT_RED_TAC ``(stmt_ret e)`` >>
REV_FULL_SIMP_TAC (srw_ss()) [] ) >>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def]) >>
FULL_SIMP_TAC (srw_ss()) [det_exp_def] >>
RES_TAC >>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def] >>
IMP_RES_TAC lemma_v_red_forall>>
ASSUME_TAC P4_exp_det >>
fs [det_exp_def] >>
RES_TAC >>
fs [same_frame_exp_def]
,
(*****************************)
(* stmt_seq *)
(*****************************)
(NTAC 2 (SIMP_TAC (srw_ss()) [det_stmt_def] >>
REPEAT STRIP_TAC >>
OPEN_STMT_RED_TAC ``(stmt_seq stm stm')`` >>
REV_FULL_SIMP_TAC (srw_ss()) []) >>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def]) >>
FULL_SIMP_TAC (srw_ss()) [det_stmt_def] >>
RES_TAC >>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def, Once same_frame_exp_def ] >>
OPEN_STMT_RED_TAC ``stmt_empty`` >>
REV_FULL_SIMP_TAC (srw_ss()) []
,
(*****************************)
(* stmt_verify *)
(*****************************)
(*(NTAC 2 (SIMP_TAC (srw_ss()) [det_stmt_def] >>
REPEAT STRIP_TAC >>
OPEN_STMT_RED_TAC ``(stmt_verify e e')`` >>
REV_FULL_SIMP_TAC (srw_ss()) []) >>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def]) >>
IMP_RES_TAC lemma_v_red_forall>>
FULL_SIMP_TAC (srw_ss()) [det_exp_def,lemma_v_red_forall] >>
RES_TAC>>
FULL_SIMP_TAC (srw_ss()) [Once same_frame_exp_def]>>
ASSUME_TAC P4_exp_det >>
fs [det_exp_def] >>
RES_TAC >>
fs [same_frame_exp_def]
,*)
(*****************************)
(* stmt_trans *)
(*****************************)
(NTAC 2 (SIMP_TAC (srw_ss()) [det_stmt_def] >>
REPEAT STRIP_TAC >>
OPEN_STMT_RED_TAC ``(stmt_trans e)`` >>
REV_FULL_SIMP_TAC (srw_ss()) []) >>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def]) >>
IMP_RES_TAC lemma_v_red_forall>>
FULL_SIMP_TAC (srw_ss()) [det_exp_def,lemma_v_red_forall] >>
RES_TAC>>
FULL_SIMP_TAC (srw_ss()) [Once same_frame_exp_def] >>
ASSUME_TAC P4_exp_det >>
fs [det_exp_def] >>
RES_TAC >>
fs [same_frame_exp_def]
,
(*****************************)
(* stmt_app *)
(*****************************)
SIMP_TAC (srw_ss()) [det_stmt_def] >>
REPEAT STRIP_TAC >>
NTAC 2 (OPEN_STMT_RED_TAC ``stmt_app s l``) >>
REV_FULL_SIMP_TAC (srw_ss()) [] >>
rw [] >> fs [] >>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def] >>
ASSUME_TAC lconst_const_imp >>
RES_TAC >>
rfs [] >>
`SOME i' =
SOME i
`
by METIS_TAC[] >> rfs [] >> rw[] >> RES_TAC >> rw[] >>
ASSUME_TAC P4_exp_det >>
fs [det_exp_def] >>
RES_TAC >>
fs [same_frame_exp_def]
,
(*****************************)
(* stmt_extern *)
(*****************************)
(NTAC 2 (SIMP_TAC (srw_ss()) [det_stmt_def] >>
REPEAT STRIP_TAC >>
OPEN_STMT_RED_TAC ``(stmt_ext)`` >>
REV_FULL_SIMP_TAC (srw_ss()) []) >>
FULL_SIMP_TAC (srw_ss()) [Once same_state_def] ) >>
rw[] >>
Cases_on `lookup_ext_fun f ext_map` >>
rw[] >>
Cases_on `ext_fun (ascope,g_scope_list,sl)`>>
rw[]
]
QED
(**************************************************)
(**************************************************)
(*determinism proof statement list*)
(**************************************************)
(**************************************************)
Theorem P4_stmtl_det:
! stmtl ty. det_stmtl stmtl ty
Proof
Cases_on `stmtl` >| [
FULL_SIMP_TAC (srw_ss()) [det_stmtl_def] >>
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases]
,
FULL_SIMP_TAC (srw_ss()) [det_stmtl_def] >>
REPEAT STRIP_TAC >>
Cases_on `t` >| [
ASSUME_TAC P4_stmt_det >>
fs [det_stmt_def, same_state_def] >>
RES_TAC >>
fs []
,
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases] >> fs [same_state_def] >>
rw[] >>
ASSUME_TAC P4_stmt_det >>
rfs [det_stmt_def, same_state_def] >>
(PAT_ASSUM `` ∀stmt._``
( STRIP_ASSUME_TAC o (Q.SPECL [`h`,
`(apply_table_f,ext_map,func_map,b_func_map,pars_map,tbl_map)`,
`ascope`,
`ss`,
`(ascope'³',g_scope_list'³',
frame_list'' ⧺ [(f,stmt_stack'',scope_list'')],status'³')`,
`(ascope'',g_scope_list'',
frame_list' ⧺ [(f,stmt_stack',scope_list')],status'')`,
`g_scope_list`,
`f`,
`status`
])) >>
RES_TAC >>
fs [])
]]
QED
val ret_lemma1a =
prove(``
! e v c g_scope_list funn scopes_stack ctrl stmt' ascope.
stmt_red c
(ascope, g_scope_list,[(funn,[stmt_ret e],scopes_stack)],
status_running)
(ascope, g_scope_list,[(funn,[stmt'],scopes_stack)],status_returnv v) ==> (e = e_v v) ``,
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases]
);
val ret_lemma1b =
prove(``
! e v c g_scope_list funn scopes_stack scopes_stack' stmt' ascope.
stmt_red c
(ascope, g_scope_list,[(funn,[stmt_ret e],scopes_stack)],
status_running)
(ascope, g_scope_list,[(funn,[stmt'],scopes_stack')],status_returnv v) ==> (e = e_v v /\ scopes_stack = scopes_stack') ``,
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases]
);
val ret_lemma1c =
prove(``
! e v c g_scope_list g_scope_list' funn scopes_stack scopes_stack' stmt' ascope.
stmt_red c
(ascope, g_scope_list,[(funn,[stmt_ret e],scopes_stack)],
status_running)
(ascope, g_scope_list',[(funn,[stmt'],scopes_stack')],status_returnv v) ==> (e = e_v v /\ scopes_stack = scopes_stack' /\ g_scope_list = g_scope_list') ``,
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases]
);
val ret_lemma1d =
prove(``
! e v c g_scope_list g_scope_list' funn scopes_stack scopes_stack' stmt' ascope.
stmt_red c
(ascope, g_scope_list,[(funn,[stmt_ret e],scopes_stack)],
status_running)
(ascope, g_scope_list',[(funn,[stmt'],scopes_stack')],status_returnv v) ==> (e = e_v v /\ scopes_stack = scopes_stack' /\ g_scope_list = g_scope_list') ``,
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases]
);
val not_ret_status =
prove(``
! c g_scope_list g_scope_list' f stmt stmt' ss frame_list ctrl ctrl' v ascope.
~ stmt_red c
(ascope, g_scope_list,[(f,[stmt_ret (e_v v)],ss)], status_running)
(ascope, g_scope_list', frame_list,status_running)``
,
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases] >>
rw[] >>
rfs[lemma_v_red_forall]
);
val is_ret_status =
prove(``
! c g_scope_list g_scope_list' f stmt stmt' ss frame_list ctrl ctrl' v status ascope.
stmt_red c
(ascope, g_scope_list,[(f,[stmt_ret (e_v v)],ss)], status_running)
(ascope, g_scope_list', frame_list,status) ==> (status = status_returnv v)``
,
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases] >>
rw[] >>
rfs[lemma_v_red_forall]
);
val not_trans_status =
prove(``
! c g_scope_list g_scope_list' f stmt stmt' ss frame_list v v' ascope.
~ stmt_red c
(ascope, g_scope_list,[(f,[stmt_trans (e_v (v_str v'))],ss)], status_running)
(ascope, g_scope_list', frame_list,status_returnv v)
``
,
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases] >>
rw[] >>
rfs[lemma_v_red_forall]
);
(*
We can never reach the status error... shall we remove it?
*)
val not_trans_status =
prove(``
! c g_scope_list g_scope_list' f stmt stmt' ss frame_list v v' ascope.
~ stmt_red c
(ascope, g_scope_list,[(f,[stmt_trans (e_v (v_str v'))],ss)], status_running)
(ascope, g_scope_list', frame_list,status_returnv v)
``
,
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases] >>
rw[] >>
rfs[lemma_v_red_forall]
);
val not_ret_run_is_trans_or_error =
prove(``
! status.
status ≠ status_running /\ notret status ==> ((?v. status = status_trans v)) ``
,
Cases_on `status` >>
rw [notret_def]
);
(**************************************************)
(**************************************************)
(*determinism proof frame list*)
(**************************************************)
(**************************************************)
val P4_frame_det =
prove ( `` ! frame ty. det_frame frame ty ``,
Cases_on `frame` >>
Cases_on `r`>>
Cases_on `q'` >| [
FULL_SIMP_TAC (srw_ss()) [det_frame_def] >>
FULL_SIMP_TAC (srw_ss()) [Once frames_red_cases, same_state_def] >>
REPEAT STRIP_TAC >>
FULL_SIMP_TAC (srw_ss()) [Once stmt_red_cases, same_state_def]
,
FULL_SIMP_TAC (srw_ss()) [det_frame_def] >>
REPEAT STRIP_TAC >>
FULL_SIMP_TAC (srw_ss()) [Once frames_red_cases] >>
rw[] >>
Cases_on ‘map_to_pass q b_func_map’ >> gvs[] >>
Cases_on ‘scopes_to_pass q func_map b_func_map g_scope_list ’>> gvs[] >>
Cases_on ‘tbl_to_pass q b_func_map tbl_map’ >> gvs[] >>
rw[] >>
ASSUME_TAC P4_stmtl_det >>
fs [det_stmtl_def] >>
rfs[same_state_def] >>
RES_TAC >>
fs [] >>