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p4_frames_subject_reductionScript.sml
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p4_frames_subject_reductionScript.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;
open p4_auxTheory;
open p4_deterTheory;
open p4_e_subject_reductionTheory;
open p4_e_progressTheory;
open p4_stmt_subject_reductionTheory;
open p4_stmt_progressTheory;
open bitstringTheory;
open wordsTheory;
open optionTheory;
open sumTheory;
open stringTheory;
open ottTheory;
open pairTheory;
open rich_listTheory;
open arithmeticTheory;
open alistTheory;
open numeralTheory;
fun OPEN_LVAL_TYP_TAC lval_term =
(Q.PAT_X_ASSUM `lval_typ (g1,q1) t (^lval_term) (tp)` (fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once lval_typ_cases] thm)))
val _ = new_theory "p4_frames_subject_reduction";
val t_scopes_consistent_list_def = Define ‘
t_scopes_consistent_list funnl tsll t_scope_list_g (delta_g, delta_b, delta_x, delta_t) order =
(∀ i . i+1 < LENGTH funnl ⇒
∃ passed_delta_b passed_delta_t passed_tslg.
t_passed_elem (EL (i+1) funnl) delta_g delta_b delta_t t_scope_list_g = (SOME passed_delta_b, SOME passed_delta_t , SOME passed_tslg) ∧
t_scopes_consistent (order, (EL (i+1) funnl), (delta_g, passed_delta_b, delta_x, passed_delta_t)) (EL (i+1) tsll) passed_tslg (EL i tsll) )
’;
val (WT_state_rules, WT_state_ind, WT_state_cases) = Hol_reln`
(* defn WT_state *)
( (* WT_state_state *)
! (funnl: funn list) (tsll : t_scope_list list) (scll: scope_list list) (stmtll: stmt_stack list)
(ctx:'a ctx) (ascope:'a) (g_scope_list:g_scope_list) (status:status) (Prs_n:Prs_n) (order:order) (t_scope_list_g:t_scope_list_g) (delta_g:delta_g) (delta_b:delta_b) (delta_x:delta_x) (delta_t:delta_t) apply_table_f ext_map func_map b_func_map pars_map tbl_map .
( LENGTH funnl = LENGTH tsll /\ LENGTH tsll = LENGTH scll /\ LENGTH tsll = LENGTH stmtll ) /\
(WF_ft_order funnl delta_g delta_b delta_x order /\
t_scopes_consistent_list funnl tsll t_scope_list_g (delta_g, delta_b, delta_x, delta_t) order /\
type_state_tsll scll tsll /\
type_scopes_list g_scope_list t_scope_list_g /\
(ctx = ( apply_table_f , ext_map , func_map , b_func_map , pars_map , tbl_map ) ) ∧
WT_c ctx order t_scope_list_g delta_g delta_b delta_x delta_t Prs_n /\
( ( type_frames g_scope_list (ZIP(funnl,ZIP(stmtll,scll))) Prs_n order t_scope_list_g tsll delta_g delta_b delta_x delta_t func_map b_func_map) ))
==>
( ( WT_state ctx ( ascope , g_scope_list , (ZIP(funnl,ZIP(stmtll,scll)) ) , status ) Prs_n order t_scope_list_g ( tsll) ( delta_g , delta_b , delta_x , delta_t ) )))
`;
val sr_state_def = Define `
sr_state (framel) (ty:'a itself) =
∀ framel' (c:'a ctx) ascope ascope' gscope gscope' status status' tslg order delta_g delta_b delta_t delta_x Prs_n tsll.
(WT_state c ( ascope , gscope , framel , status) Prs_n order tslg tsll (delta_g, delta_b, delta_x, delta_t)) ∧
(frames_red c ( ascope , gscope , framel , status)
( ascope', gscope' , framel' , status')) ⇒
∃ tsll' . (WT_state c ( ascope' , gscope' , framel' , status') Prs_n order tslg tsll' (delta_g, delta_b, delta_x, delta_t))
`;
Theorem scopes_to_pass_imp_typed_lemma:
∀ gscope tslg funn func_map b_func_map delta_g delta_b g_scope_passed.
dom_b_eq delta_b b_func_map ∧
dom_g_eq delta_g func_map ∧
dom_tmap_ei delta_g delta_b ∧
LENGTH tslg = 2 ∧
type_scopes_list gscope tslg ⇒
scopes_to_pass funn func_map b_func_map gscope = SOME g_scope_passed ⇒
∃ tslg_passed .
t_scopes_to_pass funn delta_g delta_b tslg = SOME tslg_passed ∧
type_scopes_list g_scope_passed tslg_passed
Proof
gvs[scopes_to_pass_def, t_scopes_to_pass_def] >>
REPEAT STRIP_TAC >>
Cases_on ‘funn’ >> gvs[] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[dom_g_eq_def, dom_eq_def, is_lookup_defined_def] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s’])) >>
gvs[] >>
simp_tac list_ss [Once type_scopes_list_normalize] >>
srw_tac [][type_scopes_list_EL] >>
simp_tac list_ss [type_scopes_list_def, similarl_def, similar_def] >>
gvs[dom_b_eq_def, dom_eq_def, is_lookup_defined_def] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s’])) >>
gvs[]
QED
Theorem typed_imp_scopes_to_pass_lemma:
∀ gscope tslg funn func_map b_func_map delta_g delta_b tslg_passed .
dom_b_eq delta_b b_func_map ∧
dom_g_eq delta_g func_map ∧
dom_tmap_ei delta_g delta_b ∧
LENGTH tslg = 2 ∧
type_scopes_list gscope tslg ∧
t_scopes_to_pass funn delta_g delta_b tslg = SOME tslg_passed ⇒
∃ g_scope_passed . scopes_to_pass funn func_map b_func_map gscope = SOME g_scope_passed ∧
type_scopes_list g_scope_passed tslg_passed
Proof
gvs[scopes_to_pass_def, t_scopes_to_pass_def] >>
REPEAT STRIP_TAC >>
Cases_on ‘funn’ >> gvs[] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[dom_g_eq_def, dom_eq_def, is_lookup_defined_def] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s’])) >>
gvs[] >>
simp_tac list_ss [Once type_scopes_list_normalize] >>
srw_tac [][type_scopes_list_EL] >>
simp_tac list_ss [type_scopes_list_def, similarl_def, similar_def] >>
gvs[dom_b_eq_def, dom_eq_def, is_lookup_defined_def] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s’])) >>
gvs[]
QED
val frame_to_multi_frame_transition1 = prove (“
∀ c ascope ascope' gscope gscope' funn stmtl scope_list frame_list status status'.
stmt_red c (ascope , gscope , [(funn,stmtl,scope_list)], status)
(ascope', gscope', frame_list , status') ⇒
∃new_frame stmtl' scope_list'. frame_list = new_frame++[(funn,stmtl',scope_list')] ”,
REPEAT STRIP_TAC >>
gvs[Once stmt_red_cases] >> gvs[]
);
val frame_to_multi_frame_transition2 = prove (“
∀ c ascope ascope' gscope gscope' funn stmtl scope_list frame_list status status'.
stmt_red c (ascope, gscope,[(funn,stmtl,scope_list)],status)
(ascope',gscope',frame_list,status') ⇒
∃new_frame stmtl' scope_list'. frame_list = [(funn,stmtl',scope_list')] ∨
frame_list = new_frame++[(funn,stmtl',scope_list')] ”,
STRIP_TAC >>
gvs[Once stmt_red_cases] >> gvs[] >>
srw_tac [SatisfySimps.SATISFY_ss][] >>
METIS_TAC []
);
val MAP_comp_quad_trio = prove (
“∀ l .
MAP (λ(a,b,c,d). (a,b,c)) l =
ZIP(MAP (λ(a,b,c,d). a) l,
ZIP (MAP (λ(a,b,c,d). b) l,
MAP (λ(a,b,c,d). c) l))”,
Induct >> gvs[] >> REPEAT STRIP_TAC >>
PairCases_on ‘h’ >> gvs[]
);
val t_scopes_lookup_empty_ctx_lemma = prove (“
∀ delta_g delta_b e1 e2 s t_scope_list_g q r.
ALOOKUP delta_g s = SOME (q,r) ∧
dom_tmap_ei delta_g delta_b ∧
t_scopes_to_pass (funn_name s) delta_g delta_b [e1; e2] = SOME t_scope_list_g ⇒
t_scopes_to_pass (funn_name s) delta_g [] [[]; e2] = SOME t_scope_list_g”,
REPEAT STRIP_TAC >>
gvs[t_scopes_to_pass_def] >>
gvs[dom_tmap_ei_def, dom_empty_intersection_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s’])) >> gvs[]
);
Theorem find_star_of_globals_ctx_lemma:
∀ delta_g delta_b func_map b_func_map e1 e2 q r x.
dom_map_ei func_map b_func_map ∧
dom_tmap_ei delta_g delta_b ∧
dom_g_eq delta_g func_map ∧
dom_b_eq delta_b b_func_map ∧
Fb_star_defined b_func_map [e1; e2] ∧
Fg_star_defined func_map [e1; e2] ∧
ALOOKUP delta_g x = SOME (q,r) ∧
find_star_in_globals [e1; e2] (varn_star (funn_name x)) = SOME r ⇒
find_star_in_globals [[]; e2] (varn_star (funn_name x)) = SOME r
Proof
REPEAT STRIP_TAC >>
gvs[find_star_in_globals_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
fs[dom_tmap_ei_def, dom_map_ei_def ,dom_empty_intersection_def] >>
fs[dom_g_eq_def, dom_b_eq_def, dom_eq_def, is_lookup_defined_def] >>
fs[Fg_star_defined_def, Fb_star_defined_def, is_lookup_defined_def] >>
NTAC 6 (LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘x’])) >> gvs[] ) >| [
IMP_RES_TAC lookup_map_none_lemma1 >> gvs[]
,
fs[lookup_map_def, topmost_map_def, find_topmost_map_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[INDEX_FIND_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[])
]
QED
Theorem find_star_of_inst_ctx_lemma:
∀ ext_map r x e1 e2 sig.
X_star_not_defined [e1; e2] ∧
ALOOKUP ext_map x = SOME sig ∧
X_star_defined ext_map [e1; e2] ∧
find_star_in_globals [e1; e2] (varn_star (funn_inst x)) = SOME r ⇒
find_star_in_globals [[]; e2] (varn_star (funn_inst x)) = SOME r
Proof
REPEAT STRIP_TAC >>
gvs[find_star_in_globals_def, X_star_not_defined_def, X_star_defined_def, is_lookup_defined_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
REPEAT (LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘funn_inst x’, ‘x’])) >> gvs[]) >>
IMP_RES_TAC lookup_map_none_lemma1 >> gvs[] >>
gvs[lookup_map_def, topmost_map_def, find_topmost_map_def, INDEX_FIND_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[])
QED
Theorem find_star_of_ext_ctx_lemma:
∀ ext_map r x x' e1 e2 sig.
X_star_not_defined [e1; e2] ∧
ALOOKUP ext_map x = SOME sig ∧
X_star_defined ext_map [e1; e2] ∧
find_star_in_globals [e1; e2] (varn_star (funn_ext x x')) = SOME r ⇒
find_star_in_globals [[]; e2] (varn_star (funn_ext x x')) = SOME r
Proof
REPEAT STRIP_TAC >>
gvs[find_star_in_globals_def, X_star_not_defined_def, X_star_defined_def, is_lookup_defined_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
REPEAT (LAST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘funn_ext x x'’, ‘x’])) >> gvs[]) >>
IMP_RES_TAC lookup_map_none_lemma1 >> gvs[] >>
gvs[lookup_map_def, topmost_map_def, find_topmost_map_def, INDEX_FIND_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[])
QED
Theorem WTFg_empty_empty_db:
∀ func_map b_func_map order g1 g2 delta_g delta_b delta_x Prs_n.
dom_tmap_ei delta_g delta_b ∧ dom_map_ei func_map b_func_map ∧
dom_g_eq delta_g func_map ∧ dom_b_eq delta_b b_func_map ∧
Fb_star_defined b_func_map [g1; g2] ∧
Fg_star_defined func_map [g1; g2] ∧
WTFg func_map order [g1; g2] delta_g delta_b delta_x Prs_n⇒
WTFg func_map order [[]; g2] delta_g [] delta_x Prs_n
Proof
gvs[WTFg_cases, func_map_typed_def] >>
REPEAT STRIP_TAC >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘stmt’, ‘xdl’, ‘x’, ‘lol’])) >> gvs[] >>
Q.EXISTS_TAC ‘tau’ >> gvs[] >>
Q.EXISTS_TAC ‘txdl’ >> gvs[] >>
Q.EXISTS_TAC ‘t_scope_list_g'’ >>
gvs[] >>
gvs[t_lookup_funn_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
IMP_RES_TAC t_scopes_lookup_empty_ctx_lemma >> gvs[] >>
IMP_RES_TAC find_star_of_globals_ctx_lemma >> gvs[]
QED
Theorem WTFX_empty_empty_db:
∀ func_map (ext_map: 'a ext_map ) order g1 g2 delta_g delta_b delta_x .
X_star_not_defined [g1; g2] ∧
X_star_defined ext_map [g1; g2] ∧
WTX ext_map order [g1; g2] delta_g delta_b delta_x ⇒
WTX ext_map order [[]; g2] delta_g [] delta_x
Proof
REPEAT STRIP_TAC >>
gvs[WTX_cases] >>
CONJ_TAC >| [
gvs[extern_map_IoE_typed_def] >>
REPEAT STRIP_TAC >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘xdl’, ‘x’, ‘IoE’, ‘MoE’])) >> gvs[] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘a’, ‘g_scope_list’, ‘local_scopes’])) >> gvs[] >>
qexistsl_tac [‘txdl’, ‘tau’ ,‘a'’, ‘scope_list'’,‘status’, ‘t_scope_list_g'’] >> gvs[] >>
gvs[t_scopes_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_lookup_funn_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
IMP_RES_TAC find_star_of_inst_ctx_lemma
,
gvs[extern_MoE_typed_def] >>
REPEAT STRIP_TAC >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘xdl’, ‘x’,‘x'’ ,‘IoEsig’, ‘MoE’, ‘MoE_map’])) >> gvs[] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘a’, ‘g_scope_list’, ‘local_scopes’])) >> gvs[] >>
qexistsl_tac [‘txdl’,‘tau’,‘a'’, ‘scope_list'’,‘status’, ‘t_scope_list_g'’] >> gvs[] >>
gvs[t_scopes_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_lookup_funn_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
IMP_RES_TAC find_star_of_ext_ctx_lemma
]
QED
Theorem WT_c_empty_db:
∀ f delta_b delta_g delta_x delta_t passed_delta_b passed_delta_t
apply_table_f (ext_map: 'a ext_map) func_map b_func_map tbl_map pars_map order tau
txdl gscope g_scope_passed tslg passed_tslg Prs_n.
t_lookup_funn f delta_g passed_delta_b delta_x = SOME (txdl, tau)∧
t_tbl_to_pass f delta_b delta_t = SOME passed_delta_t ∧
t_map_to_pass f delta_b = SOME passed_delta_b ∧
t_scopes_to_pass f delta_g delta_b tslg = SOME passed_tslg ∧
scopes_to_pass f func_map b_func_map gscope = SOME g_scope_passed ∧
WT_c (apply_table_f,ext_map,func_map,b_func_map,pars_map,tbl_map)
order tslg delta_g delta_b delta_x delta_t Prs_n ⇒
∃passed_b_func_map passed_tbl_map.
map_to_pass f b_func_map = SOME passed_b_func_map ∧
tbl_to_pass f b_func_map tbl_map = SOME passed_tbl_map ∧
WT_c
(apply_table_f,ext_map,func_map,passed_b_func_map,pars_map,
passed_tbl_map) order passed_tslg delta_g passed_delta_b delta_x
passed_delta_t Prs_n
Proof
REPEAT STRIP_TAC >>
gvs[t_tbl_to_pass_def, t_map_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[tbl_to_pass_def, map_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[scopes_to_pass_def, t_scopes_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_lookup_funn_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
TRY ( gvs[WT_c_cases] >>
IMP_RES_TAC dom_eq_imp_NONE >> gvs[]
) >>
‘dom_map_ei func_map [] ∧ dom_tmap_ei delta_g [] ∧
typying_domains_ei delta_g [] delta_x ∧ dom_b_eq [] [] ∧
dom_t_eq [] [] ∧ WTFb [] order [[]; EL 1 tslg] delta_g [] delta_x [] Prs_n ’ by
(gvs [dom_map_ei_def, dom_tmap_ei_def, typying_domains_ei_def] >>
gvs [dom_empty_intersection_def] >> gvs[] >>
gvs [dom_b_eq_def, dom_t_eq_def, dom_eq_def, is_lookup_defined_def] >>
SIMP_TAC list_ss [WTFb_cases, func_map_blk_typed_def, clause_name_def] >> gvs[]) >> gvs[] >>
‘table_map_typed [] apply_table_f delta_g [] order ∧
f_in_apply_tbl [] apply_table_f’ by ( gvs [f_in_apply_tbl_def] >>
gvs[table_map_typed_def] >>
gvs[]) >> gvs[] >>
‘ Fg_star_defined func_map [[]; EL 1 tslg] ∧
Fb_star_defined [] [[]; EL 1 tslg] ’ by
(gvs [Fb_star_defined_def, is_lookup_defined_def] >> gvs[] >>
gvs [Fg_star_defined_def, is_lookup_defined_def] >> gvs[]) >> gvs[] >>
(subgoal ‘X_star_defined ext_map [[]; EL 1 tslg]’ >- (
fs[] >> gvs[] >>
gvs[X_star_defined_def, is_lookup_defined_def] >> REPEAT STRIP_TAC >> gvs[]
) >> gvs[]
) >>
fs[] >> gvs[] >>
(subgoal ‘X_star_not_defined [[]; EL 1 tslg]’ >- (
gvs[X_star_not_defined_def] >> REPEAT STRIP_TAC >> gvs[] >>
gvs[is_lookup_defined_def]
)
) >> gvs[] >> rw[] >>
Cases_on ‘tslg’ >> gvs[] >>
Cases_on ‘t’ >> gvs[] >>
IMP_RES_TAC WTFg_empty_empty_db >>
IMP_RES_TAC WTFX_empty_empty_db >>
fs[]
QED
Theorem WT_state_HD_of_list:
∀ ascope gscope f stmtl locale status Prs_n order tslg tsll delta_g delta_b
delta_x delta_t apply_table_f ext_map func_map b_func_map pars_map
tbl_map t.
WT_state ( apply_table_f , ext_map , func_map , b_func_map , pars_map , tbl_map )
(ascope,gscope,(f,stmtl,locale)::t,status) Prs_n order tslg
tsll (delta_g,delta_b,delta_x,delta_t) ⇒
∃ passed_tslg passed_gscope passed_delta_b passed_b_func_map passed_tbl_map passed_delta_t.
t_scopes_to_pass f delta_g delta_b tslg = SOME passed_tslg ∧
scopes_to_pass f func_map b_func_map gscope = SOME passed_gscope ∧
map_to_pass f b_func_map = SOME passed_b_func_map ∧
t_map_to_pass f delta_b = SOME passed_delta_b ∧
tbl_to_pass f b_func_map tbl_map = SOME passed_tbl_map ∧
t_tbl_to_pass f delta_b delta_t = SOME passed_delta_t ∧
WT_c ( apply_table_f , ext_map , func_map , passed_b_func_map , pars_map , passed_tbl_map ) order passed_tslg delta_g passed_delta_b delta_x passed_delta_t Prs_n ∧
type_scopes_list passed_gscope passed_tslg ∧
(frame_typ ( passed_tslg , (HD tsll) ) (order, f, (delta_g, passed_delta_b, delta_x, passed_delta_t)) Prs_n passed_gscope locale stmtl )
Proof
REPEAT GEN_TAC >>
STRIP_TAC >>
gvs[WT_state_cases] >>
gvs[type_state_tsll_def] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`0`])) >> gvs[] >>
subgoal ‘0 < LENGTH scll’ >- (Cases_on ‘scll’ >> gvs[] ) >> gvs[] >>
subgoal ‘locale = HD scll ∧ f = HD funnl ∧ stmtl = HD stmtll’ >- (Cases_on ‘scll’ >> gvs[] >>
Cases_on ‘stmtll’ >> gvs[] >>
Cases_on ‘funnl’ >> gvs[] ) >>
lfs[] >> gvs[] >>
gvs[type_frame_tsl_def] >>
gvs[type_frames_def, type_frame_tsl_def] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`0`])) >> gvs[] >>
gvs[map_distrub] >>
‘ ∃ g_scope_passed . scopes_to_pass (HD funnl) func_map b_func_map gscope = SOME g_scope_passed
∧ type_scopes_list g_scope_passed passed_tslg’ by (gvs[Once WT_c_cases] >> IMP_RES_TAC typed_imp_scopes_to_pass_lemma >> gvs[]) >>
Cases_on ‘scopes_to_pass (HD funnl) func_map b_func_map gscope’ >> gvs[] >>
gvs[frame_typ_cases] >>
IMP_RES_TAC WT_c_empty_db >> gvs[] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘tau_x_d_list’, ‘tau’])) >> gvs[] >>
Cases_on ‘tsll’ >> gvs[]
QED
Theorem ZIP_tri_id1:
(∀l . l = ZIP (MAP (λx. FST x) l, ZIP (MAP (λx. FST (SND x)) l,MAP (λx. SND (SND x)) l)))
Proof
Induct >>
gvs[]
QED
Theorem ZIP_tri_id2:
∀l. l = ZIP (MAP (λ(f,stmt,sc). f) l, ZIP (MAP (λ(f,stmt,sc). stmt) l,MAP (λ(f,stmt,sc). sc) l))
Proof
Induct >>
gvs[] >>
REPEAT STRIP_TAC >>
PairCases_on ‘h’ >> gvs[]
QED
Theorem WT_state_frame_len:
∀ ascope gscope status Prs_n order tslg tsll delta_g delta_b delta_x
delta_t c funn stmt_stack scope_list t .
WT_state c (ascope,gscope,(funn,stmt_stack,scope_list)::t,status) Prs_n order
tslg tsll (delta_g,delta_b,delta_x,delta_t) ⇒
LENGTH ((funn,stmt_stack,scope_list)::t) = LENGTH tsll
Proof
REPEAT STRIP_TAC >>
gvs[Once WT_state_cases] >>
Cases_on ‘funnl’ >>
Cases_on ‘stmtll’ >>
Cases_on ‘scll’ >>
gvs[]
QED
Theorem type_state_tsll_normalization:
∀ scll tsll.
LENGTH scll = LENGTH tsll ∧
LENGTH scll > 0 ∧
type_state_tsll scll tsll ⇒
type_state_tsll [HD scll] [HD tsll] ∧
type_state_tsll (TL scll) (TL tsll)
Proof
Cases_on ‘scll’ >>
Cases_on ‘tsll’ >>
REPEAT STRIP_TAC >>
gvs[type_state_tsll_def] >>
REPEAT STRIP_TAC >| [
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`0`])) >>
gvs[]
,
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [`SUC i`])) >>
gvs[] >> lfs[]
]
QED
val WT_state_imp_tl_tsll = prove (“
∀ ascope gscope status Prs_n order tslg tsll delta_g delta_b delta_x
delta_t c funn stmt_stack scope_list t .
WT_state c (ascope,gscope,(funn,stmt_stack,scope_list)::t,status) Prs_n order tslg tsll (delta_g,delta_b,delta_x,delta_t) ⇒
type_state_tsll (MAP (λ(f,stmt,sc). sc) t) (TL tsll)”,
REPEAT STRIP_TAC >>
gvs[Once WT_state_cases] >>
subgoal ‘(MAP (λ(f,stmt,sc). sc) t) = TL scll’ >-
(ASSUME_TAC (INST_TYPE [``:'a`` |-> ``:funn``, ``:'b`` |-> ``:stmt list``, ``:'c`` |-> ``:scope_list``] ZIP_tri_sep_ind) >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘t’,‘funnl’,‘stmtll’,‘scll’, ‘funn’,‘stmt_stack’,‘scope_list’])) >> gvs[] >>
gvs[ELIM_UNCURRY]
) >>
IMP_RES_TAC type_state_tsll_normalization >>
gvs[] >>
Cases_on ‘scll’ >>
Cases_on ‘tsll’ >>
gvs[]
);
val res_fr_typ_imp_typ_tsl = prove (“
∀ T_e Prs_n tslg tsl gscope locale f stmtl func_map b_func_map tsl_fr.
res_frame_typ T_e Prs_n tslg tsl gscope [(f,stmtl,locale)] func_map b_func_map tsl_fr ⇒
type_frame_tsl locale tsl ”,
REPEAT STRIP_TAC >>
PairCases_on ‘T_e’ >>
gvs[res_frame_typ_def] >>
gvs[frame_typ_cases]
);
val type_tsll_hd_l = prove (“
∀ s1 scl ts1 tsl .
type_frame_tsl s1 ts1 ∧
type_state_tsll scl tsl ⇒
type_state_tsll (s1::scl) (ts1::tsl)”,
gvs[type_state_tsll_def] >>
REPEAT STRIP_TAC >>
SIMP_TAC list_ss [Once EL_compute] >> CASE_TAC >- gvs[EL_CONS] >>
gvs[ADD1, PRE_SUB1] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [ ‘i-1’ ])) >>
gvs[] >>
gvs[EL_CONS, PRE_SUB1]
);
val type_tsll_hd_hd_l = prove (“
∀ s1 s2 scl ts1 ts2 tsl .
type_frame_tsl s1 ts1 ∧
type_frame_tsl s2 ts2 ∧
type_state_tsll scl tsl ⇒
type_state_tsll (s1::s2::scl) (ts1::ts2::tsl)”,
gvs[type_state_tsll_def] >>
REPEAT STRIP_TAC >>
SIMP_TAC list_ss [Once EL_compute] >> CASE_TAC >- gvs[EL_CONS] >>
SIMP_TAC list_ss [Once EL_compute] >> CASE_TAC >- gvs[EL_CONS] >>
gvs[ADD1, PRE_SUB1] >>
FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [ ‘i-2’ ])) >>
gvs[] >>
gvs[EL_CONS, PRE_SUB1]
);
val EL_PRE = prove (“
∀i l t h. i > 0 ⇒
EL i (h::t) = EL (i-1) (t)”,
Induct >>
gvs[Once EL_compute]
);
Theorem LIST_LENGTH_2_simp:
∀ l . LENGTH l = 2 ⇔ ∃ a b . l = [a;b]
Proof
Induct >> gvs[] >>
Cases_on ‘l’ >> gvs[]
QED
Theorem scopes_to_ret_is_wt:
∀ funn delta_g delta_b func_map b_func_map tslg passed_gscope gscope passed_tslg ret_gscope .
dom_b_eq delta_b b_func_map ∧
dom_g_eq delta_g func_map ∧
dom_tmap_ei delta_g delta_b ∧
LENGTH tslg = 2 ∧
t_scopes_to_pass funn delta_g delta_b tslg = SOME passed_tslg ∧
type_scopes_list passed_gscope passed_tslg ∧
type_scopes_list gscope tslg ∧
SOME ret_gscope = scopes_to_retrieve funn func_map b_func_map gscope passed_gscope ==>
type_scopes_list ret_gscope tslg
Proof
gvs[t_scopes_to_pass_def, scopes_to_retrieve_def] >>
REPEAT STRIP_TAC >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
IMP_RES_TAC dom_eq_imp_NONE >> gvs[] >>
IMP_RES_TAC type_scopes_list_LENGTH >>
gvs[LIST_LENGTH_2_simp] >>
simp_tac list_ss [Once type_scopes_list_normalize] >>
IMP_RES_TAC type_scopes_list_normalize >> gvs[]
QED
val block_exit_implic = prove (“
∀ ascope ascope' g_scope_list g_scope_list' funn stmtl stmtl' scope_list scope_list' status status' framel (c: 'a ctx).
stmt_red c (ascope, g_scope_list, [(funn,stmtl,scope_list)], status)
(ascope' , g_scope_list' ,framel ⧺ [(funn,stmtl',scope_list')], status') ∧
LENGTH stmtl > LENGTH stmtl' ⇒
(LENGTH stmtl − LENGTH stmtl' = 1 ∧ framel = [] ∧ g_scope_list = g_scope_list' )”,
REPEAT STRIP_TAC >>
gvs[Once stmt_red_cases] >>
gvs[ADD1] >>
gvs[Once stmt_red_cases]
);
fun OPEN_ANY_STMT_RED_TAC a =
(Q.PAT_X_ASSUM `stmt_red ct (ab, gsl,[(f,[stm_term],gam)],st) stat`
(fn thm => ASSUME_TAC (SIMP_RULE (srw_ss()) [Once stmt_red_cases] thm)))
Theorem return_imp_same_g_base_case:
∀ stmt stmtl' c ascope ascope' gscope gscope' f v framel locale locale'.
stmt_red c (ascope,gscope,[(f,[stmt],locale)],status_running)
(ascope',gscope',[(f,stmtl',locale')],status_returnv v) ⇒
gscope = gscope'
Proof
Induct >>
REPEAT GEN_TAC >>
STRIP_TAC >>
OPEN_ANY_STMT_RED_TAC “anystmt” >>
METIS_TAC []
QED
Theorem return_imp_same_g:
∀ stmtl stmtl' c ascope ascope' gscope gscope' f v framel locale locale'.
stmt_red c (ascope,gscope,[(f,stmtl,locale)],status_running)
(ascope',gscope',[(f,stmtl',locale')],status_returnv v) ⇒
gscope = gscope'
Proof
REPEAT STRIP_TAC >>
gvs[Once stmt_red_cases] >>
IMP_RES_TAC return_imp_same_g_base_case
QED
Theorem create_frame_in_stmt_same_g:
∀stmt stmt' stmt_called (c:'a ctx) ascope ascope' gscope gscope' f locale locale' f_called stmt_stack copied_in_scope.
stmt_red c (ascope,gscope,[(f,[stmt],locale)],status_running)
(ascope',gscope', [(f_called,[stmt_called],copied_in_scope); (f,stmt_stack ⧺ [stmt'],locale')],status_running) ⇒
gscope = gscope'
Proof
Induct >>
REPEAT GEN_TAC >>
STRIP_TAC >>
OPEN_ANY_STMT_RED_TAC “anystmt” >>
METIS_TAC []
QED
Theorem create_frame_in_stmt_same_g2:
∀stmt stmt_called (c:'a ctx) ascope ascope' gscope gscope' f locale locale'
f_called copied_in_scope stmt_stack' status status'.
stmt_red c (ascope,gscope,[(f,[stmt],locale)],status)
(ascope',gscope', [(f_called,[stmt_called],copied_in_scope); (f,stmt_stack',locale')],status') ⇒
gscope = gscope'
Proof
Induct >>
REPEAT GEN_TAC >>
STRIP_TAC >>
OPEN_ANY_STMT_RED_TAC “anystmt” >>
IMP_RES_TAC create_frame_in_stmt_same_g >> gvs[] >> METIS_TAC[]
QED
Theorem create_frame_in_stmt_same_g3:
∀ stmtl stmtl' (c:'a ctx) f f_called stmt_called copied_in_scope ascope ascope' gscope gscope' scope_list scope_list' status status' .
stmt_red c (ascope, gscope, [(f,stmtl,scope_list)],status)
(ascope',gscope', [(f_called,[stmt_called],copied_in_scope); (f,stmtl',scope_list')],status') ⇒
gscope = gscope'
Proof
REPEAT STRIP_TAC >>
gvs[Once stmt_red_cases] >>
IMP_RES_TAC create_frame_in_stmt_same_g >>
IMP_RES_TAC create_frame_in_stmt_same_g2
QED
val WF_ft_order_called_f = prove (“
∀ order f_called f fl delta_g delta_b delta_x.
order (order_elem_f f_called) (order_elem_f f) ∧
WF_ft_order (f::fl) delta_g delta_b delta_x order ⇒
WF_ft_order (f_called::f::fl) delta_g delta_b delta_x order”,
gvs[WF_ft_order_cases] >>
gvs[ordered_list_def] >>
REPEAT STRIP_TAC >>
SIMP_TAC list_ss [Once EL_compute] >>
CASE_TAC >>
gvs[EL_CONS]
);
val t_map_to_pass_twice = prove (“
∀ f f_called delta_b passed_delta_b passed_delta_b' delta_g delta_x txdl txdl' tau tau'.
SOME (txdl',tau') = t_lookup_funn f delta_g passed_delta_b delta_x ∧
SOME (txdl,tau) = t_lookup_funn f_called delta_g passed_delta_b' delta_x ∧
typying_domains_ei delta_g delta_b delta_x ∧
typying_domains_ei delta_g passed_delta_b delta_x ∧
t_map_to_pass f delta_b = SOME passed_delta_b ∧
t_map_to_pass f_called passed_delta_b = SOME passed_delta_b' ⇒
t_map_to_pass f_called delta_b = SOME passed_delta_b' ”,
REPEAT GEN_TAC >> STRIP_TAC >>
gvs[t_map_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_lookup_funn_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[typying_domains_ei_def, dom_empty_intersection_def] >>
REPEAT (FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s’])) >> fs[])
);
val t_tbl_to_pass_twice = prove (“
∀ f f_called delta_b passed_delta_b passed_delta_b' delta_t passed_delta_t passed_delta_t' delta_g delta_x txdl txdl' tau tau'.
SOME (txdl',tau') = t_lookup_funn f delta_g passed_delta_b delta_x ∧
SOME (txdl ,tau) = t_lookup_funn f_called delta_g passed_delta_b' delta_x ∧
typying_domains_ei delta_g delta_b delta_x ∧
typying_domains_ei delta_g passed_delta_b delta_x ∧
t_map_to_pass f delta_b = SOME passed_delta_b ∧
t_map_to_pass f_called passed_delta_b = SOME passed_delta_b' ∧
t_tbl_to_pass f delta_b delta_t = SOME passed_delta_t ∧
t_tbl_to_pass f_called passed_delta_b passed_delta_t = SOME passed_delta_t' ⇒
t_tbl_to_pass f_called delta_b delta_t = SOME passed_delta_t' ”,
REPEAT GEN_TAC >> STRIP_TAC >>
gvs[t_map_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_lookup_funn_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_tbl_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[typying_domains_ei_def, dom_empty_intersection_def] >>
REPEAT (FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s’])) >> fs[]) >>
gvs[]
);
val t_scopes_to_pass_twice = prove (“
∀ f f_called delta_b passed_delta_b passed_delta_b' delta_g delta_x txdl txdl' tau tau' tslg passed_tslg passed_tslg'.
SOME (txdl',tau') = t_lookup_funn f delta_g passed_delta_b delta_x ∧
SOME (txdl,tau) = t_lookup_funn f_called delta_g passed_delta_b' delta_x ∧
typying_domains_ei delta_g delta_b delta_x ∧
typying_domains_ei delta_g passed_delta_b delta_x ∧
t_map_to_pass f delta_b = SOME passed_delta_b ∧
t_map_to_pass f_called passed_delta_b = SOME passed_delta_b' ∧
t_scopes_to_pass f delta_g delta_b tslg = SOME passed_tslg ∧
t_scopes_to_pass f_called delta_g passed_delta_b passed_tslg = SOME passed_tslg'
⇒
t_scopes_to_pass f_called delta_g delta_b tslg = SOME passed_tslg' ”,
REPEAT GEN_TAC >> STRIP_TAC >>
gvs[LIST_LENGTH_2_simp] >>
gvs[t_map_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_lookup_funn_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_scopes_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[typying_domains_ei_def, dom_empty_intersection_def] >>
REPEAT (FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s’])) >> fs[]) >>
gvs[]
);
val scopes_to_pass_twice = prove (“
∀ f f_called delta_b passed_delta_b passed_delta_b' delta_g delta_x txdl txdl' tau tau' func_map b_func_map b_func_map'
gscope gscope' passed_gscope g_scope_list' .
SOME (txdl', tau') = t_lookup_funn f delta_g passed_delta_b delta_x ∧
SOME (txdl , tau ) = t_lookup_funn f_called delta_g passed_delta_b' delta_x ∧
SOME gscope' = scopes_to_retrieve f func_map b_func_map gscope g_scope_list' ∧
typying_domains_ei delta_g delta_b delta_x ∧
typying_domains_ei delta_g passed_delta_b delta_x ∧
dom_b_eq delta_b b_func_map ∧
dom_b_eq passed_delta_b b_func_map' ∧
dom_g_eq delta_g func_map ∧
dom_map_ei func_map b_func_map' ∧
dom_map_ei func_map b_func_map ∧
map_to_pass f b_func_map = SOME b_func_map' ∧
t_map_to_pass f delta_b = SOME passed_delta_b ∧
t_map_to_pass f_called passed_delta_b = SOME passed_delta_b' ∧
scopes_to_pass f func_map b_func_map gscope = SOME g_scope_list' ∧
scopes_to_pass f_called func_map b_func_map' g_scope_list' = SOME passed_gscope ⇒
scopes_to_pass f_called func_map b_func_map gscope' = SOME passed_gscope”,
REPEAT GEN_TAC >> STRIP_TAC >>
IMP_RES_TAC type_scopes_list_LENGTH >>
gvs[LIST_LENGTH_2_simp] >>
gvs[map_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_map_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_lookup_funn_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[typying_domains_ei_def, dom_empty_intersection_def] >>
REPEAT (FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s’])) >> fs[]) >>
gvs[] >>
gvs[scopes_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
IMP_RES_TAC dom_eq_imp_NONE >> gvs[] >>
gvs[t_scopes_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[scopes_to_retrieve_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[dom_b_eq_def, dom_g_eq_def, dom_eq_def, is_lookup_defined_def] >>
REPEAT (FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s'’])) >> fs[]) >>
gvs[] >>
gvs[dom_map_ei_def, dom_empty_intersection_def ] >>
REPEAT (FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s'’])) >> fs[]) >>
gvs[]
);
val retrived_scopes_can_be_passed = prove (“
∀ f txdl tau delta_g delta_b passed_delta_b func_map b_func_map delta_x gscope gscope' g_scope_list'.
SOME (txdl,tau) = t_lookup_funn f delta_g passed_delta_b delta_x ∧
t_map_to_pass f delta_b = SOME passed_delta_b ∧
scopes_to_pass f func_map b_func_map gscope = SOME g_scope_list' ∧
SOME gscope' = scopes_to_retrieve f func_map b_func_map gscope g_scope_list' ⇒
scopes_to_pass f func_map b_func_map gscope' = SOME g_scope_list' ”,
REPEAT GEN_TAC >> STRIP_TAC >>
gvs[t_map_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[t_lookup_funn_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[typying_domains_ei_def, dom_empty_intersection_def] >>
REPEAT (FIRST_X_ASSUM (STRIP_ASSUME_TAC o (Q.SPECL [‘s’])) >> fs[]) >>
gvs[] >>
gvs[scopes_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
IMP_RES_TAC dom_eq_imp_NONE >> gvs[] >>
gvs[t_scopes_to_pass_def] >>
REPEAT (BasicProvers.FULL_CASE_TAC >> gvs[]) >>
gvs[scopes_to_retrieve_def] >>