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panItreeSemEquivScript.sml
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(*
Proof of correspondence between functional big-step
and itree semantics for Pancake.
*)
open preamble
itreeTauTheory
panSemTheory
panItreePropsTheory
panItreeSemTheory
panLangTheory;
local open alignmentTheory
miscTheory (* for read_bytearray *)
wordLangTheory (* for word_op and word_sh *)
ffiTheory
panPropsTheory
in end;
val _ = new_theory "panItreeSemEquiv";
val _ = temp_set_fixity "≈" (Infixl 500);
Overload "≈" = “itree_wbisim”;
val _ = temp_set_fixity ">>=" (Infixl 500);
Overload ">>=" = “itree_bind”;
Overload "case" = “itree_CASE”;
Definition query_oracle_def[nocompute]:
query_oracle (ffis:'ffi ffi_state) (FFI_call s conf bytes) =
case call_FFI ffis s conf bytes of
FFI_return ffis' bytes' => (FFI_return ffis' bytes',bytes',ffis')
| FFI_final (Final_event name conf' bytes' outcome) =>
(FFI_final (Final_event name conf' bytes' outcome),bytes',ffis)
End
(* Path over semtrees:
- states consist of (ffi_state x 'a result option) pairs,
- transition labels have type: 'b sem_vis_event option
val t = “t:('a,'b,'c) itree”;
Definition semtree_path_def:
semtree_path f s ^t =
unfold (λ(t,s1). case t of
Ret r => (s1,SOME r)
| Tau u => (s1,NONE)
| Vis e k => let (a,s1') = (f s1 e) in (s1',NONE))
(λ(t,s1). case t of
Ret r => NONE
| Tau u => SOME ((u,s1),NONE)
| Vis e k => let (a,s1') = (f s1 e) in
SOME ((k a,s1'),SOME e))
(t,s)
End
val g = “g:('a,'b) mtree_ans -> ('a,'b) ltree”;
*)
Theorem ltree_lift_bind_left_ident:
(ltree_lift f st (mrec_sem ht)) ≈ Ret x ⇒
(ltree_lift f st (mrec_sem (ht >>= k))) ≈ (ltree_lift f (ltree_lift_state f st (mrec_sem ht)) (mrec_sem (k x)))
Proof
disch_tac >>
rw [mrec_sem_monad_law] >>
rw [ltree_lift_monad_law] >>
irule itree_wbisim_trans>>
irule_at Any itree_bind_resp_t_wbisim>>
pop_assum $ irule_at Any>>
simp[itree_bind_thm]>>
irule itree_wbisim_refl
QED
(* Main correspondence theorem *)
(* Extension for ffi$behaviour capturing evaluation result
of convergent computations *)
Datatype:
sem_behaviour =
SemDiverge (io_event llist)
| SemTerminate (('a result option) # ('a,'b) bstate)
| SemFail
End
Definition fbs_semantics_beh_def:
fbs_semantics_beh s prog =
if ∃k. FST $ panSem$evaluate (prog,(reclock s) with clock := k) ≠ SOME TimeOut
then (case some (r,s'). ∃k. evaluate (prog,(reclock s) with clock := k) = (r,s') ∧ r ≠ SOME TimeOut of
SOME (r,s') => (case r of
SOME (Return _) => SemTerminate (r,unclock s')
| SOME (FinalFFI _) => SemTerminate (r,unclock s')
| SOME Error => SemFail
| _ => SemTerminate (r,unclock s'))
| NONE => SemFail)
else SemDiverge (build_lprefix_lub
(IMAGE (λk. fromList
(SND (evaluate (prog,(reclock s) with clock := k))).ffi.io_events) UNIV))
End
Definition event_filter_def:
event_filter (FFI_return _ _) = T ∧
event_filter _ = F
End
Definition itree_semantics_beh_def:
itree_semantics_beh (s:('a,'b) bstate) (prog:'a prog) =
let lt = ltree_lift query_oracle s.ffi (mrec_sem (h_prog (prog,s))) in
case some (r,s'). lt ≈ Ret (r,s') of
| SOME (r,s') => (case r of
SOME TimeOut => SemTerminate (r,s')
| SOME (FinalFFI _) => SemTerminate (r,s')
| SOME (Return _) => SemTerminate (r,s')
| SOME Error => SemFail
| _ => SemTerminate (r,s'))
| NONE => SemDiverge (LAPPEND (fromList(s.ffi.io_events)) (stree_trace query_oracle event_filter s.ffi (to_stree (mrec_sem (h_prog (prog,s))))))
End
Theorem fbs_sem_div_compos_thm:
fbs_semantics_beh s (Dec v e prog) = SemDiverge l ∧
eval (reclock s) e = SOME x ⇒
fbs_semantics_beh (s with locals := s.locals |+ (v,x)) prog = SemDiverge l
Proof
rpt strip_tac>>
fs[fbs_semantics_beh_def,Once evaluate_def] >>
fs[bool_case_eq] >>
fs[option_case_eq,pair_case_eq,result_case_eq] >>
conj_tac>-
(strip_tac>>first_x_assum $ qspec_then ‘k’ assume_tac>>
FULL_CASE_TAC>>fs[]>>
pairarg_tac>>fs[]>>gvs[panPropsTheory.eval_upd_clock_eq,panItreeSemTheory.reclock_def])>>
rveq >> irule lprefix_lubTheory.IMP_build_lprefix_lub_EQ>>
conj_asm1_tac>-
(simp[lprefix_chain_def]>>
rpt strip_tac>>fs[]>>
Cases_on ‘k' < k’>-
(disj2_tac>>
simp[LPREFIX_def,from_toList]>>
irule IS_PREFIX_TRANS>>
irule_at Any panPropsTheory.evaluate_add_clock_io_events_mono>>
qexists_tac ‘k - k'’>>fs[])>>
fs[NOT_LESS]>>
disj1_tac>>
simp[LPREFIX_def,from_toList]>>
irule IS_PREFIX_TRANS>>
irule_at Any panPropsTheory.evaluate_add_clock_io_events_mono>>
qexists_tac ‘k' - k’>>fs[])>>
conj_asm1_tac>-
(simp[lprefix_chain_def]>>
rpt strip_tac>>fs[]>>
Cases_on ‘k' < k’>-
(disj2_tac>>
simp[LPREFIX_def,from_toList]>>
irule IS_PREFIX_TRANS>>
irule_at Any panPropsTheory.evaluate_add_clock_io_events_mono>>
qexists_tac ‘k - k'’>>fs[])>>
fs[NOT_LESS]>>
disj1_tac>>
simp[LPREFIX_def,from_toList]>>
irule IS_PREFIX_TRANS>>
irule_at Any panPropsTheory.evaluate_add_clock_io_events_mono>>
qexists_tac ‘k' - k’>>fs[])>>
conj_tac>-
(simp[lprefix_rel_def]>>
rpt strip_tac>>
simp[PULL_EXISTS]>>
simp[LPREFIX_def,from_toList]>>
simp[Once evaluate_def,
panItreeSemTheory.reclock_def,
panPropsTheory.eval_upd_clock_eq]>>
qexists_tac `k` >> fs[] >>
pairarg_tac>>fs[])>>
simp[lprefix_rel_def]>>
rpt strip_tac>>
simp[PULL_EXISTS]>>
simp[LPREFIX_def,from_toList]>>
simp[Once evaluate_def,
panItreeSemTheory.reclock_def,
panPropsTheory.eval_upd_clock_eq]>>
qexists_tac ‘k’>>
pairarg_tac>>fs[panItreeSemTheory.reclock_def]
QED
Theorem fbs_semantics_beh_simps:
fbs_semantics_beh s Skip = SemTerminate (NONE,s) ∧
fbs_semantics_beh s (Annot _ _) = SemTerminate (NONE,s) ∧
(eval (reclock s) e = NONE ⇒ fbs_semantics_beh s (Dec v e prog) ≠ SemTerminate p)
Proof
rw []
>~ [‘Dec _ _ _’]
>- (rw [fbs_semantics_beh_def,
evaluate_def] >>
rw [panPropsTheory.eval_upd_clock_eq] >>
DEEP_INTRO_TAC some_intro >> rw [] >>
FULL_CASE_TAC >> fs [])>>
(rw [fbs_semantics_beh_def,
evaluate_def] >>
DEEP_INTRO_TAC some_intro >> rw [EXISTS_PROD] >>
ntac 2 TOP_CASE_TAC >>
pairarg_tac >> gvs [panItreeSemTheory.unclock_def,panItreeSemTheory.reclock_def,
panItreeSemTheory.bstate_component_equality])
QED
Theorem itree_semantics_beh_Dec:
itree_semantics_beh s (Dec vname e prog) =
case eval (reclock s) e of
NONE => SemFail
| SOME value =>
case itree_semantics_beh (s with locals := s.locals |+ (vname,value)) prog of
| SemTerminate (res,s') =>
SemTerminate (res,s' with locals := res_var s'.locals (vname,FLOOKUP s.locals vname))
| res => res
Proof
rw[itree_semantics_beh_def] >>
Cases_on ‘eval (reclock s) e’ >>
gvs[h_prog_def,h_prog_dec_def,mrec_sem_simps,ltree_lift_cases,
wbisim_Ret_eq,
ELIM_UNCURRY
] >>
CONV_TAC SYM_CONV >>
DEEP_INTRO_TAC some_intro >>
reverse conj_tac
>- (rw[mrec_sem_monad_law,
ltree_lift_monad_law,
ltree_lift_nonret_bind,
to_stree_monad_law,
to_stree_simps,
stree_trace_simps,
ltree_lift_nonret_bind_stree
]) >>
rw[] >>
rename1 ‘Ret r’ >> Cases_on ‘r’ >> gvs[] >>
drule ltree_lift_bind_left_ident >>
qmatch_goalsub_abbrev_tac ‘_ >>= k1’ >>
disch_then $ qspec_then ‘k1’ mp_tac >>
unabbrev_all_tac >>
simp[mrec_sem_simps,ltree_lift_cases] >>
strip_tac >>
DEEP_INTRO_TAC some_intro >>
reverse conj_tac
>- (rw[] >> gvs[]) >>
rw[] >>
dxrule_then strip_assume_tac itree_wbisim_sym >>
dxrule itree_wbisim_trans >>
strip_tac >>
first_x_assum dxrule >>
rw[wbisim_Ret_eq] >>
rpt(PURE_TOP_CASE_TAC >> gvs[])
QED
Theorem itree_semantics_beh_If:
itree_semantics_beh s (If e p1 p2) =
case eval (reclock s) e of
| SOME(ValWord g) => itree_semantics_beh s (if g ≠ 0w then p1 else p2)
| _ => SemFail
Proof
rw[itree_semantics_beh_def] >>
Cases_on ‘eval (reclock s) e’ >>
gvs[h_prog_def,h_prog_cond_def,mrec_sem_simps,ltree_lift_cases,
wbisim_Ret_eq,
ELIM_UNCURRY
] >>
CONV_TAC SYM_CONV >>
PURE_TOP_CASE_TAC >> gvs[mrec_sem_simps,ltree_lift_cases,wbisim_Ret_eq] >>
PURE_TOP_CASE_TAC >> gvs[mrec_sem_simps,ltree_lift_cases,wbisim_Ret_eq] >>
rename1 ‘h_prog(p,s)’ >>
PURE_TOP_CASE_TAC >> gvs[] >>
gvs[stree_trace_simps,to_stree_simps]
QED
Theorem mrec_sem_while_unfold:
mrec_sem (h_prog (While e p,s)) =
case eval(reclock s) e of
SOME (ValWord w) =>
if w = 0w then Ret (NONE, s)
else
Tau(mrec_sem (h_prog (p,s)) >>=
λ(res,s').
case res of
NONE => Tau $ mrec_sem (h_prog (While e p, s'))
| SOME Break => Ret (NONE, s')
| SOME Continue => Tau $ mrec_sem (h_prog (While e p, s'))
| _ => Ret (res, s')
)
| _ => Ret(SOME Error, s)
Proof
rw[h_prog_def,h_prog_while_def] >>
rw[Once itree_iter_thm] >>
PURE_TOP_CASE_TAC >> gvs[mrec_sem_simps] >>
PURE_TOP_CASE_TAC >> gvs[mrec_sem_simps] >>
PURE_TOP_CASE_TAC >> gvs[mrec_sem_simps] >>
reverse $ PURE_TOP_CASE_TAC >> gvs[mrec_sem_simps] >>
rw[mrec_sem_monad_law] >>
AP_TERM_TAC >>
simp[FUN_EQ_THM] >>
PairCases >>
rw[] >>
PURE_TOP_CASE_TAC >> gvs[mrec_sem_simps] >>
PURE_TOP_CASE_TAC >> gvs[mrec_sem_simps]
QED
Theorem ltree_lift_state_lift:
ltree_lift query_oracle s.ffi (mrec_sem (h_prog (p,s))) ≈ Ret (res,s')
⇒
(ltree_lift_state query_oracle s.ffi (mrec_sem (h_prog (p,s)))) = s'.ffi
Proof
strip_tac >> dxrule itree_wbisim_Ret_FUNPOW >>
simp[PULL_EXISTS] >>
MAP_EVERY qid_spec_tac [‘p’,‘s’,‘res’,‘s'’] >>
Induct_on ‘n’ using COMPLETE_INDUCTION >>
CONV_TAC $ RESORT_FORALL_CONV rev >>
Cases
>~ [‘Dec’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_dec_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_dec_def,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau
] >>
gvs[mrec_sem_monad_law,ltree_lift_monad_law] >>
drule FUNPOW_Tau_bind_thm >>
rw[] >>
pairarg_tac >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau] >>
rename [‘ltree_lift _ _ _ = FUNPOW Tau mm _’] >>
last_x_assum $ qspec_then ‘mm’ mp_tac >>
impl_tac >- simp[] >>
disch_then $ resolve_then (Pos hd) mp_tac EQ_TRANS >>
disch_then $ drule_at $ Pos last >>
qmatch_goalsub_abbrev_tac ‘h_prog (a1,a2)’ >>
disch_then $ qspecl_then [‘a2’,‘a1’] mp_tac >>
unabbrev_all_tac >>
simp[] >>
strip_tac >>
qmatch_goalsub_abbrev_tac ‘_ >>= a1’ >>
drule_then (qspec_then ‘a1’ mp_tac) ltree_lift_state_bind_funpow >>
unabbrev_all_tac >>
rw[mrec_sem_simps,ltree_lift_state_simps])
>~ [‘If’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_cond_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_dec_def,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau
] >>
PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau
] >>
PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau
] >>
first_x_assum irule >>
first_x_assum $ irule_at Any >>
simp[])
>~ [‘While’]
>- (rw[ltree_lift_cases,mrec_sem_simps,
Once mrec_sem_while_unfold,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
PURE_REWRITE_TAC[Once mrec_sem_while_unfold] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
gvs[tau_eq_funpow_tau] >>
gvs[mrec_sem_monad_law,ltree_lift_monad_law] >>
drule FUNPOW_Tau_bind_thm >>
rw[] >>
pairarg_tac >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
FUNPOW_Tau_bind,FUNPOW_ADD] >>
last_assum $ drule_at Any >>
impl_tac >- simp[] >>
strip_tac >>
qpat_x_assum ‘ltree_lift _ s.ffi _ = _ ’ assume_tac >>
qmatch_goalsub_abbrev_tac ‘_ >>= a1’ >>
drule_then (qspec_then ‘a1’ mp_tac) ltree_lift_state_bind_funpow >>
unabbrev_all_tac >>
strip_tac >>
simp[mrec_sem_simps,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[ltree_lift_state_simps,
ltree_lift_cases,
tau_eq_funpow_tau
]
>- (last_x_assum $ irule_at $ Pos hd >>
first_x_assum $ irule_at $ Pos last >>
simp[]) >>
PURE_TOP_CASE_TAC >>
gvs[ltree_lift_state_simps,
ltree_lift_cases,
tau_eq_funpow_tau,
ret_eq_funpow_tau
] >>
last_x_assum $ irule_at $ Pos hd >>
first_x_assum $ irule_at $ Pos last >>
simp[])
>~ [‘ExtCall’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_ext_call_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
rpt(PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau]) >>
gvs[query_oracle_def,ELIM_UNCURRY,AllCaseEqs(),
tau_eq_funpow_tau,ret_eq_funpow_tau
] >>
rpt(PURE_FULL_CASE_TAC >> gvs[empty_locals_defs]))
>~ [‘ShMemLoad’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_sh_mem_load_def,nb_op_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
rpt(PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau]) >>
gvs[query_oracle_def,ELIM_UNCURRY,AllCaseEqs(),
tau_eq_funpow_tau,ret_eq_funpow_tau
] >>
rpt(PURE_FULL_CASE_TAC >> gvs[empty_locals_defs]))
>~ [‘ShMemStore’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_sh_mem_store_def,nb_op_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
rpt(PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau]) >>
gvs[query_oracle_def,ELIM_UNCURRY,AllCaseEqs(),
tau_eq_funpow_tau,ret_eq_funpow_tau
] >>
rpt(PURE_FULL_CASE_TAC >> gvs[empty_locals_defs]))
>~ [‘Call’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_call_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
tau_eq_funpow_tau] >>
gvs[mrec_sem_monad_law,ltree_lift_monad_law] >>
drule FUNPOW_Tau_bind_thm >>
rw[] >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
FUNPOW_Tau_bind] >>
irule EQ_TRANS >>
irule_at (Pos hd) ltree_lift_state_bind_funpow >>
first_assum $ irule_at $ Pos hd >>
rename [‘ltree_lift _ s.ffi _ = FUNPOW _ mm (Ret st)’] >>
Cases_on ‘st’ >>
last_assum $ qspec_then ‘mm’ mp_tac >>
impl_tac >- simp[] >>
disch_then $ resolve_then (Pos hd) mp_tac EQ_TRANS >>
disch_then $ drule_at $ Pos last >>
qmatch_goalsub_abbrev_tac ‘h_prog (a1,a2)’ >>
disch_then $ qspecl_then [‘a2’,‘a1’] mp_tac >>
unabbrev_all_tac >>
simp[] >>
strip_tac >>
gvs[] >>
gvs[oneline h_handle_call_ret_def] >>
rpt(IF_CASES_TAC ORELSE PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
tau_eq_funpow_tau,empty_locals_defs,
set_var_def,panSemTheory.set_var_def]) >>
qmatch_goalsub_abbrev_tac ‘_ _ a1.ffi (_ (_ (_, a2)))’ >>
‘a1.ffi = a2.ffi’ by(rw[Abbr ‘a1’, Abbr ‘a2’]) >>
pop_assum SUBST_ALL_TAC >>
first_x_assum irule >>
first_x_assum $ irule_at $ Pos last >>
simp[])
>~ [‘DecCall’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_deccall_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
tau_eq_funpow_tau] >>
gvs[mrec_sem_monad_law,ltree_lift_monad_law] >>
drule FUNPOW_Tau_bind_thm >>
rw[] >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
FUNPOW_Tau_bind] >>
irule EQ_TRANS >>
irule_at (Pos hd) ltree_lift_state_bind_funpow >>
first_assum $ irule_at $ Pos hd >>
rename [‘ltree_lift _ s.ffi _ = FUNPOW _ mm (Ret st)’] >>
Cases_on ‘st’ >>
last_assum $ qspec_then ‘mm’ mp_tac >>
impl_tac >- simp[] >>
disch_then $ resolve_then (Pos hd) mp_tac EQ_TRANS >>
disch_then $ drule_at $ Pos last >>
qmatch_goalsub_abbrev_tac ‘h_prog (a1,a2)’ >>
disch_then $ qspecl_then [‘a2’,‘a1’] mp_tac >>
unabbrev_all_tac >>
simp[] >>
strip_tac >>
gvs[] >>
gvs[oneline h_handle_deccall_ret_def] >>
(* TODO *)
rpt(IF_CASES_TAC ORELSE PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
tau_eq_funpow_tau,empty_locals_defs,
set_var_def,panSemTheory.set_var_def]) >>
gvs[mrec_sem_monad_law,ltree_lift_monad_law] >>
drule FUNPOW_Tau_bind_thm >>
rw[] >>
pairarg_tac >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau] >>
rename [‘ltree_lift _ _ _ = FUNPOW Tau mm _’] >>
last_x_assum $ qspec_then ‘mm’ mp_tac >>
impl_tac >- simp[] >>
disch_then $ resolve_then (Pos hd) mp_tac EQ_TRANS >>
disch_then $ drule_at $ Pos last >>
qmatch_goalsub_abbrev_tac ‘h_prog (a1,a2)’ >>
disch_then $ qspecl_then [‘a2’,‘a1’] mp_tac >>
unabbrev_all_tac >>
simp[] >>
strip_tac >>
qmatch_goalsub_abbrev_tac ‘_ >>= a1’ >>
drule_then (qspec_then ‘a1’ mp_tac) ltree_lift_state_bind_funpow >>
unabbrev_all_tac >>
rw[mrec_sem_simps,ltree_lift_state_simps])
>~ [‘Seq’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_seq_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
gvs[tau_eq_funpow_tau] >>
gvs[mrec_sem_monad_law,ltree_lift_monad_law] >>
drule FUNPOW_Tau_bind_thm >>
rw[] >>
pairarg_tac >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
FUNPOW_Tau_bind,FUNPOW_ADD] >>
last_assum $ drule_at Any >>
impl_tac >- simp[] >>
strip_tac >>
qpat_x_assum ‘ltree_lift _ s.ffi _ = _ ’ assume_tac >>
qmatch_goalsub_abbrev_tac ‘_ >>= a1’ >>
drule_then (qspec_then ‘a1’ mp_tac) ltree_lift_state_bind_funpow >>
unabbrev_all_tac >>
strip_tac >>
simp[mrec_sem_simps,ltree_lift_state_simps] >>
reverse IF_CASES_TAC >>
gvs[ltree_lift_state_simps,mrec_sem_simps,ltree_lift_cases,
ret_eq_funpow_tau,tau_eq_funpow_tau
] >>
last_x_assum irule >>
first_x_assum $ irule_at $ Pos last >>
simp[]
)
>~ [‘Raise’]
>- (pop_assum kall_tac >>
rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_raise_def,
ltree_lift_state_simps,
ret_eq_funpow_tau] >>
rpt(IF_CASES_TAC ORELSE PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
ltree_lift_state_simps,empty_locals_defs])) >>
rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_store_def,
h_prog_store_byte_def,
h_prog_assign_def,
h_prog_raise_def,
h_prog_return_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
rpt (PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau,empty_locals_defs])
QED
Theorem ltree_lift_state_lift':
ltree_lift query_oracle (s:('a,'b)state).ffi (mrec_sem (h_prog (p,t))) ≈ Ret (res,s') ∧
t.ffi = s.ffi ==>
(ltree_lift_state query_oracle s.ffi (mrec_sem (h_prog (p,t)))) = s'.ffi
Proof
metis_tac[ltree_lift_state_lift]
QED
Theorem stree_trace_bind_append:
ltree_lift f st t ≈ Ret x
⇒ stree_trace f p st (to_stree t >>= k) =
LAPPEND (stree_trace f p st (to_stree t)) (stree_trace f p (ltree_lift_state f st t) (k x))
Proof
strip_tac >> dxrule itree_wbisim_Ret_FUNPOW >>
simp[PULL_EXISTS] >>
MAP_EVERY qid_spec_tac [‘t’,‘st’] >>
Induct_on ‘n’ >>
rw[FUNPOW_SUC]
>- (Cases_on ‘t’ >> rw[] >>
gvs[ltree_lift_cases,to_stree_simps,wbisim_Ret_eq,stree_trace_simps,
ltree_lift_state_simps,ltree_lift_Vis_alt,ELIM_UNCURRY]) >>
Cases_on ‘t’ >> rw[] >>
gvs[ltree_lift_cases,to_stree_simps,wbisim_Ret_eq,stree_trace_simps,
stree_trace_Vis,ltree_lift_state_simps,ltree_lift_Vis_alt,ELIM_UNCURRY] >>
IF_CASES_TAC >> gvs[]
QED
Theorem stree_trace_ret_events:
ltree_lift query_oracle st.ffi (mrec_sem (h_prog (p,st))) ≈ Ret (res,st')
⇒ fromList st'.ffi.io_events =
LAPPEND (fromList st.ffi.io_events) (stree_trace query_oracle event_filter st.ffi (to_stree (mrec_sem (h_prog (p,st)))))
Proof
strip_tac >> dxrule itree_wbisim_Ret_FUNPOW >>
simp[PULL_EXISTS] >>
MAP_EVERY qid_spec_tac [‘p’,‘st’,‘res’,‘st'’] >>
Induct_on ‘n’ using COMPLETE_INDUCTION >>
CONV_TAC $ RESORT_FORALL_CONV rev >>
Induct
>~ [‘Dec’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_dec_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_dec_def,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau,
to_stree_simps,stree_trace_simps,
LAPPEND_NIL_2ND
] >>
gvs[mrec_sem_monad_law,ltree_lift_monad_law,
to_stree_monad_law] >>
drule FUNPOW_Tau_bind_thm >>
rw[] >>
pairarg_tac >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau] >>
rename [‘ltree_lift _ _ _ = FUNPOW Tau mm _’] >>
last_x_assum $ qspec_then ‘mm’ mp_tac >>
impl_tac >- simp[] >>
disch_then $ resolve_then (Pos hd) mp_tac EQ_TRANS >>
disch_then $ drule_at $ Pos last >>
qmatch_goalsub_abbrev_tac ‘h_prog (a1,a2)’ >>
disch_then $ qspecl_then [‘a2’,‘a1’] mp_tac >>
unabbrev_all_tac >>
simp[] >>
strip_tac >>
AP_TERM_TAC >>
CONV_TAC SYM_CONV >>
irule EQ_TRANS >>
irule_at (Pos hd) stree_trace_bind_append >>
irule_at (Pos hd) FUNPOW_Tau_imp_wbisim >>
first_x_assum $ irule_at $ Pos hd >>
simp[to_stree_simps,stree_trace_simps,mrec_sem_simps,LAPPEND_NIL_2ND]
)
>~ [‘If’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_cond_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_dec_def,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau,
to_stree_simps,stree_trace_simps,
LAPPEND_NIL_2ND
] >>
PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau,
to_stree_simps,stree_trace_simps,
LAPPEND_NIL_2ND
] >>
PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau,
to_stree_simps,stree_trace_simps,
LAPPEND_NIL_2ND
] >>
first_x_assum irule >>
first_x_assum $ irule_at Any >>
simp[])
>~ [‘While’]
>- (rw[ltree_lift_cases,mrec_sem_simps,
Once mrec_sem_while_unfold,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
PURE_REWRITE_TAC[Once mrec_sem_while_unfold] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
gvs[tau_eq_funpow_tau] >>
gvs[mrec_sem_monad_law,ltree_lift_monad_law,to_stree_monad_law] >>
drule FUNPOW_Tau_bind_thm >>
rw[] >>
pairarg_tac >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
FUNPOW_Tau_bind,FUNPOW_ADD] >>
last_assum $ drule_at Any >>
impl_tac >- simp[] >>
strip_tac >>
rename1 ‘ltree_lift _ s.ffi _ = FUNPOW _ _ (Ret (ress,ss))’ >>
qmatch_asmsub_abbrev_tac ‘ltree_lift _ a1 _ = FUNPOW _ n''' _’ >>
subgoal ‘a1 = ss.ffi’ >> unabbrev_all_tac
>- (irule ltree_lift_state_lift >>
irule_at (Pos hd) FUNPOW_Tau_imp_wbisim >>
first_x_assum $ irule_at $ Pos hd) >>
simp[] >>
gvs[] >>
Cases_on ‘ress’ >> gvs[ltree_lift_cases,tau_eq_funpow_tau] >>
TRY(rename1 ‘Ret (SOME rr, _)’ >> Cases_on ‘rr’ >>
gvs[ltree_lift_cases,tau_eq_funpow_tau]) >>
gvs[ret_eq_funpow_tau]
>~ [‘Ret (SOME Error, _)’]
>- (AP_TERM_TAC >>
CONV_TAC SYM_CONV >>
irule EQ_TRANS >>
irule_at (Pos hd) stree_trace_bind_append >>
irule_at (Pos hd) FUNPOW_Tau_imp_wbisim >>
first_x_assum $ irule_at $ Pos hd >>
simp[to_stree_simps,stree_trace_simps,mrec_sem_simps,LAPPEND_NIL_2ND]) >>
last_x_assum $ rev_drule_at $ Pos last >> rw[LAPPEND_ASSOC] >>
AP_TERM_TAC >>
CONV_TAC SYM_CONV >>
irule EQ_TRANS >>
irule_at (Pos hd) stree_trace_bind_append >>
irule_at (Pos hd) FUNPOW_Tau_imp_wbisim >>
first_x_assum $ irule_at $ Pos hd >>
simp[to_stree_simps,stree_trace_simps,mrec_sem_simps,LAPPEND_NIL_2ND])
>~ [‘ExtCall’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_ext_call_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
rpt(PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau]) >>
gvs[query_oracle_def,ELIM_UNCURRY,AllCaseEqs(),
tau_eq_funpow_tau,ret_eq_funpow_tau,
to_stree_simps,stree_trace_simps,
oneline event_filter_def,
LAPPEND_NIL_2ND
] >>
rpt(PURE_FULL_CASE_TAC >> gvs[empty_locals_defs]) >>
gvs[stree_trace_Vis,make_io_event_def,
ffiTheory.call_FFI_def,AllCaseEqs(),
query_oracle_def,to_stree_simps,mrec_sem_simps,stree_trace_simps,
GSYM LAPPEND_fromList,
oneline event_filter_def,
LAPPEND_NIL_2ND]
)
>~ [‘ShMemLoad’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_sh_mem_load_def,nb_op_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
rpt(PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau]) >>
gvs[query_oracle_def,ELIM_UNCURRY,AllCaseEqs(),
tau_eq_funpow_tau,ret_eq_funpow_tau,
to_stree_simps,stree_trace_simps,
oneline event_filter_def,
LAPPEND_NIL_2ND
] >>
rpt(PURE_FULL_CASE_TAC >> gvs[empty_locals_defs]) >>
gvs[stree_trace_Vis,make_io_event_def,
ffiTheory.call_FFI_def,AllCaseEqs(),
query_oracle_def,to_stree_simps,mrec_sem_simps,stree_trace_simps,
GSYM LAPPEND_fromList, oneline event_filter_def, LAPPEND_NIL_2ND])
>~ [‘ShMemStore’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_sh_mem_store_def,nb_op_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
rpt(PURE_TOP_CASE_TAC >>
gvs[ltree_lift_cases,h_prog_def,mrec_sem_simps,
ltree_lift_state_simps,ret_eq_funpow_tau,
tau_eq_funpow_tau]) >>
gvs[query_oracle_def,ELIM_UNCURRY,AllCaseEqs(),
tau_eq_funpow_tau,ret_eq_funpow_tau,
to_stree_simps,stree_trace_simps,
oneline event_filter_def,
LAPPEND_NIL_2ND
] >>
rpt(PURE_FULL_CASE_TAC >> gvs[empty_locals_defs]) >>
gvs[stree_trace_Vis,make_io_event_def,
ffiTheory.call_FFI_def,AllCaseEqs(),
query_oracle_def,to_stree_simps,mrec_sem_simps,stree_trace_simps,
GSYM LAPPEND_fromList, oneline event_filter_def, LAPPEND_NIL_2ND])
>~ [‘Call’]
>- (rw[ltree_lift_cases,h_prog_def,mrec_sem_simps,
h_prog_call_def,
ltree_lift_state_simps,
ret_eq_funpow_tau
] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,ltree_lift_state_simps,
tau_eq_funpow_tau,to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND
] >>
gvs[mrec_sem_monad_law,ltree_lift_monad_law,to_stree_monad_law] >>
drule FUNPOW_Tau_bind_thm >>
rw[] >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
FUNPOW_Tau_bind,FUNPOW_ADD] >>
rename1 ‘ltree_lift _ s.ffi _ = FUNPOW _ _ (Ret xx)’ >>
PairCases_on ‘xx’ >>
qmatch_asmsub_abbrev_tac ‘ltree_lift _ a1’ >>
subgoal ‘a1 = xx1.ffi’ >> unabbrev_all_tac
>- (PURE_REWRITE_TAC [Once $ GSYM $ SIMP_CONV (srw_ss()) [] “(s with locals := r).ffi”] >>
irule ltree_lift_state_lift >>
irule_at (Pos hd) FUNPOW_Tau_imp_wbisim >>
simp[] >> metis_tac[]) >>
simp[] >>
gvs[] >>
gvs[oneline h_handle_call_ret_def] >>
CONV_TAC SYM_CONV >>
irule EQ_TRANS >>
irule_at (Pos hd) $ METIS_PROVE [] “x = y ⇒ f x = f y” >>
irule_at (Pos hd) stree_trace_bind_append >>
irule_at (Pos hd) FUNPOW_Tau_imp_wbisim >>
first_assum $ irule_at $ Pos hd >>
simp[to_stree_simps,stree_trace_simps,mrec_sem_simps,LAPPEND_NIL_2ND] >>
CONV_TAC SYM_CONV >>
simp[oneline h_handle_call_ret_def] >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND]
>- (irule EQ_TRANS >>
last_x_assum $ irule_at $ Pos hd >>
irule_at Any EQ_TRANS >>
first_x_assum $ irule_at $ Pos $ hd o tl >>
qrefine ‘_ with locals := _’ >>
simp[] >>
irule_at (Pos hd) EQ_REFL >>
simp[]) >>
PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
simp[empty_locals_defs]
>- (irule EQ_TRANS >>
last_x_assum $ irule_at $ Pos hd >>
irule_at Any EQ_TRANS >>
first_x_assum $ irule_at $ Pos $ hd o tl >>
qrefine ‘_ with locals := _’ >>
simp[] >>
irule_at (Pos hd) EQ_REFL >>
simp[])
>- (irule EQ_TRANS >>
last_x_assum $ irule_at $ Pos hd >>
irule_at Any EQ_TRANS >>
first_x_assum $ irule_at $ Pos $ hd o tl >>
qrefine ‘_ with locals := _’ >>
simp[] >>
irule_at (Pos hd) EQ_REFL >>
simp[])
>- (irule EQ_TRANS >>
last_x_assum $ irule_at $ Pos hd >>
irule_at Any EQ_TRANS >>
first_x_assum $ irule_at $ Pos $ hd o tl >>
qrefine ‘_ with locals := _’ >>
simp[] >>
irule_at (Pos hd) EQ_REFL >>
simp[])
>- (irule EQ_TRANS >>
last_x_assum $ irule_at $ Pos hd >>
irule_at Any EQ_TRANS >>
first_x_assum $ irule_at $ Pos $ hd o tl >>
qrefine ‘_ with locals := _’ >>
simp[] >>
irule_at (Pos hd) EQ_REFL >>
simp[])
>- (PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
simp[empty_locals_defs]
>- (irule EQ_TRANS >>
last_x_assum $ irule_at $ Pos hd >>
irule_at Any EQ_TRANS >>
first_x_assum $ irule_at $ Pos $ hd o tl >>
qrefine ‘_ with locals := _’ >>
simp[] >>
irule_at (Pos hd) EQ_REFL >>
simp[]) >>
PURE_TOP_CASE_TAC >> gvs[] >> CASE_TAC
>- (gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,to_stree_simps,
stree_trace_simps,LAPPEND_NIL_2ND] >>
simp[set_var_def,panSemTheory.set_var_def] >>
irule EQ_TRANS >>
last_x_assum $ irule_at $ Pos hd >>
irule_at Any EQ_TRANS >>
first_x_assum $ irule_at $ Pos $ hd o tl >>
simp[] >>
qrefine ‘_ with locals := _’ >>
simp[] >>
irule_at (Pos hd) EQ_REFL >>
simp[]) >>
CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,to_stree_simps,
stree_trace_simps,LAPPEND_NIL_2ND] >>
simp[set_var_def,panSemTheory.set_var_def] >>
irule EQ_TRANS >>
last_x_assum $ irule_at $ Pos hd >>
irule_at Any EQ_TRANS >>
first_x_assum $ irule_at $ Pos $ hd o tl >>
simp[] >>
qrefine ‘_ with locals := _’ >>
simp[] >>
irule_at (Pos hd) EQ_REFL >>
simp[])
>- (PURE_TOP_CASE_TAC >>
gvs[mrec_sem_simps,ltree_lift_cases,ret_eq_funpow_tau,
to_stree_simps,stree_trace_simps,LAPPEND_NIL_2ND] >>
simp[empty_locals_defs]
>- (irule EQ_TRANS >>
last_x_assum $ irule_at $ Pos hd >>
irule_at Any EQ_TRANS >>
first_x_assum $ irule_at $ Pos $ hd o tl >>
qrefine ‘_ with locals := _’ >>
simp[] >>
irule_at (Pos hd) EQ_REFL >>