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evaluatePropsScript.sml
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(*
Properties of the operational semantics.
*)
open preamble evaluateTheory
namespaceTheory namespacePropsTheory
semanticPrimitivesTheory semanticPrimitivesPropsTheory
fpSemPropsTheory;
val _ = temp_delsimps ["lift_disj_eq", "lift_imp_disj"]
val _ = new_theory"evaluateProps";
Theorem call_FFI_LENGTH:
(call_FFI st index conf x = FFI_return new_st new_bytes) ==>
(LENGTH x = LENGTH new_bytes)
Proof
fs[ffiTheory.call_FFI_def] \\ every_case_tac \\ rw[] \\ fs[LENGTH_MAP]
QED
Definition call_FFI_rel_def:
call_FFI_rel s1 s2 <=> ?n conf bytes t. call_FFI s1 n conf bytes = FFI_return s2 t
End
Theorem call_FFI_rel_consts:
call_FFI_rel s1 s2 ⇒ (s2.oracle = s1.oracle)
Proof
rw[call_FFI_rel_def]
\\ fs[ffiTheory.call_FFI_def]
\\ fs[CaseEq"bool",CaseEq"oracle_result"]
\\ rw[]
QED
Theorem RTC_call_FFI_rel_consts:
∀s1 s2. RTC call_FFI_rel s1 s2 ⇒ (s2.oracle = s1.oracle)
Proof
once_rewrite_tac[EQ_SYM_EQ]
\\ match_mp_tac RTC_lifts_equalities
\\ rw[call_FFI_rel_consts]
QED
Definition dest_IO_event_def:
dest_IO_event (IO_event s c b) = (s,c,b)
End
val _ = export_rewrites["dest_IO_event_def"];
Definition io_events_mono_def:
io_events_mono s1 s2 ⇔
s1.io_events ≼ s2.io_events ∧
(s2.io_events = s1.io_events ⇒ s2 = s1)
End
Theorem io_events_mono_refl[simp]:
io_events_mono ffi ffi
Proof
rw[io_events_mono_def]
QED
Theorem io_events_mono_trans:
io_events_mono ffi1 ffi2 ∧ io_events_mono ffi2 ffi3 ⇒
io_events_mono ffi1 ffi3
Proof
rw[io_events_mono_def]
\\ metis_tac[IS_PREFIX_TRANS, IS_PREFIX_ANTISYM]
QED
Theorem io_events_mono_antisym:
io_events_mono s1 s2 ∧ io_events_mono s2 s1 ⇒ s1 = s2
Proof
rw[io_events_mono_def]
\\ imp_res_tac IS_PREFIX_ANTISYM
\\ rfs[]
QED
Theorem call_FFI_rel_io_events_mono:
∀s1 s2.
RTC call_FFI_rel s1 s2 ⇒ io_events_mono s1 s2
Proof
REWRITE_TAC[io_events_mono_def] \\
ho_match_mp_tac RTC_INDUCT
\\ simp[call_FFI_rel_def,ffiTheory.call_FFI_def]
\\ rpt gen_tac \\ strip_tac
\\ every_case_tac \\ fs[] \\ rveq \\ fs[]
\\ fs[IS_PREFIX_APPEND]
QED
Theorem do_app_call_FFI_rel:
do_app (r,ffi) op vs = SOME ((r',ffi'),res) ⇒
call_FFI_rel^* ffi ffi'
Proof
srw_tac[][do_app_cases] >> rw[] >>
FULL_CASE_TAC >>
fs[option_case_eq] >>
rpt (FULL_CASE_TAC \\ fs[]) >>
match_mp_tac RTC_SUBSET >> rw[call_FFI_rel_def] >> fs[] >> every_case_tac
>> fs[] >> metis_tac[]
QED
Theorem evaluate_call_FFI_rel:
(∀(s:'ffi state) e exp.
RTC call_FFI_rel s.ffi (FST (evaluate s e exp)).ffi) ∧
(∀(s:'ffi state) e v pes errv.
RTC call_FFI_rel s.ffi (FST (evaluate_match s e v pes errv)).ffi) ∧
(∀(s:'ffi state) e ds.
RTC call_FFI_rel s.ffi (FST (evaluate_decs s e ds)).ffi)
Proof
ho_match_mp_tac full_evaluate_ind >>
srw_tac[][full_evaluate_def, do_eval_res_def] >>
every_case_tac >> full_simp_tac(srw_ss())[] >>
fs[shift_fp_opts_def, astTheory.isFpBool_def] >>
imp_res_tac do_app_call_FFI_rel >>
rev_full_simp_tac(srw_ss())[dec_clock_def] >>
metis_tac[RTC_TRANSITIVE,transitive_def,FST]
QED
Theorem evaluate_call_FFI_rel_imp:
(∀s e p s' r.
evaluate s e p = (s',r) ⇒
RTC call_FFI_rel s.ffi s'.ffi) ∧
(∀s e v pes errv s' r.
evaluate_match s e v pes errv = (s',r) ⇒
RTC call_FFI_rel s.ffi s'.ffi) ∧
(∀s e p s' r.
evaluate_decs s e p = (s',r) ⇒
RTC call_FFI_rel s.ffi s'.ffi)
Proof
metis_tac[PAIR,FST,evaluate_call_FFI_rel]
QED
Triviality evaluate_decs_call_FFI_rel:
∀s e d.
RTC call_FFI_rel s.ffi (FST (evaluate_decs s e d)).ffi
Proof
ho_match_mp_tac evaluate_decs_ind >>
srw_tac[][evaluate_decs_def] >>
every_case_tac >> full_simp_tac(srw_ss())[] >>
metis_tac[RTC_TRANSITIVE,transitive_def,evaluate_call_FFI_rel,FST]
QED
Theorem evaluate_decs_call_FFI_rel_imp:
∀s e p s' r.
evaluate_decs s e p = (s',r) ⇒
RTC call_FFI_rel s.ffi s'.ffi
Proof
metis_tac[PAIR,FST,evaluate_decs_call_FFI_rel]
QED
Theorem do_app_io_events_mono:
do_app (r,ffi) op vs = SOME ((r',ffi'),res) ⇒ io_events_mono ffi ffi'
Proof
metis_tac[do_app_call_FFI_rel,call_FFI_rel_io_events_mono]
QED
Theorem evaluate_io_events_mono:
(∀(s:'ffi state) e exp.
io_events_mono s.ffi (FST (evaluate s e exp)).ffi) ∧
(∀(s:'ffi state) e v pes errv.
io_events_mono s.ffi (FST (evaluate_match s e v pes errv)).ffi) ∧
(∀s e d.
io_events_mono s.ffi (FST (evaluate_decs s e d)).ffi)
Proof
metis_tac[evaluate_call_FFI_rel,call_FFI_rel_io_events_mono]
QED
Theorem evaluate_io_events_mono_imp:
(∀s e p s' r.
evaluate s e p = (s',r) ⇒
io_events_mono s.ffi s'.ffi) ∧
(∀s e v pes errv s' r.
evaluate_match s e v pes errv = (s',r) ⇒
io_events_mono s.ffi s'.ffi) ∧
(∀s e p s' r.
evaluate_decs s e p = (s',r) ⇒
io_events_mono s.ffi s'.ffi)
Proof
metis_tac[PAIR,FST,evaluate_io_events_mono]
QED
Definition is_clock_io_mono_def:
is_clock_io_mono f s = (case f s of (s', r) =>
io_events_mono s.ffi s'.ffi
/\ s'.clock <= s.clock
/\ s.next_type_stamp <= s'.next_type_stamp
/\ s.next_exn_stamp <= s'.next_exn_stamp
/\ LENGTH s.refs <= LENGTH s'.refs
/\ (!clk. case f (s with clock := clk) of (s'', r') =>
(~ (r' = Rerr (Rabort Rtimeout_error))
==> ~ (r = Rerr (Rabort Rtimeout_error))
==> r' = r
/\ s'' = (s' with clock := clk - (s.clock - s'.clock)))
/\ (~ (r = Rerr (Rabort Rtimeout_error))
==> (clk >= s.clock - s'.clock
<=> ~ (r' = Rerr (Rabort Rtimeout_error))))
/\ (~ (r' = Rerr (Rabort Rtimeout_error))
==> clk <= s.clock
==> ~ (r = Rerr (Rabort Rtimeout_error)))
/\ (clk <= s.clock ==> io_events_mono s''.ffi s'.ffi)
))
End
Theorem is_clock_io_mono_cong:
s = t ==>
(!s. s.eval_state = t.eval_state /\ s.refs = t.refs /\ s.ffi = t.ffi ==>
f s = g s) ==>
(is_clock_io_mono f s <=> is_clock_io_mono g t)
Proof
simp [is_clock_io_mono_def]
QED
Theorem is_clock_io_mono_return:
is_clock_io_mono (\s. (s,Rval r)) s
Proof
fs [is_clock_io_mono_def]
QED
Theorem is_clock_io_mono_ret_fpOpt:
is_clock_io_mono (\s. (s with fp_state := (s.fp_state with canOpt := Strict), Rval v)) s
Proof
fs [is_clock_io_mono_def]
QED
Theorem is_clock_io_mono_err:
is_clock_io_mono (\s. (s,Rerr r)) s
Proof
fs [is_clock_io_mono_def]
QED
Theorem pair_CASE_eq_forall:
(case x of (a, b) => P a b) = (!a b. x = (a, b) ==> P a b)
Proof
Cases_on `x` \\ fs []
QED
Theorem is_clock_io_mono_bind:
is_clock_io_mono f s /\ (!s' r. f s = (s', r)
==> is_clock_io_mono (g r) s')
/\ (!s'. g (Rerr (Rabort Rtimeout_error)) s'
= (s', Rerr (Rabort Rtimeout_error)))
==> is_clock_io_mono (\s. case f s of (s', r) => g r s') s
Proof
fs [is_clock_io_mono_def]
\\ rpt (gen_tac ORELSE disch_tac)
\\ fs [pair_case_eq] \\ fs []
\\ simp_tac bool_ss [pair_CASE_eq_forall, pair_case_eq]
\\ rpt (FIRST [DISCH_TAC, GEN_TAC])
\\ full_simp_tac (bool_ss ++ pairSimps.PAIR_ss) []
\\ fsrw_tac [SATISFY_ss] [io_events_mono_trans]
\\ fs []
\\ rpt (gen_tac ORELSE disch_tac)
\\ fs [pair_CASE_eq_forall, pair_case_eq]
\\ rpt (FIRST [first_x_assum drule, disch_tac,
drule_then strip_assume_tac (METIS_PROVE [] ``(P ==> Q) ==> P \/ ~ P``)]
\\ fs [] \\ rfs [])
\\ fsrw_tac [SATISFY_ss] [io_events_mono_trans]
QED
Definition adj_clock_def:
adj_clock inc dec s = (s with clock := ((s.clock + inc) - dec))
End
Theorem is_clock_io_mono_check:
(~ (s.clock = 0) ==>
is_clock_io_mono (\s. f (adj_clock 1 0 s)) (dec_clock s))
==> is_clock_io_mono (\s. if s.clock = 0
then (s,Rerr (Rabort Rtimeout_error)) else f s) s
Proof
fs [is_clock_io_mono_def, dec_clock_def, adj_clock_def, with_same_clock]
\\ rpt (CASE_TAC ORELSE DISCH_TAC ORELSE GEN_TAC ORELSE CHANGED_TAC (fs []))
\\ fs [pair_CASE_eq_forall]
\\ first_x_assum (qspec_then `clk - 1` mp_tac)
\\ simp []
\\ rpt (CASE_TAC ORELSE DISCH_TAC ORELSE GEN_TAC ORELSE CHANGED_TAC (fs []))
\\ Cases_on `r' = Rerr (Rabort Rtimeout_error)` \\ fs []
QED
Theorem dec_inc_clock:
dec_clock (adj_clock 1 0 s) = s
Proof
simp [dec_clock_def, adj_clock_def, with_same_clock]
QED
Theorem do_app_refs_length:
do_app refs_ffi op vs = SOME res ==>
LENGTH (FST refs_ffi) <= LENGTH (FST (FST res))
Proof
rw [] \\ Cases_on `refs_ffi` \\ Cases_on `op` \\ fs [do_app_def]
\\ every_case_tac \\ fs []
\\ fs [store_assign_def,store_alloc_def]
\\ rveq \\ fs [] \\ rveq \\ fs[]
QED
Theorem is_clock_io_mono_do_app_simple:
! xs (st:'ffi state).
is_clock_io_mono (\st'.
case do_app (st'.refs, st'.ffi) op xs of
NONE => (st', Rerr (Rabort Rtype_error))
| SOME ((refs,ffi),r) =>
(st' with<| refs := refs; ffi := ffi |>, list_result r)) st
Proof
fs [is_clock_io_mono_def, shift_fp_opts_def]
\\ rpt (CASE_TAC ORELSE CHANGED_TAC (fs []) ORELSE CHANGED_TAC rveq ORELSE gen_tac)
\\ imp_res_tac do_app_refs_length \\ gs[]
\\ metis_tac [do_app_io_events_mono]
QED
Theorem is_clock_io_mono_do_app_icing:
! xs (st:'ffi state).
is_clock_io_mono (\st'.
case do_app (st'.refs,st'.ffi) op (REVERSE a) of
NONE => (st',Rerr (Rabort Rtype_error))
| SOME ((refs,ffi),r) =>
((if st'.fp_state.canOpt = FPScope Opt then
shift_fp_opts st'
else st') with <|refs := refs; ffi := ffi|>,
list_result
(if isFpBool op then
(case
if st'.fp_state.canOpt = FPScope Opt then
case
do_fprw r (st'.fp_state.opts 0)
st'.fp_state.rws
of
NONE => r
| SOME r_opt => r_opt
else r
of
Rval (Litv v21) => Rval (Litv v21)
| Rval (Conv v22 v23) => Rval (Conv v22 v23)
| Rval (Closure v24 v25 v26) =>
Rval (Closure v24 v25 v26)
| Rval (Recclosure v27 v28 v29) =>
Rval (Recclosure v27 v28 v29)
| Rval (Loc b v30) => Rval (Loc b v30)
| Rval (Vectorv v31) => Rval (Vectorv v31)
| Rval (FP_WordTree v32) => Rval (FP_WordTree v32)
| Rval (FP_BoolTree fv) =>
Rval (Boolv (compress_bool fv))
| Rval (Real v34) => Rval (Real v34)
| Rval (Env v35 v36) => Rval (Env v35 v36)
| Rerr v4 => Rerr v4)
else if st'.fp_state.canOpt = FPScope Opt then
(case
do_fprw r (st'.fp_state.opts 0) st'.fp_state.rws
of
NONE => r
| SOME r_opt => r_opt)
else r))) st
Proof
fs [is_clock_io_mono_def, shift_fp_opts_def]
\\ rpt (CASE_TAC ORELSE CHANGED_TAC (fs []) ORELSE CHANGED_TAC rveq ORELSE gen_tac)
\\ imp_res_tac do_app_refs_length \\ gs[]
\\ metis_tac [do_app_io_events_mono]
QED
Theorem is_clock_io_mono_acc_safe:
!v g. (!st clk. f (st with clock := clk) = f st) /\
(f st = v \/ f st <> v) /\
is_clock_io_mono (\st'. g (f st) st') st ==>
is_clock_io_mono (\st'. g (f st') st') st
Proof
rw [is_clock_io_mono_def]
QED
Theorem is_clock_io_mono_if_safe = is_clock_io_mono_acc_safe
|> ISPEC T |> Q.SPEC `\b st. if b then j st else k st`
|> SIMP_RULE bool_ss []
Theorem is_clock_io_mono_option_case_safe = is_clock_io_mono_acc_safe
|> Q.ISPEC `NONE` |> Q.SPEC `\v st. case v of NONE => g st | SOME x => h x st`
|> SIMP_RULE bool_ss []
Theorem is_clock_io_mono_match_case_safe = is_clock_io_mono_acc_safe
|> Q.ISPEC `No_match` |> Q.SPEC `\m st. case m of No_match => g st
| Match_type_error => h st | Match env => j env st`
|> SIMP_RULE bool_ss []
Theorem is_clock_io_mono_match_case_pair_safe = is_clock_io_mono_acc_safe
|> Q.ISPEC `No_match` |> Q.SPEC `\m st. (st, case m of No_match => g st
| Match_type_error => h st | Match env => j env st)`
|> SIMP_RULE bool_ss []
Theorem is_clock_io_mono_do_app_real:
! xs (st:'ffi state).
is_clock_io_mono (\st'.
if st'.fp_state.real_sem then
case do_app (st'.refs, st'.ffi) op xs of
NONE => (st', Rerr (Rabort Rtype_error))
| SOME ((refs,ffi),r) =>
(st' with<| refs := refs; ffi := ffi |>, list_result r)
else (shift_fp_opts st', Rerr (Rabort Rtype_error))) st
Proof
fs [is_clock_io_mono_def, shift_fp_opts_def]
\\ rpt (CASE_TAC ORELSE CHANGED_TAC (fs []) ORELSE CHANGED_TAC rveq ORELSE gen_tac)
\\ imp_res_tac do_app_refs_length \\ gs[]
\\ metis_tac [do_app_io_events_mono]
QED
Theorem is_clock_io_mono_fp_optimise:
! (s:'ffi state) env es.
is_clock_io_mono (\ s. evaluate s env [e])
(s with fp_state :=
(if s.fp_state.canOpt = Strict then s.fp_state else s.fp_state with canOpt := FPScope sc)) ==>
is_clock_io_mono (\ s. evaluate s env [FpOptimise sc e]) s
Proof
Cases_on `sc` \\ fs[is_clock_io_mono_def, evaluate_def]
\\ rpt gen_tac
\\ ntac 2 (TOP_CASE_TAC \\ fs[])
\\ rename [`evaluate _ env [e] = (s1, r1)`]
\\ Cases_on `r1` \\ fs[] \\ rveq \\ fs[]
\\ rpt strip_tac
\\ first_x_assum (qspec_then `clk` assume_tac)
\\ pop_assum mp_tac \\ ntac 2 (TOP_CASE_TAC \\ fs[])
\\ pop_assum mp_tac \\ TOP_CASE_TAC
\\ rpt strip_tac
\\ rveq \\ fs[fpState_component_equality, state_component_equality]
\\ qpat_x_assum `_.ffi = _.ffi` ( fn thm => fs[thm])
QED
val step_tac =
rpt (FIRST ([strip_tac]
@ map ho_match_mp_tac [is_clock_io_mono_bind, is_clock_io_mono_check]
@ [CHANGED_TAC (fs [Cong is_clock_io_mono_cong,
is_clock_io_mono_return, is_clock_io_mono_err,
do_eval_res_def, dec_inc_clock]), TOP_CASE_TAC]))
Theorem is_clock_io_mono_evaluate:
(!(s : 'ffi state) env es. is_clock_io_mono (\s. evaluate s env es) s) /\
(!(s : 'ffi state) env v pes err_v.
is_clock_io_mono (\s. evaluate_match s env v pes err_v) s) /\
(!(s : 'ffi state) env ds.
is_clock_io_mono (\s. evaluate_decs s env ds) s)
Proof
ho_match_mp_tac full_evaluate_ind
\\ rpt strip_tac \\ fs [full_evaluate_def, combine_dec_result_def]
\\ TRY (step_tac \\ NO_TAC)
\\ TRY (
drule (SIMP_RULE std_ss [evaluate_def] is_clock_io_mono_fp_optimise) \\fs[])
>- (
ho_match_mp_tac is_clock_io_mono_bind \\ fs[]
\\ rpt strip_tac
\\ ntac 2 (TOP_CASE_TAC
\\ fs [is_clock_io_mono_return, is_clock_io_mono_err])
>- (
fs[Cong is_clock_io_mono_cong, do_eval_res_def]
\\ ho_match_mp_tac is_clock_io_mono_bind
\\ gs[]
\\ rpt conj_tac
\\ step_tac \\ gs[is_clock_io_mono_def, fix_clock_def])
>- (
TOP_CASE_TAC \\ fs[]
\\ rpt (FIRST [CHANGED_TAC (fs[is_clock_io_mono_return, is_clock_io_mono_err,
is_clock_io_mono_do_app_simple]), CASE_TAC])
\\ ho_match_mp_tac is_clock_io_mono_check \\ gs[] \\ rpt strip_tac
\\ res_tac \\ gs[dec_inc_clock])
>- (assume_tac (SIMP_RULE std_ss [] is_clock_io_mono_do_app_simple) \\ fs[])
>- (assume_tac (SIMP_RULE std_ss [] is_clock_io_mono_do_app_icing) \\ gs[])
\\ assume_tac (SIMP_RULE std_ss [] is_clock_io_mono_do_app_real) \\ fs[])
>- (step_tac \\ fs[is_clock_io_mono_def])
>- (step_tac \\ fs[is_clock_io_mono_def])
\\ step_tac \\ fs[is_clock_io_mono_def]
\\ TRY (fs [is_clock_io_mono_def] \\ NO_TAC)
(* ho_match_mp_tac full_evaluate_ind
\\ rpt strip_tac \\ fs [full_evaluate_def,combine_dec_result_def]
\\ rpt (FIRST ([strip_tac]
@ map ho_match_mp_tac [is_clock_io_mono_bind, is_clock_io_mono_check]
@ [CHANGED_TAC (fs [Cong is_clock_io_mono_cong,
is_clock_io_mono_return, is_clock_io_mono_err,
do_eval_res_def, dec_inc_clock]), TOP_CASE_TAC]))
\\ imp_res_tac do_app_io_events_mono
\\ imp_res_tac do_app_refs_length
\\ TRY (fs [is_clock_io_mono_def] \\ NO_TAC) *)
QED
Theorem is_clock_io_mono_evaluate_decs:
!s e p. is_clock_io_mono (\s. evaluate_decs s e p) s
Proof
fs [is_clock_io_mono_evaluate]
QED
Theorem is_clock_io_mono_extra:
(!s. is_clock_io_mono f s)
==> f s = (s', r) /\ ~ (r = Rerr (Rabort Rtimeout_error))
==> f (s with clock := s.clock + extra)
= (s' with clock := s'.clock + extra,r)
Proof
DISCH_TAC
\\ FIRST_X_ASSUM (MP_TAC o Q.SPEC `s with clock := s.clock + extra`)
\\ fs [is_clock_io_mono_def]
\\ CASE_TAC
\\ rpt (DISCH_TAC ORELSE GEN_TAC)
\\ fs []
\\ FIRST_X_ASSUM (MP_TAC o Q.SPEC `s.clock`)
\\ fs [semanticPrimitivesPropsTheory.with_same_clock]
\\ rpt DISCH_TAC
\\ rpt (CHANGED_TAC (fs [semanticPrimitivesPropsTheory.with_same_clock]))
QED
Theorem is_clock_io_mono_extra_mono:
(!s. is_clock_io_mono f s)
==> io_events_mono (FST(f s)).ffi
(FST(f (s with clock := s.clock + extra))).ffi
Proof
DISCH_TAC
\\ FIRST_X_ASSUM (MP_TAC o Q.SPEC `s with clock := s.clock + extra`)
\\ fs [is_clock_io_mono_def]
\\ CASE_TAC
\\ rpt (DISCH_TAC ORELSE GEN_TAC)
\\ fs []
\\ FIRST_X_ASSUM (MP_TAC o Q.SPEC `s.clock`)
\\ fs [semanticPrimitivesPropsTheory.with_same_clock]
\\ CASE_TAC
QED
fun mk_extra_lemmas mp_rule monad_rule
= BODY_CONJUNCTS monad_rule
|> map (BETA_RULE o MATCH_MP mp_rule o Q.GEN `s`)
fun prove_extra mp_rule monad_rule
= simp_tac bool_ss (mk_extra_lemmas mp_rule monad_rule)
Theorem evaluate_add_to_clock:
!(s:'ffi state) env es s' r extra.
evaluate s env es = (s',r) ∧
r ≠ Rerr (Rabort Rtimeout_error) ⇒
evaluate (s with clock := s.clock + extra) env es =
(s' with clock := s'.clock + extra,r)
Proof
prove_extra is_clock_io_mono_extra is_clock_io_mono_evaluate
QED
Theorem evaluate_match_add_to_clock:
!(s:'ffi state) env v pes err_v s' r extra.
evaluate_match s env v pes err_v = (s', r) ∧
r ≠ Rerr (Rabort Rtimeout_error) ⇒
evaluate_match (s with clock := s.clock + extra) env v pes err_v =
(s' with clock := s'.clock + extra,r)
Proof
prove_extra is_clock_io_mono_extra is_clock_io_mono_evaluate
QED
Theorem list_result_eq_Rval[simp]:
list_result r = Rval r' ⇔ ∃v. r' = [v] ∧ r = Rval v
Proof
Cases_on`r`>>srw_tac[][list_result_def,EQ_IMP_THM]
QED
Theorem list_result_eq_Rerr[simp]:
list_result r = Rerr e ⇔ r = Rerr e
Proof
Cases_on`r`>>srw_tac[][list_result_def,EQ_IMP_THM]
QED
Theorem result_rel_list_result[simp]:
result_rel (LIST_REL R) Q (list_result r1) (list_result r2) ⇔
result_rel R Q r1 r2
Proof
Cases_on`r1`>>srw_tac[][PULL_EXISTS]
QED
Theorem list_result_inj:
list_result x = list_result y ⇒ x = y
Proof
Cases_on`x`>>Cases_on`y`>>EVAL_TAC
QED
Theorem do_fpoptimise_list_length[local]:
! vs.
LENGTH vs = n ==>
LENGTH (do_fpoptimise sc vs) = n
Proof
Induct_on `n` \\ fs[do_fpoptimise_def] \\ rpt strip_tac
\\ Cases_on `vs` \\ fs[] \\ res_tac \\ fs[do_fpoptimise_def, Once do_fpoptimise_cons]
\\ Cases_on `h` \\ fs[do_fpoptimise_def]
QED
Theorem evaluate_length:
(∀(s:'ffi state) e p s' r. evaluate s e p = (s',Rval r) ⇒ LENGTH r = LENGTH p) ∧
(∀(s:'ffi state) e v p er s' r. evaluate_match s e v p er = (s',Rval r) ⇒ LENGTH r = 1) ∧
(∀(s:'ffi state) e ds s' r. evaluate_decs s e ds = (s',Rval r) ⇒ T)
Proof
ho_match_mp_tac full_evaluate_ind >>
srw_tac[][evaluate_def,LENGTH_NIL] >> srw_tac[][] >>
every_case_tac >> full_simp_tac(srw_ss())[list_result_eq_Rval] >>
srw_tac[][] >> fs[] >>
every_case_tac >> fs[] >> rveq >> fs[do_fpoptimise_list_length]
QED
Theorem evaluate_nil[simp]:
∀(s:'ffi state) env. evaluate s env [] = (s,Rval [])
Proof
rw [evaluate_def]
QED
Theorem evaluate_sing:
∀(s:'ffi state) env e s' vs. evaluate s env [e] = (s',Rval vs) ⇒ ∃v. vs = [v]
Proof
rw []
>> imp_res_tac evaluate_length
>> Cases_on `vs`
>> fs []
>> Cases_on `t`
>> fs []
QED
Theorem evaluate_cons:
∀(s:'ffi state) env e es.
evaluate s env (e::es) =
case evaluate s env [e] of
| (s', Rval vs) =>
(case evaluate s' env es of
| (s'', Rval vs') => (s'', Rval (vs++vs'))
| err => err)
| err => err
Proof
Cases_on `es`
>> rw [evaluate_def]
>- every_case_tac
>> split_pair_case_tac
>> simp []
>> rename1 `evaluate _ _ _ = (st',r)`
>> Cases_on `r`
>> simp []
>> split_pair_case_tac
>> simp []
>> rename1 `evaluate _ _ (e'::es) = (st'',r)`
>> Cases_on `r`
>> simp []
>> drule evaluate_sing
>> rw [] >> rw[]
QED
Theorem evaluate_append:
∀(s:'ffi state) env xs ys.
evaluate s env (xs ++ ys) =
case evaluate s env xs of
| (s', Rval vs) =>
(case evaluate s' env ys of
| (s'', Rval vs') => (s'', Rval (vs++vs'))
| err => err)
| err => err
Proof
Induct_on `xs`
THEN1
(rw [] \\ Cases_on `evaluate s env ys` \\ fs []
\\ Cases_on `r` \\ fs [])
\\ fs [] \\ once_rewrite_tac [evaluate_cons]
\\ rw [] \\ Cases_on `evaluate s env [h]` \\ fs []
\\ Cases_on `r` \\ fs []
\\ every_case_tac \\ fs []
QED
Theorem evaluate_decs_nil[simp]:
∀(s:'ffi state) env.
evaluate_decs s env [] = (s,Rval <| v := nsEmpty; c := nsEmpty |>)
Proof
rw [evaluate_decs_def]
QED
Theorem evaluate_decs_cons:
∀(s:'ffi state) env d ds.
evaluate_decs s env (d::ds) =
case evaluate_decs s env [d] of
| (s1, Rval env1) =>
(case evaluate_decs s1 (extend_dec_env env1 env) ds of
| (s2, r) => (s2, combine_dec_result env1 r)
| err => err)
| err => err
Proof
Cases_on `ds`
>> rw [evaluate_decs_def]
>> split_pair_case_tac
>> simp []
>> rename1 `evaluate_decs _ _ _ = (s1,r)`
>> Cases_on `r`
>> simp [combine_dec_result_def, sem_env_component_equality]
QED
Theorem evaluate_decs_append:
∀ds1 s env ds2.
evaluate_decs s env (ds1 ++ ds2) =
case evaluate_decs s env ds1 of
(s1,Rval env1) =>
(case evaluate_decs s1 (extend_dec_env env1 env) ds2 of
(s2,r) => (s2,combine_dec_result env1 r))
| (s1,Rerr v7) => (s1,Rerr v7)
Proof
Induct \\ rw []
>- (
rw [extend_dec_env_def, combine_dec_result_def]
\\ rpt CASE_TAC)
\\ once_rewrite_tac [evaluate_decs_cons] \\ simp []
\\ gs [combine_dec_result_def, extend_dec_env_def]
\\ rpt CASE_TAC \\ gs []
QED
Theorem evaluate_match_list_result:
evaluate_match s e v p er = (s',r) ⇒
∃r'. r = list_result r'
Proof
Cases_on`r` >> srw_tac[][] >>
imp_res_tac evaluate_length >|[
Cases_on`a` >> full_simp_tac(srw_ss())[LENGTH_NIL],all_tac] >>
metis_tac[list_result_def]
QED
val evaluate_decs_lemmas
= BODY_CONJUNCTS is_clock_io_mono_evaluate_decs
|> map (BETA_RULE o MATCH_MP is_clock_io_mono_extra o Q.GEN `s`)
Theorem evaluate_decs_add_to_clock:
!s e p s' r extra.
evaluate_decs s e p = (s',r) ∧
r ≠ Rerr (Rabort Rtimeout_error) ⇒
evaluate_decs (s with clock := s.clock + extra) e p =
(s' with clock := s'.clock + extra,r)
Proof
simp_tac bool_ss evaluate_decs_lemmas
QED
Triviality add_lemma:
!(k:num) k'. ?extra. k = k' + extra ∨ k' = k + extra
Proof
intLib.ARITH_TAC
QED
Triviality with_clock_ffi:
(s with clock := k).ffi = s.ffi
Proof
EVAL_TAC
QED
Theorem evaluate_decs_clock_determ:
!s e p s1 r1 s2 r2 k1 k2.
evaluate_decs (s with clock := k1) e p = (s1,r1) ∧
evaluate_decs (s with clock := k2) e p = (s2,r2)
⇒
case (r1,r2) of
| (Rerr (Rabort Rtimeout_error), Rerr (Rabort Rtimeout_error)) =>
T
| (Rerr (Rabort Rtimeout_error), _) =>
k1 < k2
| (_, Rerr (Rabort Rtimeout_error)) =>
k2 < k1
| _ =>
s1.ffi = s2.ffi ∧ r1 = r2
Proof
rw []
>> Cases_on `r2 = Rerr (Rabort Rtimeout_error)`
>> Cases_on `r1 = Rerr (Rabort Rtimeout_error)`
>> fs []
>> fs []
>> fs []
>> rw []
>- (
`k2 < k1` suffices_by (every_case_tac >> fs [])
>> CCONTR_TAC
>> `?extra. k2 = k1 + extra` by intLib.ARITH_TAC
>> qpat_x_assum `evaluate_decs _ _ _ = _` mp_tac
>> drule evaluate_decs_add_to_clock
>> rw [])
>- (
`k1 < k2` suffices_by (every_case_tac >> fs [])
>> CCONTR_TAC
>> `?extra. k1 = k2 + extra` by intLib.ARITH_TAC
>> drule evaluate_decs_add_to_clock
>> fs []
>> qexists_tac `extra`
>> simp [])
>- (
every_case_tac >>
fs [] >>
rw [] >>
`(?extra. k1 = k2 + extra) ∨ (?extra. k2 = k1 + extra)`
by intLib.ARITH_TAC >>
rw [] >>
imp_res_tac evaluate_decs_add_to_clock >>
fs [] >>
rw [])
QED
Theorem evaluate_add_to_clock_io_events_mono:
(∀(s:'ffi state) e d extra.
io_events_mono (FST(evaluate s e d)).ffi
(FST(evaluate (s with clock := s.clock + extra) e d)).ffi) ∧
(∀(s:'ffi state) e v d er extra.
io_events_mono (FST(evaluate_match s e v d er)).ffi
(FST(evaluate_match (s with clock := s.clock + extra) e v d er)).ffi)
Proof
prove_extra is_clock_io_mono_extra_mono is_clock_io_mono_evaluate
QED
Theorem evaluate_decs_add_to_clock_io_events_mono:
∀s e d.
io_events_mono
(FST(evaluate_decs s e d)).ffi
(FST(evaluate_decs (s with clock := s.clock + extra) e d)).ffi
Proof
prove_extra is_clock_io_mono_extra_mono is_clock_io_mono_evaluate_decs
QED
Theorem evaluate_decs_ffi_mono_clock:
∀k1 k2 s e p.
k1 ≤ k2 ⇒
io_events_mono
(FST (evaluate_decs (s with clock := k1) e p)).ffi
(FST (evaluate_decs (s with clock := k2) e p)).ffi
Proof
metis_tac [is_clock_io_mono_evaluate_decs
|> Q.SPEC `s with clock := k1`
|> SIMP_RULE (srw_ss ()) [is_clock_io_mono_def, pair_CASE_def]]
QED
(* due to Eval this is no longer true
Theorem evaluate_state_unchanged:
(!(st:'ffi state) env es st' r.
evaluate st env es = (st', r)
⇒
st'.next_type_stamp = st.next_type_stamp ∧
st'.next_exn_stamp = st.next_exn_stamp) ∧
(!(st:'ffi state) env v pes err_v st' r.
evaluate_match st env v pes err_v = (st', r)
⇒
st'.next_type_stamp = st.next_type_stamp ∧
st'.next_exn_stamp = st.next_exn_stamp)
Proof
ho_match_mp_tac evaluate_ind
>> rw [evaluate_def]
>> every_case_tac
>> fs []
>> rw [dec_clock_def, shift_fp_opts_def]
QED
*)
Theorem evaluate_next_type_stamp_mono:
(evaluate (s:'ffi state) env es = (s',res1) ==>
s.next_type_stamp ≤ s'.next_type_stamp) /\
(evaluate_match (s:'ffi state) env v pes err_v = (s',res2) ==>
s.next_type_stamp ≤ s'.next_type_stamp) /\
(evaluate_decs (s:'ffi state) env ds = (s',res3) ==>
s.next_type_stamp ≤ s'.next_type_stamp)
Proof
rpt conj_tac \\ strip_tac
\\ assume_tac (is_clock_io_mono_evaluate |> CONJUNCT1 |> SPEC_ALL)
\\ assume_tac (is_clock_io_mono_evaluate |> CONJUNCT2 |> CONJUNCT1 |> SPEC_ALL)
\\ assume_tac (is_clock_io_mono_evaluate |> CONJUNCT2 |> CONJUNCT2 |> SPEC_ALL)
\\ fs [is_clock_io_mono_def] \\ rfs []
QED
Theorem evaluate_next_exn_stamp_mono:
(evaluate (s:'ffi state) env es = (s',res1) ==>
s.next_exn_stamp ≤ s'.next_exn_stamp) /\
(evaluate_match (s:'ffi state) env v pes err_v = (s',res2) ==>
s.next_exn_stamp ≤ s'.next_exn_stamp) /\
(evaluate_decs (s:'ffi state) env ds = (s',res3) ==>
s.next_exn_stamp ≤ s'.next_exn_stamp)
Proof
rpt conj_tac \\ strip_tac
\\ assume_tac (is_clock_io_mono_evaluate |> CONJUNCT1 |> SPEC_ALL)
\\ assume_tac (is_clock_io_mono_evaluate |> CONJUNCT2 |> CONJUNCT1 |> SPEC_ALL)
\\ assume_tac (is_clock_io_mono_evaluate |> CONJUNCT2 |> CONJUNCT2 |> SPEC_ALL)
\\ fs [is_clock_io_mono_def] \\ rfs []
QED
Theorem evaluate_case_eqs = LIST_CONJ
[pair_case_eq, result_case_eq, error_result_case_eq, bool_case_eq,
option_case_eq, list_case_eq, exp_or_val_case_eq, match_result_case_eq]
Theorem evaluate_set_next_stamps:
(∀(s0:'a state) env xs s1 res.
evaluate s0 env xs = (s1,res) ==>
(s1.next_type_stamp = s0.next_type_stamp ==>
!k. evaluate (s0 with next_type_stamp := k) env xs =
(s1 with next_type_stamp := k,res)) ∧
(s1.next_exn_stamp = s0.next_exn_stamp ==>
!k. evaluate (s0 with next_exn_stamp := k) env xs =
(s1 with next_exn_stamp := k,res))) ∧
(∀(s0:'a state) env v pes errv s1 res.
evaluate_match s0 env v pes errv = (s1,res) ==>
(s1.next_type_stamp = s0.next_type_stamp ==>
!k. evaluate_match (s0 with next_type_stamp := k) env v pes errv =
(s1 with next_type_stamp := k,res)) ∧
(s1.next_exn_stamp = s0.next_exn_stamp ==>
!k. evaluate_match (s0 with next_exn_stamp := k) env v pes errv =
(s1 with next_exn_stamp := k,res))) ∧
(∀(s0:'a state) env ds s1 res.
evaluate_decs s0 env ds = (s1,res) ==>
(s1.next_type_stamp = s0.next_type_stamp ==>
!k. evaluate_decs (s0 with next_type_stamp := k) env ds =
(s1 with next_type_stamp := k,res)) ∧
(s1.next_exn_stamp = s0.next_exn_stamp ==>
!k. evaluate_decs (s0 with next_exn_stamp := k) env ds =
(s1 with next_exn_stamp := k,res)))
Proof
ho_match_mp_tac full_evaluate_ind
\\ fs [full_evaluate_def]
\\ rpt conj_tac \\ rpt gen_tac \\ strip_tac \\ rpt gen_tac
\\ strip_tac
\\ fs [evaluate_case_eqs, dec_clock_def, do_eval_res_def, shift_fp_opts_def]
\\ TRY (Cases_on ‘getOpClass op’)
\\ fs [evaluate_case_eqs, dec_clock_def, do_eval_res_def]
\\ rveq \\ fs []
\\ fs [Q.ISPEC `(_, _)` EQ_SYM_EQ]
\\ rveq \\ fs []
\\ imp_res_tac evaluate_next_type_stamp_mono
\\ imp_res_tac evaluate_next_exn_stamp_mono
\\ rw []
\\ fs [build_tdefs_def]
\\ qpat_x_assum `fix_clock _ _ = _` mp_tac
\\ rpt (TOP_CASE_TAC \\ gs[fix_clock_def])
\\ rpt strip_tac \\ rveq
\\ gs[fix_clock_def]
QED
Theorem call_FFI_return_unchanged:
call_FFI ffi s conf bytes = FFI_return ffi bytes' <=>
(s = ExtCall "" /\ bytes' = bytes)
Proof
simp [ffiTheory.call_FFI_def]
\\ every_case_tac
\\ simp [ffiTheory.ffi_state_component_equality]
\\ TRY EQ_TAC
\\ simp []
QED
Theorem do_app_ffi_unchanged:
do_app (refs, ffi) op vs = SOME ((refs',ffi),r) ==>
(∀outcome. r ≠ Rerr (Rabort (Rffi_error outcome))) ==>
!ffi2. do_app (refs, ffi2) op vs = SOME ((refs',ffi2), r)
Proof
disch_then (strip_assume_tac o REWRITE_RULE [do_app_cases])
\\ rw [do_app_def] \\ rveq \\ fs []
\\ every_case_tac \\ rveq \\ fs [] \\ rveq \\ fs []
\\ fs [call_FFI_return_unchanged,
Q.SPECL [`x`, `ExtCall ""`] ffiTheory.call_FFI_def]
\\ rveq \\ fs []
\\ fs [store_assign_def, store_lookup_def]
\\ rfs [store_v_same_type_def]
QED
val trivial =
rpt strip_tac \\ rveq
\\ fs[fpState_component_equality, state_component_equality];
val by_eq =
`s1.fp_state.choices = s2.fp_state.choices`
by (imp_res_tac fpSemPropsTheory.evaluate_fp_opts_inv
\\ fs[fpState_component_equality, dec_clock_def])
\\ `s1.fp_state = s2.fp_state`
by ( (drule fpSemPropsTheory.evaluate_fp_stable \\ disch_then drule \\ fs[]) ORELSE (
imp_res_tac evaluate_fp_opts_inv \\ gs[state_component_equality, fpState_component_equality, FUN_EQ_THM]
\\ rpt strip_tac \\ qpat_x_assum `∀ x. q.fp_state.opts _ = _` $ gs o single o GSYM)
\\ ‘s1.fp_state.choices = s2.fp_state.choices’
by (imp_res_tac fpSemPropsTheory.evaluate_fp_opts_inv
\\ fs[fpState_component_equality, dec_clock_def])
\\ gs[])
\\ fs[fpState_component_equality, state_component_equality];
Theorem evaluate_fp_intro_strict:
(∀ (s:'a state) env e s' r.
evaluate s env e = (s', r) ∧
s.fp_state = s'.fp_state ∧
~ s.fp_state.real_sem ⇒
!fp_state2.
fp_state2.canOpt = Strict ⇒
evaluate (s with fp_state := fp_state2) env e = (s' with fp_state := fp_state2, r))
∧
(∀ (s:'a state) env v pes errv s' r.
evaluate_match s env v pes errv = (s', r) ∧
s.fp_state = s'.fp_state ∧
~ s.fp_state.real_sem ⇒
∀ fp_state2.
fp_state2.canOpt = Strict ⇒
evaluate_match (s with fp_state := fp_state2) env v pes errv = (s' with fp_state := fp_state2, r))
∧
(∀ (s:'a state) env decs s' r.
evaluate_decs s env decs = (s', r) ∧
s.fp_state = s'.fp_state ∧
~ s.fp_state.real_sem ⇒
∀ fp_state2.
fp_state2.canOpt = Strict ⇒
evaluate_decs (s with fp_state := fp_state2) env decs = (s' with fp_state := fp_state2, r))
Proof
ho_match_mp_tac full_evaluate_ind \\ rpt strip_tac
\\ fs[full_evaluate_def, state_component_equality, fpState_component_equality]
\\ qpat_x_assum `_ = (_, _)` mp_tac
>- (
ntac 2 (reverse TOP_CASE_TAC \\ fs[]) >- trivial
\\ ntac 2 (reverse TOP_CASE_TAC \\ fs[])
\\ rpt strip_tac \\ rveq \\ fs[]
\\ rename [`evaluate s1 env [e1] = (s2, Rval r)`,
`evaluate s2 env _ = (s3, _)`]
\\ by_eq)
>- (ntac 2 (TOP_CASE_TAC \\ fs[]) \\ trivial)
>- (ntac 2 (TOP_CASE_TAC \\ fs[]) >- trivial
\\ reverse TOP_CASE_TAC \\ fs[] >- trivial
\\ reverse TOP_CASE_TAC \\ fs[] >- trivial
\\ strip_tac
\\ rename [`evaluate s1 env [e1] = (s2, _)`,
`evaluate_match s2 env _ _ _ = (s3, _)`]
\\ by_eq)
>- (ntac 3 (reverse TOP_CASE_TAC \\ fs[]) >- trivial
\\ TOP_CASE_TAC \\ fs[state_component_equality, fpState_component_equality])
>- (TOP_CASE_TAC \\ fs[state_component_equality, fpState_component_equality])
>- (ntac 2 (reverse TOP_CASE_TAC \\ fs[]) >- trivial
\\ TOP_CASE_TAC \\ fs[]