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evaluate_skipScript.sml
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evaluate_skipScript.sml
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(*
Lemmas used in repl_typesTheory. These lemmas show that a plain
evaluate run can be replicated in a state with junk refs, extra type
stamps and unused exception definitions.
*)
open preamble
open evaluateTheory semanticPrimitivesTheory evaluatePropsTheory
open namespacePropsTheory ml_progTheory
local open helperLib in end
val _ = new_theory "evaluate_skip";
val _ = numLib.prefer_num ();
Theorem NOT_NIL_CONS:
xs ≠ [] ⇔ ∃y ys. xs = y::ys
Proof
Cases_on ‘xs’ \\ gs []
QED
Inductive stamp_rel:
(∀ft fe n m1 m2.
FLOOKUP ft m1 = SOME m2 ⇒
stamp_rel ft fe (TypeStamp n m1) (TypeStamp n m2))
∧
(∀ft fe m1 m2.
FLOOKUP fe m1 = SOME m2 ⇒
stamp_rel ft fe (ExnStamp m1) (ExnStamp m2))
End
Definition ctor_rel_def:
ctor_rel ft fe envc1 (envc2: (modN, conN, (num # stamp)) namespace) ⇔
nsAll2 (λid. ($= ### stamp_rel ft fe)) envc1 envc2
End
Inductive v_rel:
(∀(fr: num |-> num) (ft: num |-> num) (fe: num |-> num) l.
v_rel fr ft fe (Litv l) (Litv l))
∧
(∀fr ft fe fp.
v_rel fr ft fe (FP_WordTree fp) (FP_WordTree fp))
∧
(∀fr ft fe fp.
v_rel fr ft fe (FP_BoolTree fp) (FP_BoolTree fp))
∧
(∀fr ft fe r.
v_rel fr ft fe (Real r) (Real r))
∧
(∀fr ft fe t1 t2 xs ys.
LIST_REL (v_rel fr ft fe) xs ys ∧ OPTREL (stamp_rel ft fe) t1 t2 ⇒
v_rel fr ft fe (Conv t1 xs) (Conv t2 ys))
∧
(∀fr ft fe xs ys.
LIST_REL (v_rel fr ft fe) xs ys ⇒
v_rel fr ft fe (Vectorv xs) (Vectorv ys))
∧
(∀fr ft fe l1 l2.
FLOOKUP fr l1 = SOME l2 ⇒
v_rel fr ft fe (Loc l1) (Loc l2))
∧
(∀fr ft fe env1 env2 n e.
env_rel fr ft fe env1 env2 ⇒
v_rel fr ft fe (Closure env1 n e) (Closure env2 n e))
∧
(∀fr ft fe env1 env2 fns n.
env_rel fr ft fe env1 env2 ⇒
v_rel fr ft fe (Recclosure env1 fns n) (Recclosure env2 fns n))
∧
(∀fr ft fe env1 env2 ns.
env_rel fr ft fe env1 env2 ⇒
v_rel fr ft fe (Env env1 ns) (Env env2 ns))
∧
(∀fr ft fe env1 env2.
ctor_rel ft fe env1.c env2.c ∧
nsAll2 (λid. v_rel fr ft fe) env1.v env2.v ⇒
env_rel fr ft fe env1 env2)
End
Theorem v_rel_def =
[“v_rel fr ft fe (Conv opt vs) v”,
“v_rel fr ft fe (Closure env n x) v”,
“v_rel fr ft fe (Recclosure env f n) v”,
“v_rel fr ft fe (Vectorv vs) v”,
“v_rel fr ft fe (Litv lit) v”,
“v_rel fr ft fe (FP_WordTree fp) v”,
“v_rel fr ft fe (FP_BoolTree fp) v”,
“v_rel fr ft fe (Real r) v”,
“v_rel fr ft fe (Loc loc) v”,
“v_rel fr ft fe (Env env ns) v”,
“v_rel fr ft fe v (Conv opt vs)”,
“v_rel fr ft fe v (Closure env n x)”,
“v_rel fr ft fe v (Recclosure env f n)”,
“v_rel fr ft fe v (Vectorv vs)”,
“v_rel fr ft fe v (Litv lit)”,
“v_rel fr ft fe v (FP_WordTree fp)”,
“v_rel fr ft fe v (FP_BoolTree fp)”,
“v_rel fr ft fe v (Real r)”,
“v_rel fr ft fe v (Loc loc)”,
“v_rel fr ft fe v (Env env ns)”]
|> map (SIMP_CONV (srw_ss()) [Once v_rel_cases])
|> LIST_CONJ;
Theorem env_rel_def =
“env_rel fr ft fe env1 env2” |> SIMP_CONV std_ss [Once v_rel_cases];
Definition ref_rel_def:
ref_rel f (Refv v) (Refv w) = f v w ∧
ref_rel f (Varray vs) (Varray ws) = LIST_REL f vs ws ∧
ref_rel f (W8array a) (W8array b) = (a = b) ∧
ref_rel f _ _ = F
End
Theorem ref_rel_mono:
ref_rel P v w ∧
(∀v w. P v w ⇒ Q v w) ⇒
ref_rel Q v w
Proof
Cases_on ‘v’ \\ Cases_on ‘w’ \\ gs [ref_rel_def] \\ rw []
\\ irule LIST_REL_mono
\\ first_assum (irule_at Any) \\ gs []
QED
Theorem ref_rel_refl:
(∀x. P x x) ⇒
ref_rel P x x
Proof
Cases_on ‘x’ \\ rw [ref_rel_def]
\\ Induct_on ‘l’ \\ gs []
QED
Definition state_rel_def:
state_rel l fr ft fe (s:'ffi semanticPrimitives$state)
(t:'ffi semanticPrimitives$state) ⇔
INJ ($' fr) (FDOM fr) (count (LENGTH t.refs)) ∧
INJ ($' ft) (FDOM ft) (count t.next_type_stamp) ∧
INJ ($' fe) (FDOM fe) (count t.next_exn_stamp) ∧
FDOM fr = count (LENGTH s.refs) ∧
FDOM ft = count s.next_type_stamp ∧
FDOM fe = count s.next_exn_stamp ∧
t.clock = s.clock ∧
s.eval_state = NONE ∧
t.eval_state = NONE ∧
s.fp_state.canOpt = Strict ∧ (* No FP optimizations allowed *)
s.fp_state.real_sem = F ∧
s.fp_state = t.fp_state ∧
t.ffi = s.ffi ∧
(∀n. n < l ⇒ FLOOKUP fr n = SOME (n:num) ∧ n < LENGTH s.refs) ∧
FLOOKUP ft bool_type_num = SOME bool_type_num ∧
FLOOKUP ft list_type_num = SOME list_type_num ∧
FLOOKUP fe 0 = SOME 0 ∧ (* Bind *)
FLOOKUP fe 1 = SOME 1 ∧ (* Chr *)
FLOOKUP fe 2 = SOME 2 ∧ (* Div *)
FLOOKUP fe 3 = SOME 3 ∧ (* Subscript *)
(∀n. if n < LENGTH s.refs then
(∃m. FLOOKUP fr n = SOME m ∧
m < LENGTH t.refs ∧
ref_rel (v_rel fr ft fe) (EL n s.refs) (EL m t.refs))
else FLOOKUP fr n = NONE)
End
Definition res_rel_def[simp]:
(res_rel f g (Rval e) (Rval e1) ⇔ f e e1) ∧
(res_rel f g (Rerr (Rraise e)) (Rerr (Rraise e1)) ⇔ g e e1) ∧
(res_rel f g (Rerr (Rabort e)) (Rerr (Rabort e1)) ⇔ e = e1) ∧
(res_rel f g x y ⇔ F)
End
Theorem res_rel_abort[simp]:
res_rel R Q (Rerr (Rabort x)) r ⇔ r = Rerr (Rabort x)
Proof
Cases_on ‘r’ \\ gs []
\\ Cases_on ‘e’ \\ gs []
\\ rw [EQ_SYM_EQ]
QED
Theorem stamp_rel_update:
∀ft fe x y.
stamp_rel ft fe x y ⇒
∀ft1 fe1.
ft ⊑ ft1 ∧ fe ⊑ fe1 ⇒
stamp_rel ft1 fe1 x y
Proof
ho_match_mp_tac stamp_rel_ind \\ rw []
\\ gs [stamp_rel_rules, FLOOKUP_SUBMAP, SF SFY_ss]
QED
Theorem ctor_rel_update:
ctor_rel ft fe envc1 envc2 ∧
ft ⊑ ft1 ∧
fe ⊑ fe1 ⇒
ctor_rel ft1 fe1 envc1 envc2
Proof
rw [ctor_rel_def]
\\ irule nsAll2_mono
\\ first_assum (irule_at Any)
\\ simp [FORALL_PROD] \\ rw []
\\ irule stamp_rel_update
\\ gs [SF SFY_ss]
QED
Theorem v_rel_update_lemma:
(∀fr ft fe v w.
v_rel fr ft fe v w ⇒
∀fr1 ft1 fe1.
fr ⊑ fr1 ∧
ft ⊑ ft1 ∧
fe ⊑ fe1 ⇒
v_rel fr1 ft1 fe1 v w) ∧
(∀fr ft fe env1 env2.
env_rel fr ft fe env1 env2 ⇒
∀fr1 ft1 fe1.
fr ⊑ fr1 ∧
ft ⊑ ft1 ∧
fe ⊑ fe1 ⇒
env_rel fr1 ft1 fe1 env1 env2)
Proof
ho_match_mp_tac v_rel_ind \\ rw []
\\ FIRST (map irule (CONJUNCTS v_rel_rules)) \\ gs []
>- ((* Conv *)
irule_at Any LIST_REL_mono
\\ first_assum (irule_at Any) \\ gs []
\\ irule OPTREL_MONO
\\ first_assum (irule_at Any) \\ gs [] \\ rw []
\\ irule stamp_rel_update
\\ gs [SF SFY_ss])
>- ((* Vectorv *)
irule_at Any LIST_REL_mono
\\ first_assum (irule_at Any) \\ gs [])
>- ((* Loc *)
irule FLOOKUP_SUBMAP
\\ first_assum (irule_at Any) \\ gs [])
\\ irule_at Any nsAll2_mono
\\ first_assum (irule_at Any)
\\ simp [FORALL_PROD]
\\ irule ctor_rel_update
\\ gs [SF SFY_ss]
QED
Theorem env_rel_update = CONJUNCT2 v_rel_update_lemma;
Theorem v_rel_update = CONJUNCT1 v_rel_update_lemma;
Theorem ctor_rel_nsAppend:
ctor_rel ft fe x1.c x2.c ∧
ctor_rel ft fe y1.c y2.c ⇒
ctor_rel ft fe (nsAppend x1.c y1.c) (nsAppend x2.c y2.c)
Proof
rw [ctor_rel_def]
\\ irule nsAll2_nsAppend \\ gs []
QED
Theorem env_rel_nsAppend:
env_rel fr ft fe x1 x2 ∧
env_rel fr ft fe y1 y2 ⇒
env_rel fr ft fe <|v:= nsAppend x1.v y1.v; c:= nsAppend x1.c y1.c|>
<|v:= nsAppend x2.v y2.v; c:= nsAppend x2.c y2.c|>
Proof
simp [env_rel_def] \\ strip_tac
\\ irule_at Any ctor_rel_nsAppend \\ gs []
\\ irule nsAll2_nsAppend \\ gs []
QED
Theorem env_rel_extend_dec_env:
env_rel fr ft fe env1 env2 ∧
env_rel fr ft fe env1' env2' ⇒
env_rel fr ft fe (extend_dec_env env1' env1)
(extend_dec_env env2' env2)
Proof
rw [extend_dec_env_def]
\\ irule env_rel_nsAppend
\\ gs []
QED
Theorem env_rel_nsBind:
env_rel fr ft fe env1 env2 ∧
v_rel fr ft fe v w ⇒
env_rel fr ft fe (env1 with v := nsBind n v env1.v)
(env2 with v := nsBind n w env2.v)
Proof
rw [env_rel_def]
\\ irule nsAll2_nsBind
\\ gs []
QED
Theorem fmap_greatest[local]:
∀m. ∃y. ∀x. x ∈ FRANGE m ⇒ x < y
Proof
Induct \\ rw []
\\ qmatch_goalsub_rename_tac ‘_ = z ∨ _’
\\ qexists_tac ‘SUC y + z’
\\ qx_gen_tac ‘k’
\\ rw [] \\ gs []
\\ mp_tac (Q.ISPECL [‘x’,‘m: 'a |-> num’, ‘λx. x < y’]
(GEN_ALL IN_FRANGE_DOMSUB_suff))
\\ rw [] \\ gs []
\\ first_x_assum drule \\ gs []
QED
(* --------------------------------------------------------------------------
*
* -------------------------------------------------------------------------- *)
Definition match_res_rel_def[simp]:
(match_res_rel R (Match env1) (Match env2) ⇔ R env1 env2) ∧
(match_res_rel R No_match No_match ⇔ T) ∧
(match_res_rel R Match_type_error Match_type_error ⇔ T) ∧
(match_res_rel R _ _ ⇔ F)
End
Theorem pmatch_update:
(∀envC s p v ws res.
pmatch envC s p v ws = res ⇒
∀envC' s' v' ws' res' ns ms.
ctor_rel ft fe envC envC' ∧
v_rel fr ft fe v v' ∧
INJ ($' ft) (FDOM ft) ns ∧
INJ ($' fe) (FDOM fe) ms ∧
LIST_REL (λ(n,v) (m,w). n = m ∧ v_rel fr ft fe v w) ws ws' ∧
pmatch envC' s' p v' ws' = res' ∧
(∀n. if n < LENGTH s then
∃m. FLOOKUP fr n = SOME m ∧ m < LENGTH s' ∧
ref_rel (v_rel fr ft fe) (EL n s) (EL m s')
else FLOOKUP fr n = NONE) ⇒
match_res_rel (λenv env'.
LIST_REL (λ(n,v) (m,w). n = m ∧ v_rel fr ft fe v w)
env env') res res') ∧
(∀envC s ps vs ws res.
pmatch_list envC s ps vs ws = res ⇒
∀envC' s' vs' ws' res' ns ms.
INJ ($' ft) (FDOM ft) ns ∧
INJ ($' fe) (FDOM fe) ms ∧
ctor_rel ft fe envC envC' ∧
LIST_REL (v_rel fr ft fe) vs vs' ∧
LIST_REL (λ(n,v) (m,w). n = m ∧ v_rel fr ft fe v w) ws ws' ∧
pmatch_list envC' s' ps vs' ws' = res' ∧
(∀n. if n < LENGTH s then
∃m. FLOOKUP fr n = SOME m ∧ m < LENGTH s' ∧
ref_rel (v_rel fr ft fe) (EL n s) (EL m s')
else FLOOKUP fr n = NONE) ⇒
match_res_rel (λenv env'.
LIST_REL (λ(n,v) (m,w). n = m ∧ v_rel fr ft fe v w)
env env') res res')
Proof[exclude_simps = option.OPTREL_NONE]
ho_match_mp_tac pmatch_ind \\ rw [] \\ gvs [pmatch_def, v_rel_def, SF SFY_ss]
>- (rw [] \\ gs [])
>- (
gvs [CaseEqs ["bool", "option", "prod"], v_rel_def, pmatch_def]
\\ rename1 ‘pmatch _ _ _ (Conv tt _)’
\\ Cases_on ‘tt’ \\ gvs [pmatch_def, CaseEqs ["prod", "option", "bool"]]
\\ CASE_TAC \\ gs []
>- (
gs [ctor_rel_def]
\\ drule_all_then assume_tac nsAll2_nsLookup_none \\ gs [])
\\ CASE_TAC \\ gs [ctor_rel_def]
\\ drule_all_then strip_assume_tac nsAll2_nsLookup1 \\ gs []
\\ PairCases_on ‘v2’ \\ gvs []
\\ imp_res_tac LIST_REL_LENGTH \\ gs []
\\ gvs [stamp_rel_cases, same_ctor_def, same_type_def]
\\ rw [] \\ gs []
\\ TRY (first_x_assum irule \\ gs [SF SFY_ss])
\\ fs [INJ_DEF] \\ fs [flookup_thm] \\ gs [])
>- (
rename1 ‘pmatch _ _ _ (Conv tt _)’
\\ Cases_on ‘tt’ \\ gvs [pmatch_def, CaseEqs ["prod", "option", "bool"]]
\\ rw [] \\ gs []
\\ imp_res_tac LIST_REL_LENGTH \\ gs []
\\ first_x_assum irule
\\ gs [SF SFY_ss])
>- (
CASE_TAC \\ gs [store_lookup_def] \\ gvs []
>- (
first_x_assum (qspec_then ‘lnum’ assume_tac) \\ gs [])
\\ first_assum (qspec_then ‘lnum’ mp_tac)
\\ IF_CASES_TAC \\ simp_tac std_ss [] \\ rw [] \\ gs []
\\ rpt CASE_TAC \\ gs [ref_rel_def]
\\ first_x_assum irule \\ gs [SF SFY_ss])
>- (
rename1 ‘pmatch _ _ _ (Conv tt _)’
\\ Cases_on ‘tt’ \\ gvs [pmatch_def, CaseEqs ["prod", "option", "bool"]])
>- (
rename1 ‘pmatch _ _ _ (Conv tt _)’
\\ Cases_on ‘tt’ \\ gvs [pmatch_def, CaseEqs ["prod", "option", "bool"]])
>- (
Cases_on ‘pmatch envC s p v ws’ \\ gs []
\\ rpt (first_x_assum drule_all \\ rw [] \\ gs [])
\\ rpt (CASE_TAC \\ gs [])
\\ first_x_assum irule \\ gs [SF SFY_ss])
QED
local
val ind_thm =
full_evaluate_ind
|> Q.SPECL [
‘λs env xs. ∀res s1 l fr ft fe t env1.
evaluate s env xs = (s1, res) ∧
state_rel l fr ft fe s t ∧
env_rel fr ft fe env env1 ⇒
∃fr1 ft1 fe1 t1 res1.
fr ⊑ fr1 ∧ ft ⊑ ft1 ∧ fe ⊑ fe1 ∧
state_rel l fr1 ft1 fe1 s1 t1 ∧
evaluate t env1 xs = (t1, res1) ∧
res_rel (LIST_REL (v_rel fr1 ft1 fe1))
(v_rel fr1 ft1 fe1) res res1’,
‘λs env v ps errv. ∀res s1 l fr ft fe t env1 w errw.
evaluate_match s env v ps errv = (s1, res) ∧
state_rel l fr ft fe s t ∧
env_rel fr ft fe env env1 ∧
v_rel fr ft fe v w ∧
v_rel fr ft fe errv errw ⇒
∃fr1 ft1 fe1 t1 res1.
fr ⊑ fr1 ∧ ft ⊑ ft1 ∧ fe ⊑ fe1 ∧
state_rel l fr1 ft1 fe1 s1 t1 ∧
evaluate_match t env1 w ps errw = (t1, res1) ∧
res_rel (LIST_REL (v_rel fr1 ft1 fe1))
(v_rel fr1 ft1 fe1) res res1’,
‘λs env ds. ∀res s1 l fr ft fe t env1.
evaluate_decs s env ds = (s1, res) ∧
state_rel l fr ft fe s t ∧
env_rel fr ft fe env env1 ⇒
∃fr1 ft1 fe1 t1 res1.
fr ⊑ fr1 ∧ ft ⊑ ft1 ∧ fe ⊑ fe1 ∧
state_rel l fr1 ft1 fe1 s1 t1 ∧
evaluate_decs t env1 ds = (t1, res1) ∧
res_rel (λenv' env1'.
env_rel fr1 ft1 fe1 env' env1') (*
env_rel fr1 ft1 fe1 (extend_dec_env env' env)
(extend_dec_env env1' env1)) *)
(v_rel fr1 ft1 fe1) res res1’]
|> CONV_RULE (DEPTH_CONV BETA_CONV);
val ind_goals =
ind_thm |> concl |> dest_imp |> fst
|> helperLib.list_dest dest_conj
in
fun get_goal s =
first (can (find_term (can (match_term (Term [QUOTE s]))))) ind_goals
|> helperLib.list_dest dest_forall
|> last
fun evaluate_update () = ind_thm |> concl |> rand
fun the_ind_thm () = ind_thm
end
Theorem evaluate_update_Nil:
^(get_goal "[]")
Proof
rw [evaluate_def]
\\ first_assum (irule_at Any) \\ gs []
QED
Theorem evaluate_update_Cons:
^(get_goal "_::_::_")
Proof
rw [evaluate_def]
\\ gvs [CaseEqs ["result", "prod"], PULL_EXISTS]
>~ [‘evaluate _ _ [_] = (_, Rerr err)’] >- (
first_x_assum (drule_all_then strip_assume_tac)
\\ Cases_on ‘res1’ \\ gs []
\\ rename1 ‘_ _ (Rerr err) (Rerr err1)’
\\ Cases_on ‘err’ \\ Cases_on ‘err1’ \\ gs []
\\ first_assum (irule_at Any) \\ gs []
\\ first_assum (irule_at Any) \\ gs [])
\\ first_x_assum (drule_all_then strip_assume_tac)
\\ Cases_on ‘res1’ \\ gs []
\\ drule_all_then assume_tac env_rel_update
\\ first_x_assum (drule_all_then strip_assume_tac)
\\ Cases_on ‘res1’ \\ gs []
\\ first_assum (irule_at Any) \\ gs []
\\ gs [SUBMAP_TRANS, SF SFY_ss]
\\ drule_then strip_assume_tac evaluate_sing \\ gvs []
\\ irule v_rel_update
\\ gs [SUBMAP_TRANS, SF SFY_ss]
\\ first_assum (irule_at (Pos last))
QED
Theorem evaluate_update_Lit:
^(get_goal "Lit l")
Proof
rw [evaluate_def] \\ gs []
\\ first_assum (irule_at Any)
\\ simp [v_rel_rules]
QED
Theorem evaluate_update_Raise:
^(get_goal "Raise e")
Proof
rw [evaluate_def] \\ gs []
\\ gvs [CaseEqs ["result", "prod"], PULL_EXISTS]
\\ first_x_assum (drule_all_then strip_assume_tac)
\\ Cases_on ‘res1’ \\ gs []
\\ first_assum (irule_at Any) \\ gs []
\\ drule_then strip_assume_tac evaluate_sing \\ gvs []
QED
Theorem can_pmatch_all_thm:
∀ps envc1 s v envc2 t w ms ns.
ctor_rel ft fe envc1 envc2 ∧
INJ ($' ft) (FDOM ft) ns ∧
INJ ($' fe) (FDOM fe) ms ∧
(∀n. if n < LENGTH s then
∃m. FLOOKUP fr n = SOME m ∧ m < LENGTH t ∧
ref_rel (v_rel fr ft fe) (EL n s) (EL m t)
else FLOOKUP fr n = NONE) ∧
v_rel fr ft fe v w ⇒
(can_pmatch_all envc1 s ps v ⇔ can_pmatch_all envc2 t ps w)
Proof
Induct \\ rw [can_pmatch_all_def]
\\ first_x_assum drule_all \\ rw []
\\ rw [EQ_IMP_THM] \\ gs []
>- (
‘∃res. pmatch envc1 s h v [] = res’ by gs []
\\ drule (CONJUNCT1 pmatch_update)
\\ rpt (disch_then drule) \\ gs []
\\ rpt (disch_then drule) \\ gs [] \\ rw []
\\ Cases_on ‘pmatch envc1 s h v []’ \\ Cases_on ‘pmatch envc2 t h w []’
\\ gs [])
>- (
strip_tac
\\ drule (CONJUNCT1 pmatch_update)
\\ rpt (disch_then drule) \\ gs []
\\ qexists_tac ‘t’ \\ gs []
\\ Cases_on ‘pmatch envc1 s h v []’ \\ Cases_on ‘pmatch envc2 t h w []’
\\ gs [])
QED
Theorem evaluate_update_Handle:
^(get_goal "Handle e")
Proof
rw [evaluate_def]
\\ gvs [CaseEqs ["prod", "result", "error_result", "bool"], PULL_EXISTS]
\\ first_x_assum (drule_all_then strip_assume_tac) \\ gs []
>~ [‘evaluate _ _ [_] = (_, Rerr (Rabort _))’] >- (
first_assum (irule_at Any) \\ gs [])
\\ Cases_on ‘res1’ \\ gs []
>~ [‘evaluate _ _ [_] = (_, Rval _)’] >- (
first_assum (irule_at Any) \\ gs [])
\\ rename1 ‘res_rel _ _ (Rerr _) (Rerr err)’
\\ Cases_on ‘err’ \\ gs []
\\ drule_all_then assume_tac env_rel_update
>- (
first_x_assum (drule_all_then strip_assume_tac) \\ gs []
\\ first_assum (irule_at (Pat ‘state_rel’)) \\ gs []
\\ first_assum (irule_at Any)
\\ irule_at Any SUBMAP_TRANS \\ first_assum (irule_at Any) \\ gs []
\\ irule_at Any SUBMAP_TRANS \\ first_assum (irule_at Any) \\ gs []
\\ irule_at Any SUBMAP_TRANS \\ first_assum (irule_at Any) \\ gs []
\\ gs [state_rel_def, env_rel_def]
\\ drule can_pmatch_all_thm \\ gs []
\\ rpt (disch_then drule)
\\ disch_then (qspec_then ‘MAP FST pes’ assume_tac)
\\ gs [SF SFY_ss])
\\ first_assum (irule_at (Pat ‘state_rel’)) \\ gs []
\\ gs [state_rel_def, env_rel_def]
\\ drule can_pmatch_all_thm \\ gs []
\\ rpt (disch_then drule)
\\ disch_then (qspec_then ‘MAP FST pes’ assume_tac)
\\ gs [SF SFY_ss]
QED
Theorem do_con_check_update:
env_rel fr ft fe env env1 ⇒
do_con_check env.c cn n = do_con_check env1.c cn n
Proof
strip_tac \\ eq_tac \\ gs [env_rel_def, ctor_rel_def]
\\ rw [do_con_check_def]
\\ CASE_TAC \\ gs []
\\ Cases_on ‘nsLookup env.c x’ \\ gs []
\\ Cases_on ‘nsLookup env1.c x’ \\ gs []
\\ rename1 ‘pair_CASE z’
\\ PairCases_on ‘z’ \\ gvs []
\\ imp_res_tac nsAll2_nsLookup_none
\\ imp_res_tac nsAll2_nsLookup1 \\ imp_res_tac nsAll2_nsLookup2 \\ gvs []
\\ (PairCases_on ‘v1’ ORELSE PairCases_on ‘v2’) \\ gs []
QED
Theorem evaluate_update_Con:
^(get_goal "Con cn es")
Proof
rw [evaluate_def]
\\ gvs [CaseEqs ["prod", "result", "option"], PULL_EXISTS]
\\ drule_then assume_tac do_con_check_update \\ gs []
>~ [‘¬do_con_check _ _ _’] >- (
first_assum (irule_at Any) \\ gs [])
\\ first_x_assum (drule_all_then strip_assume_tac)
\\ Cases_on ‘res1’ \\ gs []
\\ gs [env_rel_def, ctor_rel_def, build_conv_def]
\\ gvs [CaseEqs ["prod", "option"]]
\\ first_assum (irule_at Any) \\ gs [v_rel_rules]
\\ rename1 ‘nsLookup env1.c id’
>- (
dxrule_all nsAll2_nsLookup_none \\ rw []
\\ dxrule_all nsAll2_nsLookup_none \\ rw []
\\ gs [])
\\ gs [PULL_EXISTS]
\\ imp_res_tac nsAll2_nsLookup1 \\ gs[ ]
\\ PairCases_on ‘v2’ \\ gs []
\\ simp [v_rel_def]
\\ irule stamp_rel_update
\\ gs [SF SFY_ss]
QED
Theorem evaluate_update_Var:
^(get_goal "ast$Var n")
Proof
rw [evaluate_def]
\\ gvs [CaseEqs ["option"]]
\\ first_assum (irule_at Any) \\ gs [] \\ dsimp []
\\ gs [env_rel_def, ctor_rel_def]
>- (
dxrule_all nsAll2_nsLookup_none
\\ dxrule_all nsAll2_nsLookup_none \\ rw []
\\ gs [])
\\ imp_res_tac nsAll2_nsLookup1 \\ gs[ ]
QED
Theorem evaluate_update_Fun:
^(get_goal "ast$Fun n e")
Proof
rw [evaluate_def]
\\ gvs [CaseEqs ["option"]]
\\ first_assum (irule_at Any) \\ gs []
\\ simp [v_rel_def]
QED
Theorem evaluate_update_Eval:
op = Eval ⇒ ^(get_goal "App")
Proof
rw [evaluate_def]
\\ gvs [AllCaseEqs(), evaluateTheory.do_eval_res_def]
\\ first_x_assum (drule_all_then strip_assume_tac) \\ gs []
\\ first_assum (irule_at Any)
\\ Cases_on ‘res1’ \\ gs []
\\ dsimp []
\\ gs [state_rel_def, do_eval_def]
QED
Theorem state_rel_store_lookup:
state_rel l fr ft fe s t ∧
FLOOKUP fr n = SOME m ⇒
OPTREL (ref_rel (v_rel fr ft fe))
(store_lookup n s.refs)
(store_lookup m t.refs)
Proof
rw [OPTREL_def, store_lookup_def, state_rel_def]
\\ first_x_assum (qspec_then ‘n’ assume_tac) \\ gs []
QED
Theorem state_rel_store_assign:
state_rel l fr ft fe s t ∧
FLOOKUP fr n = SOME m ∧
ref_rel (v_rel fr ft fe) v w ⇒
OPTREL (λr1 r2.
state_rel l fr ft fe
(s with <| refs := r1 |>) (t with refs := r2))
(store_assign n v s.refs)
(store_assign m w t.refs)
Proof
rw [OPTREL_def, store_assign_def, state_rel_def]
\\ ‘n < LENGTH s.refs ∧ m < LENGTH t.refs’
by (qpat_x_assum ‘INJ ($' fr) _ _’ mp_tac
\\ qpat_x_assum ‘FLOOKUP fr n = _’ mp_tac
\\ rw [INJ_DEF, flookup_thm]
\\ first_x_assum drule \\ gs [])
\\ gs []
\\ first_assum (qspec_then ‘n’ mp_tac)
\\ IF_CASES_TAC \\ gs []
\\ rw []
\\ ‘store_v_same_type (EL n s.refs) (EL m t.refs)’
by (rw [store_v_same_type_def]
\\ CASE_TAC \\ gs [] \\ CASE_TAC \\ gs [ref_rel_def])
\\ ‘store_v_same_type v w’
by (rw [store_v_same_type_def]
\\ CASE_TAC \\ gs [] \\ CASE_TAC \\ gs [ref_rel_def])
\\ ‘store_v_same_type (EL n s.refs) v = store_v_same_type (EL m t.refs) w’
by (rw [EQ_IMP_THM] \\ gs [store_v_same_type_def]
\\ Cases_on ‘EL n s.refs’ \\ Cases_on ‘EL m t.refs’
\\ Cases_on ‘v’ \\ Cases_on ‘w’ \\ gs [])
\\ csimp []
\\ simp [DISJ_EQ_IMP]
\\ strip_tac \\ gs []
\\ qx_gen_tac ‘n1’
\\ first_x_assum (qspec_then ‘n1’ assume_tac)
\\ rw [] \\ gs [EL_LUPDATE]
\\ rw [] \\ gs [ref_rel_def]
\\ qpat_x_assum ‘INJ ($' fr) _ _’ mp_tac
\\ qpat_x_assum ‘FLOOKUP fr n1 = SOME _’ mp_tac
\\ qpat_x_assum ‘FLOOKUP fr n = SOME _’ mp_tac
\\ rw [flookup_thm, INJ_DEF] \\ gs []
QED
Theorem v_rel_v_to_list:
∀x1 res1 x2 res2.
v_rel fr ft fe x1 x2 ∧
v_to_list x1 = res1 ∧
v_to_list x2 = res2 ∧
INJ ($' ft) (FDOM ft) (count ns) ∧
FLOOKUP ft list_type_num = SOME list_type_num ∧
list_type_num < ns ⇒
OPTREL (LIST_REL (v_rel fr ft fe)) res1 res2
Proof
ho_match_mp_tac v_to_list_ind \\ gs []
\\ rw [] \\ gvs [v_rel_def, v_to_list_def]
\\ gvs [OPTREL_def, stamp_rel_cases, v_to_list_def, flookup_thm]
>- (
Cases_on ‘m1 = list_type_num’ \\ gs []
\\ rpt strip_tac \\ gvs []
\\ gs [INJ_DEF])
\\ gs [CaseEq "option"]
\\ Cases_on ‘m1 = list_type_num ∧ n = "::"’ \\ gvs []
>- (
first_x_assum (drule_all_then assume_tac)
\\ gs [])
\\ Cases_on ‘m1 = list_type_num’ \\ gs [] \\ rw []
\\ gs [INJ_DEF]
QED
Theorem v_rel_vs_to_string:
∀x1 res1 x2 res2.
LIST_REL (v_rel fr ft fe) x1 x2 ∧
vs_to_string x1 = res1 ∧
vs_to_string x2 = res2 ∧
INJ ($' ft) (FDOM ft) (count ns) ∧
FLOOKUP ft list_type_num = SOME list_type_num ∧
list_type_num < ns ⇒
res1 = res2
Proof
ho_match_mp_tac vs_to_string_ind \\ gs []
\\ rw [] \\ gvs [v_rel_def, vs_to_string_def]
\\ gs [CaseEq "option"]
\\ first_x_assum (drule_all_then assume_tac)
\\ gs [option_nchotomy]
QED
Theorem v_rel_list_to_v:
∀x1 xs1 x2 xs2.
v_to_list x1 = SOME xs1 ∧
v_to_list x2 = SOME xs2 ∧
LIST_REL (v_rel fr ft fe) xs1 xs2 ∧
FLOOKUP ft list_type_num = SOME list_type_num ⇒
v_rel fr ft fe (list_to_v xs1) (list_to_v xs2)
Proof
ho_match_mp_tac v_to_list_ind
\\ rw [] \\ gvs [v_to_list_def, list_to_v_def, v_rel_def, stamp_rel_cases,
CaseEq "option"]
\\ Cases_on ‘x2’ \\ gs [v_to_list_def]
\\ rename1 ‘Conv opt ws’ \\ Cases_on ‘opt’ \\ gs [v_to_list_def]
\\ rename1 ‘Conv (SOME opt) ws’ \\ Cases_on ‘ws’ \\ gs [v_to_list_def]
\\ rename1 ‘Conv (SOME opt) (w::ws)’ \\ Cases_on ‘ws’ \\ gs [v_to_list_def]
\\ Cases_on ‘t’ \\ gvs [v_to_list_def, CaseEq "option"]
\\ first_assum (irule_at Any) \\ gs [SF SFY_ss]
QED
Theorem v_rel_v_to_char_list:
∀x1 res1 x2 res2.
v_rel fr ft fe x1 x2 ∧
v_to_char_list x1 = res1 ∧
v_to_char_list x2 = res2 ∧
INJ ($' ft) (FDOM ft) (count ns) ∧
FLOOKUP ft list_type_num = SOME list_type_num ∧
list_type_num < ns ⇒
res1 = res2
Proof
ho_match_mp_tac v_to_char_list_ind \\ gs []
\\ rw [] \\ gvs [v_rel_def, v_to_char_list_def]
\\ TRY (gs[v_rel_cases] \\ NO_TAC)
\\ gvs [CaseEq "bool", OPTREL_def, stamp_rel_cases]
\\ gs [v_to_char_list_def]
>- (
Cases_on ‘m1 = list_type_num’ \\ gs []
\\ rpt strip_tac \\ gvs []
\\ gs [INJ_DEF, flookup_thm])
\\ gs [CaseEq "option"]
\\ Cases_on ‘m1 = list_type_num ∧ n = "::"’ \\ gvs []
>- (
first_x_assum (drule_all_then assume_tac)
\\ gs [option_nchotomy])
\\ Cases_on ‘m1 = list_type_num’ \\ gs [] \\ rw []
\\ gs [INJ_DEF, flookup_thm]
QED
Theorem v_to_list_list_to_v:
∀xs.
v_to_list (list_to_v xs) = SOME xs
Proof
Induct \\ rw [list_to_v_def, v_to_list_def]
QED
Theorem v_rel_do_eq:
(∀x1 y1 x2 y2.
v_rel fr ft fe x1 x2 ∧
v_rel fr ft fe y1 y2 ∧
FLOOKUP ft bool_type_num = SOME bool_type_num ∧
INJ ($' fr) (FDOM fr) (count rs) ∧
INJ ($' ft) (FDOM ft) (count ns) ∧
INJ ($' fe) (FDOM fe) (count ms) ⇒
do_eq x1 y1 = do_eq x2 y2) ∧
(∀x1 y1 x2 y2.
LIST_REL (v_rel fr ft fe) x1 x2 ∧
LIST_REL (v_rel fr ft fe) y1 y2 ∧
FLOOKUP ft bool_type_num = SOME bool_type_num ∧
INJ ($' fr) (FDOM fr) (count rs) ∧
INJ ($' ft) (FDOM ft) (count ns) ∧
INJ ($' fe) (FDOM fe) (count ms) ⇒
do_eq_list x1 y1 = do_eq_list x2 y2)
Proof
ho_match_mp_tac do_eq_ind \\ rw []
\\ gvs [v_rel_def, do_eq_def, Boolv_def]
\\ gvs [OPTREL_def]
\\ imp_res_tac LIST_REL_LENGTH \\ gs []
\\ rw [EQ_IMP_THM] \\ gs []
\\ gvs [ctor_same_type_def, same_type_def, stamp_rel_cases, flookup_thm]
\\ rpt CASE_TAC \\ gs []
\\ gs [INJ_DEF]
QED
Theorem fp_translate_alt:
fp_translate v =
case v of
|FP_WordTree fp => SOME (FP_WordTree fp)
|FP_BoolTree fp => SOME (FP_BoolTree fp)
|Litv (Word64 w) => SOME (FP_WordTree (Fp_const w))
| _ => NONE
Proof
rpt (TOP_CASE_TAC \\ gs[fp_translate_def])
QED
Theorem do_app_update:
do_app (s.refs,s.ffi) op vs = res ∧
state_rel l fr ft fe s t ∧
LIST_REL (v_rel fr ft fe) vs ws ∧
op ≠ Opapp ∧
op ≠ Eval ⇒
∃fr1 ft1 fe1 res1.
do_app (t.refs,t.ffi) op ws = res1 ∧
fr ⊑ fr1 ∧
ft ⊑ ft1 ∧
fe ⊑ fe1 ∧
OPTREL (λ((refs1,ffi1),res) ((refs1',ffi1'),res1).
∃s1 t1.
s1.refs = refs1 ∧ s1.ffi = ffi1 ∧
t1.refs = refs1' ∧ t1.ffi = ffi1' ∧
s1.next_exn_stamp = s.next_exn_stamp ∧
t1.next_exn_stamp = t.next_exn_stamp ∧
s1.next_type_stamp = s.next_type_stamp ∧
t1.next_type_stamp = t.next_type_stamp ∧
s1.fp_state = s.fp_state ∧
t1.fp_state = t.fp_state ∧
state_rel l fr1 ft1 fe1 s1 t1 ∧
res_rel (v_rel fr1 ft1 fe1) (v_rel fr1 ft1 fe1) res res1)
res res1
Proof
strip_tac
\\ Cases_on ‘op = Env_id’ \\ gs []
>- (
Cases_on ‘res’ \\ gvs [do_app_def, CaseEqs ["list", "v", "option", "prod"],
v_rel_def, OPTREL_def]
\\ rpt (irule_at Any SUBMAP_REFL) \\ gs [v_rel_def, nat_to_v_def]
\\ first_assum (irule_at Any) \\ gs [])
\\ Cases_on ‘∃chn. op = FFI chn’ \\ gs []
>- (
Cases_on ‘res’ \\ gvs [do_app_def, v_rel_def, OPTREL_def,
CaseEqs ["list", "v", "option", "prod", "lit",
"store_v"], PULL_EXISTS]
\\ rpt (irule_at Any SUBMAP_REFL) \\ gs []
\\ drule_all_then assume_tac state_rel_store_lookup
\\ ‘t.ffi = s.ffi’
by gs [state_rel_def]
\\ gs [OPTREL_def]
\\ rename1 ‘ref_rel _ _ y0’ \\ Cases_on ‘y0’ \\ gvs [ref_rel_def]
>- (
gvs [CaseEqs ["ffi_result", "option", "bool", "oracle_result"],
ffiTheory.call_FFI_def, PULL_EXISTS, state_rel_def]
\\ gs [store_assign_def, store_lookup_def, v_rel_def])
\\ gvs [CaseEqs ["ffi_result", "option", "bool", "oracle_result"],
ffiTheory.call_FFI_def, PULL_EXISTS, EXISTS_PROD, v_rel_def,
store_assign_def, store_lookup_def]
\\ Q.REFINE_EXISTS_TAC
‘<| refs := r1; ffi := f1; clock := s.clock;
next_type_stamp := nts1; next_exn_stamp := nes1;
fp_state := fp1;
eval_state := NONE |>’ \\ gs []
\\ Q.REFINE_EXISTS_TAC
‘<| refs := r2; ffi := f2; clock := t.clock;
next_type_stamp := nts2; next_exn_stamp := nes2;
fp_state := fp2;
eval_state := NONE |>’ \\ gs []
\\ gs [state_rel_def, EL_LUPDATE]
\\ qx_gen_tac ‘n1’ \\ rw [] \\ gs [ref_rel_def]
\\ first_x_assum (qspec_then ‘n1’ assume_tac) \\ gs []
\\ rw [] \\ gs []
\\ qpat_x_assum ‘INJ ($' fr) _ _’ mp_tac \\ rw [INJ_DEF]
\\ qpat_x_assum ‘FLOOKUP fr _ = _’ mp_tac \\ rw [flookup_thm]
\\ qpat_x_assum ‘FLOOKUP fr _ = _’ mp_tac \\ rw [flookup_thm]
\\ gs [])
\\ Cases_on ‘op = ConfigGC’ \\ gs []
>- (
Cases_on ‘res’ \\ gvs [do_app_def, v_rel_def, OPTREL_def,
CaseEqs ["list", "v", "option", "prod", "lit"]]
\\ rpt (irule_at Any SUBMAP_REFL) \\ gs []
\\ first_assum (irule_at Any) \\ gs [])
\\ Cases_on ‘op = ListAppend’ \\ gs []
>- (
‘FLOOKUP ft list_type_num = SOME list_type_num ∧
INJ ($' ft) (FDOM ft) (count t.next_type_stamp) ∧
list_type_num < t.next_type_stamp’
by (gs [state_rel_def]
\\ qpat_x_assum ‘INJ ($' ft) _ _’ mp_tac \\ rw [INJ_DEF]
\\ rpt (first_x_assum (qspec_then ‘list_type_num’ assume_tac))
\\ gs [flookup_thm])
\\ Cases_on ‘res’ \\ gvs [do_app_def, v_rel_def, OPTREL_def,
CaseEqs ["list", "v", "option", "prod", "lit"]]
\\ rpt (irule_at Any SUBMAP_REFL) \\ gs []
\\ dxrule v_rel_v_to_list
\\ rpt (disch_then drule) \\ gs []
\\ disch_then drule \\ rw [OPTREL_def]
\\ gs [option_nchotomy]
\\ dxrule v_rel_v_to_list
\\ rpt (disch_then drule) \\ gs []
\\ disch_then drule \\ rw [OPTREL_def]
\\ first_assum (irule_at Any)
\\ first_assum (irule_at Any) \\ gs []
\\ irule v_rel_list_to_v \\ gs []
\\ irule_at Any v_to_list_list_to_v
\\ irule_at Any v_to_list_list_to_v
\\ gs [LIST_REL_EL_EQN, EL_APPEND_EQN]
\\ rw [] \\ gs[])
\\ Cases_on ‘op = Aw8sub_unsafe’ \\ gs []
>- (
Cases_on ‘res’ \\ gvs [do_app_def, v_rel_def, OPTREL_def,
CaseEqs ["list", "v", "option", "prod", "lit",
"store_v"]]
\\ rpt (irule_at Any SUBMAP_REFL) \\ gs []
\\ drule_all_then assume_tac state_rel_store_lookup \\ gs [OPTREL_def]
\\ rename1 ‘ref_rel _ _ y0’ \\ Cases_on ‘y0’ \\ gs [ref_rel_def]
\\ rw [] \\ gs []
\\ first_assum (irule_at Any) \\ gs [v_rel_def])
\\ Cases_on ‘op = Aw8update_unsafe’ \\ gs []
>- (
Cases_on ‘res’ \\ gvs [do_app_def, v_rel_def, OPTREL_def,
CaseEqs ["list", "v", "option", "prod", "lit",
"store_v"]]
\\ rpt (irule_at Any SUBMAP_REFL) \\ gs []
\\ drule_all_then assume_tac state_rel_store_lookup \\ gs [OPTREL_def]
\\ rename1 ‘ref_rel _ _ y0’ \\ Cases_on ‘y0’ \\ gs [ref_rel_def]
\\ gvs [store_assign_def, store_lookup_def]
\\ rw [] \\ gs [v_rel_def]
\\ Q.REFINE_EXISTS_TAC
‘<| refs := r1; ffi := f1; clock := s.clock;
next_type_stamp := nts1; next_exn_stamp := nes1;
fp_state := fp1;
eval_state := NONE |>’ \\ gs []
\\ Q.REFINE_EXISTS_TAC
‘<| refs := r2; ffi := f2; clock := t.clock;
next_type_stamp := nts2; next_exn_stamp := nes2;
fp_state := fp2;
eval_state := NONE |>’ \\ gs []
\\ gs [state_rel_def, EL_LUPDATE]
\\ qx_gen_tac ‘n1’
\\ first_x_assum (qspec_then ‘n1’ assume_tac)
\\ rw [] \\ gs [ref_rel_def]
\\ qpat_x_assum ‘INJ ($' fr) _ _’ mp_tac \\ rw [INJ_DEF]
\\ qpat_x_assum ‘FLOOKUP fr _ = _’ mp_tac \\ rw [flookup_thm]
\\ qpat_x_assum ‘FLOOKUP fr _ = _’ mp_tac \\ rw [flookup_thm]
\\ gs [])
\\ Cases_on ‘op = Aupdate_unsafe’ \\ gs []
>- (
Cases_on ‘res’ \\ gvs [do_app_def, v_rel_def, OPTREL_def,
CaseEqs ["list", "v", "option", "prod", "lit",
"store_v"]]
\\ rpt (irule_at Any SUBMAP_REFL) \\ gs []
\\ drule_all_then assume_tac state_rel_store_lookup \\ gs [OPTREL_def]
\\ rename1 ‘ref_rel _ _ y0’ \\ Cases_on ‘y0’ \\ gs [ref_rel_def]
\\ gvs [store_assign_def, store_lookup_def, store_v_same_type_def]
\\ rw [] \\ gs [v_rel_def]
\\ drule_then assume_tac LIST_REL_LENGTH \\ gs []
\\ Q.REFINE_EXISTS_TAC
‘<| refs := r1; ffi := f1; clock := s.clock;
next_type_stamp := nts1; next_exn_stamp := nes1;
fp_state := fp1;
eval_state := NONE |>’ \\ gs []
\\ Q.REFINE_EXISTS_TAC
‘<| refs := r2; ffi := f2; clock := t.clock;
next_type_stamp := nts2; next_exn_stamp := nes2;
fp_state := fp2;
eval_state := NONE |>’ \\ gs []
\\ gs [state_rel_def, EL_LUPDATE]
\\ qx_gen_tac ‘n1’
\\ first_x_assum (qspec_then ‘n1’ assume_tac)
\\ rw [] \\ gs [ref_rel_def]
\\ gvs [LIST_REL_EL_EQN, EL_LUPDATE]
\\ rw [] \\ gs []
\\ qpat_x_assum ‘INJ ($' fr) _ _’ mp_tac \\ rw [INJ_DEF]
\\ qpat_x_assum ‘FLOOKUP fr _ = _’ mp_tac \\ rw [flookup_thm]
\\ qpat_x_assum ‘FLOOKUP fr _ = _’ mp_tac \\ rw [flookup_thm]
\\ gs [])
\\ Cases_on ‘op = Asub_unsafe’ \\ gs []
>- (
Cases_on ‘res’ \\ gvs [do_app_def, v_rel_def, OPTREL_def,
CaseEqs ["list", "v", "option", "prod", "lit",
"store_v"]]
\\ rpt (irule_at Any SUBMAP_REFL) \\ gs []