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smallStepPropsScript.sml
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(*
Properties about the small-step semantics.
*)
open preamble;
open semanticPrimitivesTheory ffiTheory semanticPrimitivesPropsTheory
evaluatePropsTheory smallStepTheory determTheory;
val _ = new_theory "smallStepProps";
(**
Theorem application_thm:
!op env s vs c.
application op env s vs c =
if op = Opapp then
case do_opapp vs of
| NONE => Eabort Rtype_error
| SOME (env,e) => Estep (env,s,Exp e,c)
else
case do_app s op vs of
| NONE => Eabort Rtype_error
| SOME (v1,Rval v') => return env v1 v' c
| SOME (v1,Rerr (Rraise v)) => Estep (env,v1,Exn v,c)
| SOME (v1,Rerr (Rabort a)) => Eabort a
Proof
srw_tac[][application_def] >>
cases_on `op` >>
srw_tac[][]
QED
**)
Theorem application_thm = fetch "smallStep" "application_def";
(******************** Expressions *******************)
Theorem list_end_case[local]:
!l. l = [] ∨ ?x l'. l = l' ++ [x]
Proof
Induct_on `l` >>
srw_tac[][] >>
metis_tac [APPEND]
QED
Theorem small_eval_prefix:
∀s env fp e c cenv' s' env' fp' e' c' r.
e_step_reln^* (env,s,fp,Exp e,c) (env',s',fp',Exp e',c') ∧
small_eval env' s' fp' e' c' r
⇒
small_eval env s fp e c r
Proof
srw_tac[][] >>
PairCases_on `r` >>
cases_on `r3` >>
full_simp_tac(srw_ss())[small_eval_def] >-
metis_tac [transitive_RTC, transitive_def] >>
cases_on `e''` >>
TRY (Cases_on `a`) >>
full_simp_tac(srw_ss())[small_eval_def] >>
metis_tac [transitive_RTC, transitive_def]
QED
Theorem e_single_step_add_ctxt:
!s env fp e c s' env' fp' e' c' c''.
(e_step (env,s,fp,e,c) = Estep (env',s',fp',e',c'))
⇒
(e_step (env,s,fp,e,c++c'') = Estep (env',s',fp',e',c'++c''))
Proof
srw_tac[][e_step_def] >>
cases_on `e` >>
full_simp_tac(srw_ss())[push_def, return_def] >>
srw_tac[][] >>
full_simp_tac(srw_ss())[] >>
srw_tac[][] >>
(** To avoid unnecessary cases *)
TRY TOP_CASE_TAC >> gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[] >>
srw_tac[][]
>- (full_simp_tac(srw_ss())[application_thm] >>
every_case_tac >> gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[return_def])
>- (full_simp_tac(srw_ss())[continue_def] >>
cases_on `c` >>
full_simp_tac(srw_ss())[] >>
cases_on `h` >>
full_simp_tac(srw_ss())[] >>
cases_on `q` >>
full_simp_tac(srw_ss())[] >>
every_case_tac >>
full_simp_tac(srw_ss())[push_def, return_def] >>
srw_tac[][] >>
full_simp_tac(srw_ss())[application_thm] >>
every_case_tac >> gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[return_def])
QED
Theorem fix_fp_state_no_optimise_id:
EVERY (\c. (~∃ oldSc sc env. c = (Coptimise oldSc sc (), env))) c ⇒
fix_fp_state c fp = fp
Proof
Induct_on ‘c’ >> gs[fix_fp_state_def]
>> rpt strip_tac >> Cases_on ‘h’ >> Cases_on ‘q’ >> gs[fix_fp_state_def]
QED
Theorem fix_fp_state_app:
∀ fp. fix_fp_state (c1 ++ c2) fp = fix_fp_state c2 (fix_fp_state c1 fp)
Proof
Induct_on ‘c1’ >> gs[fix_fp_state_def]
>> rpt strip_tac >> Cases_on ‘h’ >> Cases_on ‘q’ >> gs[fix_fp_state_def]
QED
Theorem fix_fp_state_no_optimise_app:
EVERY (\c. (~∃ oldSc sc env. c = (Coptimise oldSc sc (), env))) c' ⇒
fix_fp_state (c ++ c') fp = fix_fp_state c fp
Proof
gs[fix_fp_state_app, fix_fp_state_no_optimise_id]
QED
Theorem e_single_error_add_ctxt:
!env s fp e c c'.
EVERY (\c. (∀ oldSc sc env. c ≠ (Coptimise oldSc sc (), env))) c' ∧
(e_step (env,s,fp,e,c) = Eabort a)
⇒
(e_step (env,s,fp,e,c++c') = Eabort a)
Proof
srw_tac[][e_step_def] >>
cases_on `e` >>
full_simp_tac(srw_ss())[push_def, return_def] >>
srw_tac[][] >>
full_simp_tac(srw_ss())[] >>
srw_tac[][] >>
(** To avoid unnecessary cases *)
TRY TOP_CASE_TAC >> gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[] >>
srw_tac[][] >> gs[fix_fp_state_no_optimise_app]
>- (full_simp_tac(srw_ss())[application_thm] >>
every_case_tac >> gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[return_def] >> rveq >>
gs[fix_fp_state_no_optimise_app])
>- (full_simp_tac(srw_ss())[continue_def] >>
cases_on `c` >>
full_simp_tac(srw_ss())[] >>
cases_on `h` >>
full_simp_tac(srw_ss())[] >>
cases_on `q` >>
full_simp_tac(srw_ss())[] >>
every_case_tac >>
full_simp_tac(srw_ss())[push_def, return_def] >>
srw_tac[][] >>
full_simp_tac(srw_ss())[application_thm] >>
every_case_tac >> gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[return_def] >> rveq >>
gs[fix_fp_state_no_optimise_app])
QED
Theorem e_single_error_add_ctxt_optimise:
!env s fp fp2 e c c'.
(e_step (env,s,fp,e,c) = Eabort (fp2, a))
⇒
(e_step (env,s,fp,e,c++c') = Eabort (fix_fp_state c' fp2, a))
Proof
srw_tac[][e_step_def] >>
cases_on `e` >>
full_simp_tac(srw_ss())[push_def, return_def] >>
srw_tac[][] >>
full_simp_tac(srw_ss())[] >>
srw_tac[][] >>
(** To avoid unnecessary cases *)
TRY TOP_CASE_TAC >> gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[] >>
srw_tac[][] >> gs[fix_fp_state_app]
>- (full_simp_tac(srw_ss())[application_thm] >>
every_case_tac >> gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[return_def] >> rveq >>
gs[fix_fp_state_app])
>- (full_simp_tac(srw_ss())[continue_def] >>
cases_on `c` >>
full_simp_tac(srw_ss())[] >>
cases_on `h` >>
full_simp_tac(srw_ss())[] >>
cases_on `q` >>
full_simp_tac(srw_ss())[] >>
every_case_tac >>
full_simp_tac(srw_ss())[push_def, return_def] >>
srw_tac[][] >>
full_simp_tac(srw_ss())[application_thm] >>
every_case_tac >> gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[return_def] >> rveq >>
gs[fix_fp_state_app])
QED
Theorem e_step_add_ctxt_help[local]:
!st1 st2.
e_step_reln^* st1 st2 ⇒
!s1 env1 fp1 e1 c1 s2 env2 fp2 e2 c2 c'.
(st1 = (env1,s1,fp1,e1,c1)) ∧ (st2 = (env2,s2,fp2,e2,c2))
⇒
e_step_reln^* (env1,s1,fp1,e1,c1++c') (env2,s2,fp2,e2,c2++c')
Proof
HO_MATCH_MP_TAC RTC_INDUCT >>
srw_tac[][e_step_reln_def] >-
metis_tac [RTC_REFL] >>
PairCases_on `st1'` >>
full_simp_tac(srw_ss())[] >>
imp_res_tac e_single_step_add_ctxt >>
full_simp_tac(srw_ss())[] >>
srw_tac[][Once RTC_CASES1] >>
metis_tac [e_step_reln_def]
QED
Theorem e_step_add_ctxt:
!s1 env1 fp1 e1 c1 s2 env2 fp2 e2 c2 c'.
e_step_reln^* (env1,s1,fp1,e1,c1) (env2,s2,fp2,e2,c2)
⇒
e_step_reln^* (env1,s1,fp1,e1,c1++c') (env2,s2,fp2,e2,c2++c')
Proof
metis_tac [e_step_add_ctxt_help]
QED
Theorem e_step_raise_lemma:
!s env fp c v.
EVERY (\c. (¬?pes env. c = (Chandle () pes, env)) ∧
(~∃oldSc sc env. c = (Coptimise oldSc sc (), env))) c ∧
(c ≠ [])
⇒
e_step_reln^* (env,s,fp,Exn v,c) (env,s,fp,Exn v,[])
Proof
induct_on `c` >>
srw_tac[][] >>
srw_tac[][Once RTC_CASES1] >>
cases_on `c` >>
full_simp_tac(srw_ss())[] >>
srw_tac[][e_step_reln_def, e_step_def, continue_def] >>
every_case_tac >>
full_simp_tac(srw_ss())[] >>
cases_on `o'` >>
full_simp_tac(srw_ss())[]
QED
Theorem small_eval_err_add_ctxt:
!s env fp e c err c' s' fp'.
EVERY (\c. (¬?pes env. c = (Chandle () pes, env)) ∧ (~∃ oldSc sc env. c = (Coptimise oldSc sc (), env))) c'
⇒
small_eval env s fp e c (s', fp', Rerr err) ⇒ small_eval env s fp e (c++c') (s', fp', Rerr err)
Proof
srw_tac[][] >>
`?a. err = Rabort a ∨ ?v. err = Rraise v`
by (cases_on `err` >> srw_tac[][]) >>
srw_tac[][] >>
‘EVERY (\c. (~∃ oldSc sc env. c = (Coptimise oldSc sc (), env))) c'’ by (
irule MONO_EVERY >> first_x_assum $ irule_at Any >> gs[]) >>
‘EVERY (\c. (~∃ pes env. c = (Chandle () pes, env))) c'’ by (
irule MONO_EVERY >> last_x_assum $ irule_at Any >> gs[]) >>
full_simp_tac(srw_ss())[small_eval_def]
>- (Cases_on `a` >>
full_simp_tac(srw_ss())[small_eval_def] >>
`e_step_reln^* (env,s,fp,Exp e,c++c') (env',s',fp'',e',c''++c')`
by metis_tac [e_step_add_ctxt] >>
metis_tac [e_single_error_add_ctxt])
>- (
`e_step_reln^* (env,s,fp,Exp e,c++c') (env',s',fp',Exn v, c')`
by metis_tac [e_step_add_ctxt, APPEND] >>
cases_on `c'` >>
full_simp_tac(srw_ss())[] >-
metis_tac [] >>
`e_step_reln^* (env',s',fp',Exn v,h::t) (env',s',fp',Exn v,[])`
by (match_mp_tac e_step_raise_lemma >> srw_tac[][]) >>
metis_tac [transitive_RTC, transitive_def]
)
QED
Theorem small_eval_err_add_ctxt[allow_rebind] =
SIMP_RULE (srw_ss ())
[METIS_PROVE [] ``!x y z. (x ⇒ y ⇒ z) = (x ∧ y ⇒ z)``]
small_eval_err_add_ctxt;
val small_eval_step_tac =
srw_tac[][do_con_check_def] >>
every_case_tac >>
full_simp_tac(srw_ss())[] >>
PairCases_on `r` >>
cases_on `r3` >|
[all_tac,
cases_on `e`] >>
srw_tac[][small_eval_def] >>
EQ_TAC >>
srw_tac[][] >|
[pop_assum (ASSUME_TAC o SIMP_RULE (srw_ss()) [Once RTC_CASES1]) >>
full_simp_tac(srw_ss())[return_def, e_step_reln_def, e_step_def, push_def, do_con_check_def] >>
gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[bind_exn_v_def] >>
metis_tac [pair_CASES],
srw_tac[][return_def, Once RTC_CASES1, e_step_reln_def, e_step_def, push_def,REVERSE_APPEND,
do_con_check_def] >>
fs [bind_exn_v_def] >>
metis_tac [],
pop_assum (ASSUME_TAC o SIMP_RULE (srw_ss()) [Once RTC_CASES1]) >>
full_simp_tac(srw_ss())[e_step_reln_def, e_step_def, push_def, return_def, do_con_check_def, bind_exn_v_def] >>
gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[] >>
metis_tac [],
srw_tac[][return_def, Once RTC_CASES1, e_step_reln_def, Once e_step_def, push_def,
do_con_check_def] >>
full_simp_tac(srw_ss())[REVERSE_APPEND, bind_exn_v_def] >>
metis_tac [],
qpat_x_assum `e_step_reln^* spat1 spat2`
(ASSUME_TAC o
SIMP_RULE (srw_ss()) [Once RTC_CASES1,e_step_reln_def,
e_step_def, push_def]) >>
full_simp_tac(srw_ss())[bind_exn_v_def] >> gs[] >>
every_case_tac >>
full_simp_tac(srw_ss())[return_def, do_con_check_def] >>
srw_tac[][] >-
(full_simp_tac(srw_ss())[e_step_def, push_def] >> gs[] >>
pop_assum MP_TAC >>
srw_tac[][return_def, do_con_check_def, REVERSE_APPEND]) >>
full_simp_tac(srw_ss())[] >>
metis_tac [],
srw_tac[][return_def, Once RTC_CASES1, e_step_reln_def, Once e_step_def, push_def,
do_con_check_def] >>
full_simp_tac(srw_ss())[REVERSE_APPEND, bind_exn_v_def] >>
metis_tac []];
Theorem small_eval_raise:
!s env fp cn e1 pes c r.
small_eval env s fp (Raise e1) c r =
small_eval env s fp e1 ((Craise (),env)::c) r
Proof
small_eval_step_tac
QED
Theorem small_eval_handle:
!env s fp cn e1 pes c r.
small_eval env s fp (Handle e1 pes) c r =
small_eval env s fp e1 ((Chandle () pes,env)::c) r
Proof
small_eval_step_tac
QED
Theorem small_eval_con:
!env s fp cn e1 es ns c r.
do_con_check env.c cn (LENGTH (es++[e1]))
⇒
(small_eval env s fp (Con cn (es++[e1])) c r =
small_eval env s fp e1 ((Ccon cn [] () (REVERSE es),env)::c) r)
Proof
srw_tac[][do_con_check_def] >>
every_case_tac >>
full_simp_tac(srw_ss())[] >>
small_eval_step_tac
QED
Theorem small_eval_app:
!env s fp op es c r.
small_eval env s fp (App op es) c r ⇔
(es = [] ∧ small_eval env s fp (App op []) c r) ∨
(?e es'. (es = es'++[e]) ∧ small_eval env s fp e ((Capp op [] () (REVERSE es'),env)::c) r)
Proof
srw_tac[][] >>
`es = [] ∨ ?e es'. es = es' ++ [e]` by metis_tac [list_end_case] >>
srw_tac[][] >>
`(?s' fp' v. r = (s', fp', Rval v)) ∨ (?s' fp' a. r = (s', fp', Rerr (Rabort a))) ∨
(?s' fp' err. r = (s', fp', Rerr (Rraise err)))`
by metis_tac [pair_CASES, result_nchotomy, error_result_nchotomy] >>
TRY (cases_on `a`) >>
full_simp_tac(srw_ss())[small_eval_def] >>
srw_tac[][Once RTC_CASES1, e_step_reln_def, e_step_def] >>
srw_tac[][push_def, application_thm] >>
EQ_TAC >>
srw_tac[][] >>
full_simp_tac(srw_ss())[REVERSE_APPEND] >>
metis_tac []
QED
Theorem small_eval_log:
!env s fp op e1 e2 c r.
small_eval env s fp (Log op e1 e2) c r =
small_eval env s fp e1 ((Clog op () e2,env)::c) r
Proof
small_eval_step_tac
QED
Theorem small_eval_if:
!env s fp e1 e2 e3 c r.
small_eval env s fp (If e1 e2 e3) c r =
small_eval env s fp e1 ((Cif () e2 e3,env)::c) r
Proof
small_eval_step_tac
QED
Theorem small_eval_match:
!env s fp e1 pes c r err_v.
small_eval env s fp (Mat e1 pes) c r =
small_eval env s fp e1 ((Cmat_check () pes (Conv (SOME bind_stamp) []),env)::c) r
Proof
small_eval_step_tac
QED
Theorem small_eval_let:
!env s fp n e1 e2 c r.
small_eval env s fp (Let n e1 e2) c r =
small_eval env s fp e1 ((Clet n () e2,env)::c) r
Proof
small_eval_step_tac
QED
Theorem small_eval_letrec:
!menv cenv env s fp funs e1 c r.
ALL_DISTINCT (MAP (λ(x,y,z). x) funs) ⇒
(small_eval env s fp (Letrec funs e1) c r =
small_eval (env with v := build_rec_env funs env env.v) s fp e1 c r)
Proof
small_eval_step_tac
QED
Theorem small_eval_tannot:
!env s fp e1 t c r.
small_eval env s fp (Tannot e1 t) c r =
small_eval env s fp e1 ((Ctannot () t,env)::c) r
Proof
small_eval_step_tac
QED
Theorem small_eval_lannot:
!env s fp e1 l c r.
small_eval env s fp (Lannot e1 l) c r =
small_eval env s fp e1 ((Clannot () l,env)::c) r
Proof
small_eval_step_tac
QED
Theorem small_eval_fpoptimise:
!env s fp e1 sc c r.
small_eval env s fp (FpOptimise sc e1) c r =
small_eval env s
(if fp.canOpt = Strict then fp else fp with canOpt := FPScope sc)
e1 ((Coptimise fp.canOpt sc (),env)::c) r
Proof
srw_tac[][] >>
small_eval_step_tac
QED
Inductive small_eval_list:
(!env s fp. small_eval_list env s fp [] (s, fp, Rval [])) ∧
(!s1 env fp1 e es v vs s2 fp2 s3 fp3 env'.
e_step_reln^* (env,s1,fp1, Exp e,[]) (env',s2,fp2, Val v,[]) ∧
small_eval_list env s2 fp2 es (s3, fp3, Rval vs)
⇒
small_eval_list env s1 fp1 (e::es) (s3, fp3, Rval (v::vs))) ∧
(!s1 env e es env' s2 s3 fp1 fp2 fp3 v err_v.
e_step_reln^* (env,s1,fp1,Exp e,[]) (env',s3,fp3,Exn err_v,[]) ∨
(e_step_reln^* (env,s1,fp1,Exp e,[]) (env',s2,fp2,Val v,[]) ∧
small_eval_list env s2 fp2 es (s3, fp3, Rerr (Rraise err_v)))
⇒
(small_eval_list env s1 fp1 (e::es) (s3, fp3, Rerr (Rraise err_v)))) ∧
(!s1 env e es e' c' env' s2 v s3 fp1 fp3 fp2 fp4.
(e_step_reln^* (env,s1,fp1,Exp e,[]) (env',s3,fp3,e',c') ∧
(e_step (env',s3,fp3,e',c') = Eabort (fp4, a))) ∨
(e_step_reln^* (env,s1,fp1,Exp e,[]) (env',s2,fp2,Val v,[]) ∧
small_eval_list env s2 fp2 es (s3, fp4, Rerr (Rabort a)))
⇒
(small_eval_list env s1 fp1 (e::es) (s3, fp4, Rerr (Rabort a))))
End
Theorem small_eval_list_length:
!env s1 fp1 es r. small_eval_list env s1 fp1 es r ⇒
!vs s2 fp2. (r = (s2, fp2, Rval vs)) ⇒ (LENGTH es = LENGTH vs)
Proof
HO_MATCH_MP_TAC small_eval_list_ind >>
srw_tac[][] >>
srw_tac[][]
QED
Theorem small_eval_list_step:
!env s2 fp2 es r.
small_eval_list env s2 fp2 es r ⇒
(!e v vs cn vs' env' s1 fp1 s3 fp3 v_con.
do_con_check env.c cn (LENGTH vs' + 1 + LENGTH vs) ∧
(build_conv env.c cn (REVERSE (REVERSE vs'++[v]++vs)) = SOME v_con) ∧
(r = (s3, fp3, Rval vs)) ∧ e_step_reln^* (env,s1,fp1,Exp e,[]) (env',s2,fp2,Val v,[]) ⇒
e_step_reln^* (env,s1,fp1,Exp e,[(Ccon cn vs' () es,env)])
(env,s3,fp3,Val v_con,[]))
Proof
HO_MATCH_MP_TAC (fetch "-" "small_eval_list_strongind") >>
srw_tac[][] >|
[`e_step_reln^* (env,s1,fp1,Exp e,[(Ccon cn vs' () [],env)])
(env',s2,fp2,Val v,[(Ccon cn vs' () [],env)])`
by metis_tac [e_step_add_ctxt,APPEND] >>
`e_step_reln (env',s2,fp2,Val v,[(Ccon cn vs' () [],env)])
(env,s2,fp2,Val v_con,[])`
by fs[return_def, continue_def, e_step_reln_def, e_step_def, REVERSE_APPEND] >>
metis_tac [transitive_RTC, transitive_def, RTC_SINGLE, APPEND],
`LENGTH (v'::vs'') + 1 + LENGTH vs = LENGTH vs'' + 1 + SUC (LENGTH vs)`
by (full_simp_tac(srw_ss())[] >>
DECIDE_TAC) >>
`REVERSE vs'' ++ [v'] ++ v::vs = (REVERSE vs'' ++ [v']) ++ [v] ++ vs`
by metis_tac [APPEND, APPEND_ASSOC] >>
`e_step_reln^* (env,s2,fp2,Exp e,[(Ccon cn (v'::vs'') () es,env)])
(env,s3,fp3,Val v_con,[])`
by metis_tac [APPEND_ASSOC, APPEND,REVERSE_DEF] >>
`e_step_reln^* (env,s1,fp1,Exp e',[(Ccon cn vs'' () (e::es),env)])
(env'',s2,fp2,Val v',[(Ccon cn vs'' () (e::es),env)])`
by metis_tac [e_step_add_ctxt, APPEND] >>
`LENGTH es = LENGTH vs` by metis_tac [small_eval_list_length] >>
`e_step_reln (env'',s2,fp2,Val v',[(Ccon cn vs'' () (e::es),env)])
(env,s2,fp2,Exp e,[(Ccon cn (v'::vs'') () es,env)])`
by (srw_tac[][push_def,continue_def, e_step_reln_def, e_step_def] >>
full_simp_tac (srw_ss() ++ ARITH_ss) [arithmeticTheory.ADD1]) >>
full_simp_tac(srw_ss())[] >>
`LENGTH vs'' + 1 + 1 + LENGTH es = LENGTH vs'' + 1 + SUC (LENGTH es)`
by DECIDE_TAC >>
`e_step_reln^* (env,s1,fp1,Exp e',[(Ccon cn vs'' () (e::es),env)])
(env,s3,fp3,Val v_con,[])`
by metis_tac [RTC_SINGLE, transitive_RTC, transitive_def] >>
metis_tac [APPEND_ASSOC, APPEND]]
QED
Theorem small_eval_list_err:
!env s2 fp2 es r. small_eval_list env s2 fp2 es r ⇒
(!e v err_v cn vs' env' s1 s3 fp1 fp3.
do_con_check env.c cn (LENGTH vs' + 1 + LENGTH es) ∧
(r = (s3, fp3, Rerr (Rraise err_v))) ∧
e_step_reln^* (env,s1,fp1,e,[]) (env',s2,fp2,Val v,[]) ⇒
?env''. e_step_reln^* (env,s1,fp1,e,[(Ccon cn vs' () es,env)])
(env'',s3,fp3,Exn err_v,[]))
Proof
ho_match_mp_tac small_eval_list_ind >>
srw_tac[][] >>
`e_step_reln^* (env,s1,fp1,e',[(Ccon cn vs' () (e::es),env)])
(env'',s2,fp2,Val v',[(Ccon cn vs' () (e::es),env)])`
by metis_tac [e_step_add_ctxt, APPEND] >>
`LENGTH vs' + 1 + 1 + LENGTH es = LENGTH vs' + 1 + SUC (LENGTH es)`
by DECIDE_TAC >>
`e_step_reln (env'',s2,fp2,Val v',[(Ccon cn vs' () (e::es),env)])
(env,s2,fp2,Exp e,[(Ccon cn (v'::vs') () es,env)])`
by srw_tac[][push_def,continue_def, e_step_reln_def, e_step_def] >>
full_simp_tac(srw_ss())[]
>- (`e_step_reln^* (env,s2,fp2,Exp e,[(Ccon cn (v'::vs') () es,env)])
(env',s3,fp3,Exn err_v,[(Ccon cn (v'::vs') () es,env)])`
by metis_tac [e_step_add_ctxt,APPEND] >>
`e_step_reln^*
(env',s3,fp3,Exn err_v,[(Ccon cn (v'::vs') () es,env)])
(env',s3,fp3,Exn err_v,[])`
by (match_mp_tac e_step_raise_lemma >>
srw_tac[][]) >>
rpt (simp[Once RTC_CASES_RTC_TWICE] >> first_x_assum $ irule_at Any) >>
simp[])
>- (`LENGTH (v'::vs') + 1 + LENGTH es = LENGTH vs' + 1 + SUC (LENGTH es)`
by (full_simp_tac(srw_ss())[] >>
DECIDE_TAC) >>
`?env''. e_step_reln^* (env,s2,fp2,Exp e,[(Ccon cn (v'::vs') () es,env)])
(env'',s3,fp3,Exn err_v, [])`
by metis_tac [] >>
metis_tac [RTC_SINGLE, transitive_RTC, transitive_def])
QED
Theorem small_eval_list_terr:
!env s2 fp2 es r.
small_eval_list env s2 fp2 es r ⇒
(!e v err cn vs' env' s1 fp1 s3 fp4.
do_con_check env.c cn (LENGTH vs' + 1 + LENGTH es) ∧
(r = (s3, fp4, Rerr (Rabort a))) ∧
e_step_reln^* (env,s1,fp1,e,[]) (env',s2,fp2,Val v,[]) ⇒
?env'' e' c' fp3. e_step_reln^* (env,s1,fp1,e,[(Ccon cn vs' () es,env)])
(env'',s3,fp3,e',c') ∧
(e_step (env'',s3,fp3,e',c') = (Eabort (fp4,a))))
Proof
HO_MATCH_MP_TAC small_eval_list_ind >>
srw_tac[][] >>
`e_step_reln^* (env,s1,fp1,e'',[(Ccon cn vs' () (e::es),env)])
(env'',s2,fp2,Val v',[(Ccon cn vs' () (e::es),env)])`
by metis_tac [e_step_add_ctxt, APPEND] >>
`LENGTH vs' + 1 + 1 + LENGTH es = LENGTH vs' + 1 + SUC (LENGTH es)`
by DECIDE_TAC >>
`e_step_reln (env'',s2,fp2,Val v',[(Ccon cn vs' () (e::es),env)])
(env,s2,fp2,Exp e,[(Ccon cn (v'::vs') () es,env)])`
by srw_tac[][push_def,continue_def, e_step_reln_def, e_step_def] >>
full_simp_tac(srw_ss())[] >|
[`e_step_reln^* (env,s2,fp2,Exp e,[(Ccon cn (v'::vs') () es,env)])
(env',s3,fp3,e',c'++[(Ccon cn (v'::vs') () es,env)])`
by metis_tac [e_step_add_ctxt,APPEND] >>
`e_step (env',s3,fp3,e',c'++[(Ccon cn (v'::vs') () es,env)]) = Eabort (fp4, a)`
by (irule e_single_error_add_ctxt >> gs[]) >>
metis_tac [RTC_SINGLE, transitive_RTC, transitive_def],
`LENGTH (v'::vs') + 1 + LENGTH es = LENGTH vs' + 1 + SUC (LENGTH es)`
by (full_simp_tac(srw_ss())[] >>
DECIDE_TAC) >>
`?env'' e' c' fp3. e_step_reln^* (env,s2,fp2,Exp e,[(Ccon cn (v'::vs') () es,env)])
(env'',s3,fp3,e',c') ∧
(e_step (env'',s3,fp3,e',c') = Eabort (fp4,a))`
by metis_tac [] >>
metis_tac [RTC_SINGLE, transitive_RTC, transitive_def]]
QED
Inductive small_eval_match:
(!env s fp err_v v. small_eval_match env s fp v [] err_v (s, fp, Rerr (Rraise err_v))) ∧
(!env s fp p e pes r v err_v.
ALL_DISTINCT (pat_bindings p []) ∧
pmatch env.c (FST s) p v [] = Match env' ∧
small_eval (env with v := nsAppend (alist_to_ns env') env.v) s fp e [] r
⇒
small_eval_match env s fp v ((p,e)::pes) err_v r) ∧
(!env s fp e p pes r v err_v.
ALL_DISTINCT (pat_bindings p []) ∧
(pmatch env.c (FST s) p v [] = No_match) ∧
small_eval_match env s fp v pes err_v r
⇒
small_eval_match env s fp v ((p,e)::pes) err_v r) ∧
(!env s fp p e pes v err_v.
¬(ALL_DISTINCT (pat_bindings p []))
⇒
small_eval_match env s fp v ((p,e)::pes) err_v (s, fp, Rerr (Rabort Rtype_error))) ∧
(!env s fp p e pes v err_v.
(pmatch env.c (FST s) p v [] = Match_type_error)
⇒
small_eval_match env s fp v ((p,e)::pes) err_v (s, fp, Rerr (Rabort Rtype_error)))
End
Definition alt_small_eval_def:
(alt_small_eval env s1 fp1 e c (s2, fp2, Rval v) ⇔
∃env'. e_step_reln^* (env,s1,fp1,e,c) (env',s2,fp2,Val v,[])) ∧
(alt_small_eval env s1 fp1 e c (s2, fp2, Rerr (Rraise err_v)) ⇔
∃env'.
e_step_reln^* (env,s1,fp1,e,c) (env',s2,fp2,Exn err_v,[])) ∧
(alt_small_eval env s1 fp1 e c (s2, fp3, Rerr (Rabort a)) ⇔
∃env' e' c' fp2.
e_step_reln^* (env,s1,fp1,e,c) (env',s2,fp2,e',c') ∧
(e_step (env',s2,fp2,e',c') = Eabort (fp3,a)))
End
Theorem small_eval_match_thm:
!env s fp v pes err_v r.
small_eval_match env s fp v pes err_v r ⇒
!env2. alt_small_eval env2 s fp (Val v) [(Cmat () pes err_v,env)] r
Proof
HO_MATCH_MP_TAC small_eval_match_ind >>
srw_tac[][alt_small_eval_def]
>- (qexists_tac `env` >>
match_mp_tac RTC_SINGLE >>
srw_tac[][e_step_reln_def, e_step_def, continue_def])
>- (PairCases_on `r` >>
cases_on `r3` >|
[all_tac,
cases_on `e'`] >>
full_simp_tac(srw_ss())[alt_small_eval_def, small_eval_def]
>- (srw_tac[][Once RTC_CASES1, e_step_reln_def] >>
srw_tac[][e_step_def, continue_def] >>
metis_tac[])
>- (srw_tac[][Once RTC_CASES1, e_step_reln_def] >>
srw_tac[][e_step_def, continue_def] >>
metis_tac []) >>
srw_tac[][] >>
srw_tac[][Once RTC_CASES1, e_step_reln_def] >>
qexists_tac `env''` >>
qexists_tac `e'` >>
qexists_tac `c'` >>
qexists_tac ‘fp''’ >>
srw_tac[][] >>
srw_tac[][e_step_def, continue_def])
>- (PairCases_on `r` >>
cases_on `r3` >|
[all_tac,
cases_on `e'`] >>
full_simp_tac(srw_ss())[alt_small_eval_def] >>
srw_tac[][Once RTC_CASES1, e_step_reln_def] >> full_simp_tac(srw_ss())[] >|
[srw_tac[][e_step_def, push_def, continue_def] >>
metis_tac [],
srw_tac[][e_step_def, push_def, continue_def] >>
metis_tac [],
srw_tac[][] >>
pop_assum (qspec_then`env`strip_assume_tac) >>
qexists_tac `env'` >>
qexists_tac `e'` >>
qexists_tac `c'` >>
qexists_tac ‘fp2'’ >>
srw_tac[][] >>
srw_tac[][e_step_def, push_def, continue_def]])
>- (qexists_tac `env2` >>
qexists_tac `Val v` >>
qexists_tac `[(Cmat () ((p,e)::pes) err_v,env)]` >>
qexists_tac ‘fp’ >>
srw_tac[][RTC_REFL] >>
srw_tac[][e_step_def, continue_def, fix_fp_state_def]
(* PairCases_on `env` >>
full_simp_tac(srw_ss())[] >>
metis_tac []*))
>- (qexists_tac `env2` >>
qexists_tac `Val v` >>
qexists_tac `[(Cmat () ((p,e)::pes) err_v,env)]` >>
qexists_tac ‘fp’ >>
srw_tac[][RTC_REFL] >>
srw_tac[][e_step_def, continue_def, fix_fp_state_def]
(* PairCases_on `env` >>
full_simp_tac(srw_ss())[]*))
QED
Theorem small_eval_opapp_err:
∀env s fp es res.
small_eval_list env s fp es res ⇒
∀s' fp' vs.
res = (s', fp', Rval vs) ⇒
∀env0 v1 v0.
LENGTH es + LENGTH v0 ≠ 1 ⇒
∃env' e' c' fp''.
e_step_reln^* (env0,s,fp,Val v1,[Capp Opapp v0 () es,env]) (env',s',fp'',e',c') ∧
e_step (env',s',fp'',e',c') = Eabort (fp', Rtype_error)
Proof
ho_match_mp_tac small_eval_list_ind >> simp[] >> srw_tac[][] >>
srw_tac[boolSimps.DNF_ss][Once RTC_CASES1,e_step_reln_def] >- (
srw_tac[][Once e_step_def,continue_def,application_thm] >>
Cases_on `v0` >>
full_simp_tac(srw_ss())[do_opapp_def, fix_fp_state_def] >>
Cases_on`t`>>full_simp_tac(srw_ss())[]) >>
disj2_tac >>
srw_tac[][Once e_step_def,continue_def,push_def] >>
imp_res_tac e_step_add_ctxt >>
pop_assum(qspec_then`[Capp Opapp (v1::v0) () es,env]`strip_assume_tac) >>
full_simp_tac(srw_ss())[] >>
first_x_assum(qspecl_then[`env'`,`v`,`v1::v0`]mp_tac) >>
impl_tac >- simp[] >>
metis_tac[transitive_RTC,transitive_def]
QED
Theorem small_eval_app_err:
∀env s fp es res.
small_eval_list env s fp es res ⇒
∀s' fp' vs.
res = (s', fp', Rval vs) ⇒
∀op env0 v1 v0.
LENGTH es + LENGTH v0 > 2 ∧ op ≠ Opapp ∧ op ≠ AallocFixed
∧ op ≠ CopyStrStr ∧ op ≠ CopyStrAw8 ∧ op ≠ CopyAw8Str ∧ op ≠ CopyAw8Aw8
∧ getOpClass op ≠ Reals
⇒
∃env' e' c' fp''.
e_step_reln^* (env0,s,fp,Val v1,[Capp op v0 () es,env]) (env',s',fp'',e',c') ∧
e_step (env',s',fp'',e',c') = Eabort (fp', Rtype_error)
Proof
ho_match_mp_tac small_eval_list_ind >> simp[] >> srw_tac[][] >>
srw_tac[boolSimps.DNF_ss][Once RTC_CASES1,e_step_reln_def] >- (
srw_tac[][Once e_step_def,continue_def,application_thm] >>
BasicProvers.CASE_TAC >>
TRY BasicProvers.CASE_TAC >>
Cases_on`s` >> fs[do_app_cases,fix_fp_state_def] >> rw[] >> fs[] >>
rpt TOP_CASE_TAC >> gs[do_app_cases]) >>
disj2_tac >>
srw_tac[][Once e_step_def,continue_def,push_def] >>
imp_res_tac e_step_add_ctxt >>
pop_assum(qspec_then`[Capp op (v1::v0) () es,env]`strip_assume_tac) >>
full_simp_tac(srw_ss())[] >>
first_x_assum(qspecl_then[`op`,`env'`,`v`,`v1::v0`]mp_tac) >>
impl_tac >- simp[] >>
metis_tac[transitive_RTC,transitive_def]
QED
Theorem small_eval_app_err_more:
∀env s fp es res.
small_eval_list env s fp es res ⇒
∀s' fp' vs.
res = (s', fp', Rval vs) ⇒
∀op env0 v1 v0.
LENGTH es + LENGTH v0 > 4 ∧ op ≠ Opapp ∧ op ≠ AallocFixed ∧
getOpClass op ≠ Reals
⇒
∃env' e' c' fp''.
e_step_reln^* (env0,s,fp,Val v1,[Capp op v0 () es,env]) (env',s',fp'',e',c') ∧
e_step (env',s',fp'',e',c') = Eabort (fp', Rtype_error)
Proof
ho_match_mp_tac small_eval_list_ind >> simp[] >> srw_tac[][] >>
srw_tac[boolSimps.DNF_ss][Once RTC_CASES1,e_step_reln_def] >- (
srw_tac[][Once e_step_def,continue_def,application_thm] >>
BasicProvers.CASE_TAC >>
TRY BasicProvers.CASE_TAC >>
Cases_on`s` >> fs[do_app_cases,fix_fp_state_def] >> rw[] >> fs[] >>
rpt TOP_CASE_TAC >> gs[do_app_cases]) >>
disj2_tac >>
srw_tac[][Once e_step_def,continue_def,push_def] >>
imp_res_tac e_step_add_ctxt >>
pop_assum(qspec_then`[Capp op (v1::v0) () es,env]`strip_assume_tac) >>
full_simp_tac(srw_ss())[] >>
first_x_assum(qspecl_then[`op`,`env'`,`v`,`v1::v0`]mp_tac) >>
impl_tac >- simp[] >>
metis_tac[transitive_RTC,transitive_def]
QED
val _ = temp_delsimps ["getOpClass_def"]
Theorem step_e_not_timeout:
e_step (env',s3,fp3,e',c') = Eabort (fp4, a) ⇒ a ≠ Rtimeout_error
Proof
full_simp_tac(srw_ss())[e_step_def] >>
gs[push_def, return_def, continue_def, application_thm] >>
rpt (TOP_CASE_TAC >> gs[]) >>
imp_res_tac do_app_not_timeout >> strip_tac >>
gs[] >>
every_case_tac >> rveq >> gs[do_fprw_def] >>
imp_res_tac do_app_not_timeout >>
every_case_tac >> fs[]
QED
Theorem small_eval_list_not_timeout:
∀env s fp es res. small_eval_list env s fp es res ⇒
SND (SND res) ≠ Rerr (Rabort Rtimeout_error)
Proof
ho_match_mp_tac small_eval_list_ind >> srw_tac[][] >>
metis_tac [step_e_not_timeout]
QED
Theorem small_eval_list_app_type_error:
∀env s fp es res.
small_eval_list env s fp es res ⇒
∀s' fp' err.
res = (s', fp', Rerr (Rabort a)) ⇒
∀op env0 v1 v0.
∃env' e' c' fp''.
e_step_reln^* (env0,s,fp,Val v1,[Capp op v0 () es,env]) (env',s',fp'',e',c') ∧
e_step (env',s',fp'',e',c') = Eabort (fp', a)
Proof
ho_match_mp_tac (theorem"small_eval_list_strongind") >> simp[] >> srw_tac[][] >- (
srw_tac[][Once RTC_CASES1,e_step_reln_def,Once e_step_def,continue_def,push_def] >>
srw_tac[boolSimps.DNF_ss][] >> disj2_tac >>
imp_res_tac e_step_add_ctxt >>
Q.PAT_ABBREV_TAC`ctx = [(Capp A B C D,env)]` >>
first_x_assum(qspec_then`ctx`strip_assume_tac) >> full_simp_tac(srw_ss())[] >>
first_assum(match_exists_tac o concl) >> srw_tac[][] >>
irule e_single_error_add_ctxt >> unabbrev_all_tac >> gs[]) >>
srw_tac[][Once RTC_CASES1,e_step_reln_def,Once e_step_def,continue_def,push_def] >>
srw_tac[boolSimps.DNF_ss][] >> disj2_tac >>
srw_tac[][Once RTC_CASES_RTC_TWICE] >>
imp_res_tac e_step_add_ctxt >>
Q.PAT_ABBREV_TAC`ctx = [(Capp X Y Z A,env)]` >>
first_x_assum(qspec_then`ctx`strip_assume_tac) >> full_simp_tac(srw_ss())[] >>
simp[PULL_EXISTS] >>
first_assum(match_exists_tac o concl) >> srw_tac[][] >>
srw_tac[][Once RTC_CASES1,e_step_reln_def,e_step_def,continue_def,Abbr`ctx`]
QED
Theorem small_eval_list_app_error:
∀env s fp es res.
small_eval_list env s fp es res ⇒
∀s' fp' v.
res = (s',fp',Rerr (Rraise v)) ⇒
∀op env0 v1 v0.
∃env' env''.
e_step_reln^* (env0,s,fp,Val v1,[Capp op v0 () es,env]) (env',s',fp',Exn v,[])
Proof
ho_match_mp_tac (theorem"small_eval_list_strongind") >> simp[] >> srw_tac[][]
>- (
srw_tac[][Once RTC_CASES1,e_step_reln_def,Once e_step_def,continue_def,push_def] >>
imp_res_tac e_step_add_ctxt >>
Q.PAT_ABBREV_TAC`ctx = [(Capp A B C D,env)]` >>
first_x_assum(qspec_then`ctx`strip_assume_tac) >> full_simp_tac(srw_ss())[] >>
srw_tac[][Once RTC_CASES_RTC_TWICE] >>
first_assum(match_exists_tac o concl) >> srw_tac[][] >>
srw_tac[][Once RTC_CASES1,e_step_reln_def,e_step_def,continue_def,Abbr`ctx`] >>
srw_tac[][Once RTC_CASES1,e_step_reln_def,e_step_def,continue_def] >>
metis_tac[RTC_REFL]) >>
srw_tac[][Once RTC_CASES1,e_step_reln_def,Once e_step_def,continue_def,push_def] >>
srw_tac[][Once RTC_CASES_RTC_TWICE] >>
imp_res_tac e_step_add_ctxt >>
Q.PAT_ABBREV_TAC`ctx = [(Capp X Y Z A,env)]` >>
first_x_assum(qspec_then`ctx`strip_assume_tac) >> full_simp_tac(srw_ss())[] >>
first_assum(match_exists_tac o concl) >> srw_tac[][] >>
srw_tac[][Once RTC_CASES1,e_step_reln_def,e_step_def,continue_def,Abbr`ctx`]
QED
(** Does not hold anymore with floating-point optimization semantics
Theorem e_step_exp_err_any_ctxt:
e_step (x,y,fp,Exp z,c1) = Eabort (fp2, a) ⇒
e_step (x,y,fp,Exp z,c2) = Eabort (fp2, a)
Proof
srw_tac[][e_step_def] >>
rpt (TOP_CASE_TAC >> gs[]) >>
full_simp_tac(srw_ss())[push_def,return_def,continue_def,application_thm] >>
gs[] >>
every_case_tac >> full_simp_tac(srw_ss())[]
QED **)
Inductive e_step_to_match:
(ALL_DISTINCT (pat_bindings p []) ∧
pmatch env.c (FST s) p v [] = Match env' ∧
RTC e_step_reln (env with v := nsAppend (alist_to_ns env') env.v, s, fp, Exp e, [])
(env'', s', fp', ev, cs)
⇒ e_step_to_match env s fp v ((p,e)::pes) s' fp') ∧
(ALL_DISTINCT (pat_bindings p []) ∧
pmatch env.c (FST s) p v [] = No_match ∧
e_step_to_match env s fp v pes s' fp'
⇒ e_step_to_match env s fp v ((p,e)::pes) s' fp')
End
Theorem e_step_to_match_Cmat:
∀env s fp v pes s' fp'. e_step_to_match env s fp v pes s' fp' ⇒
∀env''. ∃ev env' cs.
RTC e_step_reln (env'', s, fp, Val v, [(Cmat () pes v, env)]) (env', s', fp', ev, cs)
Proof
ntac 4 strip_tac >> ho_match_mp_tac e_step_to_match_ind >> rw[] >>
irule_at Any $ cj 2 RTC_rules >>
simp[e_step_reln_def, e_step_def, continue_def, SF SFY_ss] >>
first_x_assum $ qspec_then ‘env’ strip_assume_tac >>
first_x_assum $ irule_at Any
QED
Theorem e_step_to_Con:
∀left mid right env s fp e sm fpr vals cn vs.
small_eval_list env s fp (e::left) (sm,fpr,Rval vals) ∧
do_con_check env.c cn (LENGTH left + LENGTH right + LENGTH vs + 2)
⇒
RTC e_step_reln
(env,s,fp,Exp e,[Ccon cn vs () (left ++ [mid] ++ right),env])
(env,sm,fpr,Exp mid,[Ccon cn (REVERSE vals ++ vs) () right,env])
Proof
Induct >> rw[] >> gvs[ADD1]
>- (
ntac 2 $ gvs[Once small_eval_list_cases] >>
drule e_step_add_ctxt >> simp[] >>
disch_then $ qspec_then `[Ccon cn vs () (mid::right),env]` assume_tac >>
simp[Once RTC_CASES_RTC_TWICE] >> goal_assum drule >>
irule_at Any $ cj 2 RTC_rules >>
simp[e_step_reln_def, e_step_def, continue_def, push_def]
) >>
qpat_x_assum `small_eval_list _ _ _ _ _` mp_tac >>
simp[Once small_eval_list_cases] >> rw[] >>
drule e_step_add_ctxt >> simp[] >>
disch_then $ qspec_then
`[Ccon cn vs () (h::(left ++ [mid] ++ right)),env]` assume_tac >>
simp[Once RTC_CASES_RTC_TWICE] >> goal_assum dxrule >>
irule_at Any $ cj 2 RTC_rules >>
simp[e_step_reln_def, e_step_def, continue_def, push_def] >>
last_x_assum drule >>
disch_then $ qspecl_then [`mid`,`right`,`cn`,`v::vs`] mp_tac >>
simp[ADD1, APPEND_ASSOC_CONS]
QED
Theorem e_step_to_App_mid:
∀left mid right env s fp e sm fpr vals op vs.
small_eval_list env s fp (e::left) (sm, fpr, Rval vals) ⇒
RTC e_step_reln
(env,s,fp,Exp e,[Capp op vs () (left ++ [mid] ++ right),env])
(env,sm,fpr,Exp mid,[Capp op (REVERSE vals ++ vs) () right,env])
Proof
Induct >> rw[] >> gvs[]
>- (
ntac 2 $ gvs[Once small_eval_list_cases] >>
drule e_step_add_ctxt >> simp[] >>
disch_then $ qspec_then `[Capp op vs () (mid::right),env]` assume_tac >>
simp[Once RTC_CASES_RTC_TWICE] >> goal_assum drule >>
irule_at Any $ cj 2 RTC_rules >>
simp[e_step_reln_def, e_step_def, continue_def, push_def]
) >>
qpat_x_assum `small_eval_list _ _ _ _ _` mp_tac >>
simp[Once small_eval_list_cases] >> rw[] >>
drule e_step_add_ctxt >> simp[] >>
disch_then $ qspec_then
`[Capp op vs () (h::(left ++ [mid] ++ right)),env]` assume_tac >>
simp[Once RTC_CASES_RTC_TWICE] >> goal_assum dxrule >>
irule_at Any $ cj 2 RTC_rules >>
simp[e_step_reln_def, e_step_def, continue_def, push_def] >>
last_x_assum drule >>
disch_then $ qspecl_then [`mid`,`right`,`op`,`v::vs`] mp_tac >>
simp[ADD1, APPEND_ASSOC_CONS]
QED
Theorem small_eval_list_Rval_APPEND:
small_eval_list env s fp (left ++ right) (s', fp', Rval vs) ⇔
∃lvs rvs sl fpl. vs = lvs ++ rvs ∧
small_eval_list env s fp left (sl, fpl, Rval lvs) ∧
small_eval_list env sl fpl right (s', fp', Rval rvs)
Proof
reverse eq_tac >> rw[]
>- (
ntac 2 $ pop_assum mp_tac >>
qid_spec_tac `lvs` >> Induct_on `small_eval_list` >> rw[] >> gvs[] >>
simp[Once small_eval_list_cases, SF SFY_ss]
) >>
pop_assum mp_tac >>
map_every qid_spec_tac [`vs`,`right`,`left`] >>
Induct_on `small_eval_list` >> rw[]
>- (ntac 2 $ simp[Once small_eval_list_cases]) >>
Cases_on `left` >> gvs[]
>- (ntac 2 $ simp[Once small_eval_list_cases] >> goal_assum drule >> simp[]) >>
simp[Once small_eval_list_cases, PULL_EXISTS] >>
goal_assum $ drule_at Any >>
pop_assum $ qspecl_then [`t`,`right`] mp_tac >> rw[] >>
irule_at Any EQ_REFL >> gvs[SF SFY_ss]
QED
Theorem e_step_over_App_Opapp:
∀env s fp e es s' fp' vals vs.
small_eval_list env s fp (e::es) (s', fp', Rval vals) ∧
do_opapp (REVERSE vals ++ vs) = SOME (env', ea) ⇒