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|
open HolKernel bossLib boolLib combinTheory listTheory rich_listTheory optionTheory pairTheory sortingTheory;
open wordsTheory markerTheory;
open ottTheory ottLib caml_typedefTheory;
open utilTheory basicTheory environmentTheory shiftTheory validTheory strengthenTheory;
open weakenTheory type_substTheory remv_tyvarTheory teqTheory;
val _ = new_theory "substs";
val _ = Parse.hide "S";
val subst_nexp_thm = Q.store_thm ("subst_nexp_thm",
`!nexp v x. is_non_expansive_of_expr nexp /\ is_non_expansive_of_expr v ==>
is_non_expansive_of_expr (subst_value_name_expr v x nexp)`,
recInduct is_non_expansive_of_expr_ind THEN
SRW_TAC [] [is_non_expansive_of_expr_def, subst_value_name_letrec_binding_def,
EVERY_MAP, EVERY_MEM, LAMBDA_PROD2] THENL
[Cases_on `z` THEN METIS_TAC [SND],
Cases_on `expr1` THEN
FULL_SIMP_TAC list_ss [is_binary_prim_app_value_of_expr_def, subst_value_name_letrec_binding_def],
METIS_TAC []]);
val substs_nexp_thm = Q.store_thm ("substs_nexp_thm",
`!nexp subs. is_non_expansive_of_expr nexp /\ EVERY (\x. is_non_expansive_of_expr (SND x)) subs ==>
is_non_expansive_of_expr (substs_value_name_expr subs nexp)`,
recInduct is_non_expansive_of_expr_ind THEN
SRW_TAC [] [is_non_expansive_of_expr_def, substs_value_name_letrec_binding_def, EVERY_MAP, EVERY_MEM,
LAMBDA_PROD2] THEN
FULL_SIMP_TAC list_ss [MEM_FILTER] THENL
[Cases_on `list_assoc value_name subs` THEN SRW_TAC [] [is_non_expansive_of_expr_def] THEN
METIS_TAC [list_assoc_mem, SND],
Cases_on `z` THEN METIS_TAC [SND],
Cases_on `expr1` THEN
FULL_SIMP_TAC list_ss [is_binary_prim_app_value_of_expr_def, substs_value_name_letrec_binding_def]]);
local
val lem1 = Q.prove (
`!f. (case f of NONE -> NONE || SOME EB -> SOME EB) = f`,
Cases THEN SRW_TAC [] []);
val lem2 = Q.prove (
`!l x_t_list z. ~MEM (FST z) l /\ (MAP name_vn l = MAP (\z. name_vn (FST z)) x_t_list) ==>
~MEM z x_t_list`,
Induct THEN SRW_TAC [] [] THEN Cases_on `x_t_list` THEN FULL_SIMP_TAC list_ss [name_11] THEN METIS_TAC []);
val lem3 = Q.prove (
`JTpat (shiftTsig 0 (num_tv E1) S') (E1 ++ [EB_vn x (TS_forall t'')] ++ E2) p t E /\
JTe (shiftTsig 0 (num_tv E1) S') (E ++ E1 ++ [EB_vn x (TS_forall t'')] ++ E2) e t' /\
( is_non_expansive_of_expr v /\
(E ++ E1 ++ [EB_vn x (TS_forall t'')] ++ E2 =
(E++E1) ++ [EB_vn x (TS_forall t'')] ++ E2) /\
(shiftTsig 0 (num_tv E1) S' = shiftTsig 0 (num_tv (E++E1)) S') /\
~MEM (name_vn x) (MAP domEB (E++E1)) /\
closed_env E2 /\
JTe (shiftTsig 0 1 S') (EB_tv::E2) v t'' ==>
JTe (shiftTsig 0 (num_tv (E++E1)) S') ((E++E1) ++ E2) (subst_value_name_expr v x e) t') /\
is_non_expansive_of_expr v /\
~MEM (name_vn x) (MAP domEB E1) /\
closed_env E2 /\
JTe (shiftTsig 0 1 S') (EB_tv::E2) v t'' ==>
JTe (shiftTsig 0 (num_tv E1) S') (E ++ E1 ++ E2)
(if MEM x (aux_xs_pattern_of_pattern p) then
e
else
subst_value_name_expr v x e)
t'`,
SRW_TAC [] [] THEN FULL_SIMP_TAC list_ss [] THEN
IMP_RES_TAC aux_xs_pattern_of_pattern_thm THEN SRW_TAC [] [] THEN
`num_tv E = 0` by METIS_TAC [pat_env_lem, value_env_num_tv_thm] THEN
FULL_SIMP_TAC list_ss [] THENL
[MATCH_MP_TAC (GEN_ALL (hd (CONJUNCTS (SIMP_RULE list_ss [AND_IMP_INTRO] value_env_str_thm)))) THEN
Q.EXISTS_TAC `[EB_vn x (TS_forall t'')]` THEN
SRW_TAC [] [value_env_def, domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP] THEN
METIS_TAC [MEM_MAP, MEM_REVERSE],
FULL_SIMP_TAC list_ss [num_tv_append_thm] THEN
Q.PAT_ASSUM `~MEM (name_vn x) (MAP domEB E) ==> JTe
(shiftTsig 0 (num_tv E1) S') (E ++ E1 ++ E2) (subst_value_name_expr v x e) t'`
MATCH_MP_TAC THEN
METIS_TAC [MEM_MAP, name_11, MEM_REVERSE]]);
val lem4 = Q.prove (
`MEM x (MAP FST l) /\
JTpat_matching (shiftTsig 0 (num_tv E1) S')
(REVERSE (MAP (\z. EB_vn (FST z)
(TS_forall (shiftt 0 1 (TE_arrow (FST (SND (SND z)))
(SND (SND (SND z))))))) l) ++ E1 ++
[EB_vn x (TS_forall t'')] ++ E2) pm t t' ==>
JTpat_matching (shiftTsig 0 (num_tv E1) S')
(REVERSE (MAP (\z. EB_vn (FST z)
(TS_forall (shiftt 0 1 (TE_arrow (FST (SND (SND z)))
(SND (SND (SND z))))))) l) ++ E1 ++
E2) pm t t'`,
Cases_on `pm` THEN SRW_TAC [] [JTe_fun] THEN
Q.EXISTS_TAC `pattern_e_E_list` THEN SRW_TAC [] [] THEN FULL_SIMP_TAC list_ss [EVERY_MEM] THENL
[METIS_TAC [value_env_pat_str_thm, value_env_def],
SRW_TAC [] [] THEN
MATCH_MP_TAC (GEN_ALL (hd (CONJUNCTS (SIMP_RULE list_ss [AND_IMP_INTRO] value_env_str_thm)))) THEN
Q.EXISTS_TAC `[EB_vn x (TS_forall t'')]` THEN
SRW_TAC [] [value_env_def, MAP_REVERSE, MAP_MAP, domEB_def] THEN
Q.PAT_ASSUM `MEM x (MAP FST l)` MP_TAC THEN
REPEAT (POP_ASSUM (K ALL_TAC)) THEN Induct_on `l` THEN SRW_TAC [] [] THEN METIS_TAC []]);
val lem5 = Q.prove (
`!l f g. num_tv (MAP (\x. EB_vn (f x) (g x)) l) = 0`,
METIS_TAC [value_env_map_thm, value_env_num_tv_thm]);
val lem6 = Q.prove (
`(!t n m. shiftt n m (shiftt 0 n t) = shiftt 0 (n + m) t) /\
(!tl n m. MAP (\t. shiftt n m (shiftt 0 n t)) tl = MAP (\t. shiftt 0 (n + m) t) tl)`,
Induct THEN SRW_TAC [] [shiftt_def, MAP_MAP] THEN DECIDE_TAC);
val lem7 = Q.prove (
`!S n m. shiftTsig n m (shiftTsig 0 n S) = shiftTsig 0 (n + m) S`,
Induct THEN SRW_TAC [] [shiftTsig_def, LAMBDA_PROD2, MAP_MAP, lem6]);
val lem8 = Q.prove (
`!S e t E E' t_list.
JTe (shiftTsig 0 1 S) (EB_tv::E) e t /\
closed_env E /\
Eok (E'++E) /\
EVERY (tkind (E'++E)) t_list ==>
JTe (shiftTsig 0 (num_tv E') S) (E'++E) e (idxsub t_list (shiftt 1 (num_tv E') t))`,
SRW_TAC [] [GSYM idxsubn0_thm] THEN MATCH_MP_TAC type_subst_thm THEN SRW_TAC [] [] THEN
IMP_RES_TAC ((SIMP_RULE list_ss [Eok_def, num_tv_def] o
Q.SPECL [`shiftTsig 0 1 S`, `e`, `t`, `E'`, `E`]) weak_thm) THEN
FULL_SIMP_TAC list_ss [shiftTsig_add_thm, lem7]);
in
val subst_lem = Q.store_thm ("subst_lem",
`(!S E e t. JTe S E e t ==>
!S' E1 E2 x v t'.
is_non_expansive_of_expr v /\
(E = E1 ++ [EB_vn x (TS_forall t')] ++ E2) /\
(S = shiftTsig 0 (num_tv E1) S') /\
~MEM (name_vn x) (MAP domEB E1) /\
closed_env E2 /\
JTe (shiftTsig 0 1 S') (EB_tv::E2) v t' ==>
JTe S (E1++E2) (subst_value_name_expr v x e) t) /\
(!S E pm t t'. JTpat_matching S E pm t t' ==>
!S' E1 E2 x v t''.
is_non_expansive_of_expr v /\
(E = E1 ++ [EB_vn x (TS_forall t'')] ++ E2) /\
(S = shiftTsig 0 (num_tv E1) S') /\
~MEM (name_vn x) (MAP domEB E1) /\
closed_env E2 /\
JTe (shiftTsig 0 1 S') (EB_tv::E2) v t'' ==>
JTpat_matching S (E1++E2) (subst_value_name_pattern_matching v x pm) t t') /\
(!S E lb E'. JTlet_binding S E lb E' ==>
!S' E1 E2 x v t''.
is_non_expansive_of_expr v /\
(E = E1 ++ [EB_vn x (TS_forall t'')] ++ E2) /\
(S = shiftTsig 0 (num_tv E1) S') /\
~MEM (name_vn x) (MAP domEB E1) /\
closed_env E2 /\
JTe (shiftTsig 0 1 S') (EB_tv::E2) v t'' ==>
JTlet_binding S (E1++E2) (subst_value_name_let_binding v x lb) E') /\
(!S E lrbs E'. JTletrec_binding S E lrbs E' ==>
!S' x E1 E2 v t''.
is_non_expansive_of_expr v /\
(E = E1 ++ [EB_vn x (TS_forall t'')] ++ E2) /\
(S = shiftTsig 0 (num_tv E1) S') /\
~MEM (name_vn x) (MAP domEB E1) /\
closed_env E2 /\
JTe (shiftTsig 0 1 S') (EB_tv::E2) v t'' ==>
if MEM x (aux_xs_letrec_bindings_of_letrec_bindings lrbs) then
JTletrec_binding S (E1++E2) lrbs E'
else
JTletrec_binding S (E1++E2) (subst_value_name_letrec_bindings v x lrbs) E')`,
RULE_INDUCT_TAC JTe_sind [subst_value_name_letrec_binding_def, JTe_fun]
[([``"JTe_uprim"``, ``"JTe_bprim"``, ``"JTe_constant"``, ``"JTe_typed"``],
FULL_SIMP_TAC list_ss [JTuprim_cases, JTbprim_cases, JTconst_cases, JTconstr_c_cases] THEN
SRW_TAC [] [] THEN
FULL_SIMP_TAC list_ss [lookup_def, domEB_def, name_distinct, EVERY_MEM] THEN
METIS_TAC [value_env_lookup_str_thm, value_env_ok_str_thm, APPEND, value_env_def,
value_env_inst_any_thm, value_env_teq_str_thm]),
([``"JTe_apply"``, ``"JTe_match"``], METIS_TAC []),
([``"JTe_ident"``],
SRW_TAC [] [] THEN SRW_TAC [] [JTe_fun] THEN
FULL_SIMP_TAC list_ss [JTvalue_name_cases, lookup_def, domEB_def, name_11, lookup_append_thm,
lookup_dom_thm] THEN
SRW_TAC [] [] THENL
[FULL_SIMP_TAC list_ss [JTinst_cases, shiftEB_def, shiftts_def] THEN SRW_TAC [] [] THEN
`JTe (shiftTsig 0 (num_tv E1) S'') (E1 ++ E2) v (idxsub t_list (shiftt 1 (num_tv E1) t'))`
by (MATCH_MP_TAC lem8 THEN SRW_TAC [] [] THENL
[METIS_TAC [value_env_ok_str_thm, value_env_def, ok_ok_thm],
FULL_SIMP_TAC list_ss [EVERY_MEM] THEN
METIS_TAC [value_env_ok_str_thm, value_env_def]]) THEN
METIS_TAC [teq_thm, value_env_def, value_env_teq_str_thm],
FULL_SIMP_TAC list_ss [lem1] THEN Cases_on `lookup E1 (name_vn value_name)` THEN
FULL_SIMP_TAC list_ss [option_case_def] THEN SRW_TAC [] [] THENL
[Cases_on `EB` THEN
FULL_SIMP_TAC list_ss [environment_binding_distinct, shiftEB_def, num_tv_append_thm,
num_tv_def] THEN
SRW_TAC [] [] THEN
METIS_TAC [value_env_inst_str_thm, value_env_ok_str_thm, value_env_def,
value_env_teq_str_thm],
METIS_TAC [value_env_inst_str_thm, value_env_ok_str_thm, value_env_def,
value_env_teq_str_thm]]]),
([``"JTe_tuple"``, ``"JTe_construct"``],
SRW_TAC [] [] THEN
Q.EXISTS_TAC `MAP (\e_t. (subst_value_name_expr v x (FST e_t), SND e_t)) e_t_list` THEN
SRW_TAC [] [MAP_MAP, ETA_THM, EVERY_MAP] THEN FULL_SIMP_TAC list_ss [EVERY_MEM] THEN
METIS_TAC [value_env_def, value_env_constr_p_str_thm, APPEND, value_env_teq_str_thm]),
([``"JTe_cons"``, ``"JTe_and"``, ``"JTe_or"``, ``"JTe_while"``, ``"JTe_function"``],
METIS_TAC [value_env_teq_str_thm, value_env_def]),
([``"JTe_record_constr"``],
SRW_TAC [] [] THEN
MAP_EVERY Q.EXISTS_TAC [`field_name'_list`, `t'_list`,
`MAP (\fn_e_t. (FST fn_e_t,
subst_value_name_expr v x (FST (SND fn_e_t)),
SND (SND fn_e_t)))
field_name_e_t_list`,
`typeconstr_name`,
`kind`] THEN
SRW_TAC [] [MAP_MAP, ETA_THM, EVERY_MAP] THEN FULL_SIMP_TAC list_ss [EVERY_MEM] THEN1
METIS_TAC [] THEN1
METIS_TAC [value_env_def, value_env_field_str_thm, APPEND] THEN1
METIS_TAC [value_env_def, value_env_lookup_str_thm, APPEND] THEN1
METIS_TAC [value_env_teq_str_thm, value_env_def]),
([``"JTe_record_with"``],
SRW_TAC [] [] THEN
Q.EXISTS_TAC `MAP (\fn_e_t. (FST fn_e_t,
subst_value_name_expr v x (FST (SND fn_e_t)),
SND (SND fn_e_t)))
field_name_e_t_list` THEN
SRW_TAC [] [MAP_MAP, ETA_THM, EVERY_MAP] THEN FULL_SIMP_TAC list_ss [EVERY_MEM] THEN
METIS_TAC [value_env_def, value_env_field_str_thm, APPEND, value_env_teq_str_thm]),
([``"JTe_record_proj"``],
METIS_TAC [value_env_def, value_env_field_str_thm, APPEND, value_env_teq_str_thm]),
([``"JTe_assert"``, ``"JTe_assertfalse"``],
FULL_SIMP_TAC list_ss [JTconst_cases] THEN SRW_TAC [] [] THEN
METIS_TAC [last (CONJUNCTS value_env_ok_str_thm), value_env_def, APPEND, value_env_teq_str_thm]),
([``"JTe_location"``],
METIS_TAC [value_env_def, value_env_ok_str_thm, value_env_lookup_str_thm, APPEND, value_env_teq_str_thm]),
([``"JTe_for"``],
SRW_TAC [] [shiftt_def] THEN SRW_TAC [] [] THENL
[MATCH_MP_TAC (SIMP_RULE list_ss [AND_IMP_INTRO]
(Q.SPECL [`E2`, `[EB_vn (VN_id lowercase_ident) (TS_forall t')]`,
`shiftTsig 0 (num_tv E1) S'`, `e''`,
`TE_constr [] TC_unit`,
`EB_vn (VN_id lowercase_ident) (TS_forall (TE_constr [] TC_int))::E1`]
(GEN_ALL (hd (CONJUNCTS value_env_str_thm))))) THEN
SRW_TAC [] [value_env_def, domEB_def],
Q.PAT_ASSUM `!S'' E1' E2' x' v t''. P S'' E1' E2' x' v t'' ==>
JTe (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2') (subst_value_name_expr v x' e'')
(TE_constr [] TC_unit)`
(MATCH_MP_TAC o
SIMP_RULE list_ss [num_tv_def] o
Q.SPECL [`S'`, `EB_vn (VN_id lowercase_ident)
(TS_forall (TE_constr [] TC_int)) :: E1`]) THEN
SRW_TAC [] [domEB_def, DISJOINT_RIGHT, MEM_MAP] THEN METIS_TAC [],
METIS_TAC [value_env_def, value_env_teq_str_thm]]),
([``"JTe_let_mono"``],
SRW_TAC [] [] THEN
FULL_SIMP_TAC list_ss [REVERSE_EQ, EB_vn_list_thm, aux_xs_let_binding_of_let_binding_def] THEN
IMP_RES_TAC aux_xs_pattern_of_pattern_thm THEN
FULL_SIMP_TAC list_ss [REVERSE_REVERSE, domEB_def, MAP_MAP] THEN
SRW_TAC [] [] THEN DISJ1_TAC THEN Q.EXISTS_TAC `x_t_list` THEN
SRW_TAC [] [] THENL
[MATCH_MP_TAC (GEN_ALL (hd (CONJUNCTS (SIMP_RULE list_ss [AND_IMP_INTRO] value_env_str_thm)))) THEN
Q.EXISTS_TAC `[EB_vn x (TS_forall t'')]` THEN
SRW_TAC [] [value_env_def, domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP] THEN
METIS_TAC [MEM_MAP],
Q.PAT_ASSUM `!S'' E1' E2' x' v' t'. P S'' E1' E2' x' v' t' ==>
JTe (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2') (subst_value_name_expr v' x' e) t`
(MATCH_MP_TAC o
SIMP_RULE list_ss [MAP_REVERSE] o
SIMP_RULE list_ss [num_tv_append_thm, lem5, GSYM MAP_REVERSE] o
Q.SPECL [`S'`,
`REVERSE (MAP (\z. EB_vn (FST z) (TS_forall (shiftt 0 1 (SND z)))) x_t_list) ++
E1`])
THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP] THEN
METIS_TAC [MEM_MAP, name_11]]),
([``"JTe_let_poly"``],
SRW_TAC [] [] THEN
FULL_SIMP_TAC list_ss [REVERSE_EQ, EB_vn_list_thm, aux_xs_let_binding_of_let_binding_def] THEN
IMP_RES_TAC aux_xs_pattern_of_pattern_thm THEN
FULL_SIMP_TAC list_ss [REVERSE_REVERSE, domEB_def, MAP_MAP] THEN
SRW_TAC [] [] THEN DISJ2_TAC THEN Q.EXISTS_TAC `x_t_list` THEN
SRW_TAC [ARITH_ss] [shiftTsig_add_thm] THENL
[METIS_TAC [subst_nexp_thm],
Q.PAT_ASSUM `!S'' E1' E2' x' v' t'''. P S'' E1' E2' x' v' t''' ==>
?t'. JTpat (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2') pat t' E /\
JTe (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2')
(subst_value_name_expr v' x' nexp) t'`
(MATCH_MP_TAC o
SIMP_RULE list_ss [num_tv_def] o
Q.SPECL [`S'`, `EB_tv::E1`])
THEN
SRW_TAC [] [MAP_MAP, domEB_def, MAP_REVERSE, MEM_REVERSE] THEN
SRW_TAC [ARITH_ss] [MEM_MAP, shiftTsig_add_thm],
MATCH_MP_TAC (GEN_ALL (hd (CONJUNCTS (SIMP_RULE list_ss [AND_IMP_INTRO] value_env_str_thm)))) THEN
Q.EXISTS_TAC `[EB_vn x (TS_forall t'')]` THEN
SRW_TAC [] [value_env_def, domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP] THEN
METIS_TAC [MEM_MAP],
METIS_TAC [subst_nexp_thm],
Q.PAT_ASSUM `!S'' E1' E2' x' v' t'''. P S'' E1' E2' x' v' t''' ==>
?t'. JTpat (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2') pat t' E /\
JTe (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2')
(subst_value_name_expr v' x' nexp) t'`
(MATCH_MP_TAC o
SIMP_RULE list_ss [num_tv_def] o
Q.SPECL [`S'`, `EB_tv::E1`])
THEN
SRW_TAC [] [MAP_MAP, domEB_def, MAP_REVERSE, MEM_REVERSE] THEN
SRW_TAC [ARITH_ss] [MEM_MAP, shiftTsig_add_thm],
Q.PAT_ASSUM `!S'' E1' E2' x' v' t'. P S'' E1' E2' x' v' t' ==>
JTe (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2') (subst_value_name_expr v' x' e) t`
(MATCH_MP_TAC o
SIMP_RULE list_ss [MAP_REVERSE] o
SIMP_RULE list_ss [num_tv_def, GSYM MAP_REVERSE, lem5, num_tv_append_thm] o
Q.SPECL [`S'`, `REVERSE (MAP (\z. EB_vn (FST z) (TS_forall (SND z))) x_t_list) ++E1`])
THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP] THEN
METIS_TAC [MEM_MAP, name_11]]),
([``"JTe_letrec"``],
SRW_TAC [] [] THEN IMP_RES_TAC aux_xs_letrec_bindings_of_letrec_bindings_thm THEN
FULL_SIMP_TAC list_ss [REVERSE_REVERSE, MAP_MAP, domEB_def] THEN
Q.EXISTS_TAC `x_t_list` THEN SRW_TAC [] [] THENL
[Q.PAT_ASSUM `!S'' x' E1' E2' v' t''. P S'' x' E1' E2' v' t'' ==>
(if MEM x' (aux_xs_letrec_bindings_of_letrec_bindings lrbs) then
JTletrec_binding (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2') lrbs E'
else
JTletrec_binding (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2')
(subst_value_name_letrec_bindings v' x' lrbs) E'')`
(MP_TAC o
Q.SPECL [`S''`, `x`, `EB_tv::E1`]) THEN
SRW_TAC [] [num_tv_def, shiftTsig_add_thm] THEN POP_ASSUM MATCH_MP_TAC THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP, value_env_def] THEN
SRW_TAC [] [MEM_MAP] THEN METIS_TAC [],
Q.PAT_ASSUM `!S'' x' E1' E2' v' t''. P S'' x' E1' E2' v' t'' ==>
(if MEM x' (aux_xs_letrec_bindings_of_letrec_bindings lrbs) then
JTletrec_binding (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2') lrbs E'
else
JTletrec_binding (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2')
(subst_value_name_letrec_bindings v' x' lrbs) E'')`
(MP_TAC o
Q.SPECL [`S''`, `x`, `EB_tv::E1`]) THEN
SRW_TAC [] [num_tv_def, shiftTsig_add_thm] THEN POP_ASSUM MATCH_MP_TAC THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP, value_env_def] THEN
SRW_TAC [] [MEM_MAP] THEN METIS_TAC [],
MATCH_MP_TAC (GEN_ALL (hd (CONJUNCTS (SIMP_RULE list_ss [AND_IMP_INTRO] value_env_str_thm)))) THEN
Q.EXISTS_TAC `[EB_vn x (TS_forall t')]` THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP, value_env_def] THEN
FULL_SIMP_TAC list_ss [] THEN METIS_TAC [MEM_MAP],
Q.PAT_ASSUM `!S'' E1' E2' x' v t''. P E1' E2' x' v t'' ==>
JTe (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2') (subst_value_name_expr v x' e) t`
(MATCH_MP_TAC o
SIMP_RULE list_ss [MAP_REVERSE] o
SIMP_RULE list_ss [num_tv_def, GSYM MAP_REVERSE, lem5, num_tv_append_thm] o
Q.SPECL [`S'`, `REVERSE (MAP (\z. EB_vn (FST z) (TS_forall (SND z))) x_t_list) ++E1`])
THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP, value_env_def] THEN
SRW_TAC [] [MEM_MAP] THEN Cases_on `x = FST z` THEN SRW_TAC [] [] THEN METIS_TAC [lem2]]),
([``"JTpat_matching_pm"``],
SRW_TAC [] [] THEN
Q.EXISTS_TAC `MAP (\z. (FST z,
(if MEM x (aux_xs_pattern_of_pattern (FST z)) then
FST (SND z)
else subst_value_name_expr v x (FST (SND z))),
(SND (SND z))))
pattern_e_E_list` THEN
SRW_TAC [] [MAP_MAP, subst_value_name_letrec_binding_def, EVERY_MAP] THEN
FULL_SIMP_TAC list_ss [EVERY_MEM] THENL
[METIS_TAC [value_env_pat_str_thm, value_env_def],
METIS_TAC [lem3]]),
([``"JTlet_binding_poly"``],
SRW_TAC [] [] THEN MAP_EVERY Q.EXISTS_TAC [`x_t_list`, `t`] THEN SRW_TAC [] [] THEN1
METIS_TAC [value_env_pat_str_thm, value_env_def]),
([``"JTletrec_binding_equal_function"``],
SRW_TAC [] [aux_xs_letrec_bindings_of_letrec_bindings_def, MAP_MAP,
aux_xs_letrec_binding_of_letrec_binding_def, FLAT_MAP_SING] THEN
SRW_TAC [] [] THENL
[FULL_SIMP_TAC list_ss [EVERY_MEM] THEN METIS_TAC [lem4],
Q.EXISTS_TAC `MAP (\(a, b, c, d). (a, subst_value_name_pattern_matching v x b, c, d))
value_name_pattern_matching_t_t'_list` THEN
SRW_TAC [] [subst_value_name_letrec_binding_def, MAP_MAP, LAMBDA_PROD2,
EVERY_MAP] THEN
FULL_SIMP_TAC list_ss [EVERY_MEM, AND_IMP_INTRO, GSYM RIGHT_FORALL_IMP_THM] THEN
SRW_TAC [] [] THEN
Q.PAT_ASSUM `!x' S'' E1' E2' x'' v' t'''. P x' S'' E1' E2' x'' v' t''' ==>
JTpat_matching (shiftTsig 0 (num_tv E1') S'') (E1' ++ E2')
(subst_value_name_pattern_matching v x'' (FST (SND x')))
(FST (SND (SND x'))) (SND (SND (SND x')))`
(MATCH_MP_TAC o
SIMP_RULE list_ss [num_tv_append_thm, lem5, MAP_REVERSE] o
Q.SPECL [`x'`, `S'`,
`MAP (\z. EB_vn (FST z)
(TS_forall (shiftt 0 1 (TE_arrow (FST (SND (SND z)))
(SND (SND (SND z)))))))
(REVERSE value_name_pattern_matching_t_t'_list) ++ E1`])
THEN
SRW_TAC [] [MAP_MAP, domEB_def, MAP_REVERSE, MEM_REVERSE] THEN
Q.PAT_ASSUM `~MEM x (MAP FST value_name_pattern_matching_t_t'_list)` MP_TAC THEN
REPEAT (POP_ASSUM (K ALL_TAC)) THEN
Induct_on `value_name_pattern_matching_t_t'_list` THEN SRW_TAC [] []])]
);
(* This is a slight variant on the substitution lemma needed for substitution
* for definitions. The difference is all in the S argument (which is used for
* type variable annotations in the source), and I haven't found a way to unite
* the two lemmas into one. It was copied from the proof above and modified
* slightly. *)
val lem9 = Q.prove (
`JTpat S (E1 ++ [EB_vn x (TS_forall t'')] ++ E2) p t E /\
JTe S (E ++ E1 ++ [EB_vn x (TS_forall t'')] ++ E2) e t' /\
( is_non_expansive_of_expr v /\
(E ++ E1 ++ [EB_vn x (TS_forall t'')] ++ E2 =
(E++E1) ++ [EB_vn x (TS_forall t'')] ++ E2) /\
~MEM (name_vn x) (MAP domEB (E++E1)) /\
closed_env E2 /\
(?S'. JTe S' (EB_tv::E2) (remv_tyvar_expr v) t'') ==>
JTe S ((E++E1) ++ E2) (subst_value_name_expr (remv_tyvar_expr v) x e) t') /\
is_non_expansive_of_expr v /\
~MEM (name_vn x) (MAP domEB E1) /\
closed_env E2 /\
JTe S' (EB_tv::E2) (remv_tyvar_expr v) t'' ==>
JTe S (E ++ E1 ++ E2)
(if MEM x (aux_xs_pattern_of_pattern p) then
e
else
subst_value_name_expr (remv_tyvar_expr v) x e)
t'`,
SRW_TAC [] [] THEN FULL_SIMP_TAC list_ss [] THEN
IMP_RES_TAC aux_xs_pattern_of_pattern_thm THEN SRW_TAC [] [] THENL
[MATCH_MP_TAC (GEN_ALL (hd (CONJUNCTS (SIMP_RULE list_ss [AND_IMP_INTRO] value_env_str_thm)))) THEN
Q.EXISTS_TAC `[EB_vn x (TS_forall t'')]` THEN
SRW_TAC [] [value_env_def, domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP] THEN
METIS_TAC [MEM_MAP, MEM_REVERSE],
METIS_TAC [MEM_MAP, name_11, MEM_REVERSE]]);
val lem10 = Q.prove (
`MEM x (MAP FST l) /\
JTpat_matching S
(REVERSE (MAP (\z. EB_vn (FST z)
(TS_forall (shiftt 0 1 (TE_arrow (FST (SND (SND z)))
(SND (SND (SND z))))))) l) ++ E1 ++
[EB_vn x (TS_forall t'')] ++ E2) pm t t' ==>
JTpat_matching S
(REVERSE (MAP (\z. EB_vn (FST z)
(TS_forall (shiftt 0 1 (TE_arrow (FST (SND (SND z)))
(SND (SND (SND z))))))) l) ++ E1 ++
E2) pm t t'`,
Cases_on `pm` THEN SRW_TAC [] [JTe_fun] THEN
Q.EXISTS_TAC `pattern_e_E_list` THEN SRW_TAC [] [] THEN FULL_SIMP_TAC list_ss [EVERY_MEM] THENL
[METIS_TAC [value_env_pat_str_thm, value_env_def],
SRW_TAC [] [] THEN
MATCH_MP_TAC (GEN_ALL (hd (CONJUNCTS (SIMP_RULE list_ss [AND_IMP_INTRO] value_env_str_thm)))) THEN
Q.EXISTS_TAC `[EB_vn x (TS_forall t'')]` THEN
SRW_TAC [] [value_env_def, MAP_REVERSE, MAP_MAP, domEB_def] THEN
Q.PAT_ASSUM `MEM x (MAP FST l)` MP_TAC THEN
REPEAT (POP_ASSUM (K ALL_TAC)) THEN Induct_on `l` THEN SRW_TAC [] [] THEN METIS_TAC []]);
val lem11 = Q.prove (
`!S v t E E' t_list.
JTe (shiftTsig 0 1 S) (EB_tv::E) (remv_tyvar_expr v) t /\ closed_env E /\ Eok (E' ++ E) /\
EVERY (tkind (E' ++ E)) t_list ==>
JTe S (E' ++ E) (remv_tyvar_expr v) (idxsub t_list (shiftt 1 (num_tv E') t))`,
METIS_TAC [lem8, remv_tyvar_thm, remv_tyvar_idem_thm]);
val subst_for_def_lem = Q.store_thm ("subst_for_def_lem",
`(!S E e t. JTe S E e t ==>
!E1 E2 x v t'.
is_non_expansive_of_expr v /\
(E = E1 ++ [EB_vn x (TS_forall t')] ++ E2) /\
~MEM (name_vn x) (MAP domEB E1) /\
closed_env E2 /\
(?S'. JTe S' (EB_tv::E2) (remv_tyvar_expr v) t') ==>
JTe S (E1++E2) (subst_value_name_expr (remv_tyvar_expr v) x e) t) /\
(!S E pm t t'. JTpat_matching S E pm t t' ==>
!E1 E2 x v t''.
is_non_expansive_of_expr v /\
(E = E1 ++ [EB_vn x (TS_forall t'')] ++ E2) /\
~MEM (name_vn x) (MAP domEB E1) /\
closed_env E2 /\
(?S'. JTe S' (EB_tv::E2) (remv_tyvar_expr v) t'') ==>
JTpat_matching S (E1++E2) (subst_value_name_pattern_matching (remv_tyvar_expr v) x pm) t t') /\
(!S E lb E'. JTlet_binding S E lb E' ==>
!E1 E2 x v t''.
is_non_expansive_of_expr v /\
(E = E1 ++ [EB_vn x (TS_forall t'')] ++ E2) /\
~MEM (name_vn x) (MAP domEB E1) /\
closed_env E2 /\
(?S'. JTe S' (EB_tv::E2) (remv_tyvar_expr v) t'') ==>
JTlet_binding S (E1++E2) (subst_value_name_let_binding (remv_tyvar_expr v) x lb) E') /\
(!S E lrbs E'. JTletrec_binding S E lrbs E' ==>
!x E1 E2 v t''.
is_non_expansive_of_expr v /\
(E = E1 ++ [EB_vn x (TS_forall t'')] ++ E2) /\
~MEM (name_vn x) (MAP domEB E1) /\
closed_env E2 /\
(?S'. JTe S' (EB_tv::E2) (remv_tyvar_expr v) t'') ==>
if MEM x (aux_xs_letrec_bindings_of_letrec_bindings lrbs) then
JTletrec_binding S (E1++E2) lrbs E'
else
JTletrec_binding S (E1++E2) (subst_value_name_letrec_bindings (remv_tyvar_expr v) x lrbs) E')`,
RULE_INDUCT_TAC JTe_sind [subst_value_name_letrec_binding_def, JTe_fun]
[([``"JTe_uprim"``, ``"JTe_bprim"``, ``"JTe_constant"``, ``"JTe_typed"``],
FULL_SIMP_TAC list_ss [JTuprim_cases, JTbprim_cases, JTconst_cases, JTconstr_c_cases] THEN
SRW_TAC [] [] THEN
FULL_SIMP_TAC list_ss [lookup_def, domEB_def, name_distinct, EVERY_MEM] THEN
METIS_TAC [value_env_lookup_str_thm, value_env_ok_str_thm, APPEND, value_env_def,
value_env_inst_any_thm, value_env_teq_str_thm]),
([``"JTe_apply"``, ``"JTe_match"``], METIS_TAC []),
([``"JTe_ident"``],
SRW_TAC [] [] THEN SRW_TAC [] [JTe_fun] THEN
FULL_SIMP_TAC list_ss [JTvalue_name_cases, lookup_def, domEB_def, name_11, lookup_append_thm,
lookup_dom_thm] THEN
SRW_TAC [] [] THENL
[FULL_SIMP_TAC list_ss [JTinst_cases, shiftEB_def, shiftts_def] THEN SRW_TAC [] [] THEN
`JTe S' (E1 ++ E2) (remv_tyvar_expr v) (idxsub t_list (shiftt 1 (num_tv E1) t'))`
by (MATCH_MP_TAC lem11 THEN SRW_TAC [] [] THENL
[METIS_TAC [remv_tyvar_idem_thm, remv_tyvar_thm],
METIS_TAC [value_env_ok_str_thm, value_env_def, ok_ok_thm],
FULL_SIMP_TAC list_ss [EVERY_MEM] THEN
METIS_TAC [value_env_ok_str_thm, value_env_def]]) THEN
METIS_TAC [value_env_def, value_env_teq_str_thm, teq_thm],
FULL_SIMP_TAC list_ss [lem1] THEN Cases_on `lookup E1 (name_vn value_name)` THEN
FULL_SIMP_TAC list_ss [option_case_def] THEN SRW_TAC [] [] THENL
[Cases_on `EB` THEN
FULL_SIMP_TAC list_ss [environment_binding_distinct, shiftEB_def, num_tv_append_thm,
num_tv_def] THEN
SRW_TAC [] [] THEN
METIS_TAC [value_env_inst_str_thm, value_env_ok_str_thm, value_env_def,
value_env_teq_str_thm],
METIS_TAC [value_env_inst_str_thm, value_env_ok_str_thm, value_env_def,
value_env_teq_str_thm]]]),
([``"JTe_tuple"``, ``"JTe_construct"``],
SRW_TAC [] [] THEN
Q.EXISTS_TAC `MAP (\e_t. (subst_value_name_expr (remv_tyvar_expr v) x (FST e_t), SND e_t))
e_t_list` THEN
SRW_TAC [] [MAP_MAP, ETA_THM, EVERY_MAP] THEN FULL_SIMP_TAC list_ss [EVERY_MEM] THEN
METIS_TAC [value_env_def, value_env_constr_p_str_thm, APPEND, value_env_teq_str_thm]),
([``"JTe_cons"``, ``"JTe_and"``, ``"JTe_or"``, ``"JTe_while"``, ``"JTe_function"``],
METIS_TAC [value_env_teq_str_thm, value_env_def]),
([``"JTe_record_constr"``],
SRW_TAC [] [] THEN
MAP_EVERY Q.EXISTS_TAC [`field_name'_list`, `t'_list`,
`MAP (\fn_e_t. (FST fn_e_t,
subst_value_name_expr (remv_tyvar_expr v) x (FST (SND fn_e_t)),
SND (SND fn_e_t)))
field_name_e_t_list`,
`typeconstr_name`,
`kind`] THEN
SRW_TAC [] [MAP_MAP, ETA_THM, EVERY_MAP] THEN FULL_SIMP_TAC list_ss [EVERY_MEM] THEN1
METIS_TAC [] THEN1
METIS_TAC [value_env_def, value_env_field_str_thm, APPEND] THEN
METIS_TAC [value_env_def, value_env_lookup_str_thm, APPEND, value_env_teq_str_thm]),
([``"JTe_record_with"``],
SRW_TAC [] [] THEN
Q.EXISTS_TAC `MAP (\fn_e_t. (FST fn_e_t,
subst_value_name_expr (remv_tyvar_expr v) x (FST (SND fn_e_t)),
SND (SND fn_e_t)))
field_name_e_t_list` THEN
SRW_TAC [] [MAP_MAP, ETA_THM, EVERY_MAP] THEN FULL_SIMP_TAC list_ss [EVERY_MEM] THEN
METIS_TAC [value_env_def, value_env_field_str_thm, APPEND, value_env_teq_str_thm]),
([``"JTe_record_proj"``],
METIS_TAC [value_env_def, value_env_field_str_thm, APPEND, value_env_teq_str_thm]),
([``"JTe_assert"``, ``"JTe_assertfalse"``],
FULL_SIMP_TAC list_ss [JTconst_cases] THEN SRW_TAC [] [] THEN
METIS_TAC [last (CONJUNCTS value_env_ok_str_thm), value_env_def, APPEND, value_env_teq_str_thm]),
([``"JTe_location"``],
METIS_TAC [value_env_def, value_env_ok_str_thm, value_env_lookup_str_thm, APPEND, value_env_teq_str_thm]),
([``"JTe_for"``],
SRW_TAC [] [shiftt_def] THEN SRW_TAC [] [] THENL
[MATCH_MP_TAC (SIMP_RULE list_ss [AND_IMP_INTRO]
(Q.SPECL [`E2`, `[EB_vn (VN_id lowercase_ident) (TS_forall t')]`,
`S`, `e''`,
`TE_constr [] TC_unit`,
`EB_vn (VN_id lowercase_ident) (TS_forall (TE_constr [] TC_int))::E1`]
(GEN_ALL (hd (CONJUNCTS value_env_str_thm))))) THEN
SRW_TAC [] [value_env_def, domEB_def],
Q.PAT_ASSUM `!E1' E2' x' v t''. P E1' E2' x' v t'' ==>
JTe S (E1' ++ E2')
(subst_value_name_expr (remv_tyvar_expr v) x' e'')
(TE_constr [] TC_unit)`
(MATCH_MP_TAC o
SIMP_RULE list_ss [num_tv_def] o
Q.SPECL [`EB_vn (VN_id lowercase_ident)
(TS_forall (TE_constr [] TC_int)) :: E1`]) THEN
SRW_TAC [] [domEB_def, DISJOINT_RIGHT, MEM_MAP] THEN METIS_TAC [],
METIS_TAC [value_env_def, value_env_teq_str_thm]]),
([``"JTe_let_mono"``],
SRW_TAC [] [] THEN
FULL_SIMP_TAC list_ss [REVERSE_EQ, EB_vn_list_thm, aux_xs_let_binding_of_let_binding_def] THEN
IMP_RES_TAC aux_xs_pattern_of_pattern_thm THEN
FULL_SIMP_TAC list_ss [REVERSE_REVERSE, domEB_def, MAP_MAP] THEN
SRW_TAC [] [] THEN DISJ1_TAC THEN Q.EXISTS_TAC `x_t_list` THEN
SRW_TAC [] [] THENL
[METIS_TAC [],
MATCH_MP_TAC (GEN_ALL (hd (CONJUNCTS (SIMP_RULE list_ss [AND_IMP_INTRO] value_env_str_thm)))) THEN
Q.EXISTS_TAC `[EB_vn x (TS_forall t'')]` THEN
SRW_TAC [] [value_env_def, domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP] THEN
METIS_TAC [MEM_MAP],
METIS_TAC [],
Q.PAT_ASSUM `!E1' E2' x' v' t'. P E1' E2' x' v' t' ==>
JTe S (E1' ++ E2') (subst_value_name_expr (remv_tyvar_expr v') x' e) t`
(MATCH_MP_TAC o
SIMP_RULE list_ss [MAP_REVERSE] o
SIMP_RULE list_ss [num_tv_append_thm, lem5, GSYM MAP_REVERSE] o
Q.SPECL [`REVERSE (MAP (\z. EB_vn (FST z) (TS_forall (shiftt 0 1 (SND z)))) x_t_list) ++
E1`])
THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP] THEN
METIS_TAC [MEM_MAP, name_11]]),
([``"JTe_let_poly"``],
SRW_TAC [] [] THEN
FULL_SIMP_TAC list_ss [REVERSE_EQ, EB_vn_list_thm, aux_xs_let_binding_of_let_binding_def] THEN
IMP_RES_TAC aux_xs_pattern_of_pattern_thm THEN
FULL_SIMP_TAC list_ss [REVERSE_REVERSE, domEB_def, MAP_MAP] THEN
SRW_TAC [] [] THEN DISJ2_TAC THEN Q.EXISTS_TAC `x_t_list` THEN
SRW_TAC [ARITH_ss] [shiftTsig_add_thm] THENL
[METIS_TAC [subst_nexp_thm, nexp_remv_tyvar_thm],
Q.PAT_ASSUM `!E1' E2' x' v' t'''. P E1' E2' x' v' t''' ==>
?t'. JTpat (shiftTsig 0 1 S) (E1' ++ E2') pat t' E /\
JTe (shiftTsig 0 1 S) (E1' ++ E2')
(subst_value_name_expr (remv_tyvar_expr v') x' nexp) t'`
(MATCH_MP_TAC o
SIMP_RULE list_ss [num_tv_def] o
Q.SPECL [`EB_tv::E1`])
THEN
SRW_TAC [] [MAP_MAP, domEB_def, MAP_REVERSE, MEM_REVERSE] THEN
SRW_TAC [ARITH_ss] [MEM_MAP, shiftTsig_add_thm] THEN METIS_TAC [],
MATCH_MP_TAC (GEN_ALL (hd (CONJUNCTS (SIMP_RULE list_ss [AND_IMP_INTRO] value_env_str_thm)))) THEN
Q.EXISTS_TAC `[EB_vn x (TS_forall t'')]` THEN
SRW_TAC [] [value_env_def, domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP] THEN
METIS_TAC [MEM_MAP],
METIS_TAC [subst_nexp_thm, nexp_remv_tyvar_thm],
Q.PAT_ASSUM `!E1' E2' x' v' t'''. P E1' E2' x' v' t''' ==>
?t'. JTpat (shiftTsig 0 1 S) (E1' ++ E2') pat t' E /\
JTe (shiftTsig 0 1 S) (E1' ++ E2')
(subst_value_name_expr (remv_tyvar_expr v') x' nexp) t'`
(MATCH_MP_TAC o
SIMP_RULE list_ss [num_tv_def] o
Q.SPECL [`EB_tv::E1`])
THEN
SRW_TAC [] [MAP_MAP, domEB_def, MAP_REVERSE, MEM_REVERSE] THEN
SRW_TAC [ARITH_ss] [MEM_MAP, shiftTsig_add_thm] THEN METIS_TAC [],
Q.PAT_ASSUM `!E1' E2' x' v' t'. P E1' E2' x' v' t' ==>
JTe S (E1' ++ E2') (subst_value_name_expr (remv_tyvar_expr v') x' e) t`
(MATCH_MP_TAC o
SIMP_RULE list_ss [MAP_REVERSE] o
SIMP_RULE list_ss [num_tv_def, GSYM MAP_REVERSE, lem5, num_tv_append_thm] o
Q.SPECL [`REVERSE (MAP (\z. EB_vn (FST z) (TS_forall (SND z))) x_t_list) ++E1`])
THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP] THEN
METIS_TAC [MEM_MAP, name_11]]),
([``"JTe_letrec"``],
SRW_TAC [] [] THEN IMP_RES_TAC aux_xs_letrec_bindings_of_letrec_bindings_thm THEN
FULL_SIMP_TAC list_ss [REVERSE_REVERSE, MAP_MAP, domEB_def] THEN
Q.EXISTS_TAC `x_t_list` THEN SRW_TAC [] [] THENL
[Q.PAT_ASSUM `!x' E1' E2' v' t''. P x' E1' E2' v' t'' ==>
(if MEM x' (aux_xs_letrec_bindings_of_letrec_bindings lrbs) then
JTletrec_binding (shiftTsig 0 1 S) (E1' ++ E2') lrbs E'
else
JTletrec_binding (shiftTsig 0 1 S) (E1' ++ E2')
(subst_value_name_letrec_bindings (remv_tyvar_expr v') x' lrbs) E'')`
(MP_TAC o
Q.SPECL [`x`, `EB_tv::E1`]) THEN
SRW_TAC [] [num_tv_def, shiftTsig_add_thm] THEN POP_ASSUM MATCH_MP_TAC THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP, value_env_def] THEN
SRW_TAC [] [MEM_MAP] THEN METIS_TAC [],
Q.PAT_ASSUM `!x' E1' E2' v' t''. P x' E1' E2' v' t'' ==>
(if MEM x' (aux_xs_letrec_bindings_of_letrec_bindings lrbs) then
JTletrec_binding (shiftTsig 0 1 S'') (E1' ++ E2') lrbs E'
else
JTletrec_binding (shiftTsig 0 1 S'') (E1' ++ E2')
(subst_value_name_letrec_bindings (remv_tyvar_expr v') x' lrbs) E'')`
(MP_TAC o
Q.SPECL [`x`, `EB_tv::E1`]) THEN
SRW_TAC [] [num_tv_def, shiftTsig_add_thm] THEN POP_ASSUM MATCH_MP_TAC THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP, value_env_def] THEN
SRW_TAC [] [MEM_MAP] THEN METIS_TAC [],
MATCH_MP_TAC (GEN_ALL (hd (CONJUNCTS (SIMP_RULE list_ss [AND_IMP_INTRO] value_env_str_thm)))) THEN
Q.EXISTS_TAC `[EB_vn x (TS_forall t')]` THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP, value_env_def] THEN
FULL_SIMP_TAC list_ss [] THEN METIS_TAC [MEM_MAP],
Q.PAT_ASSUM `!E1' E2' x' v t''. P E1' E2' x' v t'' ==>
JTe S (E1' ++ E2') (subst_value_name_expr (remv_tyvar_expr v) x' e) t`
(MATCH_MP_TAC o
SIMP_RULE list_ss [MAP_REVERSE] o
SIMP_RULE list_ss [num_tv_def, GSYM MAP_REVERSE, lem5, num_tv_append_thm] o
Q.SPECL [`REVERSE (MAP (\z. EB_vn (FST z) (TS_forall (SND z))) x_t_list) ++E1`])
THEN
SRW_TAC [] [domEB_def, MAP_REVERSE, MEM_REVERSE, MAP_MAP, value_env_def] THEN
SRW_TAC [] [MEM_MAP] THEN Cases_on `x = FST z` THEN SRW_TAC [] [] THEN METIS_TAC [lem2]]),
([``"JTpat_matching_pm"``],
SRW_TAC [] [] THEN
Q.EXISTS_TAC `MAP (\z. (FST z,
(if MEM x (aux_xs_pattern_of_pattern (FST z)) then
FST (SND z)
else subst_value_name_expr (remv_tyvar_expr v) x (FST (SND z))),
(SND (SND z))))
pattern_e_E_list` THEN
SRW_TAC [] [MAP_MAP, subst_value_name_letrec_binding_def, EVERY_MAP] THEN
FULL_SIMP_TAC list_ss [EVERY_MEM] THENL
[METIS_TAC [value_env_pat_str_thm, value_env_def],
METIS_TAC [lem9]]),
([``"JTlet_binding_poly"``],
SRW_TAC [] [] THEN MAP_EVERY Q.EXISTS_TAC [`x_t_list`, `t`] THEN SRW_TAC [] [] THEN
METIS_TAC [value_env_pat_str_thm, value_env_def]),
([``"JTletrec_binding_equal_function"``],
SRW_TAC [] [aux_xs_letrec_bindings_of_letrec_bindings_def, MAP_MAP,
aux_xs_letrec_binding_of_letrec_binding_def, FLAT_MAP_SING] THEN
SRW_TAC [] [] THENL
[FULL_SIMP_TAC list_ss [EVERY_MEM] THEN METIS_TAC [lem10],
Q.EXISTS_TAC `MAP (\(a, b, c, d). (a, subst_value_name_pattern_matching (remv_tyvar_expr v) x b, c, d))
value_name_pattern_matching_t_t'_list` THEN
SRW_TAC [] [subst_value_name_letrec_binding_def, MAP_MAP, LAMBDA_PROD2,
EVERY_MAP] THEN
FULL_SIMP_TAC list_ss [EVERY_MEM, AND_IMP_INTRO, GSYM RIGHT_FORALL_IMP_THM] THEN
SRW_TAC [] [] THEN
Q.PAT_ASSUM `!x' E1' E2' x'' v' t'''. P x' E1' E2' x'' v' t''' ==>
JTpat_matching S (E1' ++ E2')
(subst_value_name_pattern_matching
(remv_tyvar_expr v) x'' (FST (SND x')))
(FST (SND (SND x'))) (SND (SND (SND x')))`
(MATCH_MP_TAC o
SIMP_RULE list_ss [num_tv_append_thm, lem5, MAP_REVERSE] o
Q.SPECL [`x'`,
`MAP (\z. EB_vn (FST z)
(TS_forall (shiftt 0 1 (TE_arrow (FST (SND (SND z)))
(SND (SND (SND z)))))))
(REVERSE value_name_pattern_matching_t_t'_list) ++ E1`])
THEN
SRW_TAC [] [MAP_MAP, domEB_def, MAP_REVERSE, MEM_REVERSE] THEN
Q.PAT_ASSUM `~MEM x (MAP FST value_name_pattern_matching_t_t'_list)` MP_TAC THENL
[REPEAT (POP_ASSUM (K ALL_TAC)) THEN
Induct_on `value_name_pattern_matching_t_t'_list` THEN SRW_TAC [] [],
METIS_TAC []]])]
);
end;
val substs_empty_thm = Q.store_thm ("substs_empty_thm",
`(!lrb. substs_value_name_letrec_binding [] lrb = lrb) /\
(!lrbs. substs_value_name_letrec_bindings [] lrbs = lrbs) /\
(!lb. substs_value_name_let_binding [] lb = lb) /\
(!pe. substs_value_name_pat_exp [] pe = pe) /\
(!pm. substs_value_name_pattern_matching [] pm = pm) /\
(!e. substs_value_name_expr [] e = e) /\
(!lrb_list. MAP (substs_value_name_letrec_binding []) lrb_list = lrb_list) /\
(!pe_list. MAP (substs_value_name_pat_exp []) pe_list = pe_list) /\
(!fe_list. MAP (\(f:field, e). (f, substs_value_name_expr [] e)) fe_list = fe_list) /\
(!e_list. MAP (substs_value_name_expr []) e_list = e_list) /\
(!fe:(field#expr). (FST fe, substs_value_name_expr [] (SND fe)) = fe)`,
Induct THEN SRW_TAC [] [substs_value_name_letrec_binding_def, list_assoc_def] THENL
[METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
Cases_on `fe` THEN FULL_SIMP_TAC list_ss []]);
val substs_DISJOINT_thm = Q.prove (
`(!subs lrb. DISJOINT (fv_letrec_binding lrb) (MAP FST subs) ==>
(substs_value_name_letrec_binding subs lrb = lrb)) /\
(!subs lrbs. DISJOINT (fv_letrec_bindings lrbs) (MAP FST subs) ==>
(substs_value_name_letrec_bindings subs lrbs = lrbs)) /\
(!subs lb. DISJOINT (fv_let_binding lb) (MAP FST subs) ==>
(substs_value_name_let_binding subs lb = lb)) /\
(!subs pe. DISJOINT (fv_pat_exp pe) (MAP FST subs) ==>
(substs_value_name_pat_exp subs pe = pe)) /\
(!subs pm. DISJOINT (fv_pattern_matching pm) (MAP FST subs) ==>
(substs_value_name_pattern_matching subs pm = pm)) /\
(!subs e. DISJOINT (fv_expr e) (MAP FST subs) ==>
(substs_value_name_expr subs e = e))`,
HO_MATCH_MP_TAC substs_value_name_letrec_binding_ind THEN
SRW_TAC [] [substs_value_name_letrec_binding_def, fv_letrec_binding_def, FLAT_EQ_EMPTY, EVERY_MAP,
EVERY_MEM, DISJOINT_MEM] THEN
FULL_SIMP_TAC list_ss [LAMBDA_PROD2] THENL
[Induct_on `letrec_binding_list` THEN SRW_TAC [] [],
FULL_SIMP_TAC list_ss [list_minus_thm, MEM_MAP, MEM_FILTER] THEN METIS_TAC [],
Induct_on `pat_exp_list` THEN SRW_TAC [] [],
IMP_RES_TAC not_mem_list_assoc THEN SRW_TAC [] [],
Induct_on `expr_list` THEN SRW_TAC [] [],
Induct_on `expr_list` THEN SRW_TAC [] [],
Induct_on `field_expr_list` THEN SRW_TAC [] [] THENL
[Cases_on `h` THEN SRW_TAC [] [],
FULL_SIMP_TAC list_ss [MEM_MAP, MEM_FILTER] THEN METIS_TAC []],
Induct_on `field_expr_list` THEN SRW_TAC [] [] THENL
[Cases_on `h` THEN SRW_TAC [] [],
FULL_SIMP_TAC list_ss [MEM_MAP, MEM_FILTER] THEN METIS_TAC []],
FULL_SIMP_TAC list_ss [list_minus_thm, MEM_MAP, MEM_FILTER] THEN METIS_TAC [],
FULL_SIMP_TAC list_ss [list_minus_thm, MEM_MAP, MEM_FILTER] THEN METIS_TAC [],
FULL_SIMP_TAC list_ss [list_minus_thm, MEM_MAP, MEM_FILTER] THEN METIS_TAC [],
FULL_SIMP_TAC list_ss [list_minus_thm, MEM_MAP, MEM_FILTER] THEN METIS_TAC []]);
val substs_closed_thm = Q.store_thm ("substs_closed_thm",
`!e subs. (fv_expr e = []) ==> (substs_value_name_expr subs e = e)`,
METIS_TAC [DISJOINT_MEM, MEM, substs_DISJOINT_thm, EVERY_MEM]);
local
val lem1 = Q.prove (
`(!lrbs subs. aux_xs_letrec_bindings_of_letrec_bindings (substs_value_name_letrec_bindings subs lrbs) =
aux_xs_letrec_bindings_of_letrec_bindings lrbs) /\
(!lrb subs. aux_xs_letrec_binding_of_letrec_binding (substs_value_name_letrec_binding subs lrb) =
aux_xs_letrec_binding_of_letrec_binding lrb) /\
(!lrb_list subs. MAP (\lrb. aux_xs_letrec_binding_of_letrec_binding
(substs_value_name_letrec_binding subs lrb)) lrb_list =
MAP aux_xs_letrec_binding_of_letrec_binding lrb_list)`,
Induct THEN
SRW_TAC [] [aux_xs_letrec_bindings_of_letrec_bindings_def, substs_value_name_letrec_binding_def,
aux_xs_letrec_binding_of_letrec_binding_def, MAP_MAP]);
in
val substs_iter_thm = Q.store_thm ("substs_iter_thm",
`(!lrb subs. (fv_expr v = []) ==>
(substs_value_name_letrec_binding ((x,v)::subs) lrb =
substs_value_name_letrec_binding subs (subst_value_name_letrec_binding v x lrb))) /\
(!lrbs subs. (fv_expr v = []) ==>
(substs_value_name_letrec_bindings ((x,v)::subs) lrbs =
substs_value_name_letrec_bindings subs (subst_value_name_letrec_bindings v x lrbs))) /\
(!lb subs. (fv_expr v = []) ==>
(substs_value_name_let_binding ((x,v)::subs) lb =
substs_value_name_let_binding subs (subst_value_name_let_binding v x lb))) /\
(!pe subs. (fv_expr v = []) ==>
(substs_value_name_pat_exp ((x,v)::subs) pe =
substs_value_name_pat_exp subs (subst_value_name_pat_exp v x pe))) /\
(!pm subs. (fv_expr v = []) ==>
(substs_value_name_pattern_matching ((x,v)::subs) pm =
substs_value_name_pattern_matching subs (subst_value_name_pattern_matching v x pm))) /\
(!e subs. (fv_expr v = []) ==>
(substs_value_name_expr ((x,v)::subs) e =
substs_value_name_expr subs (subst_value_name_expr v x e))) /\
(!lrb_list subs. (fv_expr v = []) ==>
(MAP (substs_value_name_letrec_binding ((x,v)::subs)) lrb_list =
MAP (\lrb. substs_value_name_letrec_binding subs (subst_value_name_letrec_binding v x lrb))
lrb_list)) /\
(!pe_list subs. (fv_expr v = []) ==>
(MAP (substs_value_name_pat_exp ((x,v)::subs)) pe_list =
MAP (\pe. substs_value_name_pat_exp subs (subst_value_name_pat_exp v x pe)) pe_list)) /\
(!fe_list subs. (fv_expr v = []) ==>
(MAP (\(f:field, e). (f, substs_value_name_expr ((x,v)::subs) e)) fe_list =
MAP (\(f, e). (f, substs_value_name_expr subs (subst_value_name_expr v x e))) fe_list)) /\
(!e_list subs. (fv_expr v = []) ==>
(MAP (substs_value_name_expr ((x,v)::subs)) e_list =
MAP (\e. substs_value_name_expr subs (subst_value_name_expr v x e)) e_list)) /\
(!fe:(field#expr) subs. (fv_expr v = []) ==>
((FST fe, substs_value_name_expr ((x,v)::subs) (SND fe)) =
(FST fe, substs_value_name_expr subs (subst_value_name_expr v x (SND fe)))))`,
Induct THEN
SRW_TAC [] [substs_value_name_letrec_binding_def, subst_value_name_letrec_binding_def, MAP_MAP,
list_assoc_def] THEN
FULL_SIMP_TAC list_ss [LAMBDA_PROD2] THENL
[METIS_TAC [],
METIS_TAC [],
METIS_TAC [substs_closed_thm],
METIS_TAC [],
METIS_TAC [],
Cases_on `lb` THEN
FULL_SIMP_TAC list_ss [subst_value_name_letrec_binding_def, aux_xs_let_binding_of_let_binding_def],
Cases_on `lb` THEN
FULL_SIMP_TAC list_ss [subst_value_name_letrec_binding_def, aux_xs_let_binding_of_let_binding_def],
METIS_TAC [lem1],
METIS_TAC [lem1]]);
end;
local
val lem5 = Q.prove (
`!l f g. num_tv (MAP (\x. EB_vn (f x) (g x)) l) = 0`,
METIS_TAC [value_env_map_thm, value_env_num_tv_thm]);
in
val substs_lem = Q.store_thm ("substs_lem",
`!S e t E x_t_list x_v_list.
(LENGTH x_t_list = LENGTH x_v_list) /\
JTe S (REVERSE (MAP (\(x,t). EB_vn x (TS_forall t)) x_t_list) ++ E) e t /\
EVERY (\(x, v). is_non_expansive_of_expr v) x_v_list /\
closed_env E /\
ALL_DISTINCT (MAP FST x_t_list) /\
EVERY (\((x,t),x',v). (x = x') /\ JTe (shiftTsig 0 1 S) (EB_tv::E) v t)
(ZIP (x_t_list,x_v_list)) ==>
JTe S E (substs_value_name_expr x_v_list e) t`,
Induct_on `x_t_list` THEN Cases_on `x_v_list` THEN SRW_TAC [] [substs_empty_thm] THEN
Cases_on `h'` THEN Cases_on `h` THEN FULL_SIMP_TAC list_ss [] THEN SRW_TAC [] [] THEN
`fv_expr r' = []` by (FULL_SIMP_TAC list_ss [GSYM MAP_REVERSE, ELIM_UNCURRY] THEN
METIS_TAC [closed_env_fv_thm, closed_env_tv_lem]) THEN
SRW_TAC [] [substs_iter_thm] THEN
Q.PAT_ASSUM `!S' e' t' E' x_v_list. P S' e' t' E' x_v_list ==>
JTe S' E' (substs_value_name_expr x_v_list e') t'` MATCH_MP_TAC THEN
SRW_TAC [] [LAMBDA_PROD2] THEN
MATCH_MP_TAC ((SIMP_RULE list_ss [MAP_REVERSE] o
SIMP_RULE list_ss [lem5, LAMBDA_PROD2, shiftTsig_add_thm] o
(*Q.SPECL [`e`, `t'`, `S`, `MAP (\(x,t). EB_vn x (TS_forall t)) (REVERSE x_t_list)`] o*)
Q.SPECL [`shiftTsig 0 (num_tv (MAP (\(x,t). EB_vn x (TS_forall t)) (REVERSE x_t_list)))
S`,
`e`, `t'`, `S`, `MAP (\(x,t). EB_vn x (TS_forall t)) (REVERSE x_t_list)`] o
SIMP_RULE list_ss [GSYM RIGHT_FORALL_IMP_THM, AND_IMP_INTRO] o hd o CONJUNCTS) subst_lem) THEN
SRW_TAC [] [MAP_REVERSE, MEM_REVERSE, MAP_MAP, domEB_def] THEN
FULL_SIMP_TAC list_ss [LAMBDA_PROD2, domEB_def] THEN
Q.EXISTS_TAC `r` THEN SRW_TAC [] [] THEN
FULL_SIMP_TAC list_ss [MEM_MAP, name_11]);
val substs_pm_lem = Q.store_thm ("substs_pm_lem",
`!S pm t t' E x_t_list x_v_list.
(LENGTH x_t_list = LENGTH x_v_list) /\
JTpat_matching S (REVERSE (MAP (\(x,t). EB_vn x (TS_forall t)) x_t_list) ++ E) pm t t' /\
EVERY (\(x, v). is_non_expansive_of_expr v) x_v_list /\
closed_env E /\
ALL_DISTINCT (MAP FST x_t_list) /\
EVERY (\((x,t),x',v). (x = x') /\ JTe (shiftTsig 0 1 S) (EB_tv::E) v t) (ZIP (x_t_list,x_v_list)) ==>
JTpat_matching S E (substs_value_name_pattern_matching x_v_list pm) t t'`,
Induct_on `x_t_list` THEN Cases_on `x_v_list` THEN SRW_TAC [] [substs_empty_thm] THEN
Cases_on `h'` THEN Cases_on `h` THEN FULL_SIMP_TAC list_ss [] THEN SRW_TAC [] [] THEN
`fv_expr r' = []` by (FULL_SIMP_TAC list_ss [GSYM MAP_REVERSE, ELIM_UNCURRY] THEN
METIS_TAC [closed_env_fv_thm, closed_env_tv_lem]) THEN
SRW_TAC [] [substs_iter_thm] THEN
Q.PAT_ASSUM `!S' pm' t' t''' E' x_v_list. P S' pm' t' t''' E' x_v_list ==>
JTpat_matching S' E' (substs_value_name_pattern_matching x_v_list pm') t' t'''` MATCH_MP_TAC THEN
SRW_TAC [] [LAMBDA_PROD2] THEN
MATCH_MP_TAC ((SIMP_RULE list_ss [MAP_REVERSE] o
SIMP_RULE list_ss [lem5, LAMBDA_PROD2, shiftTsig_add_thm] o
(*Q.SPECL [`pm`, `t'`, `t''`, `S`, `MAP (\(x,t). EB_vn x (TS_forall t)) (REVERSE x_t_list)`] o*)
Q.SPECL [`shiftTsig 0 (num_tv (MAP (\(x,t). EB_vn x (TS_forall t)) (REVERSE x_t_list)))
S`,
`pm`, `t'`, `t''`, `S`, `MAP (\(x,t). EB_vn x (TS_forall t)) (REVERSE x_t_list)`] o
SIMP_RULE list_ss [GSYM RIGHT_FORALL_IMP_THM, AND_IMP_INTRO] o hd o tl o CONJUNCTS) subst_lem)
THEN
SRW_TAC [] [MAP_REVERSE, MEM_REVERSE, MAP_MAP, domEB_def] THEN
FULL_SIMP_TAC list_ss [LAMBDA_PROD2, domEB_def] THEN
Q.EXISTS_TAC `r` THEN SRW_TAC [] [] THEN
FULL_SIMP_TAC list_ss [MEM_MAP, name_11]);
end;
val substs_ftv_thm = Q.store_thm ("substs_ftv_thm",
`(!subs lrb. EVERY (\xv. ftv_expr (SND xv) = []) subs /\ (ftv_letrec_binding lrb = []) ==>
(ftv_letrec_binding (substs_value_name_letrec_binding subs lrb) = [])) /\
(!subs lrbs. EVERY (\xv. ftv_expr (SND xv) = []) subs /\ (ftv_letrec_bindings lrbs = []) ==>
(ftv_letrec_bindings (substs_value_name_letrec_bindings subs lrbs) = [])) /\
(!subs lb. EVERY (\xv. ftv_expr (SND xv) = []) subs /\ (ftv_let_binding lb = []) ==>
(ftv_let_binding (substs_value_name_let_binding subs lb) = [])) /\
(!subs pe. EVERY (\xv. ftv_expr (SND xv) = []) subs /\ (ftv_pat_exp pe = []) ==>
(ftv_pat_exp (substs_value_name_pat_exp subs pe) = [])) /\
(!subs pm. EVERY (\xv. ftv_expr (SND xv) = []) subs /\ (ftv_pattern_matching pm = []) ==>
(ftv_pattern_matching (substs_value_name_pattern_matching subs pm) = [])) /\
(!subs e. EVERY (\xv. ftv_expr (SND xv) = []) subs /\ (ftv_expr e = []) ==>
(ftv_expr (substs_value_name_expr subs e) = []))`,
HO_MATCH_MP_TAC substs_value_name_letrec_binding_ind THEN
SRW_TAC [] [substs_value_name_letrec_binding_def, ftv_letrec_binding_def, FLAT_EQ_EMPTY, EVERY_MAP,
EVERY_MEM, MEM_FILTER] THEN
FULL_SIMP_TAC list_ss [LAMBDA_PROD2, ftv_letrec_binding_def] THENL
[Cases_on `list_assoc value_name subs` THEN SRW_TAC [] [ftv_letrec_binding_def] THEN
METIS_TAC [list_assoc_mem, SND],
Cases_on `x` THEN METIS_TAC [SND],
Cases_on `x` THEN METIS_TAC [SND]]);
val _ = export_theory ();
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