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open bossLib HolKernel boolLib listTheory rich_listTheory optionTheory combinTheory arithmeticTheory pairTheory;
open sortingTheory wordsTheory;
open ottTheory ottLib caml_typedefTheory;
open utilTheory basicTheory;
val _ = new_theory "matching_fun";
val ALL_DISTINCT_MAP = Q.prove (
`!l f. ALL_DISTINCT (MAP f l) ==> ALL_DISTINCT l`,
Induct THEN SRW_TAC [] [MEM_MAP] THEN METIS_TAC []);
val list_assoc_MAP_11 = Q.prove (
`!l f k. (!x x'. (f x = f x') = (x = x')) ==>
(list_assoc (f k) (MAP (\x. (f (FST x), SND x)) l) = list_assoc k l)`,
Induct THEN SRW_TAC [] [list_assoc_def] THEN Cases_on `h` THEN SRW_TAC [] [list_assoc_def] THEN
FULL_SIMP_TAC list_ss []);
val MEM_SPLIT = Q.prove (
`!x l. MEM x l = ?l1 l2. l = l1++x::l2`,
Induct_on `l` THEN SRW_TAC [] [] THEN EQ_TAC THEN SRW_TAC [] [] THENL
[METIS_TAC [APPEND],
METIS_TAC [APPEND],
Cases_on `l1` THEN FULL_SIMP_TAC list_ss [] THEN METIS_TAC []]);
val pat_match_def = tDefine "pat_match"
`(pat_match (P_var vn) e = SOME [(vn, e)]) /\
(pat_match P_any e = SOME []) /\
(pat_match (P_constant constant) e =
if (e = Expr_constant constant) then
SOME []
else
NONE) /\
(pat_match (P_alias pat vn) e =
OPTION_MAP (\s. s++[(vn, e)]) (pat_match pat e)) /\
(pat_match (P_typed pat t) e = pat_match pat e) /\
(pat_match (P_or pat1 pat2) e =
case pat_match pat1 e of
SOME s -> SOME s
|| NONE -> pat_match pat2 e) /\
(pat_match (P_construct c plist) e =
case e of
Expr_construct c' elist ->
if ~(c' = c) then
NONE
else
pat_match_list plist elist
|| _ -> NONE) /\
(pat_match (P_construct_any c) e =
case e of
Expr_construct c' elist ->
if (c = c') then
SOME []
else
NONE
|| _ -> NONE) /\
(pat_match (P_tuple plist) e =
case e of
Expr_tuple elist ->
pat_match_list plist elist
|| _ -> NONE) /\
(pat_match (P_record fplist) e =
case e of
Expr_record felist ->
if ~ALL_DISTINCT (MAP FST felist) \/ ~(ALL_DISTINCT (MAP FST fplist)) then
NONE
else
pat_match_rec_list fplist felist
|| _ -> NONE) /\
(pat_match (P_cons pat1 pat2) e =
case e of
Expr_cons e1 e2 ->
(case pat_match pat1 e1 of
SOME s1 ->
(case pat_match pat2 e2 of
SOME s2 -> SOME (s1++s2)
|| NONE -> NONE)
|| NONE -> NONE)
|| _ -> NONE) /\
(pat_match_list [] [] = SOME []) /\
(pat_match_list (p::plist) (e::elist) =
case pat_match p e of
SOME s1 ->
(case pat_match_list plist elist of
SOME s2 -> SOME (s1++s2)
|| NONE -> NONE)
|| NONE -> NONE) /\
(pat_match_list _ _ = NONE) /\
(pat_match_rec_list [] _ = SOME []) /\
(pat_match_rec_list ((f, p)::fplist) felist =
case list_assoc f felist of
SOME e ->
(case pat_match p e of
SOME s1 ->
(case pat_match_rec_list fplist felist of
SOME s2 -> SOME (s1++s2)
|| NONE -> NONE)
|| NONE -> NONE)
|| NONE -> NONE)`
(WF_REL_TAC `measure (sum_case (pattern_size o FST)
(sum_case (pattern1_size o FST)
(pattern2_size o FST)))`);
val pat_match_ind = fetch "-" "pat_match_ind";
val pat_match_is_val_thm = Q.store_thm ("pat_match_is_val_thm",
`(!p e s. is_value_of_expr e /\ (pat_match p e = SOME s) ==> EVERY is_value_of_expr (MAP SND s)) /\
(!pl el s. EVERY is_value_of_expr el /\ (pat_match_list pl el = SOME s) ==>
EVERY is_value_of_expr (MAP SND s)) /\
(!fpl fel s. EVERY is_value_of_expr (MAP SND fel) /\ (pat_match_rec_list fpl fel = SOME s) ==>
EVERY is_value_of_expr (MAP SND s))`,
HO_MATCH_MP_TAC pat_match_ind THEN
SRW_TAC [] [pat_match_def, is_value_of_expr_def, EVERY_MAP] THEN
SRW_TAC [] [] THENL
[Cases_on `pat_match p e` THEN FULL_SIMP_TAC (srw_ss()) [],
Cases_on `e` THEN FULL_SIMP_TAC (srw_ss()) [COND_EXPAND_EQ, is_value_of_expr_def] THEN METIS_TAC [],
Cases_on `e` THEN FULL_SIMP_TAC (srw_ss()) [COND_EXPAND_EQ, is_value_of_expr_def] THEN METIS_TAC [EVERY_DEF],
Cases_on `e` THEN FULL_SIMP_TAC (srw_ss()) [COND_EXPAND_EQ, is_value_of_expr_def] THEN METIS_TAC [],
Cases_on `e` THEN FULL_SIMP_TAC (srw_ss()) [COND_EXPAND_EQ, is_value_of_expr_def, LAMBDA_PROD2] THEN
METIS_TAC [],
Cases_on `e` THEN FULL_SIMP_TAC (srw_ss()) [COND_EXPAND_EQ, is_value_of_expr_def] THEN
Cases_on `pat_match p e'` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
Cases_on `pat_match p' e0` THEN FULL_SIMP_TAC (srw_ss()) [] THEN METIS_TAC [EVERY_APPEND],
Cases_on `pat_match p e` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
Cases_on `pat_match_list pl el` THEN FULL_SIMP_TAC (srw_ss()) [] THEN METIS_TAC [EVERY_APPEND],
Cases_on `list_assoc f fel` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
Cases_on `pat_match p x` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
Cases_on `pat_match_rec_list fpl fel` THEN FULL_SIMP_TAC (srw_ss()) [] THEN SRW_TAC [] [] THEN
IMP_RES_TAC list_assoc_mem THEN FULL_SIMP_TAC list_ss [EVERY_MEM] THEN METIS_TAC [SND]]);
local
val lem1 = Q.prove (
`!s. (substs_x_xs case s of substs_x_xs l -> l) = s`,
Cases THEN SRW_TAC [] []);
val lem2 = Q.prove (
`!l1 l2 l3. (LENGTH l1 = LENGTH l2) /\ (LENGTH l2 = LENGTH l3) /\
EVERY (\x. JM_match (FST x) (FST (SND x)) (substs_x_xs (SND (SND x)))) (ZIP (l1,ZIP (l2,l3))) ==>
EVERY (\x. JM_match (FST x) (FST (SND x)) (SND (SND x))) (ZIP (l1,ZIP (l2,MAP substs_x_xs l3)))`,
Induct THEN Cases_on `l2` THEN Cases_on `l3` THEN FULL_SIMP_TAC list_ss []);
val lem3 = Q.prove (
`!l. EVERY (\x. P (SND (SND (SND x))) (FST (SND x)) (FST (SND (SND x)))) l /\ MEM p l /\
ALL_DISTINCT (MAP FST l) ==>
case list_assoc (FST p) (MAP (\z. (FST z,SND (SND (SND z)))) l ++ fn_v''_list) of
NONE -> F
|| SOME e -> P e (FST (SND p)) (FST (SND (SND p)))`,
Induct THEN SRW_TAC [] [list_assoc_def] THEN FULL_SIMP_TAC list_ss [] THEN
Cases_on `list_assoc (FST h) (MAP (\z. (FST z,SND (SND (SND z)))) l ++ fn_v''_list)` THEN
FULL_SIMP_TAC (srw_ss()) [MEM_MAP] THEN METIS_TAC []);
val field_to_fn_11 = Q.store_thm ("field_to_fn_11",
`!x x'. (field_to_fn x = field_to_fn x') = (x = x')`,
Cases THEN Cases THEN SRW_TAC [] [field_to_fn_def]);
val lem4 = Q.prove (
`!fplist slist l.
EVERY (\z. is_value_of_expr (SND z)) l /\
ALL_DISTINCT (MAP FST l) /\
ALL_DISTINCT (MAP FST fplist) /\
(LENGTH slist = LENGTH fplist) /\
EVERY (\z. case list_assoc (FST (FST z)) l of
NONE -> F
|| SOME e -> JM_match e (SND (FST z)) (substs_x_xs (SND z)))
(ZIP (fplist,slist)) ==>
?fn_v''_list field_name'_pat_substs_x_v'_list field_name_v_list.
(l = MAP (\z. (F_name (FST z),SND z)) field_name_v_list) /\
(fplist =
MAP (\z. (F_name (FST z),FST (SND z)))
field_name'_pat_substs_x_v'_list) /\
(FLAT slist =
FLAT
(MAP (\x. case x of substs_x_xs l' -> l')
(MAP (UNCURRY (\field_name_'. UNCURRY (\pat_. FST)))
field_name'_pat_substs_x_v'_list))) /\
EVERY (\z. is_value_of_expr (SND z)) fn_v''_list /\
EVERY (\z. is_value_of_expr (SND (SND (SND z))))
field_name'_pat_substs_x_v'_list /\
EVERY (\z. is_value_of_expr (SND z)) field_name_v_list /\
PERM
(MAP (\z. (FST z,SND (SND (SND z))))
field_name'_pat_substs_x_v'_list ++ fn_v''_list)
field_name_v_list /\
EVERY
(\x. JM_match (SND (SND (SND x))) (FST (SND x)) (FST (SND (SND x))))
field_name'_pat_substs_x_v'_list /\
ALL_DISTINCT (MAP (\z. name_fn (FST z)) field_name_v_list)`,
Induct THEN SRW_TAC [] [] THEN Cases_on `slist` THEN FULL_SIMP_TAC list_ss [] THENL
[MAP_EVERY Q.EXISTS_TAC [`MAP (\x. (field_to_fn (FST x), (SND x))) l`,
`MAP (\x. (field_to_fn (FST x), (SND x))) l`] THEN
SRW_TAC [] [MAP_MAP, field_to_fn_thm, MAP_I, EVERY_MAP] THEN
METIS_TAC [MAP_11_ALL_DISTINCT, name_11, field_to_fn_11],
Cases_on `list_assoc (FST h) l` THEN FULL_SIMP_TAC list_ss [] THEN
Q.PAT_ASSUM `!slist l'. P slist l' ==> Q slist l'` (MP_TAC o Q.SPECL [`t`, `l`]) THEN
SRW_TAC [] [] THEN
IMP_RES_TAC list_assoc_mem THEN FULL_SIMP_TAC list_ss [EVERY_MAP, MAP_MAP] THEN
Cases_on `h` THEN Cases_on `q` THEN FULL_SIMP_TAC (srw_ss()) [] THEN
`MEM (f',x) (MAP (\z. (FST z,SND z)) field_name_v_list)` by
(FULL_SIMP_TAC (srw_ss()) [MEM_MAP] THEN METIS_TAC []) THEN
FULL_SIMP_TAC (srw_ss()) [MAP_I] THEN IMP_RES_TAC PERM_MEM_EQ THEN FULL_SIMP_TAC list_ss [] THEN1
(FULL_SIMP_TAC list_ss [MEM_MAP] THEN METIS_TAC []) THEN
FULL_SIMP_TAC list_ss [MEM_SPLIT] THEN
MAP_EVERY Q.EXISTS_TAC [`l1''++l2''`,
`(f', r, substs_x_xs h', x)::
field_name'_pat_substs_x_v'_list`,
`field_name_v_list`] THEN
SRW_TAC [] [field_to_fn_thm] THEN FULL_SIMP_TAC list_ss [EVERY_MAP] THEN
METIS_TAC [PERM_REFL, CONS_PERM, APPEND, PERM_SYM, PERM_TRANS]]);
val list_TAC =
Cases_on `e` THEN SRW_TAC [] [] THEN FULL_SIMP_TAC list_ss [is_value_of_expr_def, ETA_THM, ELIM_UNCURRY] THEN
EQ_TAC THEN SRW_TAC [] [] THENL
[FULL_SIMP_TAC list_ss [LENGTH_MAP] THEN
Q.PAT_ASSUM `!s. f ==> ((?slist. P slist) = g)`
(MP_TAC o GSYM o
Q.SPEC `FLAT (MAP (\x. case SND (SND x) of substs_x_xs l -> l)
(v_pat_substs_x_list: (expr#pattern#substs_x) list))`) THEN
SRW_TAC [] [MAP_MAP] THEN
Q.EXISTS_TAC `MAP (\x. case SND (SND x) of substs_x_xs l -> l)
(v_pat_substs_x_list: (expr#pattern#substs_x) list)` THEN
SRW_TAC [] [ZIP_MAP, LENGTH_MAP, LENGTH_ZIP, MAP_ZIP_SAME] THEN
SRW_TAC [] [MAP_MAP, EVERY_MAP, lem1],
RES_TAC THEN SRW_TAC [] [] THEN
Q.EXISTS_TAC `ZIP (l,ZIP (plist,MAP substs_x_xs slist))` THEN
SRW_TAC [] [MAP_FST_ZIP, LENGTH_ZIP, GSYM MAP_MAP, GSYM EVERY_MAP, MAP_SND_ZIP] THEN
SRW_TAC [] [MAP_MAP, lem1, MAP_I, lem2]];
in
val match_thm = Q.store_thm ("match_thm",
`(!p e s. is_value_of_expr e ==> (JM_match e p (substs_x_xs s) = (pat_match p e = SOME s))) /\
(!plist elist s. EVERY is_value_of_expr elist ==>
((LENGTH plist = LENGTH elist) /\
(?slist. (LENGTH slist = LENGTH elist) /\ (s = FLAT slist) /\
EVERY (\(e, p, s). JM_match e p (substs_x_xs s)) (ZIP (elist, ZIP (plist, slist))))
=
(pat_match_list plist elist = SOME s))) /\
(!fplist felist s. EVERY is_value_of_expr (MAP SND felist) ==>
((?slist. (LENGTH slist = LENGTH fplist) /\ (s = FLAT slist) /\
EVERY (\((f, p), s).
case list_assoc f felist of NONE -> F || SOME e -> JM_match e p (substs_x_xs s))
(ZIP (fplist, slist)))
=
(pat_match_rec_list fplist felist = SOME s)))`,
HO_MATCH_MP_TAC pat_match_ind THEN SRW_TAC [] [pat_match_def, JM_match_fun, is_value_of_expr_def] THEN
IMP_RES_TAC pat_match_is_val_thm THEN FULL_SIMP_TAC list_ss [EVERY_MAP] THENL
[METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
METIS_TAC [],
SRW_TAC [] [JM_matchP_thm] THEN Cases_on `pat_match p e` THEN FULL_SIMP_TAC (srw_ss()) [] THEN METIS_TAC [],
list_TAC,
Cases_on `e` THEN SRW_TAC [] [] THEN FULL_SIMP_TAC list_ss [is_value_of_expr_def] THEN METIS_TAC [],
list_TAC,
Cases_on `e` THEN SRW_TAC [] [] THENL
[CCONTR_TAC THEN FULL_SIMP_TAC list_ss [EVERY_MAP, MAP_MAP] THEN SRW_TAC [] [] THEN
METIS_TAC [MAP_11_ALL_DISTINCT, name_11, field_11],
CCONTR_TAC THEN FULL_SIMP_TAC list_ss [EVERY_MAP, MAP_MAP] THEN SRW_TAC [] [] THEN
`PERM (MAP FST (MAP (\z. (FST z,SND (SND (SND z))))
field_name'_pat_substs_x_v'_list ++ fn_v''_list))
(MAP FST field_name_v_list)` by METIS_TAC [PERM_MAP] THEN
FULL_SIMP_TAC list_ss [MAP_MAP] THEN
IMP_RES_TAC PERM_ALL_DISTINCT THEN FULL_SIMP_TAC list_ss [ALL_DISTINCT_APPEND] THEN
METIS_TAC [MAP_11_ALL_DISTINCT, name_11, field_11],
FULL_SIMP_TAC list_ss [is_value_of_expr_def, LAMBDA_PROD2] THEN POP_ASSUM (MP_TAC o GSYM) THEN
SRW_TAC [] [] THEN EQ_TAC THEN SRW_TAC [] [] THENL
[Q.EXISTS_TAC `MAP (\x. case FST (SND (SND x)) of substs_x_xs l -> l)
(field_name'_pat_substs_x_v'_list:(field_name#pattern#substs_x#expr) list)` THEN
SRW_TAC [] [ELIM_UNCURRY, MAP_MAP, ZIP_MAP, MAP_ZIP_SAME] THEN
SRW_TAC [] [EVERY_MAP, EVERY_MEM, lem1, list_assoc_MAP_11] THEN
FULL_SIMP_TAC (srw_ss()) [MAP_MAP, MAP_11_ALL_DISTINCT, ETA_THM] THEN
`PERM (MAP FST (MAP (\z. (FST z,SND (SND (SND z))))
field_name'_pat_substs_x_v'_list ++ fn_v''_list))
(MAP FST field_name_v_list)`
by METIS_TAC [PERM_MAP] THEN
`ALL_DISTINCT (MAP FST (MAP (\z. (FST z,SND (SND (SND z))))
field_name'_pat_substs_x_v'_list ++ fn_v''_list))`
by METIS_TAC [PERM_ALL_DISTINCT, MAP_MAP, FST, MAP_APPEND] THEN
METIS_TAC [lem3, PERM_list_assoc],
METIS_TAC [lem4]]],
Cases_on `e` THEN SRW_TAC [] [] THEN Cases_on `pat_match p e'` THEN SRW_TAC [] [] THENL
[CCONTR_TAC THEN FULL_SIMP_TAC list_ss [] THEN Cases_on `substs_x1` THEN FULL_SIMP_TAC list_ss [] THEN
METIS_TAC [],
Cases_on `pat_match p' e0` THEN SRW_TAC [] [] THENL
[CCONTR_TAC THEN FULL_SIMP_TAC list_ss [] THEN Cases_on `substs_x2` THEN FULL_SIMP_TAC list_ss [] THEN
METIS_TAC [],
FULL_SIMP_TAC list_ss [is_value_of_expr_def] THEN
EQ_TAC THEN SRW_TAC [] [] THENL
[Cases_on `substs_x1` THEN Cases_on `substs_x2` THEN SRW_TAC [] [] THEN METIS_TAC [],
MAP_EVERY Q.EXISTS_TAC [`substs_x_xs x`, `substs_x_xs x'`] THEN SRW_TAC [] []]]],
Cases_on `s` THEN SRW_TAC [] [] THENL
[Q.EXISTS_TAC `[]` THEN SRW_TAC [] [],
Cases_on `slist` THEN SRW_TAC [] []],
Cases_on `pat_match p e` THEN SRW_TAC [] [] THENL
[CCONTR_TAC THEN FULL_SIMP_TAC list_ss [] THEN SRW_TAC [] [] THEN Cases_on `slist` THEN
FULL_SIMP_TAC list_ss [] THEN METIS_TAC [],
Cases_on `pat_match_list plist elist` THEN SRW_TAC [] [] THENL
[CCONTR_TAC THEN FULL_SIMP_TAC list_ss [] THEN Cases_on `slist` THEN
FULL_SIMP_TAC list_ss [EVERY_MEM, EXISTS_MEM] THEN METIS_TAC [],
EQ_TAC THEN SRW_TAC [] [] THENL
[Cases_on `slist` THEN FULL_SIMP_TAC list_ss [] THEN METIS_TAC [],
METIS_TAC [],
POP_ASSUM (MP_TAC o Q.SPEC `x'`) THEN SRW_TAC [] [] THEN
Q.EXISTS_TAC `x::slist` THEN SRW_TAC [] []]]],
Cases_on `s` THEN SRW_TAC [] [] THENL
[Q.EXISTS_TAC `[]` THEN SRW_TAC [] [],
Cases_on `slist` THEN SRW_TAC [] []],
Cases_on `list_assoc f felist` THEN SRW_TAC [] [] THENL
[Cases_on `slist` THEN SRW_TAC [] [],
Cases_on `pat_match p x` THEN SRW_TAC [] [] THENL
[CCONTR_TAC THEN Cases_on `slist` THEN FULL_SIMP_TAC list_ss [] THEN
REPEAT (POP_ASSUM MP_TAC) THEN SRW_TAC [] [EVERY_MEM] THEN
METIS_TAC [SND, list_assoc_mem],
Cases_on `pat_match_rec_list fplist felist` THEN SRW_TAC [] [] THENL
[CCONTR_TAC THEN Cases_on `slist` THEN FULL_SIMP_TAC list_ss [] THEN
REPEAT (POP_ASSUM MP_TAC) THEN SRW_TAC [] [EVERY_MEM, o_DEF] THEN
METIS_TAC [],
EQ_TAC THEN SRW_TAC [] [] THENL
[Cases_on `slist` THEN FULL_SIMP_TAC list_ss [] THEN
REPEAT (POP_ASSUM MP_TAC) THEN SRW_TAC [] [EVERY_MEM] THEN
METIS_TAC [SND, list_assoc_mem],
POP_ASSUM (MP_TAC o Q.SPEC `x''`) THEN SRW_TAC [] [] THEN
Q.EXISTS_TAC `x'::slist` THEN SRW_TAC [] [] THEN
FULL_SIMP_TAC list_ss [EVERY_MEM] THEN METIS_TAC [SND, list_assoc_mem]]]]]]);
end;
val _ = export_theory ();
|
