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pure_typingProofScript.sml
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open HolKernel Parse boolLib bossLib BasicProvers dep_rewrite;
open pairTheory arithmeticTheory integerTheory stringTheory optionTheory
listTheory rich_listTheory alistTheory finite_mapTheory
pred_setTheory
open pure_miscTheory pure_configTheory pure_expTheory pure_exp_lemmasTheory
pure_semanticsTheory pure_evalTheory pure_tcexpTheory pure_tcexp_lemmasTheory
pure_typingTheory pure_typingPropsTheory;
val _ = new_theory "pure_typingProof";
(* TODO replace existing get_atoms_SOME_SOME_eq *)
Theorem get_atoms_SOME_SOME_eq:
∀ls as. get_atoms ls = SOME (SOME as) ⇔ ls = MAP wh_Atom as
Proof
rw[get_atoms_SOME_SOME_eq] >>
rw[LIST_EQ_REWRITE, LIST_REL_EL_EQN] >> eq_tac >> gvs[EL_MAP]
QED
Theorem eval_op_type_safe:
(type_atom_op op ts t ∧ t ≠ Bool ∧
LIST_REL type_lit ls ts ⇒
∃res.
eval_op op ls = SOME (INL res) ∧
type_lit res t) ∧
(type_atom_op op ts Bool ∧
LIST_REL type_lit ls ts ⇒
∃res.
eval_op op ls = SOME (INR res))
Proof
rw[type_atom_op_cases, type_lit_cases] >> gvs[type_lit_cases, PULL_EXISTS]
>- (
ntac 2 $ last_x_assum mp_tac >> map_every qid_spec_tac [`ts`,`ls`] >>
Induct_on `LIST_REL` >> rw[] >> gvs[type_lit_cases, concat_def]
)
>- (
ntac 2 $ last_x_assum mp_tac >> map_every qid_spec_tac [`ts`,`ls`] >>
Induct_on `LIST_REL` >> rw[] >> gvs[type_lit_cases, implode_def]
)
>- (IF_CASES_TAC >> gvs[])
QED
Inductive type_wh:
(type_tcexp ns db st env (Prim (Cons $ implode s) ces) t ∧
MAP exp_of ces = es ⇒
type_wh ns db st env (wh_Constructor s es) t) ∧
(type_tcexp ns db st env (Lam [implode s] ce) t ∧
exp_of ce = e ⇒
type_wh ns db st env (wh_Closure s e) t) ∧
(type_tcexp ns db st env (Prim (AtomOp $ Lit l) []) t ⇒
type_wh ns db st env (wh_Atom l) t) ∧
(type_ok (SND ns) db t ⇒ type_wh ns db st env wh_Diverge t)
End
Triviality type_wh_PrimTy_eq_wh_Atom:
type_wh ns db st env wh (PrimTy pt) ∧ pt ≠ Bool ⇒
wh = wh_Diverge ∨ ∃a. wh = wh_Atom a
Proof
rw[type_wh_cases] >> gvs[Once type_tcexp_cases] >>
Cases_on `arg_tys` >> gvs[Functions_def]
QED
Triviality type_wh_PrimTy_Bool_eq_wh_Constructor:
type_wh ns db st env wh (PrimTy Bool) ⇒
wh = wh_Diverge ∨ wh = wh_Constructor "True" [] ∨
wh = wh_Constructor "False" []
Proof
rw[type_wh_cases] >> gvs[Once type_tcexp_cases, mlstringTheory.implode_def]
>- (Cases_on `arg_tys` >> gvs[Functions_def])
>- (gvs[get_PrimTys_def, type_atom_op_cases, type_lit_cases])
QED
Triviality type_wh_Function_eq_wh_Closure:
type_wh ns db st env wh (Function ft rt) ⇒
wh = wh_Diverge ∨ ∃x body. wh = wh_Closure x body
Proof
rw[type_wh_cases] >> gvs[Once type_tcexp_cases]
QED
Triviality type_wh_TypeCons_eq_wh_Constructor:
type_wh ns db st env wh (TypeCons id ts) ⇒
wh = wh_Diverge ∨ ∃cname es. wh = wh_Constructor cname es
Proof
rw[type_wh_cases] >> gvs[Once type_tcexp_cases, exp_of_def] >>
Cases_on `arg_tys` >> gvs[Functions_def]
QED
Triviality type_wh_Array_eq_Loc:
type_wh ns db st env wh (Array t) ⇒
wh = wh_Diverge ∨ ∃a n. wh = wh_Atom (Loc n) ∧ oEL n st = SOME t
Proof
rw[type_wh_cases] >> gvs[Once type_tcexp_cases] >>
Cases_on `arg_tys` >> gvs[Functions_def]
QED
Triviality type_wh_Tuple_eq_wh_Constructor:
type_wh ns db st env wh (Tuple ts) ⇒
wh = wh_Diverge ∨ ∃es. wh = wh_Constructor "" es
Proof
rw[type_wh_cases] >>
gvs[Once type_tcexp_cases, exp_of_def, mlstringTheory.implode_def] >>
Cases_on `arg_tys` >> gvs[Functions_def]
QED
Triviality type_wh_Exception_eq_wh_Constructor:
type_wh ns db st env wh Exception ⇒
wh = wh_Diverge ∨ ∃cn es. wh = wh_Constructor cn es
Proof
rw[type_wh_cases] >> gvs[Once type_tcexp_cases, exp_of_def] >>
Cases_on `arg_tys` >> gvs[Functions_def]
QED
Theorem eval_wh_to_Case_wh_Diverge:
closed (exp_of e) ∧ eval_wh_to k (exp_of e) = wh_Diverge ∧ es ≠ [] ⇒
eval_wh_to k (exp_of (Case e v es eopt)) = wh_Diverge
Proof
rw[exp_of_def, eval_wh_to_def, bind1_def] >>
Cases_on `es` >> gvs[rows_of_def] >>
PairCases_on `h` >> gvs[rows_of_def, subst1_def] >>
rw[eval_wh_to_def] >>
IF_CASES_TAC >> gvs[] >>
qsuff_tac `eval_wh_to (k - 3) (exp_of e) = wh_Diverge` >> gvs[] >>
CCONTR_TAC >> drule eval_wh_inc >> simp[] >> qexists_tac `k` >> simp[]
QED
Theorem eval_wh_to_lets_for:
∀vs e k cn v b.
closed e ∧ vs ≠ [] ∧ ¬ MEM v vs ⇒
∃res.
eval_wh_to k
(subst1 (explode v) e
(lets_for cn ar (explode v) (MAPi (λi v. (i,explode v)) vs) b)) =
res ∧
(res = wh_Diverge ∨
k ≠ 0 ∧
res =
eval_wh_to (k - 1)
(subst
(FEMPTY |++
MAPi (λi v. (explode v, If (IsEq cn ar T e) (Proj cn i e) Bottom)) vs)
(subst1 (explode v) e b)))
Proof
Induct using SNOC_INDUCT >> rw[SNOC_APPEND, lets_for_def, lets_for_APPEND] >>
Cases_on `vs = []` >> gvs[]
>- (
simp[lets_for_def, bind1_def, subst1_def, eval_wh_to_def, Bottom_def] >>
IF_CASES_TAC >> gvs[] >> simp[FUPDATE_LIST_THM]
) >>
last_x_assum drule >> disch_then drule >>
strip_tac >> gvs[] >>
simp[lets_for_APPEND, indexedListsTheory.MAPi_APPEND, lets_for_def] >>
pop_assum $ qspecl_then
[`k`,`cn`,
‘Let (explode x)
(If (IsEq cn ar T (Var $ explode v))
(Proj cn (LENGTH vs) (Var $ explode v)) Bottom) b’]
assume_tac >>
gvs[] >>
simp[subst_def, FLOOKUP_UPDATE, DOMSUB_FUPDATE_NEQ] >>
simp[FUPDATE_LIST_APPEND, GSYM FUPDATE_EQ_FUPDATE_LIST] >>
qmatch_goalsub_abbrev_tac `FEMPTY |++ m` >>
simp[eval_wh_to_def] >> IF_CASES_TAC >> gvs[] >> simp[bind1_def] >>
reverse $ rw[DISJ_EQ_IMP] >- gvs[Bottom_def, subst_def] >>
drule eval_wh_inc >>
disch_then $ qspec_then `k - 1` $ mp_tac o GSYM >> rw[] >>
AP_TERM_TAC >> DEP_REWRITE_TAC[subst_subst_FUNION] >> simp[] >> conj_tac
>- (
simp[DOMSUB_FUPDATE_LIST] >>
ho_match_mp_tac IN_FRANGE_FUPDATE_LIST_suff >> simp[] >>
simp[MEM_MAP, EXISTS_PROD, PULL_EXISTS, MEM_FILTER] >>
unabbrev_all_tac >> simp[indexedListsTheory.MEM_MAPi, PULL_EXISTS, Bottom_def]
) >>
AP_THM_TAC >> AP_TERM_TAC >>
rw[fmap_eq_flookup, FLOOKUP_FUNION, FLOOKUP_UPDATE, DOMSUB_FLOOKUP_THM] >>
every_case_tac >> gvs[Bottom_def, subst_def]
QED
Theorem MAPi_MAP_o:
∀f g. MAPi f (MAP g l) = MAPi (flip ($o o f) g) l
Proof
Induct_on ‘l’ >> simp[combinTheory.o_DEF, combinTheory.C_DEF]
QED
Theorem eval_wh_to_Case:
∀css c ce v k e es cname vs.
eval_wh_to k (exp_of e) = wh_Constructor (explode cname) es ∧
closed (exp_of e) ∧
ALOOKUP css cname = SOME (vs, ce) ∧
¬ MEM v vs ∧
explode cname ∉ monad_cns ∧
LENGTH vs = LENGTH es
⇒ ∃res.
eval_wh_to k (exp_of (pure_tcexp$Case e v css eopt)) = res ∧
(res = wh_Diverge ∨
k ≠ 0 ∧
res =
eval_wh_to (k - 1)
(subst (FEMPTY |++
MAPi (λi v. (explode v,
exp_of (SafeProj cname (LENGTH es) i e))) vs)
(subst1 (explode v) (exp_of e) (exp_of ce))))
Proof
Induct >> rw[exp_of_def, eval_wh_to_def, bind1_def] >>
PairCases_on `h` >> gvs[] >>
qmatch_goalsub_abbrev_tac ‘rows_of _ _ eoptcase’ >>
gvs[AllCaseEqs()] >>
simp[rows_of_def, subst1_def, eval_wh_to_def] >>
IF_CASES_TAC >> gvs[] >> simp[eval_wh_to_def] >> IF_CASES_TAC >> gvs[] >>
Cases_on `eval_wh_to (k − 3) (exp_of e) = wh_Diverge` >> gvs[] >>
drule eval_wh_inc >> disch_then $ qspec_then `k` $ mp_tac o GSYM >> rw[]
>- (
Cases_on `h1 = []` >> simp[lets_for_def, FUPDATE_LIST_THM]
>- (
rw[DISJ_EQ_IMP] >>
drule eval_wh_inc >> disch_then $ irule o GSYM >> simp[]
) >>
drule_all eval_wh_to_lets_for >>
disch_then $
qspecl_then [`LENGTH es`,`k - 2`,`explode cname`,`exp_of ce`] mp_tac >>
gvs[] >>
rw[] >> gvs[MAPi_MAP_o, combinTheory.o_ABS_R, combinTheory.C_ABS_L] >>
gvs[combinTheory.o_DEF] >>
rw[DISJ_EQ_IMP] >>
drule eval_wh_inc >> disch_then $ irule o GSYM >> simp[]
)
>- (
`eval_wh_to (k - 1) (exp_of e) = wh_Constructor (explode cname) es` by (
drule eval_wh_inc >> simp[]) >>
last_x_assum drule >> simp[] >> disch_then drule >>
gvs[exp_of_def, eval_wh_to_def, bind1_def] >>
rw[] >> gvs[] >> rw[DISJ_EQ_IMP] >>
drule eval_wh_inc >> disch_then $ irule o GSYM >> simp[]
)
QED
Definition Disj'_def:
Disj' ve [] = Cons "False" [] ∧
Disj' ve ((cn,l)::xs) =
If (IsEq cn l T ve) (Cons "True" []) (Disj' ve xs)
End
Theorem subst1_Disj_Disj':
subst1 v e (Disj v cn_ars) = Disj' e cn_ars
Proof
Induct_on `cn_ars` >> rw[pureLangTheory.Disj_def, Disj'_def] >>
PairCases_on `h` >> rw[subst1_def, pureLangTheory.Disj_def, Disj'_def]
QED
Theorem eval_wh_to_Disj':
eval_wh e = wh_Constructor cname es ∧ cname ∉ monad_cns ⇒
∃res.
eval_wh_to k (Disj' e cn_ars) = res ∧
(res = wh_Diverge ∨
res =
case ALOOKUP cn_ars cname of
| NONE => wh_False
| SOME ar => if ar = LENGTH es then wh_True else wh_Error)
Proof
qid_spec_tac `k` >>
Induct_on `cn_ars` >> rw[] >> simp[Disj'_def, eval_wh_to_def] >>
PairCases_on `h` >> simp[Disj'_def, eval_wh_to_def] >> Cases_on `k = 0` >- gvs[] >>
simp[eval_wh_to_def] >> Cases_on `k ≤ 1` >- gvs[] >>
simp[] >> Cases_on `eval_wh_to (k - 2) e = wh_Diverge` >> simp[] >>
qspecl_then [`eval_wh_to (k - 2) e`,`k - 2`,`e`]
mp_tac $ GEN_ALL $ GSYM eval_wh_to_IMP_eval_wh >>
simp[] >> strip_tac >> IF_CASES_TAC >> simp[] >> fs[] >> IF_CASES_TAC >> simp[]
QED
Theorem eval_wh_to_Case_catchall:
∀css k.
eval_wh_to k (exp_of e) = wh_Constructor cname es ∧
closed (exp_of e) ∧ cname ∉ monad_cns ∧
ALOOKUP css (implode cname) = NONE ⇒
∃res.
eval_wh_to k (exp_of (pure_tcexp$Case e v css eopt)) = res ∧
(res = wh_Diverge ∨
k ≠ 0 ∧
res =
eval_wh_to (k - 1)
(subst1 (explode v) (exp_of e)
(case eopt of | NONE => Fail | SOME (pats,cae) =>
case ALOOKUP pats (implode cname) of
| NONE => Fail | SOME ar =>
if ar = LENGTH es then exp_of cae else Fail)))
Proof
simp[exp_of_def, eval_wh_to_def, SF CONJ_ss, AllCaseEqs()] >>
‘∀E. closed (exp_of e) ⇒
bind1 (explode v) (exp_of e) E = subst1 (explode v) (exp_of e) E’
by simp[bind_def, FLOOKUP_DEF, AllCaseEqs()] >> simp[] >>
Induct_on ‘css’ >>
simp[rows_of_def, FORALL_PROD, AllCaseEqs(), subst_def, FLOOKUP_DEF] >>
simp[eval_wh_to_def] >> rw[]
>- (
TOP_CASE_TAC >> gvs[] >> PairCases_on `x` >> gvs[] >> Cases_on `k = 0` >> gvs[] >>
simp[pureLangTheory.IfDisj_def, subst1_def, subst1_Disj_Disj'] >>
simp[eval_wh_to_def] >> IF_CASES_TAC >> gvs[] >>
`eval_wh (exp_of e) = wh_Constructor cname es` by (
qspecl_then [`eval_wh_to k (exp_of e)`,`k`,`exp_of e`]
mp_tac $ GEN_ALL $ GSYM eval_wh_to_IMP_eval_wh >> simp[]) >>
drule_all eval_wh_to_Disj' >>
disch_then $ qspecl_then [`k - 2`,`MAP (explode ## I) x0`] assume_tac >> gvs[] >>
pop_assum mp_tac >> TOP_CASE_TAC >> reverse $ rw[]
>- (
qsuff_tac `ALOOKUP x0 (implode cname) = NONE` >- simp[eval_wh_to_def] >>
gvs[ALOOKUP_NONE, MEM_MAP, FORALL_PROD]
)
>- (
qsuff_tac `ALOOKUP x0 (implode cname) = SOME x` >> simp[] >>
qpat_x_assum `ALOOKUP _ _ = _` mp_tac >> rpt $ pop_assum kall_tac >>
Induct_on `x0` >> rw[] >> PairCases_on `h` >> gvs[implodeEQ] >> rw[] >> gvs[]
) >>
`ALOOKUP x0 (implode cname) = SOME (LENGTH es)` by (
qpat_x_assum `ALOOKUP _ _ = _` mp_tac >> rpt $ pop_assum kall_tac >>
Induct_on `x0` >> rw[] >> PairCases_on `h` >> gvs[implodeEQ] >> rw[] >> gvs[]) >>
simp[] >> rw[DISJ_EQ_IMP] >>
drule eval_wh_inc >> disch_then $ qspec_then `k - 1` mp_tac >> simp[]
) >>
rename [
‘eval_wh_to (k - 2) (IsEq (explode pnm) (LENGTH pargs) T (exp_of e))’] >>
Cases_on
‘eval_wh_to (k - 2) (IsEq (explode pnm) (LENGTH pargs) T (exp_of e)) =
wh_Diverge’
>- simp[] >>
simp[eval_wh_to_def] >> Cases_on ‘k ≤ 2’ >> simp[] >>
Cases_on ‘eval_wh_to (k - 3) (exp_of e) = wh_Diverge’ >> simp[] >>
drule_then (qspec_then ‘k’ (assume_tac o SRULE[] o GSYM)) eval_wh_inc >>
simp[] >> gs[implodeEQ] >>
first_x_assum $ qspec_then ‘k - 1’ mp_tac >> simp[] >> impl_tac
>- (qspecl_then [‘k - 1’, ‘exp_of e’, ‘k - 3’] mp_tac eval_wh_inc >> simp[])>>
strip_tac >> simp[] >> simp[DECIDE “p ∨ q ⇔ ~p ⇒ q”] >> strip_tac >>
dxrule_then (qspec_then ‘k-1’ assume_tac) eval_wh_inc >> gs[]
QED
Triviality MAPi_ID[simp]:
∀l. MAPi (λn v. v) l = l
Proof
Induct >> rw[combinTheory.o_DEF]
QED
Theorem FUN_FMAP_SING:
FUN_FMAP f {k} = FEMPTY |+ (k, f k)
Proof
simp[fmap_EXT, FUN_FMAP_DEF]
QED
Theorem FUN_FMAP_IMAGE:
FINITE A ⇒
FUN_FMAP f (IMAGE explode A) = FUN_FMAP (f o explode) A f_o implode
Proof
strip_tac >>
‘∀h. FINITE { x | mlstring$implode x ∈ FDOM h }’
by (‘∀h. { x | implode x ∈ FDOM h } = IMAGE explode (FDOM h)’
by simp[EXTENSION, GSYM implodeEQ] >>
simp[]) >>
simp[fmap_EXT, PULL_EXISTS, FUN_FMAP_DEF, FAPPLY_f_o] >>
simp[EXTENSION, GSYM implodeEQ]
QED
Theorem FUN_FMAP_DOM:
FUN_FMAP (λx. g (f ' x)) (FDOM f) = g o_f f
Proof
simp[fmap_EXT, FUN_FMAP_DEF]
QED
Theorem o_f_FUDLIST_MAP:
f o_f (fm |++ MAP (λ(k,v). (k, g v)) kvs) =
(f o_f fm) |++ MAP (λ(k,v). (k, f (g v))) kvs
Proof
qid_spec_tac ‘fm’ >> Induct_on ‘kvs’ >> simp[FUPDATE_LIST_THM] >>
simp[fmap_EXT] >> simp[FDOM_FUPDATE_LIST, FORALL_PROD]
QED
Theorem FDOM_f_o_implode:
{ x | implode x ∈ FDOM fm } = IMAGE explode (FDOM fm) ∧
FDOM (fm f_o implode) = IMAGE explode (FDOM fm)
Proof
conj_asm1_tac >- simp[EXTENSION, GSYM implodeEQ] >>
simp[FDOM_f_o]
QED
Theorem FUPDATE_f_o_implode:
(fm |+ (k,v)) f_o implode = (fm f_o implode) |+ (explode k, v)
Proof
simp[FAPPLY_f_o, fmap_EXT, FAPPLY_FUPDATE_THM, FDOM_f_o_implode,
DISJ_IMP_THM, FORALL_AND_THM, PULL_EXISTS]
QED
Theorem FUPDATE_LIST_MAP_f_o:
∀fm. (fm |++ MAP (λ(k,v). (k, f v)) kvs) f_o implode =
(fm f_o implode) |++ MAP (λ(k,v). (explode k, f v)) kvs
Proof
Induct_on ‘kvs’ >>
simp[FUPDATE_LIST_THM] >>
simp[fmap_EXT, FDOM_FUPDATE_LIST, FORALL_PROD, MEM_MAP, PULL_EXISTS,
FDOM_f_o_implode, EXISTS_PROD, DISJ_IMP_THM, FORALL_AND_THM,
FUPDATE_f_o_implode]
QED
Theorem FUPDATE_LIST_f_o_implode:
∀fm.
(fm |++ kvs) f_o implode =
(fm f_o implode) |++ (MAP (explode ## I) kvs)
Proof
Induct_on ‘kvs’ >>
simp[FUPDATE_LIST_THM, FUPDATE_f_o_implode, combinTheory.o_DEF,
FORALL_PROD]
QED
Theorem monad_cns_SUBSET_reserved_cns:
monad_cns ⊆ reserved_cns ∧ "Subscript" ∉ monad_cns
Proof
simp[SUBSET_DEF, monad_cns_def, reserved_cns_def, DISJ_IMP_THM]
QED
Theorem type_soundness_up_to:
∀k ce ns db st t.
namespace_ok ns ∧
EVERY (type_ok (SND ns) db) st ∧
type_tcexp ns db st [] ce t
⇒ ∃wh. eval_wh_to k (exp_of ce) = wh ∧ type_wh ns db st [] wh t
Proof
strip_tac >> completeInduct_on `k` >>
recInduct exp_of_ind >> rw[exp_of_def]
>- ( (* Var *)
last_x_assum kall_tac >> gvs[Once type_tcexp_cases]
)
>- ( (* Prim *)
Cases_on `p` >> gvs[pure_cexpTheory.op_of_def]
>- (
simp[eval_wh_to_def, type_wh_cases, SF ETA_ss] >>
goal_assum $ drule_at Any >> simp[]
)
>- (
rgs[Once type_tcexp_cases] >> gvs[]
>- (simp[Once type_wh_cases] >> simp[Once type_tcexp_cases]) >>
imp_res_tac get_PrimTys_SOME >>
gvs[eval_wh_to_def, MAP_MAP_o, combinTheory.o_DEF] >>
IF_CASES_TAC >> gvs[]
>- (
qspec_then `xs` assume_tac $ GEN_ALL get_atoms_MAP_Diverge >>
reverse $ Cases_on `xs` >> gvs[combinTheory.K_DEF]
>- simp[type_wh_cases, type_ok] >>
simp[get_atoms_def] >>
qspecl_then [`[]`,`pt`,`a`,`[]`]
assume_tac $ GEN_ALL eval_op_type_safe >> gvs[] >>
Cases_on `pt = Bool` >> gvs[]
>- (IF_CASES_TAC >> simp[type_wh_cases] >>
simp[Once type_tcexp_cases, mlstringTheory.implode_def]) >>
simp[type_wh_cases] >> simp[Once type_tcexp_cases, get_PrimTys_def] >>
simp[type_atom_op_cases]
) >>
CASE_TAC >> gvs[] >- simp[type_wh_cases, type_ok] >>
CASE_TAC >> gvs[]
>- (
gvs[get_atoms_SOME_NONE_eq, EXISTS_MEM, LIST_REL_EL_EQN, EL_MAP, MEM_EL,
PULL_EXISTS] >>
first_x_assum drule >> strip_tac >>
last_x_assum $ qspec_then `k - 1` mp_tac >> simp[] >>
disch_then drule_all >> strip_tac >>
qmatch_asmsub_abbrev_tac `type_wh _ _ _ _ atom` >>
imp_res_tac type_atom_op_no_Bool >>
drule type_wh_PrimTy_eq_wh_Atom >> gvs[MEM_EL] >>
impl_tac >- (CCONTR_TAC >> gvs[]) >> rw[] >>
pop_assum kall_tac >> first_x_assum $ qspec_then `n` assume_tac >> gvs[]
) >>
gvs[get_atoms_SOME_SOME_eq] >> rename1 `MAP wh_Atom atoms` >>
gvs[MAP_EQ_EVERY2] >>
`LIST_REL type_lit atoms pts` by (
gvs[LIST_REL_EL_EQN, EL_MAP] >> rw[] >>
ntac 2 (first_x_assum drule >> strip_tac) >>
last_x_assum $ qspec_then `k - 1` assume_tac >> gvs[] >>
pop_assum drule_all >> gvs[] >> simp[Once type_wh_cases] >>
simp[Once type_tcexp_cases, get_PrimTys_def, type_atom_op_cases]) >>
qspecl_then [`pts`,`pt`,`a`,`atoms`]
assume_tac $ GEN_ALL eval_op_type_safe >> gvs[] >>
Cases_on `pt = Bool` >> gvs[]
>- (IF_CASES_TAC >>
simp[type_wh_cases, Once type_tcexp_cases,
mlstringTheory.implode_def]) >>
simp[type_wh_cases] >>
simp[Once type_tcexp_cases, get_PrimTys_def, type_atom_op_cases]
)
>- (
pop_assum mp_tac >> simp[Once type_tcexp_cases] >> rw[] >>
simp[eval_wh_to_def] >> reverse $ IF_CASES_TAC >> gvs[]
>- (FULL_CASE_TAC >> gvs[])
>- (
simp[type_wh_cases] >>
irule type_tcexp_type_ok >> simp[] >>
rpt $ goal_assum $ drule_at Any >> simp[]
) >>
FULL_CASE_TAC >> gvs[] >>
last_x_assum $ qspec_then `k - 1` mp_tac >> simp[] >>
ntac 2 $ disch_then drule >>
disch_then $ qspecl_then [`e1`,`t1`] mp_tac >> gvs[] >>
simp[type_wh_cases]
)
)
>- ( (* Let *)
rgs[eval_wh_to_def] >> rw[]
>- (
simp[type_wh_cases] >> irule type_tcexp_type_ok >> simp[] >>
rpt $ goal_assum $ drule_at Any >> simp[]
) >>
qpat_x_assum `type_tcexp _ _ _ _ _ _` mp_tac >> rw[Once type_tcexp_cases] >>
last_x_assum $ qspec_then `k - 1` mp_tac >> simp[] >>
ntac 2 $ disch_then drule >>
disch_then $ qspecl_then [`subst_tc1 v x y`,`t`] mp_tac >>
simp[subst_exp_of, FMAP_MAP2_FUPDATE] >> impl_tac
>- (irule type_tcexp_closing_subst1 >> simp[] >> goal_assum drule >> simp[]) >>
simp[bind1_def, FMAP_MAP2_FEMPTY] >> IF_CASES_TAC >> gvs[] >>
imp_res_tac type_tcexp_freevars_tcexp >> gvs[closed_def, freevars_exp_of] >>
simp[FUN_FMAP_SING])
>- ( (* Apps *)
qpat_x_assum `type_tcexp _ _ _ _ _ _` mp_tac >> rw[Once type_tcexp_cases] >>
rename1 `Functions _ rt` >>
qpat_x_assum `∀a. MEM a _ ⇒ _` kall_tac >> qpat_x_assum `_ ≠ _` kall_tac >>
first_x_assum drule_all >> strip_tac >>
pop_assum mp_tac >> qpat_x_assum `type_tcexp _ _ _ _ _ _` mp_tac >>
map_every qid_spec_tac [`f`,`t`,`rt`,`ts`] >>
pop_assum mp_tac >> map_every qid_spec_tac [`arg_tys`,`xs`] >>
ho_match_mp_tac LIST_REL_strongind >> rw[] >> gvs[Apps_def, Functions_def] >>
first_x_assum $ qspecl_then [`rt`,`App f [h1]`] mp_tac >>
simp[exp_of_def, Apps_def] >> disch_then irule >> rw[]
>- (simp[Once type_tcexp_cases] >> qexists_tac `[h2]` >> simp[Functions_def]) >>
simp[eval_wh_to_def] >>
drule_at (Pos last) type_tcexp_type_ok >> simp[type_ok] >> strip_tac >>
imp_res_tac type_wh_Function_eq_wh_Closure >> gvs[] >- simp[type_wh_cases] >>
IF_CASES_TAC >- simp[type_wh_cases] >>
imp_res_tac type_tcexp_freevars_tcexp >> gvs[] >>
rw[bind1_def, closed_def, freevars_exp_of] >>
qpat_x_assum `type_wh _ _ _ _ _ _` mp_tac >>
simp[Once type_wh_cases] >> strip_tac >> gvs[] >>
pop_assum mp_tac >> simp[Once type_tcexp_cases] >> strip_tac >> gvs[] >>
rename1 `ats ≠ []` >> Cases_on `ats` >> gvs[Functions_def] >>
last_x_assum $ qspec_then `k - 1` mp_tac >> simp[] >>
disch_then $ qspec_then `subst_tc1 (implode x) h1 ce` mp_tac >>
simp[subst_exp_of, FMAP_MAP2_FUPDATE, FMAP_MAP2_FEMPTY, FUN_FMAP_SING] >>
disch_then irule >> simp[] >>
irule type_tcexp_closing_subst1 >> simp[] >>
goal_assum drule >> simp[]
)
>- ( (* Lams *)
imp_res_tac type_tcexp_tcexp_wf >> gvs[tcexp_wf_def] >>
Cases_on `vs` >> gvs[Lams_def] >> simp[eval_wh_to_def] >>
simp[Once type_wh_cases] >> rename1 `Lams (MAP explode hs)` >>
Cases_on `hs` >> gvs[]
>- (gvs[Lams_def] >> irule_at Any EQ_REFL >> simp[]) >>
rename1 `v1::v2::vs` >>
qexists_tac `Lam (v2::vs) x` >> simp[exp_of_def] >>
qpat_x_assum `type_tcexp _ _ _ _ _ _` mp_tac >>
simp[Once type_tcexp_cases] >> strip_tac >> gvs[] >>
ntac 2 $ simp[Once type_tcexp_cases, PULL_EXISTS] >>
qexistsl_tac [`[HD arg_tys]`,`TL arg_tys`,`ret_ty`] >>
simp[Functions_def] >> simp[GSYM Functions_def] >>
Cases_on `arg_tys` >> gvs[] >> Cases_on `t` >> gvs[]
)
>- ( (* Letrec *)
simp[eval_wh_to_def] >> rw[]
>- (
simp[type_wh_cases] >> irule type_tcexp_type_ok >> simp[] >>
rpt $ goal_assum $ drule_at Any >> simp[]
) >>
qpat_x_assum `type_tcexp _ _ _ _ _ _` mp_tac >> rw[Once type_tcexp_cases] >>
last_x_assum $ qspec_then `k - 1` mp_tac >> simp[] >>
ntac 2 $ disch_then drule >>
disch_then $ qspecl_then
[`subst_tc (FEMPTY |++ MAP (λ(g,x). (g, Letrec rs x)) rs) x`,`t`] mp_tac >>
simp[subst_exp_of, FMAP_MAP2_FUPDATE_LIST, FMAP_MAP2_FEMPTY] >> impl_tac
>- (
irule type_tcexp_closing_subst >> simp[] >> goal_assum $ drule_at Any >>
imp_res_tac LIST_REL_LENGTH >> simp[MAP_REVERSE, MAP_ZIP] >>
simp[MAP_MAP_o, combinTheory.o_DEF, LAMBDA_PROD, GSYM FST_THM] >> rw[] >>
gvs[LIST_REL_EL_EQN, EL_MAP] >> rw[] >>
pairarg_tac >> gvs[] >> pairarg_tac >> gvs[] >>
simp[Once type_tcexp_cases] >>
qexists_tac `MAP (tshift_scheme vars) schemes` >>
gvs[MAP_REVERSE, MAP_ZIP_ALT, MAP_MAP_o, combinTheory.o_DEF, LAMBDA_PROD] >>
first_assum drule >>
pop_assum (fn th => pop_assum (fn th' => rewrite_tac[th,th'])) >>
simp[] >> strip_tac >> reverse $ rw[]
>- (
gvs[EVERY_MAP, EVERY_MEM, FORALL_PROD] >> rw[] >>
first_x_assum drule >> rw[] >> drule type_ok_shift_db >> simp[]
) >>
rw[LIST_REL_EL_EQN, EL_MAP] >> rename1 `EL m _` >>
qmatch_goalsub_abbrev_tac `_ a (_ b)` >>
PairCases_on `a` >> PairCases_on `b` >> gvs[] >>
first_x_assum drule >> rw[] >> drule type_tcexp_shift_db >> simp[] >>
disch_then $ qspecl_then [`b0`,`vars`] mp_tac >>
simp[MAP_REVERSE, MAP_ZIP_ALT, MAP_MAP_o, combinTheory.o_DEF] >>
simp[GSYM shift_db_shift_db] >> rw[] >>
irule quotientTheory.EQ_IMPLIES >> goal_assum drule >>
rpt (AP_TERM_TAC ORELSE AP_THM_TAC) >> simp[LAMBDA_PROD]
) >>
simp[bind_def, subst_funs_def] >>
simp[MAP_MAP_o, combinTheory.o_DEF, LAMBDA_PROD, exp_of_def] >>
IF_CASES_TAC >> gvs[] >>
gvs[FUN_FMAP_IMAGE, combinTheory.o_DEF, FUN_FMAP_DOM, o_f_FUDLIST_MAP]
>- (qmatch_abbrev_tac ‘type_wh _ _ _ [] (eval_wh_to _ (subst fm1 tt)) uu ⇒
type_wh _ _ _ [] (eval_wh_to _ (subst fm2 tt)) uu’ >>
‘fm1 = fm2’suffices_by simp[] >>
simp[Abbr‘fm1’, Abbr‘fm2’, exp_of_def, FUPDATE_LIST_MAP_f_o]) >>
rename1 `false` >>
gvs[flookup_fupdate_list] >> every_case_tac >> gvs[] >>
imp_res_tac ALOOKUP_MEM >> gvs[MEM_MAP] >> pairarg_tac >> gvs[] >>
gvs[MAP_MAP_o, combinTheory.o_DEF, LAMBDA_PROD, GSYM FST_THM, freevars_exp_of]
>- (
gvs[MEM_EL, LIST_REL_EL_EQN] >>
qpat_x_assum `_ = EL _ _` $ assume_tac o GSYM >>
first_x_assum drule >> strip_tac >> gvs[] >> pairarg_tac >> gvs[] >>
imp_res_tac type_tcexp_freevars_tcexp >>
gvs[ZIP_MAP, MAP_MAP_o, combinTheory.o_DEF, LAMBDA_PROD] >>
gvs[SUBSET_DEF, MEM_MAP, PULL_EXISTS, EXISTS_PROD, MEM_ZIP] >>
metis_tac[MEM_EL]
)
>- (
pop_assum kall_tac >>
gvs[EXISTS_MAP, EXISTS_MEM] >> pairarg_tac >> gvs[freevars_exp_of] >>
gvs[MEM_EL, LIST_REL_EL_EQN] >>
qpat_x_assum `_ = EL _ _` $ assume_tac o GSYM >>
first_x_assum drule >> strip_tac >> gvs[] >> pairarg_tac >> gvs[] >>
imp_res_tac type_tcexp_freevars_tcexp >>
gvs[ZIP_MAP, MAP_MAP_o, combinTheory.o_DEF, LAMBDA_PROD] >>
gvs[SUBSET_DEF, MEM_MAP, PULL_EXISTS, EXISTS_PROD, MEM_ZIP] >>
metis_tac[MEM_EL]
)
)
>- ( (* Case *)
drule type_tcexp_freevars_tcexp >> rw[] >>
drule_at (Pos last) type_tcexp_type_ok >> rw[] >>
qpat_x_assum `type_tcexp _ _ _ _ _ _` mp_tac >>
rw[Once type_tcexp_cases] >> gvs[]
>- ( (* BoolCase *)
Cases_on `eval_wh_to k (exp_of x) = wh_Diverge`
>- (
drule_at Any eval_wh_to_Case_wh_Diverge >>
gvs[closed_def, freevars_exp_of] >>
disch_then $ qspecl_then [`v`,`rs`, ‘NONE’] mp_tac >>
impl_tac >- (Cases_on `rs` >> gvs[]) >>
rw[exp_of_def, type_wh_cases]
) >>
first_x_assum $ drule_all >> strip_tac >>
‘∃cn. eval_wh_to k (exp_of x) = wh_Constructor (explode cn) []’ by (
drule type_wh_PrimTy_Bool_eq_wh_Constructor >> rw[] >> gvs[] >>
simp[GSYM implodeEQ]) >>
drule eval_wh_to_Case >>
simp[closed_def, freevars_exp_of, FUPDATE_LIST_THM] >>
disch_then $ qspecl_then [‘NONE’, ‘rs’] mp_tac >>
‘∃ce. ALOOKUP rs cn = SOME ([],ce)’ by (
gvs[EXTENSION] >>
first_x_assum $ qspec_then `cn` assume_tac >>
Cases_on `ALOOKUP rs cn` >> gvs[ALOOKUP_NONE] >>
drule type_wh_PrimTy_Bool_eq_wh_Constructor >> strip_tac >>
gvs[GSYM implodeEQ, mlstringTheory.implode_def] >>
Cases_on `x'` >> gvs[] >>
imp_res_tac ALOOKUP_MEM >> gvs[EVERY_MEM, FORALL_PROD] >>
metis_tac[]) >>
simp[exp_of_def] >> disch_then $ qspec_then `v` mp_tac >> gvs[] >>
impl_tac >> rw[] >> gvs[]
>- (
qpat_x_assum `type_wh _ _ _ _ _ _` mp_tac >>
rw[type_wh_cases, Once type_tcexp_cases, monad_cns_def] >>
simp[GSYM mlstringTheory.implode_def]
)
>- simp[type_wh_cases] >>
last_x_assum $ qspec_then `k - 1` mp_tac >> simp[] >>
ntac 2 $ disch_then drule >>
imp_res_tac ALOOKUP_MEM >> gvs[FORALL_PROD, EVERY_MEM] >>
first_x_assum drule >> strip_tac >>
drule type_tcexp_closing_subst1 >> simp[] >>
disch_then drule >> strip_tac >>
disch_then drule >>
simp[subst_exp_of, FUN_FMAP_SING]
)
>- ( (* TupleCase *)
Cases_on `eval_wh_to k (exp_of x) = wh_Diverge`
>- (
drule_at Any eval_wh_to_Case_wh_Diverge >>
gvs[closed_def, freevars_exp_of] >>
disch_then $ qspecl_then [‘v’,‘[(«»,pvars,cexp)]’, ‘NONE’] mp_tac >>
rw[exp_of_def, type_wh_cases]
) >>
last_x_assum assume_tac >>
last_x_assum $ drule_at $ Pos last >> simp[] >> strip_tac >>
drule type_wh_Tuple_eq_wh_Constructor >> rw[] >> gvs[] >>
qpat_x_assum `type_wh _ _ _ _ _ _` mp_tac >> simp[Once type_wh_cases] >>
rw[] >> gvs[] >> pop_assum mp_tac >>
rw[Once type_tcexp_cases, type_cons_def] >>
‘eval_wh_to k (exp_of x) = wh_Constructor (explode «») (MAP exp_of ces)’
by simp[] >>
dxrule eval_wh_to_Case >> simp[closed_def, freevars_exp_of] >>
disch_then $ qspecl_then [‘NONE’, ‘[(«»,pvars,cexp)]’] mp_tac >> simp[] >>
disch_then drule >> imp_res_tac LIST_REL_LENGTH >> gvs[exp_of_def] >>
impl_tac >> rw[] >> gvs[]
>- simp[monad_cns_def]
>- simp[type_wh_cases] >>
last_x_assum $ qspec_then `k - 1` mp_tac >> simp[] >>
disch_then drule >> simp[] >>
DEP_REWRITE_TAC[GSYM subst_subst1_UPDATE] >>
simp[closed_def, freevars_exp_of] >>
DEP_ONCE_REWRITE_TAC[GSYM FUPDATE_FUPDATE_LIST_COMMUTES] >>
simp[combinTheory.o_DEF, GSYM FUPDATE_LIST_THM] >>
simp[MEM_MAP] >>
disch_then $ qspec_then
‘subst_tc (FEMPTY |++ ((v,x)::
(MAPi (λi v. (v, SafeProj «» (LENGTH tyargs) i x)) pvars))) cexp’
mp_tac >>
simp[subst_exp_of, MAP_SNOC, FUN_FMAP_IMAGE,
combinTheory.o_DEF, FUN_FMAP_DOM, FUPDATE_LIST_MAP_f_o,
o_f_FUPDATE_LIST, exp_of_def] >>
simp[FUPDATE_LIST_f_o_implode, combinTheory.o_DEF] >>
disch_then irule >> simp[] >>
irule type_tcexp_closing_subst >>
rpt $ goal_assum $ drule_at Any >>
simp[MAP_REVERSE, MAP_ZIP, combinTheory.o_DEF, MAP_MAP_o] >>
gvs[LIST_REL_EL_EQN, EL_MAP] >> rw[] >>
first_x_assum drule >> strip_tac >>
simp[Once type_tcexp_cases] >>
disj1_tac >> rpt $ goal_assum $ drule_at Any >> simp[oEL_THM]
)
>- ( (* ExceptionCase *)
Cases_on `eval_wh_to k (exp_of x) = wh_Diverge`
>- (
drule_at Any eval_wh_to_Case_wh_Diverge >>
gvs[closed_def, freevars_exp_of] >>
disch_then $ qspecl_then [`v`,`rs`, ‘NONE’] mp_tac >>
impl_tac >- (CCONTR_TAC >> gvs[namespace_ok_def]) >>
rw[exp_of_def, type_wh_cases]
) >>
first_x_assum drule >> simp[] >> disch_then drule_all >> rw[] >>
drule type_wh_Exception_eq_wh_Constructor >> rw[] >> gvs[] >>
qpat_x_assum `type_wh _ _ _ _ _ _` mp_tac >> simp[Once type_wh_cases] >>
rw[] >> gvs[] >>
pop_assum mp_tac >> rw[Once type_tcexp_cases, type_exception_def] >>
gvs[] >>
`cn ∉ monad_cns` by (
imp_res_tac ALOOKUP_MEM >>
‘MEM (implode cn) (MAP FST exndef)’
by simp[MEM_MAP, EXISTS_PROD, SF SFY_ss] >>
pop_assum mp_tac >>
qpat_x_assum ‘namespace_ok (exndef, _)’ mp_tac >>
simp[namespace_ok_def, ALL_DISTINCT_APPEND,
MEM_MAP, PULL_EXISTS] >>
metis_tac[SRULE [SUBSET_DEF] monad_cns_SUBSET_reserved_cns]) >>
‘eval_wh_to k (exp_of x) =
wh_Constructor (explode (implode cn)) (MAP exp_of ces)’
by simp[] >>
dxrule eval_wh_to_Case >> simp[closed_def, freevars_exp_of] >>
disch_then $ qspecl_then [‘NONE’, ‘rs’] mp_tac >>
Cases_on `ALOOKUP rs (implode cn)` >> gvs[]
>- (
gvs[ALOOKUP_NONE, EXTENSION] >>
imp_res_tac ALOOKUP_MEM >> gvs[MEM_MAP, FORALL_PROD, EXISTS_PROD] >>
metis_tac[]
) >>
rename1 `SOME y` >> namedCases_on `y` ["vs ce"] >> simp[] >>
imp_res_tac ALOOKUP_MEM >> gvs[EVERY_MEM] >>
first_x_assum drule >> simp[] >> strip_tac >>
disch_then drule >>
imp_res_tac LIST_REL_LENGTH >> simp[exp_of_def] >> rw[] >> simp[]
>- rw[type_wh_cases] >>
last_x_assum $ qspec_then `k - 1` mp_tac >> simp[] >>
disch_then drule >> simp[] >>
DEP_REWRITE_TAC[GSYM subst_subst1_UPDATE] >>
simp[closed_def, freevars_exp_of] >>
DEP_ONCE_REWRITE_TAC[GSYM FUPDATE_FUPDATE_LIST_COMMUTES] >>
conj_tac >- (simp[MEM_MAP, FORALL_PROD, indexedListsTheory.MEM_MAPi] >>
gvs[MEM_EL]) >>
disch_then $ qspec_then
`subst_tc (FEMPTY |++ ((v,x)::
(MAPi (λi v. (v, SafeProj (implode cn) (LENGTH vs) i x)) vs))) ce`
mp_tac >>
simp[subst_exp_of, FUN_FMAP_IMAGE, exp_of_def, combinTheory.o_DEF,
FUN_FMAP_DOM, o_f_FUPDATE_LIST, FUPDATE_LIST_f_o_implode,
FUPDATE_LIST_THM, FUPDATE_f_o_implode] >>
disch_then irule >> simp[] >>
simp[GSYM FUPDATE_LIST_THM] >>
irule type_tcexp_closing_subst >>
rpt $ goal_assum $ drule_at Any >>
simp[MAP_REVERSE, MAP_ZIP, combinTheory.o_DEF, MAP_MAP_o] >>
gvs[LIST_REL_EL_EQN, EL_MAP] >> rw[] >>
first_x_assum drule >> strip_tac >>
simp[Once type_tcexp_cases, oEL_THM]
)
>- ( (* Case *)
Cases_on `eval_wh_to k (exp_of x) = wh_Diverge`
>- (
drule_at Any eval_wh_to_Case_wh_Diverge >>
gvs[closed_def, freevars_exp_of] >>
disch_then $ qspecl_then [`v`,`rs`, ‘eopt’] mp_tac >> reverse $ impl_tac
>- rw[exp_of_def, type_wh_cases] >>
reverse $ Cases_on `eopt` >> gvs[] >- (PairCases_on `x'` >> gvs[]) >>
gvs[oEL_THM, namespace_ok_def, EVERY_EL] >>
last_x_assum drule >> simp[] >> Cases_on `rs` >> gvs[] >>
every_case_tac >> gvs[]
) >>
first_x_assum drule >> simp[] >> disch_then drule_all >> rw[] >>
drule type_wh_TypeCons_eq_wh_Constructor >> rw[] >> gvs[] >>
qpat_x_assum `type_wh _ _ _ _ _ _` mp_tac >> simp[Once type_wh_cases] >>
rw[] >> gvs[] >> pop_assum mp_tac >>
rw[Once type_tcexp_cases, type_cons_def] >>
‘eval_wh_to k (exp_of x) =
wh_Constructor (explode (implode cname)) (MAP exp_of ces)’ by simp[] >>
Cases_on `ALOOKUP rs (implode cname)` >> gvs[]
>- ( (* Catch-all *)
`eopt ≠ NONE` by (
CCONTR_TAC >> gvs[ALOOKUP_NONE] >>
imp_res_tac ALOOKUP_MEM >> gvs[MEM_MAP]) >>
Cases_on `eopt` >> gvs[] >> namedCases_on `x'` ["us_cn_ars us_e"] >> gvs[] >>
dxrule eval_wh_to_Case_catchall >> simp[] >>
disch_then $ drule_at Any >>
disch_then $ qspecl_then [`v`,`SOME (us_cn_ars,us_e)`] mp_tac >> impl_keep_tac
>- (
simp[closed_def, freevars_exp_of] >>
qpat_x_assum `namespace_ok _` mp_tac >> rw[namespace_ok_def] >>
gvs[ALL_DISTINCT_APPEND] >>
first_x_assum $ qspec_then `implode cname` mp_tac >>
simp[Once MONO_NOT_EQ] >> simp[MEM_MAP, EXISTS_PROD, SF DNF_ss] >> rw[]
>- (
disj1_tac >> irule_at Any EQ_REFL >>
pop_assum mp_tac >> rw[monad_cns_def, reserved_cns_def]
) >>
simp[MEM_FLAT, MEM_MAP, PULL_EXISTS] >>
simp[Once MEM_EL, PULL_EXISTS] >> gvs[oEL_THM] >>
goal_assum $ drule_at Any >> simp[] >>
imp_res_tac ALOOKUP_MEM >> goal_assum drule
) >>
simp[exp_of_def, eval_wh_to_def] >> Cases_on `k = 0` >> gvs[]
>- simp[type_wh_cases] >>
simp[bind1_def] >> strip_tac >> gvs[] >- simp[type_wh_cases] >>
`ALOOKUP us_cn_ars (implode cname) ≠ NONE` by (
CCONTR_TAC >> gvs[ALOOKUP_NONE] >>
qpat_x_assum `_ ∪ _ = _` $ assume_tac >> gvs[EXTENSION] >>
pop_assum $ assume_tac o iffRL >> imp_res_tac ALOOKUP_MEM >>
gvs[MEM_MAP, PULL_EXISTS, EXISTS_PROD] >> first_x_assum drule >> simp[]) >>
Cases_on `ALOOKUP us_cn_ars (implode cname)` >> gvs[] >>
`x' = LENGTH ces` by (
imp_res_tac ALOOKUP_MEM >> gvs[EVERY_MEM] >>
first_x_assum drule >> simp[] >> rw[] >> gvs[LIST_REL_EL_EQN]) >>
simp[] >> last_x_assum $ qspec_then `k - 1` mp_tac >> simp[] >>
disch_then $ qspec_then `subst_tc1 v x us_e` mp_tac >>
simp[subst_exp_of, FUN_FMAP_SING] >> disch_then irule >> simp[] >>
irule type_tcexp_closing_subst1 >> simp[] >>
goal_assum $ drule_at Any >> simp[]
)
>- ( (* Non-catch all *)
namedCases_on `x'` ["vs ce"] >>
drule_at Any eval_wh_to_Case >> simp[] >> disch_then dxrule >>
disch_then $ qspecl_then [`eopt`,`v`] mp_tac >> simp[] >> impl_keep_tac
>- (
simp[closed_def, freevars_exp_of] >>
imp_res_tac ALOOKUP_MEM >> gvs[EVERY_MEM] >>
first_x_assum drule >> simp[] >> strip_tac >> gvs[LIST_REL_EL_EQN] >>
qpat_x_assum `namespace_ok _` mp_tac >> rw[namespace_ok_def] >>
gvs[ALL_DISTINCT_APPEND] >>
first_x_assum $ qspec_then `implode cname` mp_tac >>
simp[Once MONO_NOT_EQ] >> simp[MEM_MAP, EXISTS_PROD, SF DNF_ss] >> rw[]
>- (
disj1_tac >> irule_at Any EQ_REFL >>
pop_assum mp_tac >> rw[monad_cns_def, reserved_cns_def]
) >>
simp[MEM_FLAT, MEM_MAP, PULL_EXISTS] >>
simp[Once MEM_EL, PULL_EXISTS] >> gvs[oEL_THM] >>
goal_assum $ drule_at Any >> simp[] >>
imp_res_tac ALOOKUP_MEM >> goal_assum drule
) >>
simp[exp_of_def, eval_wh_to_def] >> Cases_on `k = 0` >> gvs[]
>- simp[type_wh_cases] >>
simp[bind1_def] >> strip_tac >> gvs[] >- simp[type_wh_cases] >>
pop_assum kall_tac >>
last_x_assum $ qspec_then `k - 1` mp_tac >> simp[] >>
disch_then drule >> simp[] >>
DEP_REWRITE_TAC[GSYM subst_subst1_UPDATE] >>
simp[closed_def, freevars_exp_of] >>
DEP_ONCE_REWRITE_TAC[GSYM FUPDATE_FUPDATE_LIST_COMMUTES] >>
conj_tac
>- (simp[MEM_MAP, FORALL_PROD, indexedListsTheory.MEM_MAPi] >>
metis_tac[MEM_EL]) >>
simp[combinTheory.o_DEF, GSYM FUPDATE_LIST_THM] >>
disch_then $ qspec_then
‘subst_tc
(FEMPTY |++
((v,x):: (MAPi
(λi v. (v, SafeProj (implode cname) (LENGTH schemes) i x))
vs)))
ce’
mp_tac >>
simp[subst_exp_of, FUN_FMAP_IMAGE, exp_of_def, combinTheory.o_DEF,
FUN_FMAP_DOM, o_f_FUPDATE_LIST, FUPDATE_LIST_f_o_implode,
FUPDATE_LIST_THM, FUPDATE_f_o_implode] >>
imp_res_tac LIST_REL_LENGTH >> gvs[] >> disch_then irule >> simp[] >>
simp[GSYM FUPDATE_LIST_THM] >>
imp_res_tac ALOOKUP_MEM >> gvs[EVERY_MEM] >>
first_x_assum drule >> simp[] >> strip_tac >>
irule type_tcexp_closing_subst >> rpt $ goal_assum $ drule_at Any >>
simp[MAP_REVERSE, MAP_ZIP, combinTheory.o_DEF, MAP_MAP_o] >>
gvs[LIST_REL_EL_EQN, EL_MAP] >> rw[] >>
first_x_assum drule >> strip_tac >>
simp[Once type_tcexp_cases] >>
disj2_tac >> disj2_tac >> rpt $ goal_assum $ drule_at Any >> simp[oEL_THM]
)
)
)
>- ( (* SafeProj *)
drule type_tcexp_freevars_tcexp >> rw[] >>
drule_at (Pos last) type_tcexp_type_ok >> rw[] >>
gvs[eval_wh_to_def] >> rw[] >- simp[type_wh_cases] >>
simp[eval_wh_to_def] >> IF_CASES_TAC >> gvs[] >- simp[type_wh_cases] >>
qpat_x_assum `type_tcexp _ _ _ _ _ _` mp_tac >> rw[Once type_tcexp_cases]
>- ( (* TupleSafeProj *)
first_x_assum $ drule_all >> strip_tac >>
drule type_wh_Tuple_eq_wh_Constructor >> rw[] >> gvs[]
>- (
qsuff_tac `eval_wh_to (k - 2) (exp_of e) = wh_Diverge` >> reverse $ rw[]
>- simp[type_wh_cases] >>
CCONTR_TAC >> drule eval_wh_inc >> simp[] >> qexists_tac `k` >> simp[]
) >>
Cases_on `eval_wh_to (k - 2) (exp_of e) = wh_Diverge` >> gvs[]
>- simp[type_wh_cases] >>
drule eval_wh_inc >> disch_then $ qspec_then `k` $ assume_tac o GSYM >>
gvs[] >> simp[monad_cns_def] >>
qpat_x_assum `type_wh _ _ _ _ _ _` mp_tac >>
simp[Once type_wh_cases] >> simp[Once type_tcexp_cases] >> strip_tac >>
gvs[LIST_REL_EL_EQN, oEL_THM, EL_MAP]
)
>- ( (* ExceptionSafeProj *)
first_x_assum $ drule_all >> strip_tac >>
drule type_wh_Exception_eq_wh_Constructor >> rw[] >> gvs[]
>- (
qsuff_tac `eval_wh_to (k - 2) (exp_of e) = wh_Diverge` >> reverse $ rw[]
>- simp[type_wh_cases] >>
CCONTR_TAC >> drule eval_wh_inc >> simp[] >> qexists_tac `k` >> simp[]
) >>
Cases_on `eval_wh_to (k - 2) (exp_of e) = wh_Diverge` >> gvs[]
>- simp[type_wh_cases] >>
drule eval_wh_inc >> disch_then $ qspec_then `k` $ assume_tac o GSYM >>
gvs[] >>
`cn' ∉ monad_cns` by (
qpat_x_assum `type_wh _ _ _ _ _ _` mp_tac >>
simp[Once type_wh_cases, Once type_tcexp_cases] >>
rw[] >> gvs[type_exception_def] >>
drule ALOOKUP_MEM >> rw[] >>
qpat_x_assum ‘namespace_ok _ ’ mp_tac >>
simp[namespace_ok_def, ALL_DISTINCT_APPEND, MEM_MAP, PULL_EXISTS,
EXISTS_PROD, FORALL_PROD] >>
metis_tac[SUBSET_DEF, monad_cns_SUBSET_reserved_cns]) >>
simp[] >> qpat_x_assum `type_wh _ _ _ _ _ _` mp_tac >>
simp[Once type_wh_cases] >> simp[Once type_tcexp_cases] >> strip_tac >>
gvs[type_exception_def] >> IF_CASES_TAC >> gvs[]
>- (
assume_tac eval_wh_Bottom >> gvs[eval_wh_eq_Diverge] >>
simp[type_wh_cases]
) >>
gvs[LIST_REL_EL_EQN, oEL_THM, EL_MAP]
)
>- ( (* SafeProj *)
first_x_assum $ drule_all >> strip_tac >>
drule type_wh_TypeCons_eq_wh_Constructor >> rw[] >> gvs[]
>- (
qsuff_tac `eval_wh_to (k - 2) (exp_of e) = wh_Diverge` >> reverse $ rw[]
>- simp[type_wh_cases] >>
CCONTR_TAC >> drule eval_wh_inc >> simp[] >> qexists_tac `k` >> simp[]
) >>
Cases_on `eval_wh_to (k - 2) (exp_of e) = wh_Diverge` >> gvs[]
>- simp[type_wh_cases] >>
drule eval_wh_inc >> disch_then $ qspec_then `k` $ assume_tac o GSYM >>
gvs[] >>
`cname ∉ monad_cns` by (
qpat_x_assum `type_wh _ _ _ _ _ _` mp_tac >>
rw[Once type_wh_cases, Once type_tcexp_cases] >> gvs[type_cons_def] >>
drule ALOOKUP_MEM >> rw[] >>
qpat_x_assum ‘namespace_ok _’ mp_tac >>
simp[namespace_ok_def, ALL_DISTINCT_APPEND, MEM_MAP, PULL_EXISTS,
FORALL_PROD, EXISTS_PROD] >> gvs[oEL_THM] >>
simp[MEM_FLAT, MEM_MAP, PULL_EXISTS, FORALL_PROD] >>
gvs[MEM_EL, DISJ_IMP_THM, FORALL_AND_THM, PULL_EXISTS] >>
rpt strip_tac >>
‘cname ≠ "Subscript" ∧ cname ∈ reserved_cns’
by metis_tac[monad_cns_SUBSET_reserved_cns, SUBSET_DEF] >>
metis_tac[]) >>
qpat_x_assum `type_wh _ _ _ _ _ _` mp_tac >>
simp[Once type_wh_cases] >> simp[Once type_tcexp_cases] >> strip_tac >> gvs[] >>
IF_CASES_TAC >> gvs[]
>- (
assume_tac eval_wh_Bottom >> gvs[eval_wh_eq_Diverge] >>
simp[type_wh_cases]
) >>
gvs[type_cons_def] >> imp_res_tac LIST_REL_LENGTH >> gvs[oEL_THM, EL_MAP] >>
last_x_assum irule >> simp[] >>
gvs[LIST_REL_EL_EQN] >> first_x_assum drule >> simp[EL_MAP]
)
)
QED
Theorem type_soundness_eval_wh: