/
pegSoundScript.sml
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/
pegSoundScript.sml
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(*
Soundness proof for the parser. If the parser returns a parse tree,
then it is valid w.r.t. the specification grammar.
*)
open preamble pegTheory cmlPEGTheory gramTheory gramPropsTheory
grammarTheory
val _ = new_theory "pegSound";
val _ = set_grammar_ancestry ["cmlPEG", "gramProps"]
val d = let
val d0 = TypeBase.distinct_of ``:(α,β,γ)pegsym``
in
CONJ d0 (GSYM d0)
end
val i = TypeBase.one_one_of ``:(α,β,γ)pegsym``
infix >*
fun t1 >* t2 = (t1 >> conj_tac) >- t2
Theorem peg_eval_choicel_NIL[simp]
`peg_eval G (i0, choicel []) x = (x = NONE)`
(simp[choicel_def, Once peg_eval_cases]);
Theorem peg_eval_choicel_CONS
`∀x. peg_eval G (i0, choicel (h::t)) x ⇔
peg_eval G (i0, h) x ∧ x <> NONE ∨
peg_eval G (i0,h) NONE ∧ peg_eval G (i0, choicel t) x`
(simp[choicel_def, SimpLHS, Once peg_eval_cases] >>
simp[sumID_def, pairTheory.FORALL_PROD, optionTheory.FORALL_OPTION]);
Theorem peg_eval_seql_NIL[simp]
`peg_eval G (i0, seql [] f) x ⇔ (x = SOME(i0,f []))`
(simp[seql_def, pegf_def] >> simp[Once peg_eval_cases]);
Theorem peg_eval_try
`∀x. peg_eval G (i0, try s) x ⇔
peg_eval G (i0, s) NONE ∧ x = SOME(i0,[]) ∨
∃i r. peg_eval G (i0, s) (SOME(i,r)) ∧ x = SOME(i,r)`
(simp[Once peg_eval_cases, try_def, SimpLHS, choicel_def,
peg_eval_choice] >> simp[sumID_def] >> metis_tac[]);
Theorem peg_eval_seql_CONS
`∀x. peg_eval G (i0, seql (h::t) f) x ⇔
peg_eval G (i0, h) NONE ∧ x = NONE ∨
(∃rh i1. peg_eval G (i0,h) (SOME(i1,rh)) ∧
peg_eval G (i1, seql t I) NONE ∧ x = NONE) ∨
(∃rh i1 i rt. peg_eval G (i0, h) (SOME(i1,rh)) ∧
peg_eval G (i1, seql t I) (SOME(i,rt)) ∧
x = SOME(i,f(rh ++ rt)))`
(simp[seql_def, pegf_def] >>
simp[SimpLHS, Once peg_eval_cases] >>
simp[optionTheory.FORALL_OPTION, pairTheory.FORALL_PROD] >>
conj_tac
>- (simp[peg_eval_seq_NONE] >> metis_tac[]) >>
simp[peg_eval_seq_SOME] >> dsimp[] >> metis_tac[]);
val peg_eval_choicel_SING = save_thm(
"peg_eval_choicel_SING",
CONJ
(``peg_eval G (i0, choicel [sym]) (SOME x)``
|> SIMP_CONV (srw_ss()) [peg_eval_choicel_CONS, peg_eval_choicel_NIL])
(``peg_eval G (i0, choicel [sym]) NONE``
|> SIMP_CONV (srw_ss()) [peg_eval_choicel_CONS, peg_eval_choicel_NIL]));
Theorem not_peg0_LENGTH_decreases
`¬peg0 G s ⇒ peg_eval G (i0, s) (SOME(i,r)) ⇒ LENGTH i < LENGTH i0`
(metis_tac[peg_eval_suffix', lemma4_1a])
val _ = augment_srw_ss [rewrites [
peg_eval_seql_CONS, peg_eval_tok_SOME, tokeq_def, bindNT_def, mktokLf_def,
peg_eval_choicel_CONS, pegf_def, peg_eval_seq_SOME, pnt_def, peg_eval_try,
try_def]]
Theorem peg_eval_TypeDec_wrongtok
`FST tk ≠ DatatypeT ⇒ ¬peg_eval cmlPEG (tk::i, nt (mkNT nTypeDec) f) (SOME x)`
(simp[Once peg_eval_cases, cmlpeg_rules_applied, FDOM_cmlPEG,
peg_TypeDec_def, peg_eval_seq_SOME, tokeq_def, peg_eval_tok_SOME]);
Theorem peg_eval_TypeAbbrevDec_wrongtok
`FST tk ≠ TypeT ⇒ ¬peg_eval cmlPEG (tk::i, nt (mkNT nTypeAbbrevDec) f) (SOME x)`
(simp[Once peg_eval_cases, cmlpeg_rules_applied, FDOM_cmlPEG,
peg_eval_seq_SOME, tokeq_def, peg_eval_tok_SOME]);
Theorem peg_eval_LetDec_wrongtok
`FST tk = SemicolonT ⇒ ¬peg_eval cmlPEG (tk::i, nt (mkNT nLetDec) f) (SOME x)`
(simp[Once peg_eval_cases, cmlpeg_rules_applied, FDOM_cmlPEG,
peg_TypeDec_def, peg_eval_seq_SOME, tokeq_def, peg_eval_tok_SOME,
peg_eval_choicel_CONS, peg_eval_seql_CONS]);
Theorem peg_eval_nUQConstructor_wrongtok
`(∀s. FST t ≠ AlphaT s) ⇒
¬peg_eval cmlPEG (t::i, nt (mkNT nUQConstructorName) f) (SOME x)`
(simp[Once peg_eval_cases, cmlpeg_rules_applied,
peg_eval_tok_SOME,
peg_UQConstructorName_def] >> Cases_on `t` >> simp[]);
Theorem peg_eval_nConstructor_wrongtok
`(∀s. FST t ≠ AlphaT s) ∧ (∀s1 s2. FST t ≠ LongidT s1 s2) ⇒
¬peg_eval cmlPEG (t::i, nt (mkNT nConstructorName) f) (SOME x)`
(simp[Once peg_eval_cases, cmlpeg_rules_applied, peg_eval_tok_SOME,
peg_eval_choicel_CONS, peg_eval_seq_NONE, pegf_def, pnt_def,
peg_eval_nUQConstructor_wrongtok, peg_eval_seq_SOME] >>
Cases_on `t` >> simp[]);
Theorem peg_eval_nV_wrongtok
`(∀s. FST t ≠ AlphaT s) ∧ (∀s. FST t ≠ SymbolT s) ⇒
¬peg_eval cmlPEG (t::i, nt (mkNT nV) f) (SOME x)`
(simp[Once peg_eval_cases, cmlpeg_rules_applied, peg_V_def,
peg_eval_seq_NONE, peg_eval_choice] >>
Cases_on `t` >> simp[]);
Theorem peg_eval_nFQV_wrongtok
`(∀s. FST t ≠ AlphaT s) ∧ (∀s. FST t ≠ SymbolT s) ∧ (∀s1 s2. FST t ≠ LongidT s1 s2) ⇒
¬peg_eval cmlPEG (t::i, nt (mkNT nFQV) f) (SOME x)`
(simp[Once peg_eval_cases, cmlpeg_rules_applied,
peg_eval_seq_NONE, peg_eval_choice, peg_eval_nV_wrongtok] >>
Cases_on `t` >> simp[peg_longV_def]);
Theorem peg_eval_rpt_never_NONE
`¬peg_eval G (i, rpt sym f) NONE`
(simp[Once peg_eval_cases]);
val _ = export_rewrites ["peg_eval_rpt_never_NONE"]
val pegsym_to_sym_def = Define`
(pegsym_to_sym (tok P f) = if f = mktokLf then { TK t | P t } else ∅) ∧
pegsym_to_sym (nt N f) = { NT N } ∧
pegsym_to_sym _ = {}
`
Theorem valid_ptree_mkNd[simp]
`valid_ptree G (mkNd N subs) ⇔
N ∈ FDOM G.rules ∧ MAP ptree_head subs ∈ G.rules ' N ∧
∀pt. MEM pt subs ⇒ valid_ptree G pt`
(simp[mkNd_def]);
Theorem ptree_head_mkNd[simp]
`ptree_head (mkNd N subs) = NT N`
(simp[mkNd_def]);
val ptree_list_loc_def = grammarTheory.ptree_list_loc_def
Theorem ptree_list_loc_SING[simp]
`ptree_list_loc [pt] = ptree_loc pt`
(simp[ptree_list_loc_def]);
Theorem ptree_fringe_mkNd[simp]
`ptree_fringe (mkNd N subs) = FLAT (MAP ptree_fringe subs)`
(simp[mkNd_def]);
Theorem valid_locs_mkNd[simp]
`valid_locs (mkNd N subs) ⇔ ∀pt. MEM pt subs ⇒ valid_locs pt`
(simp[mkNd_def, ptree_list_loc_def]);
Theorem rfringe_length_not_nullable
`∀G s. ¬nullable G [s] ⇒
∀pt. ptree_head pt = s ⇒ valid_lptree G pt ⇒
0 < LENGTH (real_fringe pt)`
(metis_tac[fringe_length_not_nullable, LENGTH_real_fringe, valid_lptree_def]);
Theorem valid_lptree_mkNd[simp]
`valid_lptree G (mkNd n children) ⇔
n ∈ FDOM G.rules ∧ MAP ptree_head children ∈ G.rules ' n ∧
∀pt. MEM pt children ⇒ valid_lptree G pt`
(simp[mkNd_def, ptree_list_loc_def]);
Theorem real_fringe_mkNd[simp]
`real_fringe (mkNd n subs) = FLAT (MAP real_fringe subs)`
(simp[mkNd_def] >> rpt (AP_TERM_TAC ORELSE AP_THM_TAC) >>
simp[FUN_EQ_THM]);
Theorem ptree_head_NT_mkNd
`ptree_head pt = NN n ∧ valid_lptree cmlG pt ∧
real_fringe pt = MAP (TK ## I) pf ⇒
∃subs. pt = mkNd (mkNT n) subs`
(Cases_on `pt`
>- (rename [`ptree_head (Lf pair)`] >> Cases_on `pair` >> simp[] >>
rw[valid_lptree_def] >> rename [`(NN _, _) = (TK ## I) pair`] >>
Cases_on `pair` >> fs[]) >>
rename [`ptree_head (Nd pair _)`] >> Cases_on `pair` >>
simp[MAP_EQ_CONS, mkNd_def, ptree_list_loc_def]);
Theorem mkNd_11[simp]
`mkNd n1 sub1 = mkNd n2 sub2 ⇔ n1 = n2 ∧ sub1 = sub2`
(csimp[mkNd_def]);
Theorem peg_linfix_correct_lemma
`∀UpperN sym sepsym i0 i pts.
peg_eval cmlPEG (i0, peg_linfix UpperN sym sepsym) (SOME(i,pts)) ⇒
(∀i0' i pts s.
s ∈ {sym;sepsym} ⇒
LENGTH i0' < LENGTH i0 ⇒
peg_eval cmlPEG (i0',s) (SOME(i,pts)) ⇒
∃pt. pts = [pt] ∧ ptree_head pt ∈ pegsym_to_sym s ∧
valid_lptree cmlG pt ∧
MAP (TK ## I) i0' = real_fringe pt ++ MAP (TK ## I) i) ∧
(∀i pts s.
s ∈ {sym; sepsym} ⇒
peg_eval cmlPEG (i0, s) (SOME(i,pts)) ⇒
∃pt. pts = [pt] ∧ ptree_head pt ∈ pegsym_to_sym s ∧
valid_lptree cmlG pt ∧
MAP (TK ## I) i0 = real_fringe pt ++ MAP (TK ## I) i) ∧
¬peg0 cmlPEG sym ∧
UpperN ∈ FDOM cmlG.rules ∧
{[gsym] | gsym ∈ pegsym_to_sym sym } ⊆ cmlG.rules ' UpperN ∧
{[NT UpperN; gsep; gsym] |
gsep ∈ pegsym_to_sym sepsym ∧ gsym ∈ pegsym_to_sym sym } ⊆
cmlG.rules ' UpperN ⇒
∃pt. pts = [pt] ∧ ptree_head pt = NT UpperN ∧ valid_lptree cmlG pt ∧
MAP (TK ## I) i0 = real_fringe pt ++ MAP (TK ## I) i`
(rpt strip_tac >> qpat_x_assum `peg_eval X Y Z` mp_tac >>
simp[peg_linfix_def, peg_eval_rpt, peg_eval_seq_SOME] >>
rpt strip_tac >> rveq >>
asm_match `peg_eval cmlPEG (i0, sym) (SOME(i1,r1))` >>
first_assum (qspecl_then [`i1`, `r1`, `sym`] mp_tac) >> simp_tac(srw_ss())[]>>
ASM_REWRITE_TAC[] >> disch_then (qxchl[`rpt1`] strip_assume_tac) >> simp[] >>
rveq >>
qpat_x_assum `peg_eval_list X Y Z` mp_tac >>
`∃i2. i2 = i1 ∧ LENGTH i2 ≤ LENGTH i1` by simp[] >>
Q.UNDISCH_THEN `i2 = i1` (SUBST1_TAC o SYM) >>
`∃acc. MAP ptree_head acc ∈ cmlG.rules ' UpperN ∧
(∀pt. MEM pt acc ⇒ valid_lptree cmlG pt) ∧
[rpt1] = acc ∧ real_fringe rpt1 = FLAT (MAP real_fringe acc)`
by (simp[] >> fs[SUBSET_DEF]) >>
ntac 2 (pop_assum (CHANGED_TAC o SUBST1_TAC)) >> ntac 2 (pop_assum mp_tac) >>
pop_assum mp_tac >> simp[AND_IMP_INTRO, GSYM CONJ_ASSOC] >>
map_every qid_spec_tac [`acc`, `i2`,`i`, `l`] >>
Induct >- simp[Once peg_eval_cases, mk_linfix_def] >>
simp[Once peg_eval_cases] >>
simp[peg_eval_seq_SOME, GSYM LEFT_EXISTS_AND_THM, GSYM RIGHT_EXISTS_AND_THM,
GSYM LEFT_FORALL_IMP_THM, GSYM RIGHT_FORALL_IMP_THM] >>
map_every qx_gen_tac [`i`, `i1'`, `acc`, `i3'`, `i2'`, `sep_r`, `sym_r`] >>
strip_tac >>
`LENGTH i1 < LENGTH i0` by metis_tac[not_peg0_LENGTH_decreases] >>
`LENGTH i1' < LENGTH i0` by decide_tac >>
first_assum (qspecl_then [`i1'`, `i2'`, `sep_r`, `sepsym`] mp_tac) >>
rpt kill_asm_guard >>
disch_then (qxchl [`sep_pt`] strip_assume_tac) >> rveq >>
`LENGTH i2' ≤ LENGTH i1'`
by metis_tac[peg_eval_suffix',
DECIDE``x:num ≤ y ⇔ x = y ∨ x < y``] >>
`LENGTH i2' < LENGTH i0` by decide_tac >>
first_x_assum (qspecl_then [`i2'`, `i3'`, `sym_r`, `sym`] mp_tac) >>
rpt kill_asm_guard >>
disch_then (qxchl [`sym_pt`] strip_assume_tac) >> rveq >>
simp[mk_linfix_def] >>
`LENGTH i3' < LENGTH i2'` by metis_tac[not_peg0_LENGTH_decreases] >>
`LENGTH i3' ≤ LENGTH i1` by decide_tac >>
first_x_assum (qspecl_then [`i`, `i3'`, `[mkNd UpperN acc; sep_pt; sym_pt]`]
mp_tac) >>
simp[DISJ_IMP_THM, FORALL_AND_THM] >>
`[NT UpperN; ptree_head sep_pt; ptree_head sym_pt] ∈ cmlG.rules ' UpperN`
by fs[SUBSET_DEF] >>
simp[]);
Theorem length_no_greater
`peg_eval G (i0, sym) (SOME(i,r)) ⇒ LENGTH i ≤ LENGTH i0`
(metis_tac[peg_eval_suffix',
DECIDE ``x ≤ y:num ⇔ x < y ∨ x = y``]);
Theorem MAP_TK_11[simp]
`MAP TK x = MAP TK y ⇔ x = y`
(eq_tac >> simp[] >> strip_tac >>
match_mp_tac
(INST_TYPE [beta |-> ``:(token,MMLnonT) grammar$symbol``]
listTheory.INJ_MAP_EQ) >>
qexists_tac `TK` >>
simp[INJ_DEF]);
Theorem peg_eval_nTyOp_wrongtok
`FST tk = LparT ⇒ ¬peg_eval cmlPEG (tk::i, nt (mkNT nTyOp) f) (SOME x)`
(simp[Once peg_eval_cases, cmlpeg_rules_applied, FDOM_cmlPEG] >>
simp[Once peg_eval_cases, cmlpeg_rules_applied, FDOM_cmlPEG]);
datatype disposition = X | KEEP
fun rlresolve d P k = let
val f = case d of X => first_x_assum | _ => first_assum
in
f (fn patth =>
first_assum (k o PART_MATCH (rand o lhand) patth o
assert P o concl))
end
fun lrresolve d P k = let
val f = case d of X => first_x_assum | _ => first_assum
in
f (fn patth =>
first_assum (k o PART_MATCH (lhand o rand) patth o
assert P o concl))
end
Theorem peg_EbaseParen_sound
`∀i0 i pts.
peg_eval cmlPEG (i0, peg_EbaseParen) (SOME(i,pts)) ⇒
(∀i0' N i pts.
LENGTH i0' < LENGTH i0 ∧
peg_eval cmlPEG (i0', nt (mkNT N) I) (SOME(i,pts)) ⇒
∃pt.
pts = [pt] ∧ ptree_head pt = NT (mkNT N) ∧
valid_lptree cmlG pt ∧
MAP (TOK ## I) i0' = real_fringe pt ++ MAP (TOK ## I) i) ⇒
∃pt.
pts = [pt] ∧ ptree_head pt = NT (mkNT nEbase) ∧
valid_lptree cmlG pt ∧
MAP (TOK ## I) i0 = real_fringe pt ++ MAP (TOK ## I) i`
(rw[peg_EbaseParen_def]
>- (rlresolve X (K true) mp_tac >> simp[] >> strip_tac >> rveq >>
simp[peg_EbaseParenFn_def, cmlG_FDOM, cmlG_applied, DISJ_IMP_THM,
pairTheory.PAIR_MAP])
>- (rlresolve KEEP (free_in ``nE``) mp_tac >>
rpt kill_asm_guard >> strip_tac >> rveq >>
rlresolve X (free_in ``nElist1``) mp_tac >>
first_assum (mp_tac o MATCH_MP length_no_greater o
assert (free_in ``nE`` o concl)) >>
simp[] >> rw[] >>
rpt (qpat_x_assum`_ = FST _`(assume_tac o SYM)) >> rw[] >>
simp[peg_EbaseParenFn_def, cmlG_applied, cmlG_FDOM,
DISJ_IMP_THM, FORALL_AND_THM, pairTheory.PAIR_MAP]) >>
rlresolve KEEP (free_in ``nE``) mp_tac >> rpt kill_asm_guard >>
strip_tac >> rveq >>
rlresolve X (free_in ``nEseq``) mp_tac >>
first_assum (mp_tac o MATCH_MP length_no_greater o
assert (free_in ``nE`` o concl)) >>
simp[] >> rpt strip_tac >>rveq >>
rpt (qpat_x_assum`_ = FST _`(assume_tac o SYM)) >>
simp[peg_EbaseParenFn_def, cmlG_applied, cmlG_FDOM,
DISJ_IMP_THM, FORALL_AND_THM, pairTheory.PAIR_MAP] >> fs[])
val PAIR_MAP_I = Q.prove(
‘(f ## I) x = (f (FST x), SND x) ⇔ T’,
simp[PAIR_MAP]);
val _ = augment_srw_ss [rewrites [PAIR_MAP_I]]
val bindNT0_lemma = REWRITE_RULE [GSYM mkNd_def] bindNT0_def
val _ = augment_srw_ss [rewrites [bindNT0_lemma]]
(* left recursive rules in the grammar turn into calls to rpt in the PEG,
and this in turn requires inductions *)
Theorem ptPapply_lemma
`∀limit.
(∀i0 i pts.
LENGTH i0 < limit ⇒
peg_eval cmlPEG (i0, nt (mkNT nPbase) I) (SOME (i, pts)) ⇒
∃pt. pts = [pt] ∧ ptree_head pt = NN nPbase ∧ valid_lptree cmlG pt ∧
MAP (TK ## I) i0 = real_fringe pt ++ MAP (TK ## I) i) ⇒
∀ptlist pt0 acc i0.
peg_eval_list cmlPEG (i0, nt (mkNT nPbase) I) (i, ptlist) ∧
ptree_head pt0 = NN nPbase ∧ valid_lptree cmlG pt0 ∧
ptree_head acc = NN nPConApp ∧ valid_lptree cmlG acc ∧
LENGTH i0 < limit ⇒
∃pt. ptPapply0 acc (pt0 :: FLAT ptlist) = [pt] ∧
ptree_head pt = NN nPapp ∧ valid_lptree cmlG pt ∧
real_fringe acc ++ real_fringe pt0 ++ MAP (TK ## I) i0 =
real_fringe pt ++ MAP (TK ## I) i`
(gen_tac >> strip_tac >> Induct
>- (simp[Once peg_eval_list] >> simp[ptPapply0_def] >>
dsimp[cmlG_FDOM, cmlG_applied]) >>
dsimp[Once peg_eval_list] >> rpt strip_tac >>
first_x_assum (erule mp_tac) >> strip_tac >> rveq >> simp[ptPapply0_def] >>
imp_res_tac (MATCH_MP not_peg0_LENGTH_decreases peg0_nPbase) >>
rename [‘peg_eval _ (i0, _) (SOME (i1, [pt1]))’,
‘peg_eval_list _ (i1, _) (i, ptlist)’] >>
first_x_assum (qspecl_then [‘pt1’, ‘mkNd (mkNT nPConApp) [acc; pt0]’, ‘i1’]
mp_tac) >> simp[] >>
disch_then irule >> dsimp[cmlG_applied, cmlG_FDOM]);
Theorem peg_sound
`∀N i0 i pts.
peg_eval cmlPEG (i0,nt N I) (SOME(i,pts)) ⇒
∃pt. pts = [pt] ∧ ptree_head pt = NT N ∧
valid_lptree cmlG pt ∧
MAP (TOK ## I) i0 = real_fringe pt ++ MAP (TOK ## I) i`
(ntac 2 gen_tac >> `?iN. iN = (i0,N)` by simp[] >> pop_assum mp_tac >>
map_every qid_spec_tac [`i0`, `N`, `iN`] >>
qispl_then [`measure (LENGTH:(token # locs) list->num) LEX measure NT_rank`]
(ho_match_mp_tac o
SIMP_RULE (srw_ss()) [pairTheory.WF_LEX,
prim_recTheory.WF_measure])
relationTheory.WF_INDUCTION_THM >>
dsimp[pairTheory.LEX_DEF] >>
map_every qx_gen_tac [`N`, `i0`, `i`, `pts`] >> strip_tac >>
simp[peg_eval_NT_SOME, cmlPEGTheory.FDOM_cmlPEG] >>
disch_then (CONJUNCTS_THEN2 strip_assume_tac mp_tac) >> rveq >>
simp[cmlPEGTheory.cmlpeg_rules_applied, ptree_list_loc_def]
>- (print_tac "nNonETopLevelDecs" >>
`NT_rank (mkNT nTopLevelDec) < NT_rank (mkNT nNonETopLevelDecs)`
by simp[NT_rank_def] >>
strip_tac >> rveq >> dsimp[cmlG_FDOM, cmlG_applied]
>- (first_x_assum (erule strip_assume_tac) >> rveq >> csimp[] >>
rename1
`peg_eval _ (inp0, nt (mkNT nTopLevelDec) _) (SOME (inp1,_))` >>
`LENGTH inp1 < LENGTH inp0`
by metis_tac[not_peg0_LENGTH_decreases, peg0_nTopLevelDec] >>
csimp[PULL_EXISTS] >> metis_tac[])
>- (fs[] >> csimp[] >> metis_tac[DECIDE ``x < SUC x``])
>- (csimp[] >> fs[] >> metis_tac[DECIDE ``x < SUC x``]))
>- (print_tac "nTopLevelDecs" >>
`NT_rank (mkNT nE) < NT_rank (mkNT nTopLevelDecs) ∧
NT_rank (mkNT nTopLevelDec) < NT_rank (mkNT nTopLevelDecs)`
by simp[NT_rank_def] >>
strip_tac >> rveq >> simp[cmlG_FDOM, cmlG_applied]
>- (dsimp[APPEND_EQ_CONS] >> csimp[] >>
first_x_assum (erule strip_assume_tac) >> rveq >> simp[] >>
rename1 `peg_eval _ (inp0, _) (SOME(h::inp1,[_]))` >>
`LENGTH (h :: inp1) < LENGTH inp0`
by metis_tac[not_peg0_LENGTH_decreases, peg0_nE] >>
fs[] >> metis_tac[DECIDE ``SUC x < y ⇒ x < y``])
>- (first_x_assum (erule strip_assume_tac) >> rveq >> dsimp[] >>
csimp[] >>
rename1
`peg_eval _ (inp0, nt (mkNT nTopLevelDec) I) (SOME(inp1,_))` >>
`LENGTH inp1 < LENGTH inp0` suffices_by metis_tac[] >>
metis_tac[not_peg0_LENGTH_decreases, peg0_nTopLevelDec])
>- (dsimp[] >> csimp[] >> fs[] >> metis_tac[DECIDE ``x < SUC x``])
>- (dsimp[] >> csimp[] >> fs[] >> metis_tac[DECIDE ``x < SUC x``])
>- (rename1 `peg_eval _ (ip0,nt (mkNT nTopLevelDec) _) (SOME(ip1,pt))`>>
first_x_assum(qspecl_then [`mkNT nTopLevelDec`, `ip1`, `pt`] mp_tac)>>
simp[] >> strip_tac >> rveq >> dsimp[] >> csimp[] >>
`LENGTH ip1 < LENGTH ip0` suffices_by metis_tac[] >>
metis_tac[not_peg0_LENGTH_decreases, peg0_nTopLevelDec])
>- (dsimp[] >> csimp[] >> fs[] >> metis_tac[DECIDE ``x < SUC x``])
>- (dsimp[] >> csimp[] >> fs[] >> metis_tac[DECIDE ``x < SUC x``])
>- (rename1 `peg_eval _ (ip0,nt (mkNT nTopLevelDec) _) (SOME(ip1,pt))`>>
first_x_assum(qspecl_then [`mkNT nTopLevelDec`, `ip1`, `pt`] mp_tac)>>
simp[] >> strip_tac >> rveq >> dsimp[] >> csimp[] >>
`LENGTH ip1 < LENGTH ip0` suffices_by metis_tac[] >>
metis_tac[not_peg0_LENGTH_decreases, peg0_nTopLevelDec])
>- (dsimp[] >> csimp[] >> fs[] >> metis_tac[DECIDE ``x < SUC x``])
>- (dsimp[] >> csimp[] >> fs[] >> metis_tac[DECIDE ``x < SUC x``]))
(* >- (print_tac "nREPLTop">>
`NT_rank (mkNT nE) < NT_rank (mkNT nREPLTop) ∧
NT_rank (mkNT nTopLevelDec) < NT_rank (mkNT nREPLTop)`
by simp[NT_rank_def] >>
strip_tac >> rveq >> simp[] >>
first_x_assum (erule strip_assume_tac) >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM]) *)
(*>- (print_tac "nREPLPhrase" >>
`NT_rank (mkNT nE) < NT_rank (mkNT nREPLPhrase) ∧
NT_rank (mkNT nTopLevelDecs) < NT_rank (mkNT nREPLPhrase)`
by simp[NT_rank_def] >>
strip_tac >> rveq >> simp[] >>
first_x_assum (erule strip_assume_tac) >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM]) *)
>- (print_tac "nTopLevelDec" >>
`NT_rank (mkNT nStructure) < NT_rank (mkNT nTopLevelDec) ∧
NT_rank (mkNT nDecl) < NT_rank (mkNT nTopLevelDec)`
by simp[NT_rank_def] >>
strip_tac >>
first_x_assum (erule mp_tac) >>
strip_tac >> simp[cmlG_FDOM, cmlG_applied])
>- (print_tac "nStructure" >> strip_tac >> rveq >>
simp[DISJ_IMP_THM, FORALL_AND_THM, cmlG_FDOM, cmlG_applied] >>
loseC ``NT_rank`` >> fs[] >>
asm_match `peg_eval cmlPEG (vi, nt (mkNT nStructName) I) (SOME(opti,vt))` >>
`LENGTH vi < SUC (LENGTH vi)` by decide_tac >>
first_assum (erule strip_assume_tac) >> rveq >> simp[] >>
`LENGTH opti < LENGTH vi`
by metis_tac[not_peg0_LENGTH_decreases, peg0_nStructName] >>
`LENGTH opti < SUC (LENGTH vi)` by decide_tac >>
first_assum (erule strip_assume_tac) >> rveq >> simp[] >>
asm_match `peg_eval cmlPEG (opti, OPTSIG)
(SOME (eqT::strT::di,[opt]))` >>
`LENGTH (eqT::strT::di) ≤ LENGTH opti`
by metis_tac[peg_eval_suffix',
DECIDE``x:num ≤ y ⇔ x < y ∨ x = y``] >> fs[] >>
`LENGTH di < SUC (LENGTH vi)` by decide_tac >>
first_x_assum (erule strip_assume_tac) >> rveq >> simp[])
>- (print_tac "nStructName" >> simp[peg_StructName_def] >>
dsimp[cmlG_applied, cmlG_FDOM, PAIR_MAP])
>- (print_tac "nOptionalSignatureAscription" >> strip_tac >> rveq >>
simp[cmlG_applied, cmlG_FDOM] >> dsimp[] >>
loseC ``NT_rank`` >> dsimp[MAP_EQ_SING] >> csimp[] >> fs[] >>
metis_tac [DECIDE ``x < SUC x``])
>- (print_tac "nSignatureValue" >>
strip_tac >> rveq >> simp[cmlG_FDOM, cmlG_applied, MAP_EQ_SING] >>
dsimp[] >> csimp[] >>
first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o concl)) >>
dsimp[])
>- (print_tac "nSpecLineList" >> strip_tac >> simp[cmlG_applied, cmlG_FDOM]
>- (`NT_rank (mkNT nSpecLine) < NT_rank (mkNT nSpecLineList)`
by simp[NT_rank_def] >>
first_x_assum (erule mp_tac) >>
asm_match
`peg_eval cmlPEG (i0, nt (mkNT nSpecLine) I) (SOME (i1,r))` >>
`LENGTH i1 < LENGTH i0`
by metis_tac[not_peg0_LENGTH_decreases, peg0_nSpecLine] >>
first_x_assum (erule mp_tac) >> rpt strip_tac >> rveq >> fs[])
>- (dsimp[MAP_EQ_SING] >> csimp[]>>fs[] >> metis_tac[DECIDE``x < SUC x``])
>> fs[cmlG_FDOM, cmlG_applied, MAP_EQ_SING] >> dsimp[] >> csimp[] >>
metis_tac[DECIDE``x< SUC x``])
>- (print_tac "nSpecLine" >> strip_tac >> rveq >>
fs[cmlG_applied, cmlG_FDOM, peg_eval_TypeDec_wrongtok]
>- (asm_match
`peg_eval cmlPEG (i1, nt (mkNT nV) I) (SOME(colonT::i2,r))` >>
`LENGTH i1 < SUC (LENGTH i1)` by DECIDE_TAC >>
first_assum (erule strip_assume_tac) >>
`LENGTH (colonT::i2) < LENGTH i1`
by metis_tac[not_peg0_LENGTH_decreases, peg0_nV] >> fs[] >>
`LENGTH i2 < SUC(LENGTH i1)` by decide_tac >>
first_x_assum (erule strip_assume_tac) >> rveq >> dsimp[])
>- (asm_match
`peg_eval cmlPEG (i1, nt (mkNT nTypeName) I) (SOME(i2, nmpts))` >>
first_assum (qspecl_then [`mkNT nTypeName`, `i1`, `i2`, `nmpts`]
mp_tac) >>
simp_tac (srw_ss()) [] >> ASM_REWRITE_TAC [] >>
disch_then (qx_choose_then `nmpt` strip_assume_tac) >>
dsimp[MAP_EQ_SING] >> csimp[] >>
`LENGTH i2 < LENGTH i1`
by metis_tac[not_peg0_LENGTH_decreases, peg0_nTypeName] >>
`LENGTH i2 < SUC (LENGTH i1)` by simp[] >>
metis_tac[])
>- (dsimp[MAP_EQ_SING] >> csimp[] >> metis_tac[DECIDE``x<SUC x``])>>
rpt(qpat_x_assum`_ = FST _`(assume_tac o SYM)) \\ fs[] >> rveq \\ fs[] >>
`NT_rank (mkNT nTypeDec) < NT_rank (mkNT nSpecLine)`
by simp[NT_rank_def] >>
first_x_assum (erule strip_assume_tac) >> rveq >> simp[])
>- (print_tac "nOptTypEqn" >> strip_tac >> rveq >>
simp[cmlG_FDOM, cmlG_applied] >> dsimp[MAP_EQ_SING] >> csimp[] >>
fs[] >> metis_tac[DECIDE ``x < SUC x``])
>- (print_tac "nDecls" >>
`NT_rank (mkNT nDecl) < NT_rank (mkNT nDecls)`
by simp[NT_rank_def] >>
strip_tac >> rveq >> fs[cmlG_applied, cmlG_FDOM]
>- (first_x_assum (erule strip_assume_tac) >>
asm_match `peg_eval cmlPEG (i0,nt (mkNT nDecl) I) (SOME(i1,r))` >>
dsimp[MAP_EQ_SING] >>
`LENGTH i1 < LENGTH i0`
by metis_tac[not_peg0_LENGTH_decreases,peg0_nDecl] >>
first_x_assum (erule strip_assume_tac) >>
dsimp[MAP_EQ_SING])
>- (dsimp[MAP_EQ_SING] >> csimp[] >> metis_tac[DECIDE``x<SUC x``]) >>
dsimp[MAP_EQ_SING] >> csimp[] >> metis_tac[DECIDE``x<SUC x``])
>- (print_tac "nTypeAbbrevDec" >>
rpt strip_tac >> rveq >> simp[cmlG_applied, cmlG_FDOM] >>
dsimp[listTheory.APPEND_EQ_CONS, MAP_EQ_SING] >> csimp[] >>
qmatch_assum_rename_tac
`peg_eval cmlPEG (inp, nt(mkNT nTypeName) I) (SOME(equalsT::inp1,t1))`>>
first_assum
(qspecl_then [`mkNT nTypeName`, `inp`, `equalsT::inp1`, `t1`] mp_tac) >>
simp_tac (srw_ss()) [] >> ASM_REWRITE_TAC[] >>
disch_then (qx_choose_then `tree1` strip_assume_tac) >> simp[] >>
qmatch_assum_rename_tac
`peg_eval cmlPEG (inp1, nt(mkNT nType) I) (SOME(inp2,t2))` >>
`LENGTH (equalsT::inp1) < LENGTH inp`
by metis_tac[not_peg0_LENGTH_decreases, peg0_nTypeName] >> fs[] >>
`LENGTH inp1 < SUC (LENGTH inp)` by simp[] >>
first_x_assum (qspecl_then [`mkNT nType`, `inp1`, `inp2`, `t2`] mp_tac) >>
simp[] >> metis_tac[])
>- (print_tac "nDecl" >>
rpt strip_tac >> rveq >>
rpt(qpat_x_assum`_ = FST _`(assume_tac o SYM)) >>
fs[peg_eval_TypeDec_wrongtok, peg_eval_TypeAbbrevDec_wrongtok] >>
rveq >> fs[]
>- (asm_match `peg_eval cmlPEG (i1, nt (mkNT nPattern) I)
(SOME(equalsT::i2,r))` >>
`LENGTH i1 < SUC (LENGTH i1)` by decide_tac >>
first_assum (erule strip_assume_tac) >> rveq >>
`LENGTH (equalsT::i2) ≤ LENGTH i1`
by metis_tac[peg_eval_suffix',
DECIDE``x≤y ⇔ x = y ∨ x < y:num``] >> fs[]>>
`LENGTH i2 < SUC (LENGTH i1)` by decide_tac >>
first_x_assum (erule strip_assume_tac) >> rveq >>
simp[cmlG_FDOM, cmlG_applied, MAP_EQ_SING] >> dsimp[PAIR_MAP])
>- (dsimp[cmlG_applied, cmlG_FDOM, MAP_EQ_SING] >> csimp[PAIR_MAP] >>
metis_tac[DECIDE ``x<SUC x``])
>- (dsimp[cmlG_FDOM, cmlG_applied, MAP_EQ_SING] >> csimp[PAIR_MAP] >>
metis_tac[DECIDE``x<SUC x``])
>- (dsimp[cmlG_FDOM, cmlG_applied, APPEND_EQ_CONS] >> csimp[PAIR_MAP] >>
rename[‘MAP (TK ## I) in0’, ‘peg_eval _ (in0, nt (mkNT nDecls) I)’] >>
‘LENGTH in0 < SUC (LENGTH in0)’ by simp[] >>
first_assum (pop_assum o mp_then (Pos hd) drule) >>
disch_then (qx_choose_then ‘decls1_pt’ strip_assume_tac) >> simp[] >>
fs[MAP_EQ_APPEND] >> rveq >> dsimp[PAIR_MAP] >>
rename [‘real_fringe decls1_pt = MAP _ in00’,
‘peg_eval _ (in00 ++ [Int] ++ in01, nt (mkNT nDecls) I)’] >>
first_x_assum (qpat_assum ‘peg_eval _ (in01, _) _’ o
mp_then (Pos (el 2)) mp_tac) >> dsimp[])
>- (`NT_rank (mkNT nTypeDec) < NT_rank (mkNT nDecl)`
by simp[NT_rank_def] >>
first_x_assum (erule strip_assume_tac) >>
dsimp[cmlG_FDOM, cmlG_applied])
>- (`NT_rank (mkNT nTypeAbbrevDec) < NT_rank (mkNT nDecl)`
by simp[NT_rank_def] >>
first_x_assum (erule strip_assume_tac) >>
dsimp[cmlG_FDOM, cmlG_applied]))
>- (print_tac "nLetDecs" >> rpt strip_tac >> rveq >>
simp[cmlG_applied, cmlG_FDOM] >> fs[peg_eval_LetDec_wrongtok]
>- (simp[] >>
`NT_rank (mkNT nLetDec) < NT_rank (mkNT nLetDecs)`
by simp[NT_rank_def] >>
first_x_assum (erule strip_assume_tac) >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM, MAP_EQ_SING] >> csimp[] >>
metis_tac[not_peg0_LENGTH_decreases, peg0_nLetDec]) >>
dsimp[MAP_EQ_SING] >> csimp[] >> metis_tac[DECIDE ``x < SUC x``])
>- (print_tac "nLetDec" >>
rpt strip_tac >> rveq >> simp[cmlG_FDOM, cmlG_applied, MAP_EQ_SING] >> fs[]
>- (dsimp[listTheory.APPEND_EQ_CONS, MAP_EQ_SING] >> csimp[] >>
asm_match
`peg_eval cmlPEG (i1,nt (mkNT nPattern) I) (SOME(equalsT::i2,r))` >>
`LENGTH i1 < SUC (LENGTH i1)` by decide_tac >>
first_assum (erule strip_assume_tac) >> rveq >> simp[] >>
`LENGTH (equalsT::i2) ≤ LENGTH i1`
by metis_tac[peg_eval_suffix',
DECIDE``x≤y ⇔ x = y ∨ x < y:num``] >> fs[]>>
`LENGTH i2 < SUC (LENGTH i1)` by decide_tac >>
metis_tac[]) >>
dsimp[] >> csimp[] >> metis_tac[DECIDE``x<SUC x``])
>- (print_tac "nPtuple" >> strip_tac >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM] >> lrresolve X (K true) mp_tac>>
simp[] >> strip_tac >> rveq >> simp[])
>- (print_tac "nPatternList" >> strip_tac >> rveq >>
TRY(qpat_x_assum`CommaT = FST _`(assume_tac o SYM) >> fs[]) >>
simp[cmlG_FDOM, cmlG_applied] >>
`NT_rank (mkNT nPattern) < NT_rank (mkNT nPatternList)`
by simp[NT_rank_def] >>
first_x_assum (erule mp_tac) >> strip_tac >> rveq >> simp[] >>
imp_res_tac length_no_greater >> fs[] >>
fs[MAP_EQ_APPEND, MAP_EQ_CONS] >>
lrresolve X (free_in ``nPatternList``) mp_tac >> simp[] >>
strip_tac >> rveq >> dsimp[] >> rfs[PAIR_MAP])
>- (print_tac "nPattern" >> strip_tac >> rveq >>
`NT_rank (mkNT nPcons) < NT_rank (mkNT nPattern)` by simp[NT_rank_def] >>
simp[cmlG_FDOM, cmlG_applied]
>- (dsimp[APPEND_EQ_CONS] >> csimp[] >>
first_x_assum (erule mp_tac) >> strip_tac >> rveq >> simp[] >>
fs[MAP_EQ_APPEND] >> rveq >> dsimp[] >> fs[] >>
rename1`peg_eval _ (inp2, nt (mkNT nType) I)` >>
first_x_assum (qspecl_then [`mkNT nType`, `inp2`] mp_tac) >>
imp_res_tac length_no_greater \\ fs[] >>
simp[] >> metis_tac[])
>- (dsimp[MAP_EQ_CONS] >> disj1_tac >>
first_x_assum (erule mp_tac) >> strip_tac >> rveq >> simp[])
>- (dsimp[MAP_EQ_CONS] >> disj1_tac >>
first_x_assum (erule mp_tac) >> strip_tac >> rveq >> simp[]))
>- (print_tac "nPcons" >>
`NT_rank (mkNT nPapp) < NT_rank (mkNT nPcons)`
by simp[NT_rank_def] >> strip_tac >> rveq >>
simp[cmlG_applied, cmlG_FDOM]
>- (lrresolve KEEP (K true) mp_tac >> rpt kill_asm_guard >>
strip_tac >> rveq >> simp[MAP_EQ_CONS] >> dsimp[] >>
csimp[] >>
imp_res_tac length_no_greater >> fs[GSYM CONJ_ASSOC] >>
rpt (loseC ``NT_rank``) >> lrresolve X (K true) mp_tac >>
simp[] >> metis_tac[])
>- (lrresolve X (K true) mp_tac >> rpt kill_asm_guard >>
strip_tac >> rveq >> simp[]) >>
lrresolve X (K true) mp_tac >> rpt kill_asm_guard >>
strip_tac >> rveq >> simp[])
>- (print_tac "nPapp" >>
`NT_rank (mkNT nConstructorName) < NT_rank (mkNT nPapp) ∧
NT_rank (mkNT nPbase) < NT_rank (mkNT nPapp)`
by simp[NT_rank_def] >>
reverse strip_tac >> rveq >> simp[cmlG_FDOM, cmlG_applied]
>- (first_x_assum (erule mp_tac) >> strip_tac >> rveq >> dsimp[] >>
fs[]) >>
first_x_assum (erule mp_tac) >> strip_tac >> rveq >> dsimp[] >>
imp_res_tac
(MATCH_MP not_peg0_LENGTH_decreases peg0_nConstructorName) >>
fs[peg_eval_rpt, Once peg_eval_list]
>- simp[cmlG_FDOM, cmlG_applied] >>
first_assum (erule mp_tac) >> strip_tac >> rveq >> dsimp[] >>
simp[ptPapply_def] >>
first_assum (mp_then (Pos (el 2)) mp_tac ptPapply_lemma) >>
ntac 2 (disch_then (first_assum o mp_then (Pos (el 2)) mp_tac)) >>
rename [‘peg_eval _ (i0, nt (mkNT nConstructorName) I)’] >>
disch_then (qspec_then ‘LENGTH i0’ mp_tac) >> simp[] >>
rename [‘mkNd (mkNT nPConApp) [pt]’] >>
disch_then (qspec_then ‘mkNd (mkNT nPConApp) [pt]’ mp_tac) >>
simp[cmlG_applied, cmlG_FDOM] >> disch_then irule >>
imp_res_tac (MATCH_MP not_peg0_LENGTH_decreases peg0_nPbase) >>
simp[])
>- (print_tac "nPbase" >>
`NT_rank (mkNT nV) < NT_rank (mkNT nPbase) ∧
NT_rank (mkNT nConstructorName) < NT_rank (mkNT nPbase) ∧
NT_rank (mkNT nPtuple) < NT_rank (mkNT nPbase)`
by simp[NT_rank_def] >>
strip_tac >> rveq
>- (first_x_assum (erule mp_tac) >> strip_tac >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM])
>- (first_x_assum (erule mp_tac) >> strip_tac >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM])
>- (simp[cmlG_FDOM, cmlG_applied] >> asm_match `isInt (FST h)` >>
Cases_on `FST h` >> fs[])
>- (simp[cmlG_FDOM, cmlG_applied] >> asm_match `isString (FST h)` >>
Cases_on `FST h` >> fs[])
>- (simp[cmlG_FDOM, cmlG_applied] >> asm_match `isCharT (FST h)` >>
Cases_on `FST h` >> fs[])
>- (first_x_assum (erule mp_tac) >> strip_tac >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM])
>- simp[cmlG_FDOM, cmlG_applied]
>- (simp[cmlG_FDOM, cmlG_applied] >>
lrresolve KEEP (K true) mp_tac >> kill_asm_guard >>
ASM_REWRITE_TAC [] >> strip_tac >> rveq >> dsimp[])
>- dsimp[cmlG_applied, cmlG_FDOM]
>- (simp[cmlG_FDOM, cmlG_applied] >>
qpat_x_assum `OpT = FST _` (ASSUME_TAC o SYM) >> dsimp[] >>
csimp[] >> fs[] >> metis_tac[DECIDE ``x < SUC x``])
>- (fs[] >> qpat_x_assum `OpT = FST _` (ASSUME_TAC o SYM) >> rveq >>
fs[])
>- (fs[] >> qpat_x_assum `OpT = FST _` (ASSUME_TAC o SYM) >> rveq >>
fs[]))
>- (print_tac "nConstructorName" >>
simp[pairTheory.EXISTS_PROD] >>
`NT_rank (mkNT nUQConstructorName) < NT_rank (mkNT nConstructorName)`
by simp[NT_rank_def] >>
strip_tac >> rveq >> simp[]
>- (first_x_assum (erule strip_assume_tac) >> rveq >>
simp[cmlG_applied, cmlG_FDOM] >> NO_TAC) >>
simp[cmlG_FDOM, cmlG_applied] >>
asm_match `destLongidT t = SOME(m,v)` >> Cases_on `t` >> fs[])
>- (print_tac "nUQConstructorName" >>
simp[peg_UQConstructorName_def] >>
dsimp[cmlG_applied, cmlG_FDOM, FORALL_PROD])
>- (print_tac "nDconstructor" >>
`NT_rank (mkNT nUQConstructorName) < NT_rank (mkNT nDconstructor)`
by simp[NT_rank_def] >>
strip_tac >>
rveq >> simp[cmlG_FDOM, cmlG_applied, listTheory.APPEND_EQ_CONS,
MAP_EQ_SING] >>
first_x_assum (qpat_x_assum ‘NT_rank _ < NT_rank _’ o
mp_then (Pos hd) mp_tac) >>
disch_then (first_assum o
mp_then (Pos hd) strip_assume_tac) >>
simp[] >> rveq >> dsimp[] >> csimp[] >>
first_x_assum (qpat_assum ‘peg_eval _ (_, nt (mkNT nTbaseList) I) _’o
mp_then Any mp_tac) >>
metis_tac[not_peg0_LENGTH_decreases, peg0_nUQConstructorName,
LENGTH, DECIDE``SUC x < y ⇒ x < y``, MAP])
>- (print_tac "nDtypeDecl" >>
`NT_rank (mkNT nTypeName) < NT_rank (mkNT nDtypeDecl)`
by simp[NT_rank_def] >>
strip_tac >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM, listTheory.APPEND_EQ_CONS, MAP_EQ_SING] >>
csimp[] >>
first_x_assum (erule strip_assume_tac) >> rveq >> simp[] >> csimp[] >>
`∀x:mlptree.
NN nDtypeCons = ptree_head x ⇔ ptree_head x = NN nDtypeCons`
by simp[EQ_SYM_EQ] >>
pop_assum (fn th => simp[th]) >>
first_x_assum (match_mp_tac o MATCH_MP peg_linfix_correct_lemma) >>
simp[DISJ_IMP_THM, FORALL_AND_THM, peg_eval_tok_SOME, pegsym_to_sym_def,
mktokLf_def, cmlG_applied, cmlG_FDOM, SUBSET_DEF] >>
(peg0_nTypeName |> Q.INST[`f` |-> `I`] |> assume_tac) >>
erule mp_tac (GEN_ALL not_peg0_LENGTH_decreases)>>
simp[] >> rw[])
>- (print_tac "nTypeDec" >> simp[peg_TypeDec_def] >> strip_tac >> rveq >>
dsimp[cmlG_FDOM, cmlG_applied, mktokLf_def, MAP_EQ_SING] >> csimp[] >>
fs[] >> pop_assum (mp_tac o MATCH_MP peg_linfix_correct_lemma) >>
simp[pegsym_to_sym_def, cmlG_applied, cmlG_FDOM, SUBSET_DEF,
DISJ_IMP_THM, FORALL_AND_THM, EXISTS_PROD] >>
dsimp[ptree_list_loc_def])
>- (print_tac "nTyVarList" >> strip_tac >>
`NT_rank (mkNT nTyvarN) < NT_rank (mkNT nTyVarList)`
by simp[NT_rank_def] >>
first_x_assum (mp_tac o MATCH_MP peg_linfix_correct_lemma) >>
simp[DISJ_IMP_THM, pegsym_to_sym_def, cmlG_FDOM, cmlG_applied,
SUBSET_DEF, FORALL_AND_THM, mktokLf_def] >> dsimp[])
>- (print_tac "nTypeName" >>
simp[] >> strip_tac >> rveq >>
dsimp[cmlG_FDOM, cmlG_applied, MAP_EQ_SING]
>- (first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o concl)) >>
simp[NT_rank_def] >> strip_tac >> rveq >> dsimp[])
>- (simp[listTheory.APPEND_EQ_CONS] >> dsimp[MAP_EQ_CONS] >>
csimp[] >> loseC ``NT_rank`` >>
first_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o
assert (free_in ``nTyVarList``) o concl)) >>
rpt kill_asm_guard >>
disch_then (qxchl [`tyvl_pt`] strip_assume_tac) >>
rveq >> simp[] >>
first_x_assum (assume_tac o MATCH_MP length_no_greater o
assert (free_in ``nTyVarList`` o concl)) >> fs[] >>
first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o concl)) >>
rpt kill_asm_guard >> metis_tac[])
>- (first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o concl)) >>
simp[] >> strip_tac >> rveq >> simp[] >> asm_match `isTyvarT HH` >>
qpat_x_assum `isTyvarT HH` mp_tac >> Cases_on `HH` >> simp[])
>- (rename [`LparT = FST sometok`] >> Cases_on `sometok` >>
fs[] >> rveq >> fs[])
>- (rename [`LparT = FST sometok`] >> Cases_on `sometok` >>
fs[] >> rveq >> fs[]) >>
rename [`LparT = FST sometok`] >> Cases_on `sometok` >>
fs[] >> rveq >> fs[])
>- (print_tac "nPType" >>
`NT_rank (mkNT nDType) < NT_rank (mkNT nPType)` by simp[NT_rank_def] >>
strip_tac >> rveq >> simp[cmlG_applied, cmlG_FDOM]
>- (first_x_assum (erule mp_tac) >>
simp[MAP_EQ_APPEND, MAP_EQ_CONS] >>
strip_tac >> rveq >> lrresolve X (free_in ``nPType``) mp_tac >>
simp[] >> strip_tac >> rveq >>
fs[MAP_EQ_APPEND, MAP_EQ_CONS] >>
dsimp[] >> metis_tac[])
>- (first_x_assum (erule mp_tac) >> strip_tac >> rveq >> simp[]) >>
first_x_assum (erule mp_tac) >> strip_tac >> rveq >> simp[])
>- (print_tac "nUQTyOp" >>
dsimp[cmlG_FDOM, cmlG_applied, EXISTS_PROD, FORALL_PROD] >>
qx_gen_tac `h` >> Cases_on `h` >> simp[])
>- (print_tac "nTyOp" >> strip_tac >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM, MAP_EQ_SING]
>- (first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o concl)) >>
simp[NT_rank_def] >> strip_tac >> rveq >> simp[])
>- (asm_match `isLongidT (FST h)` >> Cases_on `FST h` >> fs[]))
>- (print_tac "nTbaseList" >> strip_tac >> rveq >>
dsimp[cmlG_FDOM, cmlG_applied] >>
‘NT_rank (mkNT nPTbase) < NT_rank (mkNT nTbaseList)’
by simp[NT_rank_def] >>
first_x_assum (pop_assum o mp_then (Pos hd) mp_tac) >>
disch_then (first_assum o mp_then (Pos hd) strip_assume_tac) >> rveq >>
simp[] >> dsimp[] >> csimp[] >>
metis_tac[not_peg0_LENGTH_decreases, peg0_nPTbase])
>- (print_tac "nTypeList1" >>
CONV_TAC (LAND_CONV (SIMP_CONV (srw_ss() ++ DNF_ss) [EXISTS_PROD])) >>
strip_tac >> rveq >>
dsimp[cmlG_FDOM, cmlG_applied, listTheory.APPEND_EQ_CONS, MAP_EQ_SING] >>
csimp[] >>
`NT_rank (mkNT nType) < NT_rank (mkNT nTypeList1)` by simp[NT_rank_def]
>- (first_x_assum (erule mp_tac) >>
erule assume_tac
(length_no_greater |> Q.GEN `sym`
|> Q.ISPEC `nt (mkNT nType) I` |> GEN_ALL) >>
first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o concl)) >>
fs[] >> simp[] >> rpt strip_tac >> rveq >> simp[])
>- (first_x_assum (erule strip_assume_tac) >> rveq >> simp[]) >>
first_x_assum (erule strip_assume_tac) >> rveq >> simp[])
>- (print_tac "nTypeList2" >>
CONV_TAC (LAND_CONV (SIMP_CONV (srw_ss() ++ DNF_ss) [EXISTS_PROD])) >>
strip_tac >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM, listTheory.APPEND_EQ_CONS, MAP_EQ_SING]>>
csimp[] >>
`NT_rank (mkNT nType) < NT_rank (mkNT nTypeList2)` by simp[NT_rank_def]>>
first_x_assum (erule strip_assume_tac) >> rveq >> simp[] >>
first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o concl)) >>
erule assume_tac
(length_no_greater |> Q.GEN `sym` |> Q.ISPEC `nt (mkNT nType) I`
|> GEN_ALL) >> fs[] >> simp[] >>
strip_tac >> rveq >> simp[])
>- (print_tac "nPTbase" >>
CONV_TAC (LAND_CONV (SIMP_CONV (srw_ss() ++ DNF_ss) [EXISTS_PROD])) >>
strip_tac >> rveq >>
fs[cmlG_FDOM, cmlG_applied, peg_eval_nTyOp_wrongtok] >>
rveq >> fs[]
>- (first_x_assum (first_assum o mp_then Any mp_tac) >> simp[] >>
strip_tac >> rveq >> dsimp[])
>- (rename [‘isTyvarT tk’] >> Cases_on ‘tk’ >> fs[])
>- (‘NT_rank (mkNT nTyOp) < NT_rank (mkNT nPTbase)’ by simp[NT_rank_def]>>
first_x_assum (pop_assum o mp_then Any mp_tac) >>
disch_then (first_assum o mp_then (Pos hd) strip_assume_tac) >>
rveq >> simp[]))
>- (print_tac "nTbase" >>
CONV_TAC (LAND_CONV (SIMP_CONV (srw_ss() ++ DNF_ss) [EXISTS_PROD])) >>
strip_tac >> rveq >>
fs[cmlG_FDOM, cmlG_applied, peg_eval_nTyOp_wrongtok] >>
rveq >> fs[]
>- (lrresolve X (K true) mp_tac >> simp[] >> strip_tac >> rveq >>
dsimp[])
>- (lrresolve KEEP (free_in ``nTypeList2``) mp_tac >>
rpt kill_asm_guard >> strip_tac >> rveq >> dsimp[] >>
imp_res_tac length_no_greater >>
lrresolve X (free_in ``nTyOp``) mp_tac >> fs[] >> simp[] >>
strip_tac >> rveq >> dsimp[])
>- (asm_match `isTyvarT h` >> Cases_on`h` >> fs[])
>- (`NT_rank (mkNT nTyOp) < NT_rank (mkNT nTbase)` by simp[NT_rank_def] >>
first_x_assum (erule mp_tac) >> strip_tac >> rveq >> simp[])
>- (lrresolve KEEP (free_in ``nTypeList2``) mp_tac >>
rpt kill_asm_guard >> strip_tac >> rveq >> simp[] >>
imp_res_tac length_no_greater >> fs[] >>
lrresolve X (free_in ``nTyOp``) mp_tac >> simp[] >>
strip_tac >> rveq >> dsimp[]) >>
lrresolve KEEP (free_in ``nTypeList2``) mp_tac >>
rpt kill_asm_guard >> strip_tac >> rveq >> simp[] >>
imp_res_tac length_no_greater >> fs[] >>
lrresolve X (free_in ``nTyOp``) mp_tac >> simp[] >>
strip_tac >> rveq >> dsimp[])
>- (print_tac "nDType" >> strip_tac >> rveq >> simp[] >>
`NT_rank (mkNT nTbase) < NT_rank (mkNT nDType)`
by simp[NT_rank_def] >>
first_x_assum (erule mp_tac) >>
disch_then (qxchl [`base_pt`] strip_assume_tac) >> rveq >> simp[] >>
erule strip_assume_tac
(MATCH_MP not_peg0_LENGTH_decreases peg0_nTbase) >>
qpat_x_assum `peg_eval cmlPEG (II, rpt XX FF) YY` mp_tac >>
simp[peg_eval_rpt] >> disch_then (qxchl [`tyops`] strip_assume_tac) >>
rveq >> simp[] >>
asm_match `peg_eval_list cmlPEG (i1, nt (mkNT nTyOp) I) (i,tyops)`>>
pop_assum mp_tac >>
`∃i2. LENGTH i2 < LENGTH i0 ∧ i1 = i2` by simp[] >>
pop_assum SUBST1_TAC >> pop_assum mp_tac >>
`∃acc.
ptree_head acc = NN nDType ∧ valid_lptree cmlG acc ∧
real_fringe base_pt ++ MAP (TK ## I) i2 =
real_fringe acc ++ MAP (TK ## I) i2 ∧
mkNd (mkNT nDType) [base_pt] = acc`
by (simp[cmlG_FDOM, cmlG_applied]) >>
ntac 2 (pop_assum SUBST1_TAC) >> ntac 2 (pop_assum mp_tac) >>
map_every qid_spec_tac [`acc`, `i2`, `i`, `tyops`] >> Induct
>- (simp[Once peg_eval_cases] >>
simp[cmlG_FDOM, cmlG_applied]) >>
map_every qx_gen_tac [`tyop`, `i`, `i2`, `acc`] >>
simp[Once peg_eval_cases] >> ntac 3 strip_tac >>
disch_then (qxchl [`i3`] strip_assume_tac) >>
first_x_assum (erule mp_tac) >>
disch_then (qxchl [`tyop_pt2`] strip_assume_tac) >> rveq >>
simp[] >>
`LENGTH i3 < LENGTH i2`
by metis_tac[not_peg0_LENGTH_decreases, peg0_nTyOp] >>
`LENGTH i3 < LENGTH i0` by decide_tac >>
first_x_assum
(qspecl_then [`i`, `i3`, `mkNd (mkNT nDType) [acc; tyop_pt2]`]
mp_tac)>>
simp[cmlG_applied, cmlG_FDOM, DISJ_IMP_THM, FORALL_AND_THM])
>- (print_tac "nType" >> simp[peg_eval_choice] >>
`NT_rank (mkNT nPType) < NT_rank (mkNT nType)` by simp[NT_rank_def] >>
strip_tac >> rveq >> simp[]
>- (first_x_assum (erule strip_assume_tac) >> rveq >> simp[] >>
fs[MAP_EQ_CONS, MAP_EQ_APPEND] >> rveq >>
lrresolve X (free_in ``nType``) mp_tac >>
simp[] >> strip_tac >> rveq >> dsimp[cmlG_FDOM, cmlG_applied] >>
metis_tac[]) >>
first_x_assum (erule strip_assume_tac) >> rveq >>
dsimp[cmlG_applied, cmlG_FDOM])
>- (print_tac "nPbaseList1" >> strip_tac >> rveq >>
simp[cmlG_FDOM, cmlG_applied] >>
`NT_rank (mkNT nPbase) < NT_rank (mkNT nPbaseList1)`
by simp[NT_rank_def] >>
first_x_assum (erule strip_assume_tac) >> rveq >>
dsimp[MAP_EQ_CONS] >> csimp[] >>
fs[MAP_EQ_APPEND] >> disj2_tac >>
erule assume_tac (MATCH_MP not_peg0_LENGTH_decreases peg0_nPbase) >>
first_x_assum (erule strip_assume_tac) >> simp[] >> metis_tac[])
>- (print_tac "nFDecl" >> strip_tac >> rveq >> simp[] >>
`NT_rank (mkNT nV) < NT_rank (mkNT nFDecl)` by simp[NT_rank_def] >>
first_x_assum (erule strip_assume_tac) >> rveq >> simp[] >>
erule assume_tac
(MATCH_MP not_peg0_LENGTH_decreases peg0_nV |> GEN_ALL) >>
first_assum (erule strip_assume_tac) >> rveq >> dsimp[] >>
first_assum (assume_tac o MATCH_MP length_no_greater o
assert (free_in ``nPbaseList1`` o concl)) >> fs[] >>
first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o
assert (free_in ``nE``) o concl)) >>
simp[] >> strip_tac >> rveq >> simp[cmlG_FDOM, cmlG_applied])
>- (print_tac "nAndFDecls" >>
disch_then (match_mp_tac o MATCH_MP peg_linfix_correct_lemma) >>
dsimp[SUBSET_DEF, pegsym_to_sym_def, DISJ_IMP_THM, FORALL_AND_THM,
cmlG_applied, cmlG_FDOM, EXISTS_PROD] >>
first_x_assum match_mp_tac >>
simp[NT_rank_def])
>- (print_tac "nPE'" >> strip_tac >> rveq >> simp[] >>
`NT_rank (mkNT nPattern) < NT_rank (mkNT nPE')` by simp[NT_rank_def] >>
first_x_assum (erule strip_assume_tac) >> rveq >> simp[] >>
first_assum (assume_tac o MATCH_MP length_no_greater o
assert (free_in ``nPattern`` o concl)) >> fs[] >>
first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o
assert (free_in ``nE'``) o concl)) >>
simp[] >> strip_tac >> rveq >> dsimp[cmlG_FDOM, cmlG_applied])
>- (print_tac "nPE" >> strip_tac >> rveq >> simp[] >>
`NT_rank (mkNT nPattern) < NT_rank (mkNT nPE)` by simp[NT_rank_def] >>
first_x_assum (erule strip_assume_tac) >> rveq >> simp[] >>
first_assum (assume_tac o MATCH_MP length_no_greater o
assert (free_in ``nPattern`` o concl)) >> fs[] >>
first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o
assert (free_in ``nE``) o concl)) >>
simp[] >> strip_tac >> rveq >> dsimp[cmlG_FDOM, cmlG_applied])
>- (print_tac "nPEs" >>
`NT_rank (mkNT nPE') < NT_rank (mkNT nPEs) ∧
NT_rank (mkNT nPE) < NT_rank (mkNT nPEs)`
by simp[NT_rank_def] >>
strip_tac >> rveq >> simp[] >> first_x_assum (erule strip_assume_tac) >>
rveq >> simp[cmlG_applied, cmlG_FDOM] >>
first_assum (assume_tac o MATCH_MP length_no_greater o
assert (free_in ``nPE'`` o concl)) >> fs[] >>
first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o
assert (free_in ``nPEs``) o concl)) >>
simp[] >> strip_tac >> rveq >> dsimp[])
>- (print_tac "nE'" >> strip_tac >> rveq >> simp[] >> fs[]
>- ((* raise case *)
loseC ``NT_rank`` >>
first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o concl)) >>
simp[] >> strip_tac >> rveq >> dsimp[cmlG_FDOM, cmlG_applied])
>- ((* ElogicOR case *)
loseC ``LENGTH`` >>
first_x_assum (fn patth =>
first_assum (mp_tac o PART_MATCH (lhand o rand) patth o concl)) >>
simp[NT_rank_def] >> strip_tac >> rveq >>
dsimp[cmlG_FDOM, cmlG_applied])
>- ((* if-then-else case *) loseC ``NT_rank`` >>
rename [`IfT = FST hdtoken`] >> Cases_on `hdtoken` >> fs[] >> rveq >>
rename [`ThenT = FST hdtoken`] >> Cases_on `hdtoken` >> fs[] >>
rveq >>