# source:src/joint/linearise.ma@2211

Last change on this file since 2211 was 2200, checked in by tranquil, 9 years ago
• updated joint semantics: generation of linear and graph semantics
• opened two axioms in BitVectorZ
File size: 25.1 KB
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1include "joint/TranslateUtils_paolo.ma".
2include "utilities/hide.ma".
3
4definition graph_to_lin_statement :
5  ∀p : unserialized_params.∀globals.
6  joint_statement (mk_graph_params p) globals → joint_statement (mk_lin_params p) globals ≝
7  λp,globals,stmt.match stmt return λ_.joint_statement (mk_lin_params ?) ? with
8  [ sequential c _ ⇒ sequential … c it
9  | final c ⇒ final … c
10  ].
11
12lemma graph_to_lin_labels :
13  ∀p : unserialized_params.∀globals,s.
14  stmt_labels … (graph_to_lin_statement p globals s) =
15  stmt_explicit_labels … s.
16#p#globals * [//] * //
17qed.
18
19(* discard all elements passing test, return first element failing it *)
20(* and the rest of the list, if any. *)
21let rec chop A (test : A → bool) (l : list A) on l : option (A × (list A)) ≝
22  match l with
23  [ nil ⇒ None ?
24  | cons x l' ⇒ if test x then chop A test l' else return 〈x, l'〉
25  ].
26
27lemma chop_ok : ∀A,f,l.
28  match chop A f l with
29  [ Some pr ⇒
30    let x ≝ \fst pr in
31    let post ≝ \snd pr in
32    ∃pre.All ? (λx.bool_to_Prop (f x)) pre ∧
33    l = pre @ x :: post ∧ ¬bool_to_Prop (f x)
34  | None ⇒ All A (λx.bool_to_Prop (f x)) l
35  ].
36#A #f #l elim l
37[ %
38| #hd #tl #IH whd in match (chop ???);
39  elim (true_or_false_Prop (f hd)) #H >H normalize nodelta
40  [ lapply IH elim (chop ???) normalize nodelta
41    [ #G %{H G}
42    | #pr * #pre ** #H1 #H2 #H3 %{(hd :: pre)} %{H3} %
43      [ %{H H1}
44      | >H2 %
45      ]
46    ]
47  | %{[ ]} %{H} %%
48  ]
49]
50qed.
51
52unification hint 0 ≔ p, globals;
53lp ≟ lin_params_to_params p,
54sp ≟ stmt_pars lp,
55js ≟ joint_statement sp globals,
56lo ≟ labelled_obj LabelTag js
57(*----------------------------*)⊢
58list lo ≡ codeT lp globals.
59
60
61definition graph_visit_ret_type ≝ λp,globals.λg : codeT (mk_graph_params p) globals.
62  λentry : label.
63  (Σ〈visited'   : identifier_map LabelTag ℕ,
64   required'  : identifier_set LabelTag,
65   generated' : codeT (mk_lin_params p) globals〉.'hide (
66      And (And (And (And
67        (lookup ?? visited' entry = Some ? 0)
68        (required' ⊆ visited'))
69        (∃last.stmt_at … generated' 0 = Some ? (final … last)))
70        (code_forall_labels … (λl.bool_to_Prop (l∈required')) (rev … generated')))
71        (∀l,n.lookup ?? visited' l = Some ? n →
72          And (bool_to_Prop (code_has_label … (rev ? generated') l))
73            (∃s.And (And
74              (lookup … g l = Some ? s)
75              (nth_opt … n (rev … generated') = Some ? 〈Some ? l, graph_to_lin_statement … s〉))
76              (opt_All ?
77                (λnext.Or (lookup … visited' next = Some ? (S n))
78                  (nth_opt … (S n) (rev … generated') = Some ? 〈None ?, GOTO … next〉))
79                (stmt_implicit_label … s)))))).
80
81unification hint 0 ≔ tag ⊢ identifier_set tag ≡ identifier_map tag unit.
82
83include alias "common/Identifiers.ma".
84
85lemma lookup_safe_elim : ∀tag,A.∀P : A → Prop.∀m,i,prf.
86  (∀x.lookup tag A m i = Some ? x → P x) → P (lookup_safe tag A m i prf).
87#tag #A #P #m #i #prf #H @H @lookup_eq_safe qed.
88
89let rec graph_visit
90  (p : unserialized_params)
91  (globals: list ident)
92  (g : codeT (mk_graph_params p) globals)
93  (* = graph (joint_statement (mk_graph_params p) globals *)
94  (required: identifier_set LabelTag)
95  (visited: identifier_map LabelTag ℕ) (* the reversed index of labels in the new code *)
96  (generated: codeT (mk_lin_params p) globals)
97  (* ≝ list ((option label)×(joint_statement (mk_lin_params p) globals)) *)
98  (visiting: list label)
99  (gen_length : ℕ)
100  (n: nat)
101  (entry : label)
102  (g_prf : code_closed … g)
103  (required_prf1 : ∀i.i∈required → Or (In ? visiting i) (bool_to_Prop (i∈visited)))
104  (required_prf2 : code_forall_labels … (λl.bool_to_Prop (l ∈ required)) (rev … generated))
105  (generated_prf1 : ∀l,n.lookup … visited l = Some ? n → hide_Prop (
106    And (bool_to_Prop (code_has_label … (rev ? generated) l))
107    (∃s.And (And
108      (lookup … g l = Some ? s)
109      (nth_opt ? n (rev … generated) = Some ? 〈Some ? l, graph_to_lin_statement … s〉))
110      (opt_All ? (λnext.match lookup … visited next with
111                     [ Some n' ⇒ Or (n' = S n) (nth_opt ? (S n) (rev ? generated) = Some ? 〈None ?, GOTO … next〉)
112                     | None ⇒ And (nth_opt ? 0 visiting = Some ? next) (S n = gen_length) ]) (stmt_implicit_label … s)))))
113  (generated_prf2 : ∀l.lookup … visited l = None ? → does_not_occur … l (rev ? generated))
114  (generated_prf3 : (∃last.stmt_at … generated 0 = Some ? (final … last)) ∨
115     (¬All ? (λx.bool_to_Prop (x∈visited)) visiting))
116  (visiting_prf : All … (λl.bool_to_Prop (l∈g)) visiting)
117  (gen_length_prf : gen_length = length ? generated)
118  (entry_prf : Or (And (And (visiting = [entry]) (gen_length = 0)) (Not (bool_to_Prop (entry∈visited))))
119                  (lookup … visited entry = Some ? 0))
120  (n_prf : le (id_map_size … g) (plus n (id_map_size … visited)))
121  on n
122  : graph_visit_ret_type … g entry ≝
123  match chop ? (λx.x∈visited) visiting
124  return λx.? → graph_visit_ret_type … g entry with
125  [ None ⇒ λH.
126    «〈visited, required, generated〉, ?»
127  | Some pr ⇒ λH.
128    let vis_hd ≝ \fst pr in
129    let vis_tl ≝ \snd pr in
130    match n return λx.? → graph_visit_ret_type … g entry with
131    [ O ⇒ λn_prf'.⊥
132    | S n' ⇒ λn_prf'.
133      (* add the label to the visited ones *)
134      let visited' ≝ add … visited vis_hd gen_length in
135      (* take l's statement *)
136      let hd_vis_in_g ≝ (hide_prf ? ?) in
137      let statement ≝ lookup_safe ?? g vis_hd hd_vis_in_g in
138      (* translate it to its linear version *)
139      let translated_statement ≝ graph_to_lin_statement p globals statement in
140      (* add the translated statement to the code (which will be reversed later) *)
141      let generated' ≝ 〈Some … vis_hd, translated_statement〉 :: generated in
142      let required' ≝ union_set ??? (set_from_list … (stmt_explicit_labels … statement)) required in
143      (* put successors in the visiting worklist *)
144      let visiting' ≝ stmt_labels … statement @ vis_tl in
145      (* finally, check the implicit successor *)
146      (* if it has been already visited, we need to add a GOTO *)
148        match stmt_implicit_label … statement
149        with
150        [ Some next ⇒
151          if next ∈ visited' then
152            〈1, {(next)}, [〈None label, (GOTO … next : joint_statement ??)〉]〉
153          else 〈0, ∅, [ ]〉
154        | None ⇒ 〈0, ∅, [ ]〉
155        ] in
156      (* prepare a common utility to deal with add_req_gen *)
158        ∀P : (ℕ × (identifier_set LabelTag) × (codeT (mk_lin_params p) globals)) → Prop.
159        (opt_All ? (λnext.¬bool_to_Prop (next ∈ visited')) (stmt_implicit_label … statement) →
160         P 〈0,∅,[ ]〉) →
161        (∀next.stmt_implicit_label … statement = Some ? next →
162          next ∈ visited' →
163          P 〈1, {(next)}, [〈None ?, GOTO (mk_lin_params p) next〉]〉) →
164        P add_req_gen ≝ hide_prf ?? in
165      graph_visit ???
166        (union_set ??? (\snd (\fst add_req_gen)) required')
167        visited'
169        visiting'
170        (plus (\fst (\fst add_req_gen)) (S gen_length))
171        n' entry g_prf ?????????
172    ] n_prf
173  ] (chop_ok ? (λx.x∈visited) visiting).
174whd
175[ (* base case, visiting is all visited *)
176  %[%[%[%]]]
177  [ elim entry_prf
178    [ ** #eq_visiting #gen_length_O #entry_vis >eq_visiting in H; * >entry_vis *
179    | //
180    ]
181  | #l #l_req
182    elim (required_prf1 … l_req) #G
183    [ @(All_In … H G)
184    | assumption
185    ]
186  | cases generated_prf3 [//]
187    * #ABS @⊥ @ABS assumption
188  | assumption
189  | #l #n #H elim (generated_prf1 … H)
190    #H1 * #s ** #H2 #H3 #H4
191    % [assumption] %{s} %
192    [% assumption
193    | @(opt_All_mp … H4) -l #l
194      lapply (in_map_domain … visited l)
195      elim (true_or_false_Prop (l∈visited)) #l_vis >l_vis
196      normalize nodelta [ * #n' ] #EQlookup >EQlookup
197      normalize nodelta *
198      [ #EQn' % >EQn' %
199      | #H %2{H}
200      | #H' lapply (All_nth … H … H') >l_vis *
201      ]
202    ]
203  ]
204(* first, close goals where add_gen_req plays no role *)
205|13: (* vis_hd is in g *)
206  elim H #pre ** #_ #H2 #_
207  @(All_split … visiting_prf … H2)
208|2: (* n = 0 absrud *)
209  elim H #pre ** #_ #H2 #H3
210  @(absurd … n_prf')
211  @lt_to_not_le
212  lapply (add_size … visited vis_hd 0 (* dummy value *))
213  >H3 normalize nodelta
214  whd in match (lt ? ?);
215  whd in match (1 + ?);
217  [ @(All_split ? (λx.bool_to_Prop (x∈g)) ????? H2) @(All_mp … visiting_prf)
218    #a elim g #gm whd in ⊢ (?→?%); #H >(lookup_eq_safe … H) %
219  | #l #H
220    elim (generated_prf1 … (lookup_eq_safe … H)) #_ * #s ** #s_in_g #_ #_
221    whd in ⊢ (?%); >s_in_g %
222  ]
223|8:
224  elim H #pre ** #_ #H2 #_
225  @All_append
226  [ elim(g_prf … vis_hd statement ?) [2:@lookup_eq_safe] #G1 #G2
227    @(All_append … G1)
228    whd in G2; lapply G2 elim statement normalize nodelta #s [2: #_ %]
229    #l #G' %{G'} %
230  | >H2 in visiting_prf;
231    #H' lapply (All_append_r … H') -H' * #_ //
232  ]
233|10:
234  elim H #pre ** #_ #H2 #H3 -add_req_gen_prf
235  %2 elim entry_prf
236  [ ** >H2 cases pre
237    [2: #x' #pre' #ABS normalize in ABS; destruct(ABS)
238      cases pre' in e0; [2: #x'' #pre''] #ABS' normalize in ABS'; destruct(ABS')
239    |1: #EQ normalize in EQ; destruct(EQ)
240      #eq_gen_length #_
242    ]
243  | #lookup_entry cut (entry ≠ vis_hd)
244    [ % whd in match vis_hd; #entry_vis_hd <entry_vis_hd in H3;
245      whd in ⊢ (?(?%)→?); >lookup_entry * #ABS @ABS % ]
246    #entry_not_vis_hd >(lookup_add_miss ?????? entry_not_vis_hd) assumption
247  ]
248|11:
249  elim H #pre ** #_ #_ #H3
250  >commutative_plus
252  whd in match (1 + ?);
253  >commutative_plus
254  assumption
256  #P whd in match add_req_gen;
257  elim (stmt_implicit_label ???)
258  [ #H #_ @H % ]
259  #next normalize nodelta elim (true_or_false_Prop (next ∈ visited')) #next_vis
260  >next_vis normalize nodelta
261  [ #_ #H @H [% | assumption]
262  | #H #_ @H whd >next_vis % *
263  ]
264| elim H #pre ** #H1 #H2 #_
265  #i >mem_set_union
266  #H elim (orb_Prop_true … H) -H
267  [ @add_req_gen_prf [ #_ >mem_set_empty * ]
268    #next #_ #next_vis #H >(mem_set_singl_to_eq … H) %2 assumption
269  | >mem_set_union
270    #H elim (orb_Prop_true … H) -H
271    [ #i_expl %1 @Exists_append_l @Exists_append_r
272      @mem_list_as_set
273      @i_expl
274    | (* i was already required *)
275      #i_req
276      elim (required_prf1 … i_req)
277      [ >H2 #H elim (Exists_append … H) -H
278        [ (* i in preamble → i∈visited *)
280          lapply (All_In … H1 i_pre) #H @H
281        | *
282          [ (* i is vis_hd *)
284          | (* i in vis_tl → i∈visiting' *)
285            #i_post % @Exists_append_r assumption
286          ]
287        ]
288      | (* i in visited *)
289        #i_vis %2 >mem_set_add @orb_Prop_r assumption
290      ]
291    ]
292  ]
293|4,5,6: change with reverse in match rev;
294  >reverse_append whd in match (reverse ??); >rev_append_def
295  >associative_append
296  [ #pt #s
297    @(leb_elim (S pt) (|generated|)) #cmp
298    whd in match (stmt_at ????);
299    [ >nth_opt_append_l [2: >length_reverse assumption ]
300      change with (stmt_at ???? = ? → ?)
301      #EQ lapply(required_prf2 … EQ) @All_mp
302      #l #l_req >mem_set_union @orb_Prop_r
303      >mem_set_union @orb_Prop_r @l_req
304    | >nth_opt_append_r [2: >length_reverse @not_lt_to_le assumption ]
305      cases (pt - ?)
306      [ whd in match (nth_opt ???); whd in ⊢ (??%?→?);
307        #EQ destruct(EQ) whd
308        >graph_to_lin_labels in ⊢ (???%);
309        whd in match required';
310        generalize in match (stmt_explicit_labels … statement);
311        #l @list_as_set_All
312        #i >mem_set_union >mem_set_union
313        #i_l @orb_Prop_r @orb_Prop_l @i_l
315        [ #_ | #next #_ #next_vis *
316          [ whd in ⊢ (??%?→?);
317            #EQ' destruct(EQ') whd %{I} >mem_set_union
319        #n whd in ⊢ (??%?→?); #ABS destruct(ABS)
320      ]
321    ]
322  | elim H #pre ** #H1 #H2 #H3
323    #i whd in match visited';
324    @(eq_identifier_elim … i vis_hd)
325    [ #EQi >EQi -i #pos
326      >lookup_add_hit #EQpos (* too slow: destruct(EQpos) *)
327      cut (gen_length = pos)
328      [1,3,5: (* BUG: -graph_visit *) -visited destruct(EQpos) %]
329      -EQpos #EQpos <EQpos -pos
330      %
331      [ >lin_code_has_label
333        [ #_
334        | #next #_ #next_vis
335          change with (? @ ([?] @ [?])) in match (? @ [? ; ?]);
336          <associative_append >occurs_exactly_once_None
337        ]
338        >occurs_exactly_once_Some_eq >eq_identifier_refl
339        normalize nodelta
340        @generated_prf2
341        lapply (in_map_domain … visited vis_hd)
342        >H3 normalize nodelta //
343      | %{statement}
344        % [ % ]
345        [ @lookup_eq_safe
346        | <associative_append @nth_opt_append_hit_l
347          >nth_opt_append_r
348          >rev_length
349          <gen_length_prf
350          [<minus_n_n] %
352          [ lapply (refl … (stmt_implicit_label ?? statement))
353            cases (stmt_implicit_label ???) in ⊢ (???%→%);
354            [#_ #_ %]
355            #next #K change with (¬bool_to_Prop (?∈?)→?) #next_vis
356          | #next #K >K #next_vis
357          ] whd
359          @eq_identifier_elim
360          [1,3: #EQnext >EQnext * [#ABS elim(ABS I)]
362          |*: #NEQ >(lookup_add_miss … visited … NEQ)
363            change with (?∈?) in match (false∨?);
364            #next_vis lapply(in_map_domain … visited next) >next_vis
365            whd in ⊢ (% → ?); [2: * #s ]
366            #EQlookup >EQlookup
367          ] whd
368          [1,2: %2 <associative_append
369            >nth_opt_append_r >append_length >rev_length >commutative_plus
370            <gen_length_prf
371            [1,3: <minus_n_n ] %
372          | % [2: %] @nth_opt_append_hit_l whd in match (stmt_labels … statement);
373            >K %
374          ]
375        ]
376      ]
377    | #NEQ #n_i >(lookup_add_miss … visited … NEQ)
378      #Hlookup elim (generated_prf1 … Hlookup)
379      #G1 * #s ** #G2 #G3 #G4
380      %
381      [ >lin_code_has_label <associative_append
382        >occurs_exactly_once_append
383        @orb_Prop_l @andb_Prop
384        [ >occurs_exactly_once_Some_eq
385          >eq_identifier_false [2: % #ABS >ABS in NEQ; * #ABS' @ABS' % ]
386          normalize nodelta >lin_code_has_label in G1; #K @K
388          [ #_ % | #next #_ #_ % ]
389        ]
390      | %{s}
391        % [ % ]
392        [ assumption
393        | @nth_opt_append_hit_l assumption
394        | @(opt_All_mp … G4)
395          #x
396          @(eq_identifier_elim … x vis_hd) #Heq
397          [ >Heq
398            lapply (in_map_domain … visited vis_hd)
399            >H3 normalize nodelta in ⊢ (%→?);
400            #EQlookup >EQlookup * #nth_opt_visiting #gen_length_eq
403            lapply (in_map_domain … visited x)
404            elim (true_or_false_Prop (x∈visited)) #x_vis >x_vis
405            normalize nodelta in ⊢ (%→?); [ * #n' ]
406            #EQlookupx >EQlookupx whd in ⊢ (%→%); *
407            [ #G %1{G}
408            | #G %2 @nth_opt_append_hit_l
409              assumption
410            | #G elim(absurd ?? Heq)
411              (* BUG (but useless): -required -g -generated *)
412              >H2 in G; lapply H1 cases pre
413              [ #_
414              | #hd #tl * #hd_vis #_
415              ] normalize #EQ' destruct(EQ')
416              [ %
417              | >x_vis in hd_vis; *
418              ]
419            ]
420          ]
421        ]
422      ]
423    ]
424  | #i whd in match visited';
425    @(eq_identifier_elim … i vis_hd) #Heq
426    [ >Heq >lookup_add_hit #ABS destruct(ABS) ]
428    #i_n_vis
429    >does_not_occur_append @andb_Prop
430    [ @generated_prf2 assumption
431    | change with (bool_to_Prop (¬eq_identifier ??? ∧ ?))
432      >eq_identifier_false [2: % #ABS <ABS in Heq; * #ABS' @ABS' % ]
433      @add_req_gen_prf [ #_ | #next #_ #_ ] %
434    ]
435  ]
437  [ #K | #next #K #next_vis %1 %{(GOTO … next)} % ]
438  whd in match generated'; whd in match translated_statement;
439  lapply K whd in match visiting';
440  whd in match (stmt_labels ???); cases statement #last
441  [2: #_ %1 %{last} % ] #next whd in ⊢ (%→?); #next_vis
442  %2 % * >next_vis *
443| whd in match generated';
444  @add_req_gen_prf [ #_ | #next #_ #_ ] normalize >gen_length_prf %
445]
446qed.
447
448(* CSC: The branch compression (aka tunneling) optimization is not implemented
449   in Matita *)
450definition branch_compress
451  ≝ λp: graph_params.λglobals.λg:codeT p globals.
452    λentry : Σl.bool_to_Prop (code_has_label … g l).g.
453
454lemma branch_compress_closed : ∀p,globals,g,l.code_closed ?? g →
455  code_closed … (branch_compress p globals g l).
456#p#globals#g#l#H @H qed.
457
458lemma branch_compress_has_entry : ∀p,globals,g,l.
459  code_has_label … (branch_compress p globals g l) l.
460#p#globals#g*#l#l_prf @l_prf qed.
461
462definition filter_labels ≝ λtag,A.λtest : identifier tag → bool.λc : list (labelled_obj tag A).
463  map ??
464    (λs. let 〈l_opt,x〉 ≝ s in
465      〈! l ← l_opt ; if test l then return l else None ?, x〉) c.
466
467lemma does_not_occur_filter_labels :
468  ∀tag,A,test,id,c.
469    does_not_occur ?? id (filter_labels tag A test c) =
470      (does_not_occur ?? id c ∨ ¬ test id).
471#tag #A #test #id #c elim c
472[ //
473| ** [2: #lbl] #s #tl #IH
474  whd in match (filter_labels ????); normalize nodelta
475  whd in match (does_not_occur ????) in ⊢ (??%%);
476  [2: @IH]
477  normalize in match (! l ← ? ; ?); >IH
478  @(eq_identifier_elim ?? lbl id) #Heq [<Heq]
479  elim (test lbl) normalize nodelta
480  change with (eq_identifier ???) in match (instruction_matches_identifier ????);
481  [1,2: >eq_identifier_refl [2: >commutative_orb] normalize %
482  |*: >(eq_identifier_false ??? Heq) normalize nodelta %
483  ]
484]
485qed.
486
487lemma occurs_exactly_once_filter_labels :
488  ∀tag,A,test,id,c.
489    occurs_exactly_once ?? id (filter_labels tag A test c) =
490      (occurs_exactly_once ?? id c ∧ test id).
491#tag #A #test #id #c elim c
492[ //
493| ** [2: #lbl] #s #tl #IH
494  whd in match (filter_labels ????); normalize nodelta
495  whd in match (occurs_exactly_once ????) in ⊢ (??%%);
496  [2: @IH]
497  normalize in match (! l ← ? ; ?); >IH
498  >does_not_occur_filter_labels
499  @(eq_identifier_elim ?? lbl id) #Heq [<Heq]
500  elim (test lbl) normalize nodelta
501  change with (eq_identifier ???) in match (instruction_matches_identifier ????);
502  [1,2: >eq_identifier_refl >commutative_andb [ >(commutative_andb ? true) >commutative_orb | >(commutative_andb ? false)] normalize %
503  |*: >(eq_identifier_false ??? Heq) normalize nodelta %
504  ]
505]
506qed.
507
508lemma nth_opt_filter_labels : ∀tag,A,test,instrs,n.
509  nth_opt ? n (filter_labels tag A test instrs) =
510  ! 〈lopt, s〉 ← nth_opt ? n instrs ;
511  return 〈 ! lbl ← lopt; if test lbl then return lbl else None ?, s〉.
512#tag #A #test #instrs elim instrs
513[ * [2: #n'] %
514| * #lopt #s #tl #IH * [2: #n']
515  whd in match (filter_labels ????); normalize nodelta
516  whd in match (nth_opt ???) in ⊢ (??%%); [>IH] %
517]
518qed.
519
520lemma stmt_at_filter_labels : ∀p : lin_params.∀globals,test.∀c : codeT p globals.
521∀i.stmt_at p globals (filter_labels ?? test c) i = stmt_at p globals c i.
522#p#globals#test #c#i
523whd in ⊢ (??%%); >nth_opt_filter_labels
524elim (nth_opt ???); //
525qed.
526
527lemma option_bind_inverse : ∀A,B.∀m : option A.∀f : A → option B.∀r.
528  ! x ← m ; f x = return r →
529  ∃x.m = return x ∧ f x = return r.
530#A #B * normalize [2:#x] #f #r #EQ destruct
531%{x} %{EQ} %
532qed.
533
534lemma nth_opt_reverse_hit :
535  ∀A,l,n.n < |l| → nth_opt A n (reverse ? l) = nth_opt A (|l| - (S n)) l.
536#A #l elim l
537[ #n #ABS normalize in ABS; @⊥ -A /2 by absurd/
538| #hd #tl #IH #n #lim whd in match (reverse ??); >rev_append_def
539  @(leb_elim (S n) (|tl|)) #H
540  [ >nth_opt_append_l [2: >length_reverse @H ]
541    >(IH … H) >(minus_Sn_m … H) %
542  | >(le_to_le_to_eq … (le_S_S_to_le … lim) (not_lt_to_le … H))
543    >nth_opt_append_r >length_reverse [2: % ]
544    <minus_n_n <minus_n_n %
545  ]
546]
547qed.
548
549lemma nth_opt_reverse_hit_inv :
550  ∀A,l,n.n < |l| → nth_opt A (|l| - (S n)) (reverse ? l) = nth_opt A n l.
551#A #l #n #H <(reverse_reverse ? l) in ⊢ (???%); @sym_eq
552<length_reverse @nth_opt_reverse_hit >length_reverse @H
553qed.
554
555definition linearise_code:
556 ∀p : unserialized_params.∀globals.
557  ∀g : codeT (mk_graph_params p) globals.code_closed … g →
558  ∀entry : (Σl.bool_to_Prop (code_has_label … g l)).
559    (Σc : codeT (mk_lin_params p) globals.
560      code_closed … c ∧
561      ∃ sigma : identifier_map LabelTag ℕ.
562      lookup … sigma entry = Some ? 0 ∧
563      ∀l,n.lookup … sigma l = Some ? n →
564        ∃s. lookup … g l = Some ? s ∧
565          opt_Exists ?
566            (λls.let 〈lopt, ts〉 ≝ ls in
567              opt_All ? (eq ? l) lopt ∧
568              ts = graph_to_lin_statement … s ∧
569              opt_All ?
570                (λnext.Or (lookup … sigma next = Some ? (S n))
571                (nth_opt … (S n) c = Some ? 〈None ?, GOTO … next〉))
572                (stmt_implicit_label … s)) (nth_opt … n c))
573
574 λp,globals,g,g_prf,entry_sig.
575    let g ≝ branch_compress (mk_graph_params p) ? g entry_sig in
576    let g_prf ≝ branch_compress_closed … g entry_sig g_prf in
577    match graph_visit p globals g ∅ (empty_map …) [ ] [entry_sig] 0 (|g|)
578      (entry_sig) g_prf ?????????
579    with
580    [ mk_Sig triple H ⇒
581      let sigma ≝ \fst (\fst triple) in
582      let required ≝ \snd (\fst triple) in
583      let crev ≝ \snd triple in
584      let lbld_code ≝ rev ? crev in
585      mk_Sig ?? (filter_labels … (λl.l∈required) lbld_code) ? ].
586[ (* done later *)
587| #i >mem_set_empty *
588|3,4: #a #b whd in ⊢ (??%?→?); #ABS destruct(ABS)
589| #l #_ %
590| %2 % * >mem_set_empty *
591| % [2: %] @(branch_compress_has_entry … g entry_sig)
592| %
593| % % [% %] cases (pi1 … entry_sig) normalize #_ % //
594| >commutative_plus change with (? ≤ |g|) %
595]
596lapply (refl … triple)
597generalize in match triple in ⊢ (???%→%); **
598#visited #required #generated #EQtriple
599>EQtriple in H ⊢ %; normalize nodelta ****
600#entry_O #req_vis #last_fin #labels_in_req #sigma_prop
601% >EQtriple
602[ #i #s
603  >stmt_at_filter_labels #EQ
604  %
605  [ @stmt_forall_labels_explicit
606    @(All_mp … (labels_in_req … EQ))
607    #l #l_req >lin_code_has_label
608    >occurs_exactly_once_filter_labels >l_req
609    >commutative_andb whd in ⊢ (?%);
610    elim (sigma_prop ?? (lookup_eq_safe … (req_vis … l_req)))
611    >lin_code_has_label #H #_ @H
612  | lapply EQ cases s #s' [2: #_ % ]
613    * -EQ #EQ change with (bool_to_Prop (code_has_point ????))
614    whd in match (point_of_succ ???);
615    >lin_code_has_point @leb_elim [ #_ % ] >length_map >length_reverse
616    #INEQ
617    cut (|generated| = S i)
618    [ @(le_to_le_to_eq … (not_lt_to_le … INEQ) )
619      elim (option_bind_inverse ????? EQ) #x * #EQ1 #EQ2
620      <length_reverse
621      @(nth_opt_hit_length … EQ1)
622    ] #EQ_length
623    elim last_fin #fin #EQ'
624    lapply EQ whd in match (stmt_at ????);
625    >nth_opt_reverse_hit >EQ_length [2: % ] <minus_n_n
626    change with (stmt_at ???? = ? → ?)
627    >EQ' #ABS destruct(ABS)
628  ]
629]
630%{visited} % [assumption]
631#lbl #n #eq_lookup elim (sigma_prop ?? eq_lookup)
632#lbl_in_gen * #stmt ** #stmt_in_g #nth_opt_is_stmt #succ_is_in
633% [2: % [ assumption ] |]
634>nth_opt_filter_labels in ⊢ (???%);
635>nth_opt_is_stmt >m_return_bind whd >m_return_bind
636% [ % ]
637[ elim (lbl ∈ required) %
638| %
639| lapply succ_is_in
640  lapply (refl … (stmt_implicit_label … stmt))
641  cases (stmt_implicit_label … stmt) in ⊢ (???%→%); [#_ #_ %]
642  #next #EQ_next *
643  [ #H %1{H} ]
644  #H %2
645  >nth_opt_filter_labels >H %
646]
647qed.
648
649(* Paolo: for now I'm leaving the closed graph stuff out, waiting for it to
650   be done on all passes *)
651definition linearise_int_fun :
652  ∀p : unserialized_params.
653  ∀globals.
654    joint_internal_function globals (mk_graph_params p) →
655    joint_internal_function globals (mk_lin_params p)
656     (* ∃sigma : identifier_map LabelTag ℕ.
657        let g ≝ joint_if_code ?? (pi1 … fin) in
658        let c ≝ joint_if_code ?? (pi1 … fout) in
659        let entry ≝ joint_if_entry ?? (pi1 … fin) in
660         lookup … sigma entry = Some ? 0 ∧
661          ∀l,n.lookup … sigma l = Some ? n →
662            ∃s. lookup … g l = Some ? s ∧
663              opt_Exists ?
664                (λls.let 〈lopt, ts〉 ≝ ls in
665                  opt_All ? (eq ? l) lopt ∧
666                  ts = graph_to_lin_statement … s ∧
667                  opt_All ?
668                    (λnext.Or (lookup … sigma next = Some ? (S n))
669                    (nth_opt … (S n) c = Some ? 〈None ?, GOTO … next〉))
670                    (stmt_implicit_label … s)) (nth_opt … n c)*) ≝
671  λp,globals,f_in.
672  mk_joint_internal_function globals (mk_lin_params p)
673   (joint_if_luniverse ?? f_in) (joint_if_runiverse ?? f_in)
674   (joint_if_result ?? f_in) (joint_if_params ?? f_in) (joint_if_locals ?? f_in)
675   (joint_if_stacksize ?? f_in)
676   (linearise_code p globals (joint_if_code … f_in)
677   (match daemon in False with [ ])
678   (joint_if_entry … f_in))
679   0 0 (* exit is dummy! *).
680elim (linearise_code ?????) #c * #code_closed
681@hide_prf
682* #sigma * #O_in #sigma_prop
683>lin_code_has_point
684elim (sigma_prop … O_in) #s * #_
685cases c
686[1,3: *
687|*: #hd #tl #_ %
688]
689qed.
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