2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
8 \ / This file is distributed under the terms of the
9 \ / GNU General Public License Version 2
10 V_____________________________________________________________*)
17 include "turing/universal/marks.ma".
19 definition STape ≝ FinProd … FSUnialpha FinBool.
21 definition only_bits ≝ λl.
22 ∀c.memb STape c l = true → is_bit (\fst c) = true.
24 definition only_bits_or_nulls ≝ λl.
25 ∀c.memb STape c l = true → bit_or_null (\fst c) = true.
27 definition no_grids ≝ λl.
28 ∀c.memb STape c l = true → is_grid (\fst c) = false.
30 definition no_bars ≝ λl.
31 ∀c.memb STape c l = true → is_bar (\fst c) = false.
33 definition no_marks ≝ λl.
34 ∀c.memb STape c l = true → is_marked ? c = false.
36 lemma bit_not_grid: ∀d. is_bit d = true → is_grid d = false.
37 * // normalize #H destruct
40 lemma bit_or_null_not_grid: ∀d. bit_or_null d = true → is_grid d = false.
41 * // normalize #H destruct
44 lemma bit_not_bar: ∀d. is_bit d = true → is_bar d = false.
45 * // normalize #H destruct
48 lemma bit_or_null_not_bar: ∀d. bit_or_null d = true → is_bar d = false.
49 * // normalize #H destruct
52 definition mk_tuple ≝ λqin,cin,qout,cout,mv.
53 〈bar,false〉 :: qin @ cin :: 〈comma,false〉:: qout @ cout :: 〈comma,false〉 :: [mv].
55 (* by definition, a tuple is not marked *)
56 definition tuple_TM : nat → list STape → Prop ≝
57 λn,t.∃qin,cin,qout,cout,mv.
58 no_marks qin ∧ no_marks qout ∧
59 only_bits qin ∧ only_bits qout ∧
60 bit_or_null cin = true ∧ bit_or_null cout = true ∧ bit_or_null mv = true ∧
61 (cout = null → mv = null) ∧
62 |qin| = n ∧ |qout| = n ∧
63 t = mk_tuple qin 〈cin,false〉 qout 〈cout,false〉 〈mv,false〉.
65 inductive table_TM (n:nat) : list STape → Prop ≝
66 | ttm_nil : table_TM n []
67 | ttm_cons : ∀t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@T).
69 inductive match_in_table (n:nat) (qin:list STape) (cin: STape)
70 (qout:list STape) (cout:STape) (mv:STape)
74 tuple_TM n (mk_tuple qin cin qout cout mv) →
75 match_in_table n qin cin qout cout mv
76 (mk_tuple qin cin qout cout mv @tb)
78 ∀qin0,cin0,qout0,cout0,mv0,tb.
79 tuple_TM n (mk_tuple qin0 cin0 qout0 cout0 mv0) →
80 match_in_table n qin cin qout cout mv tb →
81 match_in_table n qin cin qout cout mv
82 (mk_tuple qin0 cin0 qout0 cout0 mv0@tb).
84 axiom append_l1_injective :
85 ∀A.∀l1,l2,l3,l4:list A. |l1| = |l2| → l1@l3 = l2@l4 → l1 = l2.
86 axiom append_l2_injective :
87 ∀A.∀l1,l2,l3,l4:list A. |l1| = |l2| → l1@l3 = l2@l4 → l3 = l4.
88 axiom cons_injective_l : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → a1 = a2.
89 axiom cons_injective_r : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → l1 = l2.
90 axiom tuple_len : ∀n,t.tuple_TM n t → |t| = 2*n+6.
91 axiom append_eq_tech1 :
92 ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → ∃la:list A.l1@a::la = l3.
93 axiom append_eq_tech2 :
94 ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false → ∃la:list A.l3 = l1@a::la.
95 (*axiom list_decompose_cases :
96 ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → ∃la,lb:list A.l3 = la@a::lb ∨ l4 = la@a::lb.
97 axiom list_decompose_l :
98 ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false →
99 ∃la,lb.l2 = la@lb ∧ l3 = l1@a::la.
100 axiom list_decompose_r :
101 ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l3 = false →
102 ∃la,lb.l1 = la@lb ∧ l4 = lb@a::l2.
103 axiom list_decompose_memb :
104 ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → memb A a l3 = true.*)
106 lemma table_invert_r : ∀n,t,T.
107 tuple_TM n t → table_TM n (t@T) → table_TM n T.
108 #n #t #T #Htuple #Htable inversion Htable
109 [ cases Htuple #qin * #cin * #qout * #cout * #mv * #_ #Ht >Ht
110 normalize #Hfalse destruct (Hfalse)
111 | #t0 #T0 #Htuple0 #Htable0 #_ #Heq
112 lapply (append_l2_injective ?????? Heq)
113 [ >(tuple_len … Htuple) >(tuple_len … Htuple0) % ]
114 -Heq #Heq destruct (Heq) // ]
117 lemma match_in_table_to_tuple :
118 ∀n,T,qin,cin,qout,cout,mv.
119 match_in_table n qin cin qout cout mv T → table_TM n T →
120 tuple_TM n (mk_tuple qin cin qout cout mv).
121 #n #T #qin #cin #qout #cout #mv #Hmatch elim Hmatch
123 | #qin0 #cin0 #qout0 #cout0 #mv0 #tb #Htuple #Hmatch #IH #Htable
124 @IH @(table_invert_r ???? Htable) @Htuple
128 axiom generic_match_to_match_in_table :
130 ∀qin,cin,qout,cout,mv.|qin| = n → |qout| = n →
131 only_bits qin → only_bits qout →
132 bit_or_null (\fst cin) = true → bit_or_null (\fst cout) = true →
133 bit_or_null (\fst mv) = true →
135 T = (t1@〈bar,false〉::qin@cin::〈comma,false〉::qout@cout::〈comma,false〉::[mv])@t2 →
136 match_in_table n qin cin qout cout mv T.
137 (*#n #T #Htable #qin #cin #qout #cout #mv #Hlenqin #Hlenqout
138 #Hqinbits #Hqoutbits #Hcin #Hcout #Hmv
140 [ * [ #t2 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
141 | #c0 #t1 #t2 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
142 | #tuple #T0 #H1 #Htable0#IH #t1 #t2 #HT cases H1 #qin0 * #cin0 * #qout0 * #cout0 * #mv0
144 #Hqin0marks #Hqout0marks #Hqin0bits #Hqout0bits #Hcin0 #Hcout0 #Hmv0 #Hcout0mv0
145 #Hlenqin0 #Hlenqout0 #Htuple >Htuple in H1; #H1
146 lapply (ttm_cons … T0 H1 Htable0) #Htable
148 [ >Htuple normalize in ⊢ (??%%→?);
149 >associative_append >associative_append #HT
150 cut (qin0 = qin ∧ (〈cin0,false〉 = cin ∧ (qout0 = qout ∧
151 (〈cout0,false〉 = cout ∧ (〈mv0,false〉 = mv ∧ 〈bar,false〉::T0 = t2)))))
152 [ lapply (append_l1_injective … HT) [ >Hlenqin @Hlenqin0 ]
153 #Hqin % [ @Hqin ] -Hqin
154 lapply (append_l2_injective … HT) [ >Hlenqin @Hlenqin0 ] -HT #HT
155 lapply (cons_injective_l ????? HT) #Hcin % [ @Hcin ] -Hcin
156 lapply (cons_injective_r ????? HT) -HT #HT
157 lapply (cons_injective_r ????? HT) -HT
158 >associative_append >associative_append #HT
159 lapply (append_l1_injective … HT) [ >Hlenqout @Hlenqout0 ]
160 #Hqout % [ @Hqout ] -Hqout
161 lapply (append_l2_injective … HT) [ >Hlenqout @Hlenqout0 ] -HT normalize #HT
162 lapply (cons_injective_l ????? HT) #Hcout % [ @Hcout ] -Hcout
163 lapply (cons_injective_r ????? HT) -HT #HT
164 lapply (cons_injective_r ????? HT) -HT #HT
165 lapply (cons_injective_l ????? HT) #Hmv % [ @Hmv ] -Hmv
166 @(cons_injective_r ????? HT) ]
167 -HT * #Hqin * #Hcin * #Hqout * #Hcout * #Hmv #HT0
168 >(?:qin0@(〈cin0,false〉::〈comma,false〉::qout0@[〈cout0,false〉;〈comma,false〉;〈mv0,false〉])@〈bar,false〉::T0
169 = mk_tuple qin cin qout cout mv@〈bar,false〉::T0)
170 [|>Hqin >Hqout >Hcin >Hcout >Hmv normalize >associative_append >associative_append
171 normalize >associative_append % ]
172 % %{qin0} %{cin0} %{qout0} %{cout0} %{mv0} % // % [|@Hlenqout0] % // %
173 [ | @Hcout0mv0 ] % // % // % // % // % // % // %
174 | #c0 #cs0 #HT cut (∃cs1.c0::cs0 = tuple@〈bar,false〉::cs1)
175 [ cases (append_eq_tech1 ?????? HT ?)
176 [ -HT #ta #Hta cases (append_eq_tech2 … Hta ?)
177 [ -Hta #tb #Htb %{tb} @Htb
179 | @le_S_S >length_append >(plus_n_O (|tuple|)) >commutative_plus @le_plus
181 | >Htuple normalize >length_append >length_append @le_plus [ >Hlenqin >Hlenqin0 % ]
182 @le_S_S @le_S_S >length_append >length_append @le_plus [ >Hlenqout >Hlenqout0 % ] %] ]
184 * #cs1 #Hcs1 >Hcs1 in HT; >associative_append >associative_append #HT
185 lapply (append_l2_injective … HT) // -HT #HT
186 lapply (cons_injective_r ????? HT) -HT
187 <associative_append #HT >Htuple %2 // @(IH … HT)
193 lemma table_invert_l : ∀n,T0,qin,cin,qout,cout,mv.
194 table_TM n (mk_tuple qin cin qout cout mv@〈bar,false〉::T0) →
195 tuple_TM n (mk_tuple qin cin qout cout mv).
196 #n #T #qin #cin #qout #cout #mv #HT inversion HT
197 [ change with (append ???) in ⊢ (??(??%?)?→?);cases qin [ #Hfalse | #t0 #ts0 #Hfalse] normalize in Hfalse; destruct (Hfalse)
198 | #t0 #T0 #Ht0 #HT0 #_
201 lemma table_invert_r : ∀n,T0,qin,cin,qout,cout,mv.
202 table n (mk_tuple qin cin qout cout mv@〈bar,false〉::T0) → table n T0.
205 lemma no_grids_in_tuple : ∀n,l.tuple_TM n l → no_grids l.
206 #n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * *
207 #_ #_ #Hqin #Hqout #Hcin #Hcout #Hmv #_ #_ #_ #Hl >Hl
208 #c #Hc cases (orb_true_l … Hc) -Hc #Hc
210 | cases (memb_append … Hc) -Hc #Hc
211 [ @bit_not_grid @(Hqin … Hc)
212 | cases (orb_true_l … Hc) -Hc #Hc
213 [ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid //
214 | cases (orb_true_l … Hc) -Hc #Hc
215 [ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
216 | cases (memb_append …Hc) -Hc #Hc
217 [ @bit_not_grid @(Hqout … Hc)
218 | cases (orb_true_l … Hc) -Hc #Hc
219 [ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) @bit_or_null_not_grid //
220 | cases (orb_true_l … Hc) -Hc #Hc
221 [ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
222 | >(memb_single … Hc) @bit_or_null_not_grid @Hmv
226 lemma no_marks_in_tuple : ∀n,l.tuple_TM n l → no_marks l.
227 #n #l * #qin * #cin * #qout * #cout * #mv * * * * * * * * * *
228 #Hqin #Hqout #_ #_ #_ #_ #_ #_ #_ #_ #Hl >Hl
229 #c #Hc cases (orb_true_l … Hc) -Hc #Hc
231 | cases (memb_append … Hc) -Hc #Hc
233 | cases (orb_true_l … Hc) -Hc #Hc
234 [ change with (c == 〈cin,false〉 = true) in Hc; >(\P Hc) %
235 | cases (orb_true_l … Hc) -Hc #Hc
236 [ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
237 | cases (memb_append … Hc) -Hc #Hc
239 | cases (orb_true_l … Hc) -Hc #Hc
240 [ change with (c == 〈cout,false〉 = true) in Hc; >(\P Hc) %
241 | cases (orb_true_l … Hc) -Hc #Hc
242 [ change with (c == 〈comma,false〉 = true) in Hc; >(\P Hc) %
243 | >(memb_single … Hc) %
247 lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
249 [normalize #c #H destruct
250 |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_grids_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
251 cases (memb_append … Hx) -Hx #Hx
256 lemma no_marks_in_table: ∀n.∀l.table_TM n l → no_marks l.
258 [normalize #c #H destruct
259 |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_marks_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
260 cases (memb_append … Hx) -Hx #Hx
265 axiom last_of_table: ∀n,l,b.¬ table_TM n (l@[〈bar,b〉]).
268 l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2
271 if current (* x *) = #
274 then move_right; ----
276 if current (* x0 *) = 0
277 then advance_mark ----
281 else x = 1 (* analogo *)
287 MARK NEXT TUPLE machine
288 (partially axiomatized)
290 marks the first character after the first bar (rightwards)
293 definition bar_or_grid ≝ λc:STape.is_bar (\fst c) ∨ is_grid (\fst c).
295 definition mark_next_tuple ≝
296 seq ? (adv_to_mark_r ? bar_or_grid)
297 (ifTM ? (test_char ? (λc:STape.is_bar (\fst c)))
298 (move_right_and_mark ?) (nop ?) tc_true).
300 definition R_mark_next_tuple ≝
303 (* c non può essere un separatore ... speriamo *)
304 t1 = midtape STape ls c (rs1@〈grid,false〉::rs2) →
305 no_marks rs1 → no_grids rs1 → bar_or_grid c = false →
306 (∃rs3,rs4,d,b.rs1 = rs3 @ 〈bar,false〉 :: rs4 ∧
308 Some ? 〈d,b〉 = option_hd ? (rs4@〈grid,false〉::rs2) ∧
309 t2 = midtape STape (〈bar,false〉::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@〈grid,false〉::rs2)))
311 (no_bars rs1 ∧ t2 = midtape ? (reverse ? rs1@c::ls) 〈grid,false〉 rs2).
315 (∀x.memb A x l = true → f x = false) ∨
316 (∃l1,c,l2.f c = true ∧ l = l1@c::l2 ∧ ∀x.memb ? x l1 = true → f x = false).
318 [ % #x normalize #Hfalse *)
320 theorem sem_mark_next_tuple :
321 Realize ? mark_next_tuple R_mark_next_tuple.
323 lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid)
324 (ifTM ? (test_char ? (λc:STape.is_bar (\fst c))) (move_right_and_mark ?) (nop ?) tc_true) ????)
325 [@sem_if [5: // |6: @sem_move_right_and_mark |7: // |*:skip]
327 |||#Hif cases (Hif intape) -Hif
328 #j * #outc * #Hloop * #ta * #Hleft #Hright
329 @(ex_intro ?? j) @ex_intro [|% [@Hloop] ]
331 #ls #c #rs1 #rs2 #Hrs #Hrs1 #Hrs1' #Hc
333 [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
334 | * #_ #Hta cases (tech_split STape (λc.is_bar (\fst c)) rs1)
335 [ #H1 lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?)
336 [ * #x #b #Hx whd in ⊢ (??%?); >(Hrs1' … Hx) >(H1 … Hx) %
338 | -Hta #Hta cases Hright
339 [ * #tb * whd in ⊢ (%→?); #Hcurrent
340 @False_ind cases (Hcurrent 〈grid,false〉 ?)
341 [ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
343 | * #tb * whd in ⊢ (%→?); #Hcurrent
344 cases (Hcurrent 〈grid,false〉 ?)
345 [ #_ #Htb whd in ⊢ (%→?); #Houtc
348 | >Houtc >Htb >Hta % ]
352 | * #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
353 % @(ex_intro ?? rs3) @(ex_intro ?? rs4)
354 lapply (Hta rs3 c0 (rs4@〈grid,false〉::rs2) ???)
355 [ #x #Hrs3' whd in ⊢ (??%?); >Hsplit in Hrs1;>Hsplit in Hrs3;
356 #Hrs3 #Hrs1 >(Hrs1 …) [| @memb_append_l1 @Hrs3'|]
357 >(Hrs3 … Hrs3') @Hrs1' >Hsplit @memb_append_l1 //
358 | whd in ⊢ (??%?); >Hc0 %
359 | >Hsplit >associative_append % ] -Hta #Hta
361 [ * #tb * whd in ⊢ (%→?); #Hta'
364 [ #_ #Htb' >Htb' in Htb; #Htb
365 generalize in match Hsplit; -Hsplit
367 [ #Hta #Hsplit >(Htb … Hta)
368 >(?:c0 = 〈bar,false〉)
369 [ @(ex_intro ?? grid) @(ex_intro ?? false)
371 [(* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ]
372 | (* Hc0 *) @daemon ]
373 | #r5 #rs5 >(eq_pair_fst_snd … r5)
374 #Hta #Hsplit >(Htb … Hta)
375 >(?:c0 = 〈bar,false〉)
376 [ @(ex_intro ?? (\fst r5)) @(ex_intro ?? (\snd r5))
377 % [ % [ % [ (* Hc0, Hsplit *) @daemon | (*Hrs3*) @daemon ] | % ]
378 | % ] | (* Hc0 *) @daemon ] ] | >Hta % ]
379 | * #tb * whd in ⊢ (%→?); #Hta'
382 [ #Hfalse @False_ind >Hfalse in Hc0;
388 definition init_current_on_match ≝
390 (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
391 (seq ? (move_r ?) (mark ?)))).
393 definition R_init_current_on_match ≝ λt1,t2.
394 ∀l1,l2,c,rs. no_grids l1 → is_grid c = false →
395 t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::l2) 〈grid,false〉 rs →
396 t2 = midtape STape (〈grid,false〉::l2) 〈c,true〉 ((reverse ? l1)@〈grid,false〉::rs).
398 lemma sem_init_current_on_match :
399 Realize ? init_current_on_match R_init_current_on_match.
401 cases (sem_seq ????? (sem_move_l ?)
402 (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
403 (sem_seq ????? (sem_move_r ?) (sem_mark ?))) intape)
404 #k * #outc * #Hloop #HR
405 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
406 #l1 #l2 #c #rs #Hl1 #Hc #Hintape
407 cases HR -HR #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape
408 generalize in match Hl1; cases l1
409 [#Hl1 whd in ⊢ ((???(??%%%))→?); #Hta
410 * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Hta
411 [* >Hc #Htemp destruct (Htemp) ]
412 * #_ #Htc lapply (Htc [ ] 〈grid,false〉 ? (refl ??) (refl …) Hl1)
413 whd in ⊢ ((???(??%%%))→?); -Htc #Htc
414 * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htc -Htd
415 whd in ⊢ ((???(??%%%))→?); #Htd
416 whd in ⊢ (%→?); #Houtc lapply (Houtc … Htd) -Houtc #Houtc
418 |#d #tl #Htl whd in ⊢ ((???(??%%%))→?); #Hta
419 * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb
420 [* >(Htl … (memb_hd …)) #Htemp destruct (Htemp)]
421 * #Hd >append_cons #Htb lapply (Htb … (refl ??) (refl …) ?)
422 [#x #membx cases (memb_append … membx) -membx #membx
423 [@Htl @memb_cons @membx | >(memb_single … membx) @Hc]]-Htb #Htb
424 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htb -Htc
425 >reverse_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
426 whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc #Houtc
427 >Houtc >reverse_cons >associative_append %
432 definition init_current_gen ≝
433 seq ? (adv_to_mark_l ? (is_marked ?))
434 (seq ? (clear_mark ?)
436 (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
437 (seq ? (move_r ?) (mark ?))))).
439 definition R_init_current_gen ≝ λt1,t2.
440 ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 →
441 Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
442 t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
443 t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
444 ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
446 lemma sem_init_current_gen : Realize ? init_current_gen R_init_current_gen.
448 cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
449 (sem_seq ????? (sem_clear_mark ?)
450 (sem_seq ????? (sem_move_l ?)
451 (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
452 (sem_seq ????? (sem_move_r ?) (sem_mark ?))))) intape)
453 #k * #outc * #Hloop #HR
454 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
455 #l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hintape
456 cases HR -HR #ta * whd in ⊢ (%→?); #Hta cases (Hta … Hintape) -Hta -Hintape
457 [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
458 * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%] -Hta #Hta
459 * #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta #Htb
460 * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb
461 generalize in match Hc; generalize in match Hl2; cases l2
462 [#_ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
463 whd in ⊢ ((???(??%%%))→?); #Htc
464 * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
465 [2: * whd in ⊢ (??%?→?); #Htemp destruct (Htemp) ]
466 * #_ #Htd >Htd in Htc; -Htd #Htd
467 * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
468 >reverse_append >reverse_cons
469 whd in ⊢ ((???(??%%%))→?); #Hte
470 whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
472 |#d #tl #Htl #Hc0 whd in ⊢ ((???(??%%%))→?); #Htc
473 * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
474 [* >(Htl … (memb_hd …)) whd in ⊢ (??%?→?); #Htemp destruct (Htemp)]
475 * #Hd #Htd lapply (Htd … (refl ??) (refl ??) ?)
476 [#x #membx @Htl @memb_cons @membx] -Htd #Htd
477 * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
478 >reverse_append >reverse_cons >reverse_cons
479 >reverse_cons in Hc0; >reverse_cons cases (reverse ? tl)
480 [normalize in ⊢ (%→?); #Hc0 destruct (Hc0) #Hte
481 whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
483 |* #c2 #b2 #tl2 normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
484 whd in ⊢ ((???(??%%%))→?); #Hte
485 whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
486 >Houtc >associative_append >associative_append >associative_append %
492 definition init_current ≝
493 seq ? (adv_to_mark_l ? (is_marked ?))
494 (seq ? (clear_mark ?)
495 (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
496 (seq ? (move_r ?) (mark ?)))).
498 definition R_init_current ≝ λt1,t2.
499 ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 → is_grid c = false →
500 Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
501 t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
502 t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
503 ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
505 lemma sem_init_current : Realize ? init_current R_init_current.
507 cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
508 (sem_seq ????? (sem_clear_mark ?)
509 (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
510 (sem_seq ????? (sem_move_r ?) (sem_mark ?)))) intape)
511 #k * #outc * #Hloop #HR
512 @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop]
513 cases HR -HR #ta * whd in ⊢ (%→?); #Hta
514 * #tb * whd in ⊢ (%→?); #Htb
515 * #tc * whd in ⊢ (%→?); #Htc
516 * #td * whd in ⊢ (%→%→?); #Htd #Houtc
517 #l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hc0 #Hintape
518 cases (Hta … Hintape) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
519 -Hta * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%]
520 -Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) [ >Hc -Hc * #Hc destruct (Hc) ]
521 -Htc * #_ #Htc lapply (Htc … (refl ??) (refl ??) ?) [@Hl2]
522 -Htc #Htc lapply (Htd … Htc) -Htd
523 >reverse_append >reverse_cons
524 >reverse_cons in Hc0; cases (reverse … l2)
525 [ normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
526 #Htd >(Houtc … Htd) %
527 | * #c2 #b2 #tl2 normalize in ⊢ (%→?);
528 #Hc0 #Htd >(Houtc … Htd)
529 whd in ⊢ (???%); destruct (Hc0)
530 >associative_append >associative_append %
534 definition match_tuple_step ≝
535 ifTM ? (test_char ? (λc:STape.¬ is_grid (\fst c)))
538 (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
540 (seq ? mark_next_tuple
541 (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
542 (mark ?) (seq ? (move_l ?) init_current) tc_true)) tc_true)))
545 definition R_match_tuple_step_true ≝ λt1,t2.
546 ∀ls,cur,rs.t1 = midtape STape ls cur rs →
548 (∀ls0,c,l1,l2,c1,l3,l4,rs0,n.
549 only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) →
550 bit_or_null c = true → bit_or_null c1 = true →
551 only_bits_or_nulls l3 → S n = |l1| → |l1| = |l3| →
552 table_TM (S n) (l2@〈c1,false〉::l3@〈comma,false〉::l4) →
553 ls = 〈grid,false〉::ls0 → cur = 〈c,true〉 →
554 rs = l1@〈grid,false〉::l2@〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs0 →
556 (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
557 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
558 (l2@〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs0))
560 (* non facciamo match e marchiamo la prossima tupla *)
561 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
562 ∃c2,l5,l6.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
563 (* condizioni su l5 l6 l7 *)
564 t2 = midtape STape (〈grid,false〉::ls0) 〈c,true〉
565 (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::
566 l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs0))
568 (* non facciamo match e non c'è una prossima tupla:
569 non specifichiamo condizioni sul nastro di output, perché
570 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
571 (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
573 definition R_match_tuple_step_false ≝ λt1,t2.
574 ∀ls,c,rs.t1 = midtape STape ls c rs → is_grid (\fst c) = true ∧ t2 = t1.
576 include alias "basics/logic.ma".
579 lemma eq_f4: ∀A1,A2,A3,A4,B.∀f:A1 → A2 →A3 →A4 →B.
580 ∀x1,x2,x3,x4,y1,y2,y3,y4. x1 = y1 → x2 = y2 →x3=y3 →x4 = y4 →
581 f x1 x2 x3 x4 = f y1 y2 y3 y4.
585 lemma some_option_hd: ∀A.∀l:list A.∀a.∃b.
586 Some ? b = option_hd ? (l@[a]) .
587 #A #l #a cases l normalize /2/
590 axiom tech_split2 : ∀A,l1,l2,l3,l4,x.
591 memb A x l1 = false → memb ? x l3 = false →
592 l1@x::l2 = l3@x::l4 → l1 = l3 ∧ l2 = l4.
594 axiom injective_append : ∀A,l.injective … (λx.append A x l).
596 lemma sem_match_tuple_step:
597 accRealize ? match_tuple_step (inr … (inl … (inr … start_nop)))
598 R_match_tuple_step_true R_match_tuple_step_false.
599 @(acc_sem_if_app … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) …
600 (sem_seq … sem_compare
601 (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
603 (sem_seq … sem_mark_next_tuple
604 (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
605 (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
607 [(* is_grid: termination case *)
608 2:#t1 #t2 #t3 whd in ⊢ (%→?); #H #H1 whd #ls #c #rs #Ht1
609 cases (H c ?) [2: >Ht1 %] #Hgrid #Heq %
610 [@injective_notb @Hgrid | <Heq @H1]
611 |#tapea #tapeout #tapeb whd in ⊢ (%→?); #Hcur
612 * #tapec * whd in ⊢ (%→?); #Hcompare #Hor
613 #ls #cur #rs #Htapea >Htapea in Hcur; #Hcur cases (Hcur ? (refl ??))
614 -Hcur #Hcur #Htapeb %
615 [ % #Hfalse >Hfalse in Hcur; normalize #Hfalse1 destruct (Hfalse1)]
616 #ls0 #c #l1 #l2 #c1 #l3 #l4 #rs0 #n #Hl1bitnull #Hl1marks #Hc #Hc1 #Hl3 #eqn
617 #eqlen #Htable #Hls #Hcur #Hrs -Htapea >Hls in Htapeb; >Hcur >Hrs #Htapeb
618 cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
619 cases (Hcompare c c1 l1 l3 l2 (l4@〈grid,false〉::rs0) eqlen Hl1bitnull Hl3 Hl1marks … (refl …) Hc ?)
621 [* #Htemp destruct (Htemp) #Htapec %1 % % [%]
622 >Htapec in Hor; -Htapec *
623 [2: * #t3 * whd in ⊢ (%→?); #H @False_ind
624 cases (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
625 |* #taped * whd in ⊢ (%→?); #Htaped cases (Htaped ? (refl …)) -Htaped *
626 #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped
629 |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
630 cut (〈c,false〉::l1 ≠ 〈c1,false〉::l3)
632 [@(not_to_not …H1) normalize #H destruct %
633 |#x #tl @not_to_not normalize #H destruct //
636 cut (bit_or_null d' = true)
638 [normalize in ⊢ (%→?); #H destruct //
639 |#x #tl #H @(Hl3 〈d',false〉)
640 normalize in H; destruct @memb_append_l2 @memb_hd
643 >Htapec in Hor; -Htapec *
644 [* #taped * whd in ⊢ (%→?); #H @False_ind
645 cases (H … (refl …)) >(bit_or_null_not_grid ? Hd') #Htemp destruct (Htemp)
646 |* #taped * whd in ⊢ (%→?); #H cases (H … (refl …)) -H #_
647 #Htaped * #tapee * whd in ⊢ (%→?); #Htapee
648 <(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped
649 cases (Htapee … Htaped ???) -Htaped -Htapee
650 [* #rs3 * * (* we proceed by cases on rs4 *)
651 [(* rs4 is empty : the case is absurd since the tape
652 cannot end with a bar *)
653 * #d * #b * * * #Heq1 @False_ind
654 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
655 >Hcut in Htable; >H3 >associative_append
656 normalize >Heq1 <associative_append >Hcut
657 <associative_append #Htable @(absurd … Htable)
660 * #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
661 cut (memb STape 〈d2,b2〉 (l2@〈c1,false〉::l3@〈comma,false〉::l4) = true)
663 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
664 >Hcut >H3 >associative_append @memb_append_l2
665 @memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable
666 cut (is_grid d2 = false)
667 [@(no_grids_in_table … Htable … 〈d2,b2〉 d2intable)] #Hd2
669 [@(no_marks_in_table … Htable … 〈d2,b2〉 d2intable)] #Hb2
670 >Hb2 in Heq1; #Heq1 -Hb2 -b2
671 whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
672 [(* we know current is not grid *)
673 * #tapef * whd in ⊢ (%→?); #Htapef
674 cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp)
675 |* #tapef * whd in ⊢ (%→?); #Htapef
676 cases (Htapef … (refl …)) #_ -Htapef #Htapef
677 * #tapeg >Htapef -Htapef *
680 #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
683 whd in ⊢ (%→?); #Htapeout
684 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
687 (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? l2@(〈grid,false〉::reverse ? lb))
688 c' (reverse ? la) false ls0 bar (〈d2,true〉::rs3'@〈grid,false〉::rs0) c00 b00 ?????) -Htapeout
689 [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
691 generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls0); #l
692 whd in ⊢ (???(???%)); >associative_append >associative_append %
693 |>reverse_cons @Hoption
695 [normalize in ⊢ (%→?); #Htemp destruct (Htemp)
696 @bit_or_null_not_grid @Hc
697 |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
698 @bit_or_null_not_grid @(Hl1bitnull 〈c',false〉) @memb_append_l2 @memb_hd
700 |cut (only_bits_or_nulls (la@(〈c',false〉::lb)))
701 [<H2 whd #c0 #Hmemb cases (orb_true_l … Hmemb)
702 [#eqc0 >(\P eqc0) @Hc |@Hl1bitnull]
703 |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
704 @memb_append_l1 @daemon
706 |@daemon] #Htapeout % %2 % //
708 cut (∃rs32.rs3 = lc@〈comma,false〉::rs32)
709 [ (*cases (tech_split STape (λc.c == 〈bar,false〉) l4)
711 | * #l41 * * #cbar #bfalse * #l42 * * #Hbar #Hl4 #Hl41
712 @(ex_intro ?? l41) >Hl4 in Heq1; #Heq1
714 cut (sublist … lc l3)
715 [ #x #Hx cases la in H3;
716 [ normalize #H3 destruct (H3) @Hx
717 | #p #la' normalize #Hla' destruct (Hla')
718 @memb_append_l2 @memb_cons @Hx ] ] #Hsublist*)
722 (〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3')
724 cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3')
725 [ >Hrs3 in Heq1; @daemon ] #Hl4
726 @(ex_intro … rs32) @(ex_intro … rs3') % [@Hl4]
729 |(*>Hrs3 *)>append_cons
730 > (?:l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs
731 = (l1@〈grid,false〉::l2@〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs)
732 [|>associative_append normalize
733 >associative_append normalize
734 >associative_append normalize
735 >associative_append normalize
737 >reverse_append >reverse_append >reverse_cons
738 >reverse_reverse >reverse_cons >reverse_reverse
739 >reverse_append >reverse_append >reverse_cons
740 >reverse_reverse >reverse_reverse >reverse_reverse
741 >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2)@la)@[〈d',false〉])@rs3
742 =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@la@[〈d',false〉]@rs3)))
743 [|>associative_append >associative_append
744 >associative_append >associative_append >associative_append
745 >associative_append % ]
746 <H2 normalize in ⊢ (??%?); >Hrs3
747 >associative_append >associative_append normalize
748 >associative_append >associative_append
750 >(?:la@(〈d',false〉::lc@〈comma,false〉::rs32)@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 =
751 (la@〈d',false〉::lc)@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs0 )
752 [| >associative_append normalize >associative_append % ]
757 |* #Hnobars #Htapee >Htapee -Htapee *
758 [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
759 cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
760 whd in ⊢ (%→?); #Htapeout %2 %
761 [% [//] whd #x #Hx @Hnobars @memb_append_l2 @memb_cons //
762 | >(Htapeout … (refl …)) % ]
763 |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
764 cases (Htapef … (refl …)) -Htapef
765 whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
767 |(* no marks in table *)
768 #x #membx @(no_marks_in_table … Htable)
770 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
771 >H3 >associative_append @memb_append_l2 @memb_cons @membx
772 |(* no grids in table *)
773 #x #membx @(no_grids_in_table … Htable)
775 cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
776 >H3 >associative_append @memb_append_l2 @memb_cons @membx
777 |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
780 |#x #membx @(no_marks_in_table … Htable)
781 @memb_append_l2 @memb_cons @memb_append_l1 @membx
782 |#x #membx @(no_marks_in_table … Htable)
783 @memb_append_l1 @membx
792 scrolls through the tuples in the transition table until one matching the
793 current configuration is found
796 definition match_tuple ≝ whileTM ? match_tuple_step (inr … (inl … (inr … start_nop))).
798 lemma is_grid_true : ∀c.is_grid c = true → c = grid.
799 * normalize [ #b ] #H // destruct (H)
802 definition R_match_tuple ≝ λt1,t2.
804 is_bit c = true → only_bits l1 → is_bit c1 = true → n = |l1| →
805 table_TM (S n) (〈c1,false〉::l2) →
806 t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
807 (l1@〈grid,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
810 〈c1,false〉::l2 = l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4 ∧
811 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
812 (l3@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv@l4@〈grid,false〉::rs))
814 (* non facciamo match su nessuna tupla;
815 non specifichiamo condizioni sul nastro di output, perché
816 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
817 (current ? t2 = Some ? 〈grid,true〉 ∧
819 〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4).
821 (* possible variante ?
822 definition weakR_match_tuple ≝ λt1,t2.
823 (∀ls,cur,rs,b. t1 = midtape STape ls 〈grid,b〉 rs → t2 = t1) ∧
824 (∀c,l1,c1,l2,l3,ls0,rs0,n.
825 t1 = midtape STape (〈grid,false〉::ls0) 〈bit c,true〉 rs
826 (l1@〈grid,false〉::l2@〈bit c1,true〉::l3@〈grid,false〉::rs0) →
827 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
828 table_TM (S n) (l2@〈c1,false〉::l3) →
831 〈c1,false〉::l3 = l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
832 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
833 (l2@l4@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
836 (* non facciamo match su nessuna tupla;
837 non specifichiamo condizioni sul nastro di output, perché
838 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
839 (current ? t2 = Some ? 〈grid,true〉 ∧
841 〈c1,false〉::l3 ≠ l4@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
844 definition weakR_match_tuple ≝ λt1,t2.
846 t1 = midtape STape ls cur rs →
847 (is_grid (\fst cur) = true → t2 = t1) ∧
848 (∀c,l1,c1,l2,l3,ls0,rs0,n.
849 ls = 〈grid,false〉::ls0 →
851 rs = l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0 →
852 is_bit c = true → is_bit c1 = true →
853 only_bits_or_nulls l1 → no_marks l1 → S n = |l1| →
854 table_TM (S n) (l2@〈bar,false〉::〈c1,false〉::l3) →
857 〈bar,false〉::〈c1,false〉::l3 = l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5 ∧
858 t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls0) 〈grid,false〉
859 (l2@l4@〈bar,false〉::〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv::l5@
862 (* non facciamo match su nessuna tupla;
863 non specifichiamo condizioni sul nastro di output, perché
864 non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
865 (current ? t2 = Some ? 〈grid,true〉 ∧
867 〈bar,false〉::〈c1,false〉::l3 ≠ l4@〈bar,false〉::〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv::l5)).
869 axiom table_bit_after_bar :
870 ∀n,l1,c,l2.table_TM n (l1@〈bar,false〉::〈c,false〉::l2) → is_bit c = true.
872 lemma wsem_match_tuple : WRealize ? match_tuple weakR_match_tuple.
873 #intape #k #outc #Hloop
874 lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
875 * #ta * #Hstar @(star_ind_l ??????? Hstar)
876 [ #tb whd in ⊢ (%→?); #Hleft
877 #ls #cur #rs #Htb cases (Hleft … Htb) #Hgrid #Houtc %
879 | #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs
880 >Hcur in Hgrid; #Hgrid >(is_grid_true … Hgrid) normalize in ⊢ (%→?);
883 | #tb #tc #td whd in ⊢ (%→?); #Htc
884 #Hstar1 #IH whd in ⊢ (%→?); #Hright lapply (IH Hright) -IH whd in ⊢ (%→?); #IH
886 [ #Hcur cases (Htc … Htb) * #Hfalse @False_ind @Hfalse @(is_grid_true … Hcur)
887 |#c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs #Hc #Hc1 #Hl1bitnull #Hl1marks
888 #Hl1len #Htable cases (Htc … Htb) -Htc -Htb * #_ #Htc
889 cut (∃la,lb,mv,lc.l3 = la@〈comma,false〉::lb@〈comma,false〉::mv::lc ∧
890 S n = |la| ∧ only_bits_or_nulls la)
891 [@daemon] * #la * #lb * #mv * #lc * * #Hl3 #Hlalen #Hlabitnull
892 >Hl3 in Htable; >append_cons #Htable
893 >(?: l2@〈bar,false〉::〈c1,true〉::l3@〈grid,false〉::rs0
894 = (l2@[〈bar,false〉])@〈c1,true〉::la@〈comma,false〉::(lb@〈comma,false〉::mv::
895 lc)@〈grid,false〉::rs0) in Hrs;
896 [| >associative_append normalize >Hl3
897 >associative_append normalize % ] #Hrs
898 cases (Htc ????????? Hl1bitnull Hl1marks ?? Hlabitnull Hl1len ? Htable Hls Hcur Hrs)
900 |4: whd in ⊢ (??%?); >Hc1 %
901 |3: whd in ⊢ (??%?); >Hc %
903 [ * #Heq #Htc % %{[]} %{lb} %{mv} %{lc} destruct (Heq) %
905 | cases (IH … Htc) -IH #Houtc #_ >(Houtc (refl ??))
906 >Htc @eq_f normalize >associative_append normalize
907 >associative_append normalize %
909 | * #Hdiff * #c2 * #l5 * #l6 * #Heqlblc #Htc
910 cases (IH ??? … Htc) -IH #_ #IH
911 lapply (IH ? l1 c2 (l2@〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5) l6 ? rs0 n (refl ??) (refl ??) ???????)
912 [ generalize in match Htable;
913 >associative_append normalize
914 >associative_append normalize >Heqlblc
915 >associative_append normalize //
921 | >associative_append normalize
922 >associative_append normalize
923 >associative_append %
925 [ * #l7 * #newc * #mv0 * #l8 * #Hl7l8 #Houtc %
926 >Heqlblc @(ex_intro ?? (〈bar,false〉::〈c1,false〉::la@〈comma,false〉::l5@l7))
927 %{newc} %{mv0} %{l8} %
928 [ normalize >Hl7l8 >associative_append normalize
929 >associative_append %
930 | >Houtc @eq_f >associative_append normalize
931 >associative_append normalize >associative_append
932 normalize >associative_append %
934 | * #Houtc #Hdiff %2 %
936 | #l50 #newc #mv0 #l51 >Heqlblc
942 | * * #Hdiff #Hnobars generalize in match (refl ? tc);
943 cases tc in ⊢ (???% → %);
944 [ #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
945 |2,3: #x #xs #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ]
946 #ls1 #cur1 #rs1 #Htc normalize in ⊢ (??%?→?); #Hcur1
947 cases (IH … Htc) -IH #IH #_ %2 %
948 [ destruct (Hcur1) >IH [ >Htc % | % ]
949 | #l4 #newc #mv0 #l5 (* difficile (sempre che sia dimostrabile)
950 dobbiamo veramente considerare di fare la table in modo più