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_____________________________________________________________*)
13 (* MOVE_CHAR (left) MACHINE
15 Sposta il carattere binario su cui si trova la testina appena prima del primo # alla sua destra.
18 (ls,cs,rs can be empty; # is a parameter)
34 include "turing/while_machine.ma".
36 definition mcl_states : FinSet → FinSet ≝ λalpha:FinSet.FinProd (initN 5) alpha.
38 definition mcl0 : initN 5 ≝ mk_Sig ?? 0 (leb_true_to_le 1 5 (refl …)).
39 definition mcl1 : initN 5 ≝ mk_Sig ?? 1 (leb_true_to_le 2 5 (refl …)).
40 definition mcl2 : initN 5 ≝ mk_Sig ?? 2 (leb_true_to_le 3 5 (refl …)).
41 definition mcl3 : initN 5 ≝ mk_Sig ?? 3 (leb_true_to_le 4 5 (refl …)).
42 definition mcl4 : initN 5 ≝ mk_Sig ?? 4 (leb_true_to_le 5 5 (refl …)).
45 λalpha:FinSet.λsep:alpha.
46 mk_TM alpha (mcl_states alpha)
49 let q' ≝ pi1 nat (λi.i<5) q' in (* perche' devo passare il predicato ??? *)
51 [ None ⇒ 〈〈mcl4,sep〉,None ?〉
56 [ true ⇒ 〈〈mcl4,sep〉,None ?〉
57 | false ⇒ 〈〈mcl1,a'〉,Some ? 〈a',R〉〉 ]
58 | S q' ⇒ match q' with
60 〈〈mcl2,a'〉,Some ? 〈b,L〉〉
61 | S q' ⇒ match q' with
63 〈〈mcl3,sep〉,Some ? 〈b,L〉〉
64 | S q' ⇒ match q' with
68 〈〈mcl4,sep〉,None ?〉 ] ] ] ] ])
70 (λq.let 〈q',a〉 ≝ q in q' == mcl3 ∨ q' == mcl4).
73 ∀alpha:FinSet.∀sep,a,ls,a0,rs.
75 step alpha (mcl_step alpha sep)
76 (mk_config ?? 〈mcl0,a〉 (mk_tape … ls (Some ? a0) rs)) =
77 mk_config alpha (states ? (mcl_step alpha sep)) 〈mcl1,a0〉
78 (tape_move_right alpha ls a0 rs).
80 [ #a0 #rs #Ha0 whd in ⊢ (??%?);
81 normalize in match (trans ???); >Ha0 %
82 | #a1 #ls #a0 #rs #Ha0 whd in ⊢ (??%?);
83 normalize in match (trans ???); >Ha0 %
88 ∀alpha:FinSet.∀sep,a,ls,a0,rs.
89 step alpha (mcl_step alpha sep)
90 (mk_config ?? 〈mcl1,a〉 (mk_tape … ls (Some ? a0) rs)) =
91 mk_config alpha (states ? (mcl_step alpha sep)) 〈mcl2,a0〉
92 (tape_move_left alpha ls a rs).
93 #alpha #sep #a #ls #a0 * //
97 ∀alpha:FinSet.∀sep,a,ls,a0,rs.
98 step alpha (mcl_step alpha sep)
99 (mk_config ?? 〈mcl2,a〉 (mk_tape … ls (Some ? a0) rs)) =
100 mk_config alpha (states ? (mcl_step alpha sep)) 〈mcl3,sep〉
101 (tape_move_left alpha ls a rs).
102 #alpha #sep #a #ls #a0 * //
105 definition Rmcl_step_true ≝
108 t1 = midtape alpha ls b (a::rs) →
110 t2 = mk_tape alpha (tail ? ls) (option_hd ? ls) (a::b::rs).
112 definition Rmcl_step_false ≝
114 right ? t1 ≠ [] → current alpha t1 ≠ None alpha →
115 current alpha t1 = Some alpha sep ∧ t2 = t1.
117 lemma mcl_trans_init_sep:
119 trans ? (mcl_step alpha sep) 〈〈mcl0,x〉,Some ? sep〉 = 〈〈mcl4,sep〉,None ?〉.
120 #alpha #sep #x normalize >(\b ?) //
123 lemma mcl_trans_init_not_sep:
124 ∀alpha,sep,x,y.y == sep = false →
125 trans ? (mcl_step alpha sep) 〈〈mcl0,x〉,Some ? y〉 = 〈〈mcl1,y〉,Some ? 〈y,R〉〉.
126 #alpha #sep #x #y #H1 normalize >H1 //
131 accRealize alpha (mcl_step alpha sep)
132 〈mcl3,sep〉 (Rmcl_step_true alpha sep) (Rmcl_step_false alpha sep).
133 #alpha #sep cut (∀P:Prop.〈mcl4,sep〉=〈mcl3,sep〉→P)
134 [#P whd in ⊢ ((??(???%?)(???%?))→?); #Hfalse destruct] #Hfalse
137 @(ex_intro … (mk_config ?? 〈mcl4,sep〉 (niltape ?)))
138 % [% [whd in ⊢ (??%?);% |@Hfalse] |#H1 #H2 @False_ind @(absurd ?? H2) %]
139 |#l0 #lt0 @(ex_intro ?? 2)
140 @(ex_intro … (mk_config ?? 〈mcl4,sep〉 (leftof ? l0 lt0)))
141 % [% [whd in ⊢ (??%?);% |@Hfalse] |#H1 #H2 #H3 @False_ind @(absurd ?? H3) %]
142 |#r0 #rt0 @(ex_intro ?? 2)
143 @(ex_intro … (mk_config ?? 〈mcl4,sep〉 (rightof ? r0 rt0)))
144 % [% [whd in ⊢ (??%?);% |@Hfalse] |#H1 #H2 #H3 @False_ind @(absurd ?? H3) %]
145 | #lt #c #rt cases (true_or_false (c == sep)) #Hc
147 @(ex_intro ?? (mk_config ?? 〈mcl4,sep〉 (midtape ? lt c rt)))
149 [ >(\P Hc) >loop_S_false // >loop_S_true
150 [ @eq_f whd in ⊢ (??%?); >mcl_trans_init_sep %
151 |>(\P Hc) whd in ⊢(??(???(???%))?); >mcl_trans_init_sep % ]
153 |#_ #H1 #H2 % // normalize >(\P Hc) % ]
154 |@(ex_intro ?? 4) cases rt
157 [ >loop_S_false // >mcl_q0_q1 //
158 | normalize in ⊢ (%→?); @Hfalse]
159 | normalize in ⊢ (%→?); #_ #H1 @False_ind @(absurd ?? H1) % ] ]
162 [ >loop_S_false // >mcl_q0_q1 //
163 | #_ #a #b #ls #rs #Hb destruct (Hb) %
165 | >mcl_q1_q2 >mcl_q2_q3 cases ls normalize // ] ]
166 | normalize in ⊢ (% → ?); * #Hfalse
174 (* the move_char (variant c) machine *)
175 definition move_char_l ≝
176 λalpha,sep.whileTM alpha (mcl_step alpha sep) 〈mcl3,sep〉.
178 definition R_move_char_l ≝
180 ∀b,a,ls,rs. t1 = midtape alpha ls b (a::rs) →
181 (b = sep → t2 = t1) ∧
182 (∀ls1,ls2.ls = ls1@sep::ls2 →
183 b ≠ sep → memb ? sep ls1 = false →
184 t2 = midtape alpha ls2 sep (a::reverse ? ls1@b::rs)).
186 lemma sem_move_char_l :
188 WRealize alpha (move_char_l alpha sep) (R_move_char_l alpha sep).
189 #alpha #sep #inc #i #outc #Hloop
190 lapply (sem_while … (sem_mcl_step alpha sep) inc i outc Hloop) [%]
191 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
192 [ #tapea whd in ⊢ (% → ?); #H1 #b #a #ls #rs #Htapea
194 [ #Hb >Htapea in H1; >Hb #H1 cases (H1 ??)
195 [#_ #H2 >H2 % |*: % #H2 normalize in H2; destruct (H2) ]
196 | #rs1 #rs2 #Hrs #Hb #Hrs1
197 >Htapea in H1; (* normalize in ⊢ (% → ?); *) #H1 cases (H1 ??)
198 [ #Hfalse normalize in Hfalse; @False_ind @(absurd ?? Hb) destruct %
199 |*:% normalize #H2 destruct (H2) ]
201 | #tapea #tapeb #tapec #Hstar1 #HRtrue #IH #HRfalse
202 lapply (IH HRfalse) -IH whd in ⊢ (%→%); #IH
203 #a0 #b0 #ls #rs #Htapea cases (Hstar1 … Htapea)
205 [ #Hfalse @False_ind @(absurd ?? Ha0) //
207 [ #ls2 whd in ⊢ (???%→?); #Hls #_ #_
208 >Hls in Htapeb; #Htapeb normalize in Htapeb;
209 cases (IH … Htapeb) #Houtc #_ >Houtc normalize //
210 | #l0 #ls0 #ls2 #Hls #_ #Hls0
211 cut (l0 ≠ sep ∧ memb … sep ls0 = false)
213 [ % #Hl0 >Hl0 in Hls0; >memb_hd #Hfalse destruct
214 | whd in Hls0:(??%?); cases (sep==l0) in Hls0; normalize #Hfalse
219 #Hl0 -Hls0 #Hls0 >Hls in Htapeb;
220 normalize in ⊢ (%→?); #Htapeb
221 cases (IH … Htapeb) -IH #_ #IH
222 >reverse_cons >associative_append @IH //
227 lemma terminate_move_char_l :
228 ∀alpha,sep.∀t,b,a,ls,rs. t = midtape alpha ls b (a::rs) →
229 (b = sep ∨ memb ? sep ls = true) → Terminate alpha (move_char_l alpha sep) t.
230 #alpha #sep #t #b #a #ls #rs #Ht #Hsep
231 @(terminate_while … (sem_mcl_step alpha sep))
233 |generalize in match Hsep; -Hsep
234 generalize in match Ht; -Ht
235 generalize in match rs; -rs
236 generalize in match a; -a
237 generalize in match b; -b
238 generalize in match t; -t
240 [#t #b #a #rs #Ht #Hsep % #tinit
241 whd in ⊢ (%→?); #H @False_ind
242 cases (H … Ht) #Hb #_ cases Hb #eqb @eqb
243 cases Hsep // whd in ⊢ ((??%?)→?); #abs destruct
244 |#l0 #ls0 #Hind #t #b #a #rs #Ht #Hsep % #tinit
246 cases (H … Ht) #Hbsep #Htinit
247 @(Hind … Htinit) cases Hsep
248 [#Hb @False_ind /2/ | #Hmemb cases (orb_true_l … Hmemb)
249 [#eqsep %1 >(\P eqsep) // | #H %2 //]