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.
39 λalpha:FinSet.λsep:alpha.
40 mk_TM alpha (mcl_states alpha)
44 [ None ⇒ 〈〈4,sep〉,None ?〉
49 [ true ⇒ 〈〈4,sep〉,None ?〉
50 | false ⇒ 〈〈1,a'〉,Some ? 〈a',R〉〉 ]
51 | S q' ⇒ match q' with
54 | S q' ⇒ match q' with
56 〈〈3,sep〉,Some ? 〈b,L〉〉
57 | S q' ⇒ match q' with
61 〈〈4,sep〉,None ?〉 ] ] ] ] ])
63 (λq.let 〈q',a〉 ≝ q in q' == 3 ∨ q' == 4).
66 ∀alpha:FinSet.∀sep,a,ls,a0,rs.
68 step alpha (mcl_step alpha sep)
69 (mk_config ?? 〈0,a〉 (mk_tape … ls (Some ? a0) rs)) =
70 mk_config alpha (states ? (mcl_step alpha sep)) 〈1,a0〉
71 (tape_move_right alpha ls a0 rs).
73 [ #a0 #rs #Ha0 whd in ⊢ (??%?);
74 normalize in match (trans ???); >Ha0 %
75 | #a1 #ls #a0 #rs #Ha0 whd in ⊢ (??%?);
76 normalize in match (trans ???); >Ha0 %
81 ∀alpha:FinSet.∀sep,a,ls,a0,rs.
82 step alpha (mcl_step alpha sep)
83 (mk_config ?? 〈1,a〉 (mk_tape … ls (Some ? a0) rs)) =
84 mk_config alpha (states ? (mcl_step alpha sep)) 〈2,a0〉
85 (tape_move_left alpha ls a rs).
86 #alpha #sep #a #ls #a0 * //
90 ∀alpha:FinSet.∀sep,a,ls,a0,rs.
91 step alpha (mcl_step alpha sep)
92 (mk_config ?? 〈2,a〉 (mk_tape … ls (Some ? a0) rs)) =
93 mk_config alpha (states ? (mcl_step alpha sep)) 〈3,sep〉
94 (tape_move_left alpha ls a rs).
95 #alpha #sep #a #ls #a0 * //
98 definition Rmcl_step_true ≝
101 t1 = midtape alpha ls b (a::rs) →
103 t2 = mk_tape alpha (tail ? ls) (option_hd ? ls) (a::b::rs).
105 definition Rmcl_step_false ≝
107 right ? t1 ≠ [] → current alpha t1 ≠ None alpha →
108 current alpha t1 = Some alpha sep ∧ t2 = t1.
110 lemma mcl_trans_init_sep:
112 trans ? (mcl_step alpha sep) 〈〈0,x〉,Some ? sep〉 = 〈〈4,sep〉,None ?〉.
113 #alpha #sep #x normalize >(\b ?) //
116 lemma mcl_trans_init_not_sep:
117 ∀alpha,sep,x,y.y == sep = false →
118 trans ? (mcl_step alpha sep) 〈〈0,x〉,Some ? y〉 = 〈〈1,y〉,Some ? 〈y,R〉〉.
119 #alpha #sep #x #y #H1 normalize >H1 //
124 accRealize alpha (mcl_step alpha sep)
125 〈3,sep〉 (Rmcl_step_true alpha sep) (Rmcl_step_false alpha sep).
128 @(ex_intro … (mk_config ?? 〈4,sep〉 (niltape ?)))
129 % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 @False_ind @(absurd ?? H2) %]
130 |#l0 #lt0 @(ex_intro ?? 2)
131 @(ex_intro … (mk_config ?? 〈4,sep〉 (leftof ? l0 lt0)))
132 % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 #H3 @False_ind @(absurd ?? H3) %]
133 |#r0 #rt0 @(ex_intro ?? 2)
134 @(ex_intro … (mk_config ?? 〈4,sep〉 (rightof ? r0 rt0)))
135 % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 #H3 @False_ind @(absurd ?? H3) %]
136 | #lt #c #rt cases (true_or_false (c == sep)) #Hc
138 @(ex_intro ?? (mk_config ?? 〈4,sep〉 (midtape ? lt c rt)))
140 [ >(\P Hc) >loop_S_false // >loop_S_true
141 [ @eq_f whd in ⊢ (??%?); >mcl_trans_init_sep %
142 |>(\P Hc) whd in ⊢(??(???(???%))?); >mcl_trans_init_sep % ]
144 |#_ #H1 #H2 % // normalize >(\P Hc) % ]
145 | @(ex_intro ?? 4) cases rt
148 [ >loop_S_false // >mcl_q0_q1 //
149 | normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
150 | normalize in ⊢ (%→?); #_ #H1 @False_ind @(absurd ?? H1) % ] ]
154 [ >loop_S_false // >mcl_q0_q1 //
155 | #_ #a #b #ls #rs #Hb destruct (Hb) %
157 | >mcl_q1_q2 >mcl_q2_q3 cases ls normalize // ] ]
158 | normalize in ⊢ (% → ?); * #Hfalse
166 (* the move_char (variant c) machine *)
167 definition move_char_l ≝
168 λalpha,sep.whileTM alpha (mcl_step alpha sep) 〈3,sep〉.
170 definition R_move_char_l ≝
172 ∀b,a,ls,rs. t1 = midtape alpha ls b (a::rs) →
173 (b = sep → t2 = t1) ∧
174 (∀ls1,ls2.ls = ls1@sep::ls2 →
175 b ≠ sep → memb ? sep ls1 = false →
176 t2 = midtape alpha ls2 sep (a::reverse ? ls1@b::rs)).
178 lemma sem_move_char_l :
180 WRealize alpha (move_char_l alpha sep) (R_move_char_l alpha sep).
181 #alpha #sep #inc #i #outc #Hloop
182 lapply (sem_while … (sem_mcl_step alpha sep) inc i outc Hloop) [%]
183 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
184 [ #tapea whd in ⊢ (% → ?); #H1 #b #a #ls #rs #Htapea
186 [ #Hb >Htapea in H1; >Hb normalize in ⊢ (%→?); #H1
189 |*: % #H2 destruct (H2) ]
190 | #rs1 #rs2 #Hrs #Hb #Hrs1
191 >Htapea in H1; normalize in ⊢ (% → ?); #H1
193 [ #Hfalse @False_ind @(absurd ?? Hb) destruct %
194 |*:% #H2 destruct (H2) ]
196 | #tapea #tapeb #tapec #Hstar1 #HRtrue #IH #HRfalse
197 lapply (IH HRfalse) -IH whd in ⊢ (%→%); #IH
198 #a0 #b0 #ls #rs #Htapea cases (Hstar1 … Htapea)
200 [ #Hfalse @False_ind @(absurd ?? Ha0) //
202 [ #ls2 whd in ⊢ (???%→?); #Hls #_ #_ normalize
203 >Hls in Htapeb; normalize #Htapeb
206 | #l0 #ls0 #ls2 #Hls #_ #Hls0
207 cut (l0 ≠ sep ∧ memb … sep ls0 = false)
209 [ % #Hl0 >Hl0 in Hls0; >memb_hd #Hfalse destruct
210 | whd in Hls0:(??%?); cases (sep==l0) in Hls0; normalize #Hfalse
215 #Hl0 -Hls0 #Hls0 >Hls in Htapeb;
216 normalize in ⊢ (%→?); #Htapeb
217 cases (IH … Htapeb) -IH #_ #IH
218 >reverse_cons >associative_append @IH //
223 lemma terminate_move_char_l :
224 ∀alpha,sep.∀t,b,a,ls,rs. t = midtape alpha ls b (a::rs) →
225 (b = sep ∨ memb ? sep ls = true) → Terminate alpha (move_char_l alpha sep) t.
226 #alpha #sep #t #b #a #ls #rs #Ht #Hsep
227 @(terminate_while … (sem_mcl_step alpha sep))
229 |generalize in match Hsep; -Hsep
230 generalize in match Ht; -Ht
231 generalize in match rs; -rs
232 generalize in match a; -a
233 generalize in match b; -b
234 generalize in match t; -t
236 [#t #b #a #rs #Ht #Hsep % #tinit
237 whd in ⊢ (%→?); #H @False_ind
238 cases (H … Ht) #Hb #_ cases Hb #eqb @eqb
239 cases Hsep // whd in ⊢ ((??%?)→?); #abs destruct
240 |#l0 #ls0 #Hind #t #b #a #rs #Ht #Hsep % #tinit
242 cases (H … Ht) #Hbsep #Htinit
243 @(Hind … Htinit) cases Hsep
244 [#Hb @False_ind /2/ | #Hmemb cases (orb_true_l … Hmemb)
245 [#eqsep %1 >(\P eqsep) // | #H %2 //]