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 //
127 accRealize alpha (mcl_step alpha sep)
128 〈3,sep〉 (Rmcl_step_true alpha sep) (Rmcl_step_false alpha sep).
131 @(ex_intro … (mk_config ?? 〈4,sep〉 (niltape ?)))
132 % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 @False_ind @(absurd ?? H2) %]
133 |#l0 #lt0 @(ex_intro ?? 2)
134 @(ex_intro … (mk_config ?? 〈4,sep〉 (rightof ? l0 lt0)))
135 % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 @False_ind @(absurd ?? H2) %]
136 |#r0 #rt0 @(ex_intro ?? 2)
137 @(ex_intro … (mk_config ?? 〈4,sep〉 (leftof ? r0 rt0)))
138 % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 #H2 #H3 @False_ind @(absurd ?? H3) %]
139 | #lt #c #rt cases (true_or_false (c == sep)) #Hc
141 @(ex_intro ?? (mk_config ?? 〈4,sep〉 (midtape ? lt c rt)))
143 [ >(\P Hc) >loop_S_false // >loop_S_true
144 [ @eq_f whd in ⊢ (??%?); >trans_init_sep %
145 |>(\P Hc) whd in ⊢(??(???(???%))?); >trans_init_sep % ]
147 |#_ #H1 #H2 % // normalize >(\P Hc) % ]
148 | @(ex_intro ?? 4) cases lt
151 [ >loop_S_false // >mcc_q0_q1 //
152 | normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
153 | normalize in ⊢ (%→?); #_ #H1 @False_ind @(absurd ?? H1) % ] ]
156 [ >loop_S_false // >mcc_q0_q1 //
157 | #_ #a #b #ls #rs #Hb destruct %
159 | >mcc_q1_q2 >mcc_q2_q3 cases rs normalize // ] ]
160 | normalize in ⊢ (% → ?); * #Hfalse
168 (* the move_char (variant c) machine *)
169 definition move_char_c ≝
170 λalpha,sep.whileTM alpha (mcc_step alpha sep) 〈3,sep〉.
172 definition R_move_char_c ≝
174 ∀b,a,ls,rs. t1 = midtape alpha (a::ls) b rs →
175 (b = sep → t2 = t1) ∧
176 (∀rs1,rs2.rs = rs1@sep::rs2 →
177 b ≠ sep → memb ? sep rs1 = false →
178 t2 = midtape alpha (a::reverse ? rs1@b::ls) sep rs2).
180 lemma sem_while_move_char :
182 WRealize alpha (move_char_c alpha sep) (R_move_char_c alpha sep).
183 #alpha #sep #inc #i #outc #Hloop
184 lapply (sem_while … (sem_mcc_step alpha sep) inc i outc Hloop) [%]
185 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
186 [ #tapea whd in ⊢ (% → ?); #H1 #b #a #ls #rs #Htapea
188 [ #Hb >Htapea in H1; >Hb normalize in ⊢ (%→?); #H1
191 |*: % #H2 destruct (H2) ]
192 | #rs1 #rs2 #Hrs #Hb #Hrs1
193 >Htapea in H1; normalize in ⊢ (% → ?); #H1
195 [ #Hfalse @False_ind @(absurd ?? Hb) destruct %
196 |*:% #H2 destruct (H2) ]
198 | #tapea #tapeb #tapec #Hstar1 #HRtrue #IH #HRfalse
199 lapply (IH HRfalse) -IH whd in ⊢ (%→%); #IH
200 #a0 #b0 #ls #rs #Htapea cases (Hstar1 … Htapea)
202 [ #Hfalse @False_ind @(absurd ?? Ha0) //
204 [ #rs2 whd in ⊢ (???%→?); #Hrs #_ #_ normalize
205 >Hrs in Htapeb; normalize #Htapeb
208 | #r0 #rs0 #rs2 #Hrs #_ #Hrs0
209 cut (r0 ≠ sep ∧ memb … sep rs0 = false)
211 [ % #Hr0 >Hr0 in Hrs0; >memb_hd #Hfalse destruct
212 | whd in Hrs0:(??%?); cases (sep==r0) in Hrs0; normalize #Hfalse
217 #Hr0 -Hrs0 #Hrs0 >Hrs in Htapeb;
218 normalize in ⊢ (%→?); #Htapeb
219 cases (IH … Htapeb) -IH #_ #IH
220 >reverse_cons >associative_append @IH //