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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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11 (* v GNU General Public License Version 2 *)
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15 include "turing/mono.ma".
19 - return its nth element
20 - return the index of a given element
22 axiom FS_crd : FinSet → nat.
23 axiom FS_nth : ∀F:FinSet.nat → option F.
24 axiom index_of_FS : ∀F:FinSet.F → nat.
26 (* unary bit representation (with a given length) of a certain number *)
27 axiom unary_of_nat : nat → nat → (list bool).
29 axiom FinVector : Type[0] → nat → FinSet.
31 definition binary_base_states ≝ initN 6.
33 definition bin0 : binary_base_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 6 (refl …)).
34 definition bin1 : binary_base_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 6 (refl …)).
35 definition bin2 : binary_base_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 6 (refl …)).
36 definition bin3 : binary_base_states ≝ mk_Sig ?? 3 (leb_true_to_le 4 6 (refl …)).
37 definition bin4 : binary_base_states ≝ mk_Sig ?? 4 (leb_true_to_le 5 6 (refl …)).
38 definition bin5 : binary_base_states ≝ mk_Sig ?? 5 (leb_true_to_le 6 6 (refl …)).
40 definition states_binaryTM : FinSet → FinSet → FinSet ≝ λsig,states.
41 FinProd (FinProd states binary_base_states)
42 (FinProd (FinOption sig) (initN (S (2 * (FS_crd sig))))).
44 axiom daemon : ∀T:Type[0].T.
46 definition to_initN : ∀n,m.n < m → initN m ≝ λn,m,Hn.mk_Sig … n ….// qed.
48 definition initN_pred : ∀n.∀m:initN n.initN n ≝ λn,m.mk_Sig … (pred (pi1 … m)) ….
49 cases m #m0 /2 by le_to_lt_to_lt/ qed.
51 (* controllare i contatori, molti andranno incrementati di uno *)
52 definition trans_binaryTM : ∀sig,states:FinSet.
53 (states × (option sig) → states × (option sig) × move) →
54 ((states_binaryTM sig states) × (option bool) →
55 (states_binaryTM sig states) × (option bool) × move)
56 ≝ λsig,states,trans,p.
58 let 〈s0,phase,ch,count〉 ≝ s in
59 let (H1 : O < S (2*FS_crd sig)) ≝ ? in
60 let (H2 : FS_crd sig < S (2*FS_crd sig)) ≝ ? in
61 match pi1 … phase with
62 [ O ⇒ (*** PHASE 0: read ***)
63 match pi1 … count with
64 [ O ⇒ 〈〈s0,bin1,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
66 [ Some a0 ⇒ if (a0 == true)
67 then 〈〈s0,bin0,FS_nth sig k,initN_pred … count〉, None ?,R〉
68 else 〈〈s0,bin0,ch,initN_pred … count〉,None ?,R〉
69 | None ⇒ (* Overflow position! *)
70 〈〈s0,bin4,None ?,to_initN 0 ? H1〉,None ?,R〉 ] ]
71 | S phase ⇒ match phase with
72 [ O ⇒ (*** PHASE 1: restart ***)
73 match pi1 … count with
74 [ O ⇒ 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
75 | S k ⇒ 〈〈s0,bin1,ch,initN_pred … count〉,None ?,L〉 ]
76 | S phase ⇒ match phase with
77 [ O ⇒ (*** PHASE 2: write ***)
78 let 〈s',a',mv〉 ≝ trans 〈s0,ch〉 in
79 match pi1 … count with
80 [ O ⇒ let mv' ≝ match mv with [ R ⇒ N | _ ⇒ L ] in
81 let count' ≝ match mv with [ R ⇒ 0 | N ⇒ FS_crd sig | L ⇒ 2*(FS_crd sig) ] in
82 〈〈s',bin3,ch,to_initN count' ??〉,None ?,mv'〉
84 [ None ⇒ 〈〈s0,bin2,ch,initN_pred … count〉,None ?,R〉
85 | Some a0' ⇒ let out ≝ (FS_nth ? k == a') in
86 〈〈s0,bin2,ch,initN_pred … count〉,Some ? out,R〉 ]
88 | S phase ⇒ match phase with
89 [ O ⇒ (*** PHASE 3: move head left ***)
90 match pi1 … count with
91 [ O ⇒ 〈〈s0,bin0,None ?,to_initN (FS_crd sig) ? H2〉, None ?,N〉 (* the end: restart *)
92 | S k ⇒ 〈〈s0,bin3,ch,initN_pred … count〉, None ?,L〉 ]
93 | S phase ⇒ match phase with
94 [ O ⇒ (*** PHASE 4: check position ***)
96 [ None ⇒ (* niltape/rightof: we can write *) 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
97 | Some _ ⇒ (* leftof *)
98 let 〈s',a',mv〉 ≝ trans 〈s0,ch〉 in
100 [ None ⇒ (* we don't write anything: go to end of 2 *) 〈〈s0,bin2,ch,to_initN 0 ? H1〉,None ?,N〉
101 | Some _ ⇒ (* extend tape *) 〈〈s0,bin5,ch,to_initN (FS_crd sig) ? H2〉,None ?,L〉 ]
103 | S _ ⇒ (*** PHASE 5: left extension ***)
104 match pi1 … count with
105 [ O ⇒ 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
106 | S k ⇒ 〈〈s0,bin5,ch,initN_pred … count〉,Some ? false,L〉 ]]]]]].
108 whd in match count'; cases mv whd in ⊢ (?%?); //
111 definition halt_binaryTM : ∀sig,M.states_binaryTM sig (states sig M) → bool ≝
112 λsig,M,s.let 〈s0,phase,ch,count〉 ≝ s in
113 pi1 … phase == O ∧ halt sig M s0.
116 * Una mk_binaryTM prende in input una macchina M e produce una macchina che:
117 * - ha per alfabeto FinBool
118 * - ha stati di tipo ((states … M) × (initN 7)) ×
119 ((option sig) × (initN (2*dimensione dell'alfabeto di M + 1))
120 * dove il primo elemento corrisponde allo stato della macchina input,
121 * il secondo identifica la fase (lettura, scrittura, spostamento)
122 * il terzo identifica il carattere oggetto letto
123 * il quarto è un contatore
124 * - la funzione di transizione viene prodotta da trans_binaryTM
125 * - la funzione di arresto viene prodotta da halt_binaryTM
127 definition mk_binaryTM ≝
129 mk_TM FinBool (states_binaryTM sig (states sig M))
130 (trans_binaryTM sig (states sig M) (trans sig M))
131 (〈start sig M,bin0,None ?,FS_crd sig〉) (halt_binaryTM sig M).// qed.
133 definition bin_char ≝ λsig,ch.unary_of_nat (FS_crd sig) (index_of_FS sig ch).
135 definition bin_current ≝ λsig,t.match current ? t with
136 [ None ⇒ [ ] | Some c ⇒ bin_char sig c ].
138 definition tape_bin_lift ≝ λsig,t.
139 let ls' ≝ flatten ? (map ?? (bin_char sig) (left ? t)) in
140 let c' ≝ option_hd ? (bin_current sig t) in
141 let rs' ≝ tail ? (bin_current sig t)@flatten ? (map ?? (bin_char sig) (right ? t)) in
142 mk_tape ? ls' c' rs'.
144 definition R_bin_lift ≝ λsig,R,t1,t2.
145 ∃u1.t1 = tape_bin_lift sig u1 →
146 ∃u2.t2 = tape_bin_lift sig u2 ∧ R u1 u2.
148 definition state_bin_lift :
149 ∀sig.∀M:TM sig.states sig M → states ? (mk_binaryTM ? M)
150 ≝ λsig,M,q.〈q,bin0,None ?,FS_crd sig〉.// qed.
152 lemma lift_halt_binaryTM :
153 ∀sig,M,q.halt sig M q = halt ? (mk_binaryTM sig M) (state_bin_lift ? M q).
156 lemma binaryTM_bin0_bin1 :
158 step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,O〉) t)
159 = mk_config ?? (〈q,bin1,ch,to_initN (FS_crd sig) ??〉) t. //
162 lemma binaryTM_bin0_bin4 :
164 current ? t = None ? → S k <S (2*FS_crd sig) →
165 step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t)
166 = mk_config ?? (〈q,bin4,None ?,to_initN 0 ??〉) (tape_move ? t R). [2,3://]
167 #sig #M #t #q #ch #k #Hcur #Hk
168 whd in match (step ???); whd in match (trans ???);
172 lemma binaryTM_bin0_true :
174 current ? t = Some ? true → S k <S (2*FS_crd sig) →
175 step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t)
176 = mk_config ?? (〈q,bin0,FS_nth sig k,to_initN k ??〉) (tape_move ? t R).[2,3:/2 by lt_S_to_lt/]
177 #sig #M #t #q #ch #k #Hcur #Hk
178 whd in match (step ???); whd in match (trans ???);
182 lemma binaryTM_bin0_false :
184 current ? t = Some ? false → S k <S (2*FS_crd sig) →
185 step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t)
186 = mk_config ?? (〈q,bin0,ch,to_initN k ??〉) (tape_move ? t R).[2,3:/2 by lt_S_to_lt/]
187 #sig #M #t #q #ch #k #Hcur #Hk
188 whd in match (step ???); whd in match (trans ???);
193 axiom binary_to_bin_char :∀sig,csl,csr,a.
194 csl@true::csr=bin_char sig a → FS_nth ? (length ? csr) = Some ? a.
196 lemma binaryTM_phase1_midtape_aux :
199 ∀csr,csl,t,ch.length ? csr < S (2*FS_crd sig) →
200 t = mk_tape ? (reverse ? csl@ls) (option_hd ? (csr@rs)) (tail ? (csr@rs)) →
201 csl@csr = bin_char sig a →
202 loopM ? (mk_binaryTM sig M) (S (length ? csr) + k)
203 (mk_config ?? (〈q,bin0,ch,length ? csr〉) t)
204 = loopM ? (mk_binaryTM sig M) k
205 (mk_config ?? (〈q,bin1,Some ? a,FS_crd sig〉)
206 (mk_tape ? (reverse ? (bin_char ? a)@ls) (option_hd ? rs) (tail ? rs))). [2,3://]
207 #sig #M #q #ls #a #rs #k #Hhalt #csr elim csr
208 [ #csl #t #ch #Hlen #Ht >append_nil #Hcsl >loopM_unfold >loop_S_false [|normalize //]
209 <loopM_unfold @eq_f >binaryTM_bin0_bin1 @eq_f >Ht
210 whd in match (step ???); whd in match (trans ???); <Hcsl %
212 [ #csr0 #IH #csl #t #ch #Hlen #Ht #Heq >loopM_unfold >loop_S_false [|normalize //]
213 <loopM_unfold lapply (binary_to_bin_char … Heq) #Ha >binaryTM_bin0_true
215 lapply (IH (csl@[true]) (tape_move FinBool t R) ????)
217 | >Ht whd in match (option_hd ??) in ⊢ (??%?); whd in match (tail ??) in ⊢ (??%?);
220 [ normalize >rev_append_def >rev_append_def >reverse_append %
221 | #r1 #rs1 normalize >rev_append_def >rev_append_def >reverse_append % ]
222 | #c1 #csr1 normalize >rev_append_def >rev_append_def >reverse_append % ]
225 #H whd in match (plus ??); >Ha >H @eq_f @eq_f2 //
230 lemma binaryTM_phase1_midtape :
231 ∀sig,M,t,q,ls,a,rs,ch.
232 t = mk_tape ? ls (option_hd ? (bin_char ? a)) (tail ? (bin_char sig a@rs)) →
233 step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,FS_crd sig〉) t)
234 = mk_config ?? (〈q,bin1,ch,to_initN (FS_crd sig) ??〉)
235 (mk_tape ? (reverse ? (bin_char ? a)@ls) (option_hd ? rs) (tail ? rs)). [2,3://]
236 #sig #M #t #q #ls #a #rs #ch #Ht
237 whd in match (step ???); whd in match (trans ???);
242 lemma binaryTM_loop :
244 loopM sig M i (mk_config ?? q t) = Some ? (mk_config ?? qf tf) →
245 ∃k.loopM ? (mk_binaryTM sig M) k
246 (mk_config ?? (state_bin_lift ? M q) (tape_bin_lift ? t)) =
247 Some ? (mk_config ?? (state_bin_lift ? M qf) (tape_bin_lift ? tf)).
249 [ #t #q #qf #tf change with (None ?) in ⊢ (??%?→?); #H destruct (H)
250 | -i #i #IH #t #q #tf #qf
252 lapply (refl ? (halt sig M (cstate ?? (mk_config ?? q t))))
253 cases (halt ?? q) in ⊢ (???%→?); #Hhalt
254 [ >(loop_S_true ??? (λc.halt ?? (cstate ?? c)) (mk_config ?? q t) Hhalt)
255 #H destruct (H) %{1} >loopM_unfold >loop_S_true // ]
256 (* interesting case: more than one step *)
257 >(loop_S_false ??? (λc.halt ?? (cstate ?? c)) (mk_config ?? q t) Hhalt)
258 <loopM_unfold >(config_expand ?? (step ???)) #Hloop
259 lapply (IH … Hloop) -IH * #k0 #IH <config_expand in Hloop; #Hloop
265 theorem sem_binaryTM : ∀sig,M.
266 mk_binaryTM sig M ⊫ R_bin_lift ? (R_TM ? M (start ? M)).
267 #sig #M #t #i generalize in match t; -t
268 @(nat_elim1 … i) #m #IH #intape #outc #Hloop