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_____________________________________________________________*)
12 include "basics/vectors.ma".
13 (* include "basics/relations.ma". *)
15 record tape (sig:FinSet): Type[0] ≝
20 inductive move : Type[0] ≝
25 (* We do not distinuish an input tape *)
27 record TM (sig:FinSet): Type[1] ≝
29 trans : states × (option sig) → states × (option (sig × move));
34 record config (sig:FinSet) (M:TM sig): Type[0] ≝
35 { cstate : states sig M;
39 definition option_hd ≝ λA.λl:list A.
45 definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
50 [ R ⇒ mk_tape sig ((\fst m1)::(left ? t)) (tail ? (right ? t))
51 | L ⇒ mk_tape sig (tail ? (left ? t)) ((\fst m1)::(right ? t))
55 definition step ≝ λsig.λM:TM sig.λc:config sig M.
56 let current_char ≝ option_hd ? (right ? (ctape ?? c)) in
57 let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
58 mk_config ?? news (tape_move sig (ctape ?? c) mv).
60 let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
63 | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
66 axiom loop_incr : ∀A,f,p,k1,k2,a1,a2.
67 loop A k1 f p a1 = Some ? a2 →
68 loop A (k2+k1) f p a1 = Some ? a2.
70 lemma loop_split : ∀A,f,p,q.(∀b. p b = false → q b = false) →
72 loop A k1 f p a1 = Some ? a2 →
73 loop A k2 f q a2 = Some ? a3 →
74 loop A (k1+k2) f q a1 = Some ? a3.
75 #Sig #f #p #q #Hpq #k1 elim k1
76 [normalize #k2 #a1 #a2 #a3 #H destruct
77 |#k1' #Hind #k2 #a1 #a2 #a3 normalize in ⊢ (%→?→?);
78 cases (true_or_false (p a1)) #pa1 >pa1 normalize in ⊢ (%→?);
79 [#eqa1a2 destruct #H @loop_incr //
80 |normalize >(Hpq … pa1) normalize
81 #H1 #H2 @(Hind … H2) //
86 definition initc ≝ λsig.λM:TM sig.λt.
87 mk_config sig M (start sig M) t.
89 definition Realize ≝ λsig.λM:TM sig.λR:relation (tape sig).
91 loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) (initc sig M t) = Some ? outc ∧
96 definition seq_trans ≝ λsig. λM1,M2 : TM sig.
100 if halt sig M1 s1 then 〈inr … (start sig M2), None ?〉
102 let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in
105 let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in
109 definition seq ≝ λsig. λM1,M2 : TM sig.
111 (FinSum (states sig M1) (states sig M2))
112 (seq_trans sig M1 M2)
113 (inl … (start sig M1))
115 [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]).
117 definition Rcomp ≝ λA.λR1,R2:relation A.λa1,a2.
118 ∃am.R1 a1 am ∧ R2 am a2.
121 definition injectRl ≝ λsig.λM1.λM2.λR.
123 inl … (cstate sig M1 c11) = cstate sig (seq sig M1 M2) c1 ∧
124 inl … (cstate sig M1 c12) = cstate sig (seq sig M1 M2) c2 ∧
125 ctape sig M1 c11 = ctape sig (seq sig M1 M2) c1 ∧
126 ctape sig M1 c12 = ctape sig (seq sig M1 M2) c2 ∧
129 definition injectRr ≝ λsig.λM1.λM2.λR.
131 inr … (cstate sig M2 c21) = cstate sig (seq sig M1 M2) c1 ∧
132 inr … (cstate sig M2 c22) = cstate sig (seq sig M1 M2) c2 ∧
133 ctape sig M2 c21 = ctape sig (seq sig M1 M2) c1 ∧
134 ctape sig M2 c22 = ctape sig (seq sig M1 M2) c2 ∧
137 definition Rlink ≝ λsig.λM1,M2.λc1,c2.
138 ctape sig (seq sig M1 M2) c1 = ctape sig (seq sig M1 M2) c2 ∧
139 cstate sig (seq sig M1 M2) c1 = inl … (halt sig M1) ∧
140 cstate sig (seq sig M1 M2) c2 = inr … (start sig M2). *)
142 interpretation "relation composition" 'compose R1 R2 = (Rcomp ? R1 R2).
144 theorem sem_seq: ∀sig,M1,M2,R1,R2.
145 Realize sig M1 R1 → Realize sig M2 R2 →
146 Realize sig (seq sig M1 M2) (R1 ∘ R2).
147 #sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t
148 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
149 cases (HR2 (ctape sig M1 outc1)) #k2 * #outc2 * #Hloop2 #HM2
150 @(ex_intro … (S(k1+k2))) @
155 definition empty_tapes ≝ λsig.λn.
156 mk_Vector ? n (make_list (tape sig) (mk_tape sig [] []) n) ?.
157 elim n // normalize //
160 definition init ≝ λsig.λM:TM sig.λi:(list sig).
163 (vec_cons ? (mk_tape sig [] i) ? (empty_tapes sig (tapes_no sig M)))
166 definition stop ≝ λsig.λM:TM sig.λc:config sig M.
167 halt sig M (state sig M c).
169 let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
172 | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
175 (* Compute ? M f states that f is computed by M *)
176 definition Compute ≝ λsig.λM:TM sig.λf:(list sig) → (list sig).
178 loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
181 (* for decision problems, we accept a string if on termination
182 output is not empty *)
184 definition ComputeB ≝ λsig.λM:TM sig.λf:(list sig) → bool.
186 loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
187 (isnilb ? (out ?? c) = false).
189 (* alternative approach.
190 We define the notion of computation. The notion must be constructive,
191 since we want to define functions over it, like lenght and size
193 Perche' serve Type[2] se sposto a e b a destra? *)
195 inductive cmove (A:Type[0]) (f:A→A) (p:A →bool) (a,b:A): Type[0] ≝
196 mk_move: p a = false → b = f a → cmove A f p a b.
198 inductive cstar (A:Type[0]) (M:A→A→Type[0]) : A →A → Type[0] ≝
199 | empty : ∀a. cstar A M a a
200 | more : ∀a,b,c. M a b → cstar A M b c → cstar A M a c.
202 definition computation ≝ λsig.λM:TM sig.
203 cstar ? (cmove ? (step sig M) (stop sig M)).
205 definition Compute_expl ≝ λsig.λM:TM sig.λf:(list sig) → (list sig).
206 ∀l.∃c.computation sig M (init sig M l) c →
207 (stop sig M c = true) ∧ out ?? c = f l.
209 definition ComputeB_expl ≝ λsig.λM:TM sig.λf:(list sig) → bool.
210 ∀l.∃c.computation sig M (init sig M l) c →
211 (stop sig M c = true) ∧ (isnilb ? (out ?? c) = false).