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_____________________________________________________________*)
14 include "turing/universal/tuples.ma".
15 include "basics/fun_graph.ma".
17 (* p < n is represented with a list of bits of lenght n with the
18 p-th bit from left set to 1 *)
20 let rec to_bitlist n p: list bool ≝
23 | S q ⇒ (eqb p q)::to_bitlist q p].
25 let rec from_bitlist l ≝
27 [ nil ⇒ 0 (* assert false *)
28 | cons b tl ⇒ if b then |tl| else from_bitlist tl].
30 lemma bitlist_length: ∀n,p.|to_bitlist n p| = n.
31 #n elim n normalize //
34 lemma bitlist_inv1: ∀n,p.p<n → from_bitlist (to_bitlist n p) = p.
35 #n elim n normalize -n
36 [#p #abs @False_ind /2/
38 cases (le_to_or_lt_eq … (le_S_S_to_le … lepn))
39 [#ltpn lapply (lt_to_not_eq … ltpn) #Hpn
40 >(not_eq_to_eqb_false … Hpn) normalize @Hind @ltpn
41 |#Heq >(eq_to_eqb_true … Heq) normalize <Heq //
46 lemma bitlist_lt: ∀l. 0 < |l| → from_bitlist l < |l|.
47 #l elim l normalize // #b #tl #Hind cases b normalize //
48 #Htl cases (le_to_or_lt_eq … (le_S_S_to_le … Htl)) -Htl #Htl
49 [@le_S_S @lt_to_le @Hind //
50 |cut (tl=[ ]) [/2 by append_l2_injective/] #eqtl >eqtl @le_n
54 definition nat_of: ∀n. Nat_to n → nat.
55 #n normalize * #p #_ @p
58 definition bits_of_state ≝ λn.λh:Nat_to n → bool.λs:Nat_to n.
59 h s::(to_bitlist n (nat_of n s)).
61 definition m_bits_of_state ≝ λn.λh.λp.
62 map ? (unialpha×bool) (λx.〈bit x,false〉) (bits_of_state n h p).
64 lemma no_marks_bits_of_state : ∀n,h,p. no_marks (m_bits_of_state n h p).
65 #n #h #p #x whd in match (m_bits_of_state n h p);
66 #H cases (orb_true_l … H) -H
68 |elim (to_bitlist n (nat_of n p))
69 [whd in ⊢ ((??%?)→?); #H destruct
70 |#b #l #Hind #H cases (orb_true_l … H) -H #H
78 lemma only_bits_bits_of_state : ∀n,h,p. only_bits (m_bits_of_state n h p).
79 #n #h #p #x whd in match (m_bits_of_state n h p);
80 #H cases (orb_true_l … H) -H
82 |elim (to_bitlist n (nat_of n p))
83 [whd in ⊢ ((??%?)→?); #H destruct
84 |#b #l #Hind #H cases (orb_true_l … H) -H #H
92 definition tuple_type ≝ λn.
93 (Nat_to n × (option FinBool)) × (Nat_to n × (option (FinBool × move))).
95 definition tuple_of_pair ≝ λn.λh:Nat_to n→bool.
99 let cin ≝ match a with [ None ⇒ null | Some b ⇒ bit b ] in
100 let 〈qn,action〉 ≝ outp in
104 | Some act ⇒ let 〈na,m〉 ≝ act in
106 [ R ⇒ 〈bit na,bit true〉
107 | L ⇒ 〈bit na,bit false〉
110 let qin ≝ m_bits_of_state n h q in
111 let qout ≝ m_bits_of_state n h qn in
112 mk_tuple qin 〈cin,false〉 qout 〈cout,false〉 〈mv,false〉.
114 definition WFTuple_conditions ≝
115 λn,qin,cin,qout,cout,mv.
116 no_marks qin ∧ no_marks qout ∧ (* queste fuori ? *)
117 only_bits qin ∧ only_bits qout ∧
118 bit_or_null cin = true ∧ bit_or_null cout = true ∧ bit_or_null mv = true ∧
119 (cout = null → mv = null) ∧
120 |qin| = n ∧ |qout| = n.
122 lemma is_tuple: ∀n,h,p. tuple_TM (S n) (tuple_of_pair n h p).
123 #n #h * * #q #a * #qn #action
124 @(ex_intro … (m_bits_of_state n h q))
125 letin cin ≝ match a with [ None ⇒ null | Some b ⇒ bit b ]
127 @(ex_intro … (m_bits_of_state n h qn))
130 [ None ⇒ null | Some act ⇒ bit (\fst act)]
132 letin mv ≝ match action with
136 [ R ⇒ bit true | L ⇒ bit false | N ⇒ null]
140 |whd in match cin ; cases a //
142 |whd in match cout; cases action //
144 |whd in match mv; cases action //
147 |whd in match cout; whd in match mv; cases action
148 [// | #act whd in ⊢ ((??%?)→?); #Hfalse destruct ]
150 |>length_map normalize @eq_f //
152 |>length_map normalize @eq_f //
154 |normalize cases a cases action normalize //
156 |* #c #m #c1 cases m %
161 definition tuple_length ≝ λn.2*n+6.
163 lemma length_of_tuple: ∀n,t. tuple_TM n t →
164 |t| = tuple_length n.
165 #n #t * #qin * #cin * #qout * #cout * #mv *** #_
166 #Hqin #Hqout #eqt >eqt whd in match (mk_tuple ?????);
167 normalize >length_append >Hqin -Hqin normalize
168 >length_append normalize >Hqout -Hqout //
171 definition move_eq ≝ λm1,m2:move.
173 [R ⇒ match m2 with [R ⇒ true | _ ⇒ false]
174 |L ⇒ match m2 with [L ⇒ true | _ ⇒ false]
175 |N ⇒ match m2 with [N ⇒ true | _ ⇒ false]].
177 definition tuples_of_pairs ≝ λn.λh.map … (λp.tuple_of_pair n h p).
179 definition flatten ≝ λA.foldr (list A) (list A) (append A) [].
181 lemma wftable: ∀n,h,l.table_TM (S n) (flatten ? (tuples_of_pairs n h l)).
182 #n #h #l elim l // -l #a #tl #Hind
183 whd in match (flatten … (tuples_of_pairs …));
187 lemma flatten_to_mem: ∀A,n,l,l1,l2.∀a:list A. 0 < n →
188 (∀x. mem ? x l → |x| = n) → |a| = n → flatten ? l = l1@a@l2 →
189 (∃q.|l1| = n*q) → mem ? a l.
191 [normalize #l1 #l2 #a #posn #Hlen #Ha #Hnil @False_ind
192 cut (|a|=0) [@daemon] /2/
193 |#hd #tl #Hind #l1 #l2 #a #posn #Hlen #Ha
194 whd in match (flatten ??); #Hflat * #q cases q
196 cut (a = hd) [@daemon] /2/
197 |#q1 #Hl1 lapply (split_exists … n l1 ?) //
198 * #l11 * #l12 * #Heql1 #Hlenl11 %2
199 @(Hind l12 l2 … posn ? Ha)
200 [#x #memx @Hlen %2 //
201 |@(append_l2_injective ? hd l11)
203 |>Hflat >Heql1 >associative_append %
205 |@(ex_intro …q1) @(injective_plus_r n)
206 <Hlenl11 in ⊢ (??%?); <length_append <Heql1 >Hl1 //
212 axiom match_decomp: ∀n,l,qin,cin,qout,cout,mv.
213 match_in_table (S n) qin cin qout cout mv l →
214 ∃l1,l2. l = l1@(mk_tuple qin cin qout cout mv)@l2 ∧
215 (∃q.|l1| = (tuple_length (S n))*q) ∧ tuple_TM (S n) (mk_tuple qin cin qout cout mv).
217 lemma match_tech: ∀n,l,qin,cin,qout,cout,mv.
218 (∀t. mem ? t l → |t| = |mk_tuple qin cin qout cout mv|) →
219 match_in_table (S n) qin cin qout cout mv (flatten ? l) →
220 ∃p. p = mk_tuple qin cin qout cout mv ∧ mem ? p l.
221 #n #l #qin #cin #qout #cout #mv #Hlen #Hmatch
222 @(ex_intro … (mk_tuple qin cin qout cout mv)) % //
225 lemma match_to_tuple: ∀n,h,l,qin,cin,qout,cout,mv.
226 match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_of_pairs n h l)) →
227 ∃p. p = mk_tuple qin cin qout cout mv ∧ mem ? p (tuples_of_pairs n h l).
228 #n #h #l #qin #cin #qout #cout #mv #Hmatch
229 @(ex_intro … (mk_tuple qin cin qout cout mv)) % //
230 cases (match_decomp … Hmatch) #l1 * #l2 * * #Hflat #Hlen #Htuple
231 @(flatten_to_mem … Hflat … Hlen)
234 |@(length_of_tuple … Htuple)
238 lemma mem_map: ∀A,B.∀f:A→B.∀l,b.
239 mem ? b (map … f l) → ∃a. mem ? a l ∧ f a = b.
241 [#b normalize @False_ind
242 |#a #tl #Hind #b normalize *
243 [#eqb @(ex_intro … a) /3/
244 |#memb cases (Hind … memb) #a * #mema #eqb
250 lemma match_to_pair: ∀n,h,l,qin,cin,qout,cout,mv.
251 match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_of_pairs n h l)) →
252 ∃p. tuple_of_pair n h p = mk_tuple qin cin qout cout mv ∧ mem ? p l.
253 #n #h #l #qin #cin #qout #cout #mv #Hmatch
254 cases (match_to_tuple … Hmatch)
256 cases(mem_map … (λp.tuple_of_pair n h p) … memb)
257 #p1 * #Hmem #H @(ex_intro … p1) % /2/
260 (* turning DeqMove into a DeqSet *)
261 lemma move_eq_true:∀m1,m2.
262 move_eq m1 m2 = true ↔ m1 = m2.
264 [* normalize [% #_ % |2,3: % #H destruct ]
265 |* normalize [1,3: % #H destruct |% #_ % ]
266 |* normalize [1,2: % #H destruct |% #_ % ]
269 definition DeqMove ≝ mk_DeqSet move move_eq move_eq_true.
271 unification hint 0 ≔ ;
273 (* ---------------------------------------- *) ⊢
276 unification hint 0 ≔ m1,m2;
278 (* ---------------------------------------- *) ⊢
279 move_eq m1 m2 ≡ eqb X m1 m2.
281 (* turning DeqMove into a FinSet *)
282 definition move_enum ≝ [L;R;N].
284 lemma move_enum_unique: uniqueb ? [L;R;N] = true.
287 lemma move_enum_complete: ∀x:move. memb ? x [L;R;N] = true.
291 mk_FinSet DeqMove [L;R;N] move_enum_unique move_enum_complete.
293 unification hint 0 ≔ ;
295 (* ---------------------------------------- *) ⊢
298 definition trans_source ≝ λn.FinProd (initN n) (FinOption FinBool).
299 definition trans_target ≝ λn.FinProd (initN n) (FinOption (FinProd FinBool FinMove)).
301 lemma match_to_trans:
302 ∀n.∀trans: trans_source n → trans_target n.
303 ∀h,qin,cin,qout,cout,mv.
304 match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_of_pairs n h (graph_enum ?? trans))) →
305 ∃s,t. tuple_of_pair n h 〈s,t〉 = mk_tuple qin cin qout cout mv
307 #n #trans #h #qin #cin #qout #cout #mv #Hmatch
308 cases (match_to_pair … Hmatch) -Hmatch * #s #t * #Heq #Hmem
309 @(ex_intro … s) @(ex_intro … t) % // @graph_enum_correct