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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "turing/multi_universal/moves.ma".
16 include "turing/if_multi.ma".
17 include "turing/inject.ma".
18 include "turing/basic_machines.ma".
20 definition compare_states ≝ initN 3.
22 definition comp0 : compare_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
23 definition comp1 : compare_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
24 definition comp2 : compare_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
28 0) (x,x) → (x,x)(R,R) → 1
35 definition trans_compare_step ≝
36 λi,j.λsig:FinSet.λn.λis_endc.
37 λp:compare_states × (Vector (option sig) (S n)).
40 [ O ⇒ match nth i ? a (None ?) with
41 [ None ⇒ 〈comp2,null_action ? n〉
42 | Some ai ⇒ match nth j ? a (None ?) with
43 [ None ⇒ 〈comp2,null_action ? n〉
44 | Some aj ⇒ if notb (is_endc ai) ∧ ai == aj
45 then 〈comp1,change_vec ? (S n)
46 (change_vec ? (S n) (null_action ? n) (Some ? 〈ai,R〉) i)
48 else 〈comp2,null_action ? n〉 ]
51 [ O ⇒ (* 1 *) 〈comp1,null_action ? n〉
52 | S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ].
54 definition compare_step ≝
56 mk_mTM sig n compare_states (trans_compare_step i j sig n is_endc)
57 comp0 (λq.q == comp1 ∨ q == comp2).
59 definition R_comp_step_true ≝
60 λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
63 current ? (nth i ? int (niltape ?)) = Some ? x ∧
64 current ? (nth j ? int (niltape ?)) = Some ? x ∧
67 (tape_move ? (nth i ? int (niltape ?)) (Some ? 〈x,R〉)) i)
68 (tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) j.
70 definition R_comp_step_false ≝
71 λi,j:nat.λsig,n,is_endc.λint,outt: Vector (tape sig) (S n).
72 ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
73 current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
74 current ? (nth i ? int (niltape ?)) = None ? ∨
75 current ? (nth j ? int (niltape ?)) = None ?) ∧ outt = int.
77 lemma comp_q0_q2_null :
78 ∀i,j,sig,n,is_endc,v.i < S n → j < S n →
79 (nth i ? (current_chars ?? v) (None ?) = None ? ∨
80 nth j ? (current_chars ?? v) (None ?) = None ?) →
81 step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v)
82 = mk_mconfig ??? comp2 v.
83 #i #j #sig #n #is_endc #v #Hi #Hj
84 whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?);
87 [ whd in ⊢ (??(???%)?); >Hcurrent %
88 | whd in ⊢ (??(???????(???%))?); >Hcurrent @tape_move_null_action ]
90 [ whd in ⊢ (??(???%)?); >Hcurrent cases (nth i ?? (None sig)) //
91 | whd in ⊢ (??(???????(???%))?); >Hcurrent
92 cases (nth i ?? (None sig)) [|#x] @tape_move_null_action ] ]
95 lemma comp_q0_q2_neq :
96 ∀i,j,sig,n,is_endc,v.i < S n → j < S n →
97 ((∃x.nth i ? (current_chars ?? v) (None ?) = Some ? x ∧ is_endc x = true) ∨
98 nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?)) →
99 step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v)
100 = mk_mconfig ??? comp2 v.
101 #i #j #sig #n #is_endc #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
102 cases (nth i ?? (None ?)) in ⊢ (???%→?);
103 [ #Hnth #_ @comp_q0_q2_null // % //
104 | #ai #Hai lapply (refl ? (nth j ?(current_chars ?? v)(None ?)))
105 cases (nth j ?? (None ?)) in ⊢ (???%→?);
106 [ #Hnth #_ @comp_q0_q2_null // %2 //
108 [ * #c * >Hai #Heq #Hendc whd in ⊢ (??%?);
109 >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
110 [ whd in match (trans ????); >Hai >Haj destruct (Heq)
111 whd in ⊢ (??(???%)?); >Hendc //
112 | whd in match (trans ????); >Hai >Haj destruct (Heq)
113 whd in ⊢ (??(???????(???%))?); >Hendc @tape_move_null_action
116 whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
117 [ whd in match (trans ????); >Hai >Haj
118 whd in ⊢ (??(???%)?); cut ((¬is_endc ai∧ai==aj)=false)
119 [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) // |#Hcut >Hcut //]
120 | whd in match (trans ????); >Hai >Haj
121 whd in ⊢ (??(???????(???%))?); cut ((¬is_endc ai∧ai==aj)=false)
122 [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) //
123 |#Hcut >Hcut @tape_move_null_action
132 ∀i,j,sig,n,is_endc,v,a.i ≠ j → i < S n → j < S n →
133 nth i ? (current_chars ?? v) (None ?) = Some ? a → is_endc a = false →
134 nth j ? (current_chars ?? v) (None ?) = Some ? a →
135 step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) =
139 (tape_move ? (nth i ? v (niltape ?)) (Some ? 〈a,R〉)) i)
140 (tape_move ? (nth j ? v (niltape ?)) (Some ? 〈a,R〉)) j).
141 #i #j #sig #n #is_endc #v #a #Heq #Hi #Hj #Ha1 #Hnotendc #Ha2
142 whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
143 [ whd in match (trans ????);
144 >Ha1 >Ha2 whd in ⊢ (??(???%)?); >Hnotendc >(\b ?) //
145 | whd in match (trans ????);
146 >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >Hnotendc >(\b ?) //
147 change with (change_vec ?????) in ⊢ (??(???????%)?);
148 <(change_vec_same … v j (niltape ?)) in ⊢ (??%?);
149 <(change_vec_same … v i (niltape ?)) in ⊢ (??%?);
150 >pmap_change >pmap_change >tape_move_null_action
151 @eq_f2 // @eq_f2 // >nth_change_vec_neq //
155 lemma sem_comp_step :
156 ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n →
157 compare_step i j sig n is_endc ⊨
158 [ comp1: R_comp_step_true i j sig n is_endc,
159 R_comp_step_false i j sig n is_endc ].
160 #i #j #sig #n #is_endc #Hneq #Hi #Hj #int
161 lapply (refl ? (current ? (nth i ? int (niltape ?))))
162 cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?);
165 [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ % <Hcuri in ⊢ (???%);
167 | normalize in ⊢ (%→?); #H destruct (H) ]
168 | #_ % // % %2 // ] ]
169 | #a #Ha lapply (refl ? (current ? (nth j ? int (niltape ?))))
170 cases (current ? (nth j ? int (niltape ?))) in ⊢ (???%→?);
173 [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2 <Hcurj in ⊢ (???%);
175 | normalize in ⊢ (%→?); #H destruct (H) ]
176 | #_ % // >Ha >Hcurj % % %2 % #H destruct (H) ] ]
178 cases (true_or_false (is_endc a)) #Haendc
181 [whd in ⊢ (??%?); >comp_q0_q2_neq //
182 % %{a} % // <Ha @sym_eq @nth_vec_map
183 | normalize in ⊢ (%→?); #H destruct (H) ]
184 | #_ % // % % % >Ha %{a} % // ]
186 |cases (true_or_false (a == b)) #Hab
189 [whd in ⊢ (??%?); >(comp_q0_q1 … a Hneq Hi Hj) //
190 [>(\P Hab) <Hb @sym_eq @nth_vec_map
191 |<Ha @sym_eq @nth_vec_map ]
192 | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) % // ]
193 | * #H @False_ind @H %
197 [whd in ⊢ (??%?); >comp_q0_q2_neq //
198 <(nth_vec_map ?? (current …) i ? int (niltape ?))
199 <(nth_vec_map ?? (current …) j ? int (niltape ?)) %2 >Ha >Hb
200 @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
201 | normalize in ⊢ (%→?); #H destruct (H) ]
202 | #_ % // % % %2 >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
209 definition compare ≝ λi,j,sig,n,is_endc.
210 whileTM … (compare_step i j sig n is_endc) comp1.
212 definition R_compare ≝
213 λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
214 ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
215 (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
216 current ? (nth i ? int (niltape ?)) = None ? ∨
217 current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
218 (∀ls,x,xs,ci,rs,ls0,rs0.
219 nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
220 nth j ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
221 (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
224 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
225 (mk_tape sig (reverse ? xs@x::ls0) (None ?) []) j) ∨
226 ∃cj,rs1.rs0 = cj::rs1 ∧
227 ((is_endc ci = true ∨ ci ≠ cj) →
229 (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
230 (midtape sig (reverse ? xs@x::ls0) cj rs1) j)).
232 lemma wsem_compare : ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n →
233 compare i j sig n is_endc ⊫ R_compare i j sig n is_endc.
234 #i #j #sig #n #is_endc #Hneq #Hi #Hj #ta #k #outc #Hloop
235 lapply (sem_while … (sem_comp_step i j sig n is_endc Hneq Hi Hj) … Hloop) //
236 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
237 [ #tc whd in ⊢ (%→?); * * [ * [ *
238 [* #curi * #Hcuri #Hendi #Houtc %
240 | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj #Hnotendc
242 >Hnthi in Hcuri; normalize in ⊢ (%→?); #H destruct (H)
243 >(Hnotendc ? (memb_hd … )) in Hendi; #H destruct (H)
247 | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj
248 >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
252 | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi >Hnthi in Hci;
253 normalize in ⊢ (%→?); #H destruct (H) ] ]
256 | #ls #x #xs #ci #rs #ls0 #rs0 #_ #Hnthj >Hnthj in Hcj;
257 normalize in ⊢ (%→?); #H destruct (H) ] ]
258 | #tc #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
260 [ >Hci >Hcj * [* #x0 * #H destruct (H) >Hendcx #H destruct (H)
261 |* [* #H @False_ind [cases H -H #H @H % | destruct (H)] | #H destruct (H)]]
262 | #ls #c0 #xs #ci #rs #ls0 #rs0 cases xs
263 [ #Hnthi #Hnthj #Hnotendc cases rs0 in Hnthj;
266 [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
267 >Hnthi in Hci;normalize #H destruct (H) %
268 | >(?:c0=x) [ >Hnthj % ]
269 >Hnthi in Hci;normalize #H destruct (H) % ]
270 | >Hd %2 %2 >nth_change_vec // >Hnthj % ]
271 | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // *
274 [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
275 >Hnthi in Hci;normalize #H destruct (H) %
276 | >(?:c0=x) [ >Hnthj % ]
277 >Hnthi in Hci;normalize #H destruct (H) % ]
278 | >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
279 >nth_change_vec // >Hnthi >Hnthj normalize % %{ci} % //
283 [ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
284 >Hnthi in Hci;normalize #H destruct (H) %
285 | >(?:c0=x) [ >Hnthj % ]
286 >Hnthi in Hci;normalize #H destruct (H) % ]
287 | >Hd %2 % % >nth_change_vec //
288 >nth_change_vec_neq [|@sym_not_eq //]
289 >nth_change_vec // >Hnthi >Hnthj normalize @(not_to_not … Hcir1)
293 |#x0 #xs0 #Hnthi #Hnthj #Hnotendc
294 cut (c0 = x) [ >Hnthi in Hci; normalize #H destruct (H) // ]
295 #Hcut destruct (Hcut) cases rs0 in Hnthj;
297 cases (IH2 (x::ls) x0 xs0 ci rs (x::ls0) [ ] ???) -IH2
298 [ * #_ #IH2 >IH2 >Hd >change_vec_commute in ⊢ (??%?); //
299 >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
301 | * #cj * #rs1 * #H destruct (H)
302 | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
304 | >Hd >nth_change_vec // >Hnthj %
305 | #c0 #Hc0 @Hnotendc @memb_cons @Hc0 ]
306 | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // #Hcir1
307 cases(IH2 (x::ls) x0 xs0 ci rs (x::ls0) (r1::rs1) ???)
309 | * #r1' * #rs1' * #H destruct (H) #Hc1r1 >Hc1r1 //
310 >Hd >change_vec_commute in ⊢ (??%?); //
311 >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
313 | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
315 | >Hd >nth_change_vec // >Hnthi >Hnthj %
316 | #c0 #Hc0 @Hnotendc @memb_cons @Hc0
320 lemma terminate_compare : ∀i,j,sig,n,is_endc,t.
321 i ≠ j → i < S n → j < S n →
322 compare i j sig n is_endc ↓ t.
323 #i #j #sig #n #is_endc #t #Hneq #Hi #Hj
324 @(terminate_while … (sem_comp_step …)) //
325 <(change_vec_same … t i (niltape ?))
326 cases (nth i (tape sig) t (niltape ?))
327 [ % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
328 |2,3: #a0 #al0 % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
329 | #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
330 [#t #ls #c % #t1 * #x * * * #Hendcx >nth_change_vec // normalize in ⊢ (%→?);
331 #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 %
332 #t2 * #x0 * * * #Hendcx0 >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
333 >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
334 |#r0 #rs0 #IH #t #ls #c % #t1 * #x * * >nth_change_vec //
335 normalize in ⊢ (%→?); #H destruct (H) #Hcur
336 >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
341 lemma sem_compare : ∀i,j,sig,n,is_endc.
342 i ≠ j → i < S n → j < S n →
343 compare i j sig n is_endc ⊨ R_compare i j sig n is_endc.
344 #i #j #sig #n #is_endc #Hneq #Hi #Hj @WRealize_to_Realize /2/