axiom index_of_FS : ∀F:FinSet.F → nat.
(* unary bit representation (with a given length) of a certain number *)
-axiom unary_of_nat : nat → nat → nat.
+axiom unary_of_nat : nat → nat → (list bool).
axiom FinVector : Type[0] → nat → FinSet.
-definition binary_base_states ≝ initN 7.
+definition binary_base_states ≝ initN 6.
-definition bin0 : binary_base_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 7 (refl …)).
-definition bin1 : binary_base_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 7 (refl …)).
-definition bin2 : binary_base_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 7 (refl …)).
-definition bin3 : binary_base_states ≝ mk_Sig ?? 3 (leb_true_to_le 4 7 (refl …)).
-definition bin4 : binary_base_states ≝ mk_Sig ?? 4 (leb_true_to_le 5 7 (refl …)).
-definition bin5 : binary_base_states ≝ mk_Sig ?? 5 (leb_true_to_le 6 7 (refl …)).
-definition bin6 : binary_base_states ≝ mk_Sig ?? 6 (leb_true_to_le 7 7 (refl …)).
+definition bin0 : binary_base_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 6 (refl …)).
+definition bin1 : binary_base_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 6 (refl …)).
+definition bin2 : binary_base_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 6 (refl …)).
+definition bin3 : binary_base_states ≝ mk_Sig ?? 3 (leb_true_to_le 4 6 (refl …)).
+definition bin4 : binary_base_states ≝ mk_Sig ?? 4 (leb_true_to_le 5 6 (refl …)).
+definition bin5 : binary_base_states ≝ mk_Sig ?? 5 (leb_true_to_le 6 6 (refl …)).
definition states_binaryTM : FinSet → FinSet → FinSet ≝ λsig,states.
FinProd (FinProd states binary_base_states)
- (FinProd (FinOption sig) (initN (2 * (FS_crd sig)))).
+ (FinProd (FinOption sig) (initN (S (2 * (FS_crd sig))))).
axiom daemon : ∀T:Type[0].T.
+definition to_initN : ∀n,m.n < m → initN m ≝ λn,m,Hn.mk_Sig … n ….// qed.
+
+definition initN_pred : ∀n.∀m:initN n.initN n ≝ λn,m.mk_Sig … (pred (pi1 … m)) ….
+cases m #m0 /2 by le_to_lt_to_lt/ qed.
+
(* controllare i contatori, molti andranno incrementati di uno *)
definition trans_binaryTM : ∀sig,states:FinSet.
(states × (option sig) → states × (option sig) × move) →
≝ λsig,states,trans,p.
let 〈s,a〉 ≝ p in
let 〈s0,phase,ch,count〉 ≝ s in
+ let (H1 : O < S (2*FS_crd sig)) ≝ ? in
+ let (H2 : FS_crd sig < S (2*FS_crd sig)) ≝ ? in
match pi1 … phase with
[ O ⇒ (*** PHASE 0: read ***)
- match a with
- [ Some a0 ⇒
- match count with
- [ O ⇒ 〈〈s0,1,ch,FS_crd sig〉,None ?,N〉
- | S k ⇒ if (a0 == true)
- then 〈〈s0,0,FS_nth sig k,k〉, None ?,R〉
- else 〈〈s0,0,ch,k〉,None ?,R〉 ]
- | None ⇒ (* Overflow position! *)
- 〈〈s0,4,None ?,0〉,None ?,R〉 ]
+ match pi1 … count with
+ [ O ⇒ 〈〈s0,bin1,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
+ | S k ⇒ match a with
+ [ Some a0 ⇒ if (a0 == true)
+ then 〈〈s0,bin0,FS_nth sig k,initN_pred … count〉, None ?,R〉
+ else 〈〈s0,bin0,ch,initN_pred … count〉,None ?,R〉
+ | None ⇒ (* Overflow position! *)
+ 〈〈s0,bin4,None ?,to_initN 0 ? H1〉,None ?,R〉 ] ]
| S phase ⇒ match phase with
[ O ⇒ (*** PHASE 1: restart ***)
- match count with
- [ O ⇒ 〈〈s0,2,ch,FS_crd sig〉,None ?,N〉
- | S k ⇒ 〈〈s0,1,ch,k〉,None ?,L〉 ]
+ match pi1 … count with
+ [ O ⇒ 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
+ | S k ⇒ 〈〈s0,bin1,ch,initN_pred … count〉,None ?,L〉 ]
| S phase ⇒ match phase with
[ O ⇒ (*** PHASE 2: write ***)
let 〈s',a',mv〉 ≝ trans 〈s0,ch〉 in
- match count with
+ match pi1 … count with
[ O ⇒ let mv' ≝ match mv with [ R ⇒ N | _ ⇒ L ] in
let count' ≝ match mv with [ R ⇒ 0 | N ⇒ FS_crd sig | L ⇒ 2*(FS_crd sig) ] in
- 〈〈s',3,ch,count'〉,None ?,mv'〉
+ 〈〈s',bin3,ch,to_initN count' ??〉,None ?,mv'〉
| S k ⇒ match a' with
- [ None ⇒ 〈〈s0,2,ch,k〉,None ?,R〉
- | Some a0' ⇒ let out ≝ (FS_nth k == a') in
- 〈〈s0,2,ch,k〉,Some ? out,R〉 ]
+ [ None ⇒ 〈〈s0,bin2,ch,initN_pred … count〉,None ?,R〉
+ | Some a0' ⇒ let out ≝ (FS_nth ? k == a') in
+ 〈〈s0,bin2,ch,initN_pred … count〉,Some ? out,R〉 ]
]
| S phase ⇒ match phase with
[ O ⇒ (*** PHASE 3: move head left ***)
- match count with
- [ O ⇒ 〈〈s0,6,ch,O〉, None ?,N〉
- | S k ⇒ 〈〈s0,3,ch,k〉, None ?,L〉 ]
+ match pi1 … count with
+ [ O ⇒ 〈〈s0,bin0,None ?,to_initN (FS_crd sig) ? H2〉, None ?,N〉 (* the end: restart *)
+ | S k ⇒ 〈〈s0,bin3,ch,initN_pred … count〉, None ?,L〉 ]
| S phase ⇒ match phase with
[ O ⇒ (*** PHASE 4: check position ***)
match a with
- [ None ⇒ (* niltape/rightof: we can write *) 〈〈s0,2,ch,FS_crd sig〉,None ?,N〉
+ [ None ⇒ (* niltape/rightof: we can write *) 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
| Some _ ⇒ (* leftof *)
let 〈s',a',mv〉 ≝ trans 〈s0,ch〉 in
match a' with
- [ None ⇒ (* we don't write anything: go to end of 2 *) 〈〈s0,2,ch,0〉,None ?,N〉
- | Some _ ⇒ (* extend tape *) 〈〈s0,5,ch,FS_crd sig〉,None ?,L〉 ]
+ [ None ⇒ (* we don't write anything: go to end of 2 *) 〈〈s0,bin2,ch,to_initN 0 ? H1〉,None ?,N〉
+ | Some _ ⇒ (* extend tape *) 〈〈s0,bin5,ch,to_initN (FS_crd sig) ? H2〉,None ?,L〉 ]
]
- | S phase ⇒ match phase with
- [ O ⇒ (*** PHASE 5: left extension ***)
- match count with
- [ O ⇒ 〈〈s0,2,ch,FS_crd sig〉,None ?,N〉
- | S k ⇒ 〈〈s0,5,ch,k〉,Some ? false,L〉 ]
- | S _ ⇒ (*** PHASE 6: stop ***) 〈s,None ?,N〉 ]]]]]].
+ | S _ ⇒ (*** PHASE 5: left extension ***)
+ match pi1 … count with
+ [ O ⇒ 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
+ | S k ⇒ 〈〈s0,bin5,ch,initN_pred … count〉,Some ? false,L〉 ]]]]]].
+[2,3: //]
+whd in match count'; cases mv whd in ⊢ (?%?); //
+qed.
+
+definition halt_binaryTM : ∀sig,M.states_binaryTM sig (states sig M) → bool ≝
+ λsig,M,s.let 〈s0,phase,ch,count〉 ≝ s in
+ pi1 … phase == O ∧ halt sig M s0.
(*
* Una mk_binaryTM prende in input una macchina M e produce una macchina che:
* - ha per alfabeto FinBool
- * - ha stati di tipo (states … M) × (initN 3) × (initN (dimensione dell'alfabeto di M))
+ * - ha stati di tipo ((states … M) × (initN 7)) ×
+ ((option sig) × (initN (2*dimensione dell'alfabeto di M + 1))
* dove il primo elemento corrisponde allo stato della macchina input,
* il secondo identifica la fase (lettura, scrittura, spostamento)
- * il terzo è un contatore
- * - (la funzione di transizione è complessa al punto di rendere discutibile
+ * il terzo identifica il carattere oggetto letto
+ * il quarto è un contatore
+ * - la funzione di transizione viene prodotta da trans_binaryTM
+ * - la funzione di arresto viene prodotta da halt_binaryTM
*)
definition mk_binaryTM ≝
- λsig.λM:TM sig.mk_TM FinBool (FinProd (states … M) (FinProd (initN 3) (initN
-{ no_states : nat;
- pos_no_states : (0 < no_states);
- ntrans : trans_source no_states → trans_target no_states;
- nhalt : initN no_states → bool
-}.
\ No newline at end of file
+ λsig.λM:TM sig.
+ mk_TM FinBool (states_binaryTM sig (states sig M))
+ (trans_binaryTM sig (states sig M) (trans sig M))
+ (〈start sig M,bin0,None ?,FS_crd sig〉) (halt_binaryTM sig M).// qed.
+
+definition bin_char ≝ λsig,ch.unary_of_nat (FS_crd sig) (index_of_FS sig ch).
+
+definition bin_current ≝ λsig,t.match current ? t with
+[ None ⇒ [ ] | Some c ⇒ bin_char sig c ].
+
+definition tape_bin_lift ≝ λsig,t.
+let ls' ≝ flatten ? (map ?? (bin_char sig) (left ? t)) in
+let c' ≝ option_hd ? (bin_current sig t) in
+let rs' ≝ tail ? (bin_current sig t)@flatten ? (map ?? (bin_char sig) (right ? t)) in
+ mk_tape ? ls' c' rs'.
+
+definition R_bin_lift ≝ λsig,R,t1,t2.
+ ∃u1.t1 = tape_bin_lift sig u1 →
+ ∃u2.t2 = tape_bin_lift sig u2 ∧ R u1 u2.
+
+definition state_bin_lift :
+ ∀sig.∀M:TM sig.states sig M → states ? (mk_binaryTM ? M)
+ ≝ λsig,M,q.〈q,bin0,None ?,FS_crd sig〉.// qed.
+
+lemma lift_halt_binaryTM :
+ ∀sig,M,q.halt sig M q = halt ? (mk_binaryTM sig M) (state_bin_lift ? M q).
+// qed.
+
+lemma binaryTM_bin0_bin1 :
+ ∀sig,M,t,q,ch.
+ step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,O〉) t)
+ = mk_config ?? (〈q,bin1,ch,to_initN (FS_crd sig) ??〉) t. //
+qed.
+
+lemma binaryTM_bin0_bin4 :
+ ∀sig,M,t,q,ch,k.
+ current ? t = None ? → S k <S (2*FS_crd sig) →
+ step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t)
+ = mk_config ?? (〈q,bin4,None ?,to_initN 0 ??〉) (tape_move ? t R). [2,3://]
+#sig #M #t #q #ch #k #Hcur #Hk
+whd in match (step ???); whd in match (trans ???);
+>Hcur %
+qed.
+
+lemma binaryTM_bin0_true :
+ ∀sig,M,t,q,ch,k.
+ current ? t = Some ? true → S k <S (2*FS_crd sig) →
+ step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t)
+ = mk_config ?? (〈q,bin0,FS_nth sig k,to_initN k ??〉) (tape_move ? t R).[2,3:/2 by lt_S_to_lt/]
+#sig #M #t #q #ch #k #Hcur #Hk
+whd in match (step ???); whd in match (trans ???);
+>Hcur %
+qed.
+
+lemma binaryTM_bin0_false :
+ ∀sig,M,t,q,ch,k.
+ current ? t = Some ? false → S k <S (2*FS_crd sig) →
+ step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t)
+ = mk_config ?? (〈q,bin0,ch,to_initN k ??〉) (tape_move ? t R).[2,3:/2 by lt_S_to_lt/]
+#sig #M #t #q #ch #k #Hcur #Hk
+whd in match (step ???); whd in match (trans ???);
+>Hcur %
+qed.
+
+(* to be checked *)
+axiom binary_to_bin_char :∀sig,csl,csr,a.
+ csl@true::csr=bin_char sig a → FS_nth ? (length ? csr) = Some ? a.
+
+lemma binaryTM_phase0_midtape_aux :
+ ∀sig,M,q,ls,a,rs,k.
+ halt sig M q=false →
+ ∀csr,csl,t,ch.length ? csr < S (2*FS_crd sig) →
+ t = mk_tape ? (reverse ? csl@ls) (option_hd ? (csr@rs)) (tail ? (csr@rs)) →
+ csl@csr = bin_char sig a →
+ |csl@csr| = FS_crd sig →
+ (index_of_FS ? a < |csl| → ch = Some ? a) →
+ loopM ? (mk_binaryTM sig M) (S (length ? csr) + k)
+ (mk_config ?? (〈q,bin0,ch,length ? csr〉) t)
+ = loopM ? (mk_binaryTM sig M) k
+ (mk_config ?? (〈q,bin1,Some ? a,FS_crd sig〉)
+ (mk_tape ? (reverse ? (bin_char ? a)@ls) (option_hd ? rs) (tail ? rs))). [2,3:/2 by O/]
+#sig #M #q #ls #a #rs #k #Hhalt #csr elim csr
+[ #csl #t #ch #Hlen #Ht >append_nil #Hcsl #Hlencsl #Hch >loopM_unfold >loop_S_false [|normalize //]
+ >Hch [| >Hlencsl (* lemmatize *) @daemon]
+ <loopM_unfold @eq_f >binaryTM_bin0_bin1 @eq_f >Ht
+ whd in match (step ???); whd in match (trans ???); <Hcsl %
+| #c cases c
+ [ #csr0 #IH #csl #t #ch #Hlen #Ht #Heq #Hcrd #Hch >loopM_unfold >loop_S_false [|normalize //]
+ <loopM_unfold lapply (binary_to_bin_char … Heq) #Ha >binaryTM_bin0_true
+ [| >Ht % ]
+ lapply (IH (csl@[true]) (tape_move FinBool t R) ??????)
+ [ //
+ | >associative_append @Hcrd
+ | >associative_append @Heq
+ | >Ht whd in match (option_hd ??) in ⊢ (??%?); whd in match (tail ??) in ⊢ (??%?);
+ cases csr0
+ [ cases rs
+ [ normalize >rev_append_def >rev_append_def >reverse_append %
+ | #r1 #rs1 normalize >rev_append_def >rev_append_def >reverse_append % ]
+ | #c1 #csr1 normalize >rev_append_def >rev_append_def >reverse_append % ]
+ | /2 by lt_S_to_lt/
+ |]
+ #H whd in match (plus ??); >H @eq_f @eq_f2 %
+ | #csr0 #IH #csl #t #ch #Hlen #Ht #Heq #Hcrd #Hch >loopM_unfold >loop_S_false [|normalize //]
+ <loopM_unfold >binaryTM_bin0_false [| >Ht % ]
+ lapply (IH (csl@[false]) (tape_move FinBool t R) ??????)
+ [6: @ch
+ | (* by cases: if index < |csl|, then Hch, else False *)
+ @daemon
+ | >associative_append @Hcrd
+ | >associative_append @Heq
+ | >Ht whd in match (option_hd ??) in ⊢ (??%?); whd in match (tail ??) in ⊢ (??%?);
+ cases csr0
+ [ cases rs
+ [ normalize >rev_append_def >rev_append_def >reverse_append %
+ | #r1 #rs1 normalize >rev_append_def >rev_append_def >reverse_append % ]
+ | #c1 #csr1 normalize >rev_append_def >rev_append_def >reverse_append % ]
+ | /2 by lt_S_to_lt/
+ |]
+ #H whd in match (plus ??); >H @eq_f @eq_f2 %
+ ]
+]
+qed.
+
+lemma binaryTM_phase0_midtape :
+ ∀sig,M,t,q,ls,a,rs,ch,k.
+ halt sig M q=false →
+ t = mk_tape ? ls (option_hd ? (bin_char ? a)) (tail ? (bin_char sig a@rs)) →
+ loopM ? (mk_binaryTM sig M) (S (length ? (bin_char ? a)) + k)
+ (mk_config ?? (〈q,bin0,ch,length ? (bin_char ? a)〉) t)
+ = loopM ? (mk_binaryTM sig M) k
+ (mk_config ?? (〈q,bin1,Some ? a,FS_crd sig〉)
+ (mk_tape ? (reverse ? (bin_char ? a)@ls) (option_hd ? rs) (tail ? rs))). [|@daemon|//]
+#sig #M #t #q #ls #a #rs #ch #k #Hhalt #Ht
+cut (∃c,cl.bin_char sig a = c::cl) [@daemon] * #c * #cl #Ha >Ha
+>(binaryTM_phase0_midtape_aux ? M q ls a rs ? ? (c::cl) [ ] t ch) //
+[| normalize #Hfalse @False_ind cases (not_le_Sn_O ?) /2/
+| <Ha (* |bin_char sig ?| = FS_crd sig *) @daemon
+| >Ha %
+| >Ht >Ha % ]
+<Ha %
+qed.
+
+lemma binaryTM_phase0_None :
+ ∀sig,M,t,q,ch,k,n.
+ n < 2*FS_crd sig →
+ halt sig M q=false →
+ current ? t = None ? →
+ loopM ? (mk_binaryTM sig M) (S k) (mk_config ?? (〈q,bin0,ch,S n〉) t)
+ = loopM ? (mk_binaryTM sig M) k
+ (mk_config ?? (〈q,bin4,None ?,to_initN O ??〉) (tape_move ? t R)). [2,3: /2 by le_to_lt_to_lt/ ]
+#sig #M #t #q #ch #k #n #Hn #Hhalt cases t
+[ >loopM_unfold >loop_S_false [|@Hhalt] //
+| #r0 #rs0 >loopM_unfold >loop_S_false [|@Hhalt] //
+| #l0 #ls0 >loopM_unfold >loop_S_false [|@Hhalt] //
+| #ls #cur #rs normalize in ⊢ (%→?); #H destruct (H) ]
+qed.
+
+lemma binaryTM_bin1_O :
+ ∀sig,M,t,q,ch.
+ step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin1,ch,O〉) t)
+ = mk_config ?? (〈q,bin2,ch,to_initN (FS_crd sig) ??〉) t. [2,3://]
+#sig #M #t #q #ch %
+qed.
+
+lemma binaryTM_bin1_S :
+ ∀sig,M,t,q,ch,k. S k <S (2*FS_crd sig) →
+ step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin1,ch,S k〉) t)
+ = mk_config ?? (〈q,bin1,ch,to_initN k ??〉) (tape_move ? t L). [2,3:/2 by lt_S_to_lt/]
+#sig #M #t #q #ch #k #HSk %
+qed.
+
+lemma binaryTM_phase1 :
+ ∀sig,M,q,ls1,ls2,cur,rs,ch,k.
+ |ls1| = FS_crd sig → (cur = None ? → rs = [ ]) →
+ loopM ? (mk_binaryTM sig M) (S (FS_crd sig) + k)
+ (mk_config ?? (〈q,bin1,ch,FS_crd sig〉) (mk_tape ? (ls1@ls2) cur rs))
+ = loopM ? (mk_binaryTM sig M) k
+ (mk_config ?? (〈q,bin2,ch,FS_crd sig〉)
+ (mk_tape ? ls2 (option_hd ? (reverse ? ls1@option_cons ? cur rs))
+ (tail ? (reverse ? ls1@option_cons ? cur rs)))). [2,3:/2 by O/]
+cut (∀sig,M,q,ls1,ls2,ch,k,n,cur,rs.
+ |ls1| = n → n<S (2*FS_crd sig) → (cur = None ? → rs = [ ]) →
+ loopM ? (mk_binaryTM sig M) (S n + k)
+ (mk_config ?? (〈q,bin1,ch,n〉) (mk_tape ? (ls1@ls2) cur rs))
+ = loopM ? (mk_binaryTM sig M) k
+ (mk_config ?? (〈q,bin2,ch,FS_crd sig〉)
+ (mk_tape ? ls2 (option_hd ? (reverse ? ls1@option_cons ? cur rs))
+ (tail ? (reverse ? ls1@option_cons ? cur rs))))) [1,2://]
+[ #sig #M #q #ls1 #ls2 #ch #k elim ls1
+ [ #n normalize in ⊢ (%→?); #cur #rs #Hn <Hn #Hcrd #Hcur >loopM_unfold >loop_S_false [| % ]
+ >binaryTM_bin1_O cases cur in Hcur;
+ [ #H >(H (refl ??)) -H %
+ | #cur' #_ % ]
+ | #l0 #ls0 #IH * [ #cur #rs normalize in ⊢ (%→?); #H destruct (H) ]
+ #n #cur #rs normalize in ⊢ (%→?); #H destruct (H) #Hlt #Hcur
+ >loopM_unfold >loop_S_false [|%] >binaryTM_bin1_S
+ <(?:mk_tape ? (ls0@ls2) (Some ? l0) (option_cons ? cur rs) =
+ tape_move FinBool (mk_tape FinBool ((l0::ls0)@ls2) cur rs) L)
+ [| cases cur in Hcur; [ #H >(H ?) // | #cur' #_ % ] ]
+ >(?:loop (config FinBool (states FinBool (mk_binaryTM sig M))) (S (|ls0|)+k)
+ (step FinBool (mk_binaryTM sig M))
+ (λc:config FinBool (states FinBool (mk_binaryTM sig M))
+ .halt FinBool (mk_binaryTM sig M)
+ (cstate FinBool (states FinBool (mk_binaryTM sig M)) c))
+ (mk_config FinBool (states FinBool (mk_binaryTM sig M))
+ 〈q,bin1,ch,to_initN (|ls0|) (S (2*FS_crd sig))
+ (lt_S_to_lt (|ls0|) (S (2*FS_crd sig)) Hlt)〉
+ (mk_tape FinBool (ls0@ls2) (Some FinBool l0) (option_cons FinBool cur rs)))
+ = loopM FinBool (mk_binaryTM sig M) k
+ (mk_config FinBool (states FinBool (mk_binaryTM sig M))
+ 〈q,bin2,〈ch,FS_crd sig〉〉
+ (mk_tape FinBool ls2
+ (option_hd FinBool (reverse FinBool ls0@l0::option_cons FinBool cur rs))
+ (tail FinBool (reverse FinBool ls0@l0::option_cons FinBool cur rs)))))
+ [| /2/
+ | >(?: l0::option_cons ? cur rs = option_cons ? (Some ? l0) (option_cons ? cur rs)) [| % ]
+ @trans_eq [|| @(IH ??? (refl ??)) [ /2 by lt_S_to_lt/ | #H destruct (H) ] ]
+ %
+ ]
+ >reverse_cons >associative_append %
+ ]
+| #Hcut #sig #M #q #ls1 #ls2 #cur #rs #ch #k #Hlen @Hcut // ]
+qed.
+
+lemma binaryTM_bin3_O :
+ ∀sig,M,t,q,ch.
+ step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin3,ch,O〉) t)
+ = mk_config ?? (〈q,bin0,None ?,to_initN (FS_crd sig) ??〉) t. [2,3://]
+#sig #M #t #q #ch %
+qed.
+
+lemma binaryTM_bin3_S :
+ ∀sig,M,t,q,ch,k. S k <S (2*FS_crd sig) →
+ step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin3,ch,S k〉) t)
+ = mk_config ?? (〈q,bin3,ch,to_initN k ??〉) (tape_move ? t L). [2,3:/2 by lt_S_to_lt/]
+#sig #M #t #q #ch #k #HSk %
+qed.
+
+lemma binaryTM_phase3 :∀sig,M,q,ls1,ls2,ch,k,n,cur,rs.
+ |ls1| = n → n<S (2*FS_crd sig) → (cur = None ? → rs = [ ]) →
+ loopM ? (mk_binaryTM sig M) (S n + k)
+ (mk_config ?? (〈q,bin3,ch,n〉) (mk_tape ? (ls1@ls2) cur rs))
+ = loopM ? (mk_binaryTM sig M) k
+ (mk_config ?? (〈q,bin0,None ?,FS_crd sig〉)
+ (mk_tape ? ls2 (option_hd ? (reverse ? ls1@option_cons ? cur rs))
+ (tail ? (reverse ? ls1@option_cons ? cur rs)))). [2,3://]
+#sig #M #q #ls1 #ls2 #ch #k elim ls1
+[ #n normalize in ⊢ (%→?); #cur #rs #Hn <Hn #Hcrd #Hcur >loopM_unfold >loop_S_false [| % ]
+ >binaryTM_bin3_O cases cur in Hcur;
+ [ #H >(H (refl ??)) -H %
+ | #cur' #_ % ]
+| #l0 #ls0 #IH * [ #cur #rs normalize in ⊢ (%→?); #H destruct (H) ]
+ #n #cur #rs normalize in ⊢ (%→?); #H destruct (H) #Hlt #Hcur
+ >loopM_unfold >loop_S_false [|%] >binaryTM_bin3_S
+ <(?:mk_tape ? (ls0@ls2) (Some ? l0) (option_cons ? cur rs) =
+ tape_move FinBool (mk_tape FinBool ((l0::ls0)@ls2) cur rs) L)
+ [| cases cur in Hcur; [ #H >(H ?) // | #cur' #_ % ] ]
+ >(?:loop (config FinBool (states FinBool (mk_binaryTM sig M))) (S (|ls0|)+k)
+ (step FinBool (mk_binaryTM sig M))
+ (λc:config FinBool (states FinBool (mk_binaryTM sig M))
+ .halt FinBool (mk_binaryTM sig M)
+ (cstate FinBool (states FinBool (mk_binaryTM sig M)) c))
+ (mk_config FinBool (states FinBool (mk_binaryTM sig M))
+ 〈q,bin3,ch,to_initN (|ls0|) (S (2*FS_crd sig))
+ (lt_S_to_lt (|ls0|) (S (2*FS_crd sig)) Hlt)〉
+ (mk_tape FinBool (ls0@ls2) (Some FinBool l0) (option_cons FinBool cur rs)))
+ = loopM FinBool (mk_binaryTM sig M) k
+ (mk_config FinBool (states FinBool (mk_binaryTM sig M))
+ 〈q,bin0,〈None ?,FS_crd sig〉〉
+ (mk_tape FinBool ls2
+ (option_hd FinBool (reverse FinBool ls0@l0::option_cons FinBool cur rs))
+ (tail FinBool (reverse FinBool ls0@l0::option_cons FinBool cur rs)))))
+ [| /2/
+ | >(?: l0::option_cons ? cur rs = option_cons ? (Some ? l0) (option_cons ? cur rs)) [| % ]
+ @trans_eq [|| @(IH ??? (refl ??)) [ /2 by lt_S_to_lt/ | #H destruct (H) ] ]
+ %
+ ]
+ >reverse_cons >associative_append %
+]
+qed.
+
+STOP
+
+lemma binaryTM_loop :
+ ∀sig,M,i,t,q,tf,qf.
+ loopM sig M i (mk_config ?? q t) = Some ? (mk_config ?? qf tf) →
+ ∃k.loopM ? (mk_binaryTM sig M) k
+ (mk_config ?? (state_bin_lift ? M q) (tape_bin_lift ? t)) =
+ Some ? (mk_config ?? (state_bin_lift ? M qf) (tape_bin_lift ? tf)).
+#sig #M #i elim i
+[ #t #q #qf #tf change with (None ?) in ⊢ (??%?→?); #H destruct (H)
+| -i #i #IH #t #q #tf #qf
+ >loopM_unfold
+ lapply (refl ? (halt sig M (cstate ?? (mk_config ?? q t))))
+ cases (halt ?? q) in ⊢ (???%→?); #Hhalt
+ [ >(loop_S_true ??? (λc.halt ?? (cstate ?? c)) (mk_config ?? q t) Hhalt)
+ #H destruct (H) %{1} >loopM_unfold >loop_S_true // ]
+ (* interesting case: more than one step *)
+ >(loop_S_false ??? (λc.halt ?? (cstate ?? c)) (mk_config ?? q t) Hhalt)
+ <loopM_unfold >(config_expand ?? (step ???)) #Hloop
+ lapply (IH … Hloop) -IH * #k0 #IH <config_expand in Hloop; #Hloop
+ %{(S k0)}
+
+
+
+(*
+theorem sem_binaryTM : ∀sig,M.
+ mk_binaryTM sig M ⊫ R_bin_lift ? (R_TM ? M (start ? M)).
+#sig #M #t #i generalize in match t; -t
+@(nat_elim1 … i) #m #IH #intape #outc #Hloop
+
+*)
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projects / helm.git / blobdiff