#sig * /2 by le_S/
qed.
-(* controllare i contatori, molti andranno incrementati di uno *)
+definition displ2_of_move ≝ λsig,mv.
+ match mv with
+ [ L ⇒ FS_crd sig
+ | N ⇒ O
+ | R ⇒ O ].
+
+lemma le_displ2_of_move : ∀sig,mv.displ2_of_move sig mv ≤ S (2*FS_crd sig).
+#sig * /2 by lt_to_le/
+qed.
+
+definition mv_tech ≝ λmv.match mv with [ N ⇒ N | _ ⇒ R ].
+
definition trans_binaryTM : ∀sig,states:FinSet.
(states × (option sig) → states × (option sig) × move) →
((states_binaryTM sig states) × (option bool) →
| None ⇒ (* Overflow position! *)
let 〈s',a',mv〉 ≝ trans 〈s0,None ?〉 in
match a' with
- [ None ⇒ (* we don't write anything: go to end of 2 *) 〈〈s0,bin2,None ?,to_initN 0 ? H1〉,None ?,N〉
+ [ None ⇒ (* we don't write anything: go to end of 3 *) 〈〈s',bin3,None ?,to_initN (displ2_of_move sig mv) ??〉,None ?,mv_tech mv〉
| Some _ ⇒ (* maybe extend tape *) 〈〈s0,bin4,None ?,to_initN O ? H1〉,None ?,R〉 ] ] ]
| S phase ⇒ match phase with
[ O ⇒ (*** PHASE 1: restart ***)
match pi1 … count with
[ O ⇒ 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,R〉
| S k ⇒ 〈〈s0,bin5,ch,initN_pred … count〉,Some ? false,L〉 ]]]]]].
-[2,3: /2 by lt_S_to_lt/] /2 by le_S_S/
+[ /2 by le_to_lt_to_lt/ | /2 by le_S_S/ |*: /2 by lt_S_to_lt/]
qed.
definition halt_binaryTM : ∀sig,M.states_binaryTM sig (states sig M) → bool ≝
let rs' ≝ (tail ? (opt_bin_char sig (current ? t))@bin_list ? (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〉./2 by lt_S_to_lt/ qed.
= mk_config ?? (〈q,bin1,ch,to_initN (FS_crd sig) ??〉) t. //
qed.
-lemma binaryTM_bin0_bin2 :
+lemma binaryTM_bin0_bin3 :
∀sig,M,t,q,ch,k,qn,mv.
current ? t = None ? → S k <S (2*FS_crd sig) →
〈qn,None ?,mv〉 = trans sig M 〈q,None ?〉 →
step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin0,ch,S k〉) t)
- = mk_config ?? (〈q,bin2,None ?,to_initN O ??〉) t. [2,3:/2 by transitive_lt/]
+ = mk_config ?? (〈qn,bin3,None ?,to_initN (displ2_of_move sig mv) ??〉) (tape_move ? t (mv_tech mv)). [|@le_S //|@le_S_S @le_displ2_of_move]
#sig #M #t #q #ch #k #qn #mv #Hcur #Hk #Htrans
whd in match (step ???); whd in match (trans ???);
>Hcur <Htrans %
lemma minus_tech : ∀a,b.a + b - a = b. // qed.
lemma binaryTM_phase0_midtape :
- ∀sig,M,t,q,ls,a,rs,ch,k.
- halt sig M q=false → S (FS_crd sig) ≤ k →
+ ∀sig,M,t,q,ls,a,rs,ch.
+ halt sig M q=false →
t = mk_tape ? ls (option_hd ? (bin_char ? a)) (tail ? (bin_char sig a)@rs) →
+ ∀k.S (FS_crd sig) ≤ k →
loopM ? (mk_binaryTM sig M) k
(mk_config ?? (〈q,bin0,ch,FS_crd sig〉) t)
= loopM ? (mk_binaryTM sig M) (k - S (FS_crd sig))
(mk_config ?? (〈q,bin1,Some ? a,FS_crd sig〉)
(mk_tape ? (reverse ? (bin_char ? a)@ls) (option_hd ? rs) (tail ? rs))). [|*:@le_S //]
-#sig #M #t #q #ls #a #rs #ch #k #Hhalt #Hk #Ht
-cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
+#sig #M #t #q #ls #a #rs #ch #Hhalt #Ht #k #Hk
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >(minus_tech (S (FS_crd sig)))
cut (∃c,cl.bin_char sig a = c::cl) [@daemon] * #c * #cl #Ha >Ha
cut (FS_crd sig = |bin_char sig a|) [@daemon] #Hlen
@(trans_eq ?? (loopM ? (mk_binaryTM ? M) (S (|c::cl|) + k0)
qed.
lemma binaryTM_phase0_None_None :
- ∀sig,M,t,q,ch,k,n,qn,mv.
- O < n → n < 2*FS_crd sig → O < k →
+ ∀sig,M,t,q,ch,n,qn,mv.
+ O < n → n < 2*FS_crd sig →
halt sig M q=false →
current ? t = None ? →
〈qn,None ?,mv〉 = trans sig M 〈q,None ?〉 →
+ ∀k.O < k →
loopM ? (mk_binaryTM sig M) k (mk_config ?? (〈q,bin0,ch,n〉) t)
= loopM ? (mk_binaryTM sig M) (k-1)
- (mk_config ?? (〈q,bin2,None ?,to_initN O ??〉) t). [2,3: /2 by transitive_lt/ ]
-#sig #M #t #q #ch #k #n #qn #mv #HOn #Hn #Hk #Hhalt
+ (mk_config ?? (〈qn,bin3,None ?,to_initN (displ2_of_move sig mv) ??〉) (tape_move ? t (mv_tech mv))). [| @le_S @le_S //|@le_S_S @le_displ2_of_move]
+#sig #M #t #q #ch #n #qn #mv #HOn #Hn #Hhalt #Hcur #Htrans #k #Hk
cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
cases (le_to_eq … HOn) #n0 #Hn0 destruct (Hn0)
-cases t
-[ >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin2 // /2 by refl, transitive_lt/
-| #r0 #rs0 >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin2 // /2 by refl, transitive_lt/
-| #l0 #ls0 >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin2 // /2 by refl, transitive_lt/
+lapply Htrans lapply Hcur -Htrans -Hcur cases t
+[ >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin3 //
+| #r0 #rs0 >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin3 //
+| #l0 #ls0 >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin3 //
| #ls #cur #rs normalize in ⊢ (%→?); #H destruct (H) ]
qed.
lemma binaryTM_phase0_None_Some :
- ∀sig,M,t,q,ch,k,n,qn,chn,mv.
- O < n → n < 2*FS_crd sig → O < k →
+ ∀sig,M,t,q,ch,n,qn,chn,mv.
+ O < n → n < 2*FS_crd sig →
halt sig M q=false →
current ? t = None ? →
〈qn,Some ? chn,mv〉 = trans sig M 〈q,None ?〉 →
+ ∀k.O < k →
loopM ? (mk_binaryTM sig M) k (mk_config ?? (〈q,bin0,ch,n〉) t)
= loopM ? (mk_binaryTM sig M) (k-1)
(mk_config ?? (〈q,bin4,None ?,to_initN O ??〉) (tape_move ? t R)). [2,3: /2 by transitive_lt/ ]
-#sig #M #t #q #ch #k #n #qn #chn #mv #HOn #Hn #Hk #Hhalt
+#sig #M #t #q #ch #n #qn #chn #mv #HOn #Hn #Hhalt #Hcur #Htrans #k #Hk
cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
cases (le_to_eq … HOn) #n0 #Hn0 destruct (Hn0)
-cases t
+lapply Htrans lapply Hcur -Hcur -Htrans cases t
[ >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin4 // /2 by refl, transitive_lt/
| #r0 #rs0 >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin4 // /2 by refl, transitive_lt/
| #l0 #ls0 >loopM_unfold >loop_S_false [|@Hhalt] #Hcur #Htrans >binaryTM_bin0_bin4 // /2 by refl, transitive_lt/
qed.
lemma binaryTM_phase1 :
- ∀sig,M,q,ls1,ls2,cur,rs,ch,k.
- S (FS_crd sig) ≤ k → |ls1| = FS_crd sig → (cur = None ? → rs = [ ]) →
+ ∀sig,M,q,ls1,ls2,cur,rs,ch.
+ |ls1| = FS_crd sig → (cur = None ? → rs = [ ]) →
+ ∀k.S (FS_crd sig) ≤ k →
loopM ? (mk_binaryTM sig M) k
(mk_config ?? (〈q,bin1,ch,FS_crd sig〉) (mk_tape ? (ls1@ls2) cur rs))
= loopM ? (mk_binaryTM sig M) (k - S (FS_crd sig))
]
>reverse_cons >associative_append %
]
-| #Hcut #sig #M #q #ls1 #ls2 #cur #rs #ch #k #Hk #Hlen
- cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech @Hcut // ]
+| #Hcut #sig #M #q #ls1 #ls2 #cur #rs #ch #Hlen #Hcur #k #Hk
+ cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech @Hcut /2/ ]
qed.
lemma binaryTM_bin2_O :
let rec iter (T:Type[0]) f n (t:T) on n ≝
match n with [ O ⇒ t | S n0 ⇒ iter T f n0 (f t) ].
-lemma binaryTM_phase2_None :∀sig,M,q,ch,qn,mv,k,n. S n ≤ k →
- ∀t.n≤S (2*FS_crd sig) →
+lemma binaryTM_phase2_None :∀sig,M,q,ch,qn,mv.
〈qn,None ?,mv〉 = trans sig M 〈q,ch〉 →
+ ∀n.n≤S (2*FS_crd sig) →
+ ∀t,k.S n ≤ k →
loopM ? (mk_binaryTM sig M) k
(mk_config ?? (〈q,bin2,ch,n〉) t)
= loopM ? (mk_binaryTM sig M) (k - S n)
(mk_config ?? (〈qn,bin3,ch,to_initN (displ_of_move sig mv) ??〉)
(iter ? (λt0.tape_move ? t0 R) n t)). [2,3: @le_S_S /2 by lt_S_to_lt/]
-#sig #M #q #ch #qn #mv #k #n #Hk
-cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
+#sig #M #q #ch #qn #mv #Htrans #n #Hn #t #k #Hk
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech lapply Hn lapply t -Hn -t
elim n
-[ #t #Hle #Htrans >loopM_unfold >loop_S_false //
+[ #t #Hle >loopM_unfold >loop_S_false //
>(binaryTM_bin2_O … Htrans) //
-| #n0 #IH #t #Hn0 #Htrans >loopM_unfold >loop_S_false //
+| #n0 #IH #t #Hn0 >loopM_unfold >loop_S_false //
>(binaryTM_bin2_S_None … Htrans) @(trans_eq ???? (IH …)) //
]
qed.
-lemma binaryTM_phase2_Some_of : ∀sig,M,q,ch,qn,chn,mv,ls,k.
- S (FS_crd sig) ≤ k → 〈qn,Some ? chn,mv〉 = trans sig M 〈q,ch〉 →
+lemma binaryTM_phase2_Some_of : ∀sig,M,q,ch,qn,chn,mv,ls.
+ 〈qn,Some ? chn,mv〉 = trans sig M 〈q,ch〉 →
+ ∀k.S (FS_crd sig) ≤ k →
loopM ? (mk_binaryTM sig M) k
(mk_config ?? (〈q,bin2,ch,FS_crd sig〉) (mk_tape ? ls (None ?) [ ]))
= loopM ? (mk_binaryTM sig M) (k - S (FS_crd sig))
cut (bin_char ? chn = reverse ? csl@(FS_nth ? n0 == Some ? chn)::fs0) [@daemon]
-Hbinchar #Hbinchar >Hbinchar @(trans_eq ???? (IH …)) //
[ %{fs0} >reverse_cons >associative_append @Hbinchar
- | whd in ⊢ (??%?); /2 by / ]
+ | whd in ⊢ (??%?); <Hcrd // ]
@eq_f @eq_f @eq_f3 //
]
]
-| #Hcut #sig #M #q #ch #qn #chn #mv #ls #k #Hk #Htrans
+| #Hcut #sig #M #q #ch #qn #chn #mv #ls #Htrans #k #Hk
@trans_eq
[3: @(trans_eq ???? (Hcut ??????? ls ? (FS_crd sig) ? Htrans …)) //
[3:@([ ]) | %{(bin_char ? chn)} % | % ]
]
qed.
-lemma binaryTM_phase2_Some_ow : ∀sig,M,q,ch,qn,chn,mv,ls,k,cs,rs.
- S (FS_crd sig) ≤ k → 〈qn,Some ? chn,mv〉 = trans sig M 〈q,ch〉 →
+lemma binaryTM_phase2_Some_ow : ∀sig,M,q,ch,qn,chn,mv,ls,cs,rs.
+ 〈qn,Some ? chn,mv〉 = trans sig M 〈q,ch〉 →
|cs| = FS_crd sig →
+ ∀k.S (FS_crd sig) ≤ k →
loopM ? (mk_binaryTM sig M) k
(mk_config ?? (〈q,bin2,ch,FS_crd sig〉)
(mk_tape ? ls (option_hd ? (cs@rs)) (tail ? (cs@rs))))
]
]
]
-| #Hcut #sig #M #q #ch #qn #chn #mv #ls #k #cs #rs #Hk #Htrans #Hcrd
+| #Hcut #sig #M #q #ch #qn #chn #mv #ls #cs #rs #Htrans #Hcrd #k #Hk
cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech @trans_eq
[3: @(trans_eq ???? (Hcut ??????? ls ?? cs Htrans [ ] …)) //
[ normalize % // | normalize @Hcrd | >Hcrd // ]
#sig #M #t #q #ch #k #HSk %
qed.
-lemma binaryTM_phase3 :∀sig,M,q,ch,k,n.
- S n ≤ k → n ≤ S (2*FS_crd sig) →
- ∀t.loopM ? (mk_binaryTM sig M) k
+lemma binaryTM_phase3 :∀sig,M,q,ch,n.
+ n ≤ S (2*FS_crd sig) →
+ ∀t,k.S n ≤ k →
+ loopM ? (mk_binaryTM sig M) k
(mk_config ?? (〈q,bin3,ch,n〉) t)
= loopM ? (mk_binaryTM sig M) (k - S n)
(mk_config ?? (〈q,bin0,None ?,FS_crd sig〉)
(iter ? (λt0.tape_move ? t0 L) n t)). [2,3: /2 by lt_S_to_lt, le_to_lt_to_lt/]
-#sig #M #q #ch #k #n #Hk
-cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech elim n
-[ #Hcrd #t >loopM_unfold >loop_S_false [| % ] >binaryTM_bin3_O //
-| #n0 #IH #Hlt #t >loopM_unfold >loop_S_false [|%] >binaryTM_bin3_S [|//]
- <IH [|/2 by lt_to_le/]
+#sig #M #q #ch #n #Hcrd #t #k #Hk
+cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >(minus_tech (S n) k0)
+lapply t lapply Hcrd -t -Hcrd elim n
+[ #Hcrd #t >loopM_unfold >loop_S_false [| % ] >binaryTM_bin3_O //
+| #n0 #IH #Hlt #t >loopM_unfold >loop_S_false [|%] >binaryTM_bin3_S [|@Hlt]
+ <IH [|@lt_to_le @Hlt ]
<loopM_unfold % ]
qed.
∀sig,M,t,q,ch.
current ? t = None ? →
step ? (mk_binaryTM sig M) (mk_config ?? (〈q,bin4,ch,O〉) t)
- = mk_config ?? (〈q,bin2,ch,to_initN (FS_crd sig) ??〉) t. [2,3: @le_S //]
+ = mk_config ?? (〈q,bin2,ch,to_initN (FS_crd sig) ??〉) t. [|@le_S_S @le_O_n | @le_S_S // ]
#sig #M #t #q #ch #Hcur whd in ⊢ (??%?); >Hcur %
qed.
-lemma binaryTM_phase4_write : ∀sig,M,q,ch,k,t.
- O < k → current ? t = None ? →
+lemma binaryTM_phase4_write : ∀sig,M,q,ch,t.current ? t = None ? →
+ ∀k.O < k →
loopM ? (mk_binaryTM sig M) k
(mk_config ?? (〈q,bin4,ch,O〉) t)
= loopM ? (mk_binaryTM sig M) (k-1)
- (mk_config ?? (〈q,bin2,ch,to_initN (FS_crd sig) ??〉) t). [2,3: @le_S //]
-#sig #M #q #ch #k #t #Hk #Hcur
+ (mk_config ?? (〈q,bin2,ch,to_initN (FS_crd sig) ??〉) t). [|@le_S_S @le_O_n|@le_S_S //]
+#sig #M #q #ch #t #Hcur #k #Hk
cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
->loopM_unfold >loop_S_false // <loopM_unfold >binaryTM_bin4_None //
+>loopM_unfold >loop_S_false // <loopM_unfold >binaryTM_bin4_None [|//] %
qed.
(* we don't get here any more! *
whd in match (trans FinBool ??); <Htrans %
qed.
-lemma binaryTM_phase4_extend : ∀sig,M,q,ch,k,t,cur,qn,an,mv.
- O < k → current ? t = Some ? cur →
- â\8c©qn,Some ? an,mvâ\8cª = trans sig M â\8c©q,châ\8cª →
+lemma binaryTM_phase4_extend : ∀sig,M,q,ch,t,cur,qn,an,mv.
+ current ? t = Some ? cur → 〈qn,Some ? an,mv〉 = trans sig M 〈q,ch〉 →
+ â\88\80k.O < k →
loopM ? (mk_binaryTM sig M) k
(mk_config ?? (〈q,bin4,ch,O〉) t)
= loopM ? (mk_binaryTM sig M) (k-1)
(mk_config ?? (〈q,bin5,ch,to_initN (FS_crd sig) ??〉) (tape_move ? t L)). [2,3: @le_S //]
-#sig #M #q #ch #k #t #cur #qn #an #mv #Hk #Hcur #Htrans
+#sig #M #q #ch #t #cur #qn #an #mv #Hcur #Htrans #k #Hk
cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
->loopM_unfold >loop_S_false // <loopM_unfold >binaryTM_bin4_extend //
+>loopM_unfold >loop_S_false // <loopM_unfold >(binaryTM_bin4_extend … Hcur) [|*://] %
qed.
lemma binaryTM_bin5_O :
(* extends the tape towards the left with an unimportant sequence that will be
immediately overwritten *)
-lemma binaryTM_phase5 :∀sig,M,q,ch,k,n. S n ≤ k →
+lemma binaryTM_phase5 :∀sig,M,q,ch,n.
∀rs.n<S (2*FS_crd sig) →
∃bs.|bs| = n ∧
+ ∀k.S n ≤ k →
loopM ? (mk_binaryTM sig M) k
(mk_config ?? (〈q,bin5,ch,n〉) (mk_tape ? [] (None ?) rs))
= loopM ? (mk_binaryTM sig M) (k - S n)
(mk_config ?? (〈q,bin2,ch,FS_crd sig〉)
(mk_tape ? [] (option_hd ? (bs@rs)) (tail ? (bs@rs)))). [2,3:@le_S //]
-#sig #M #q #ch #k #n #Hk
-cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech
-elim n
-[ #rs #Hlt %{[]} % // cases rs //
-| #n0 #IH #rs #Hn0 cases (IH (false::rs) ?) [|/2 by lt_S_to_lt/]
- #bs * #Hbs -IH #IH
- %{(bs@[false])} % [ <Hbs >length_append /2 by increasing_to_injective/ ]
+#sig #M #q #ch #n elim n
+[ #rs #Hlt %{[]} % // #k #Hk cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech -Hk0
+ cases rs //
+| #n0 #IH #rs #Hn0 cases (IH (false::rs) ?) [|/2 by lt_S_to_lt/]
+ #bs * #Hbs -IH #IH %{(bs@[false])} % [ <Hbs >length_append /2 by increasing_to_injective/ ]
+ #k #Hk cases (le_to_eq … Hk) #k0 #Hk0 >Hk0
>loopM_unfold >loop_S_false // >binaryTM_bin5_S
- >associative_append normalize in match ([false]@?); <IH
+ >associative_append normalize in match ([false]@?); <(IH (S n0 + k0)) [|//]
>loopM_unfold @eq_f @eq_f cases rs //
]
qed.
#n0 #IH #x <plus_n_Sm whd in ⊢ (??%(????%)); >IH %
qed.
+lemma iter_O : ∀T,f,x.iter T f O x = x.// qed.
+
lemma iter_tape_move_R : ∀T,n,ls,cs,rs.|cs| = n →
iter ? (λt0.tape_move T t0 R) n (mk_tape ? ls (option_hd ? (cs@rs)) (tail ? (cs@rs)))
= mk_tape ? (reverse ? cs@ls) (option_hd ? rs) (tail ? rs).
#T * normalize // #l2 #Hfalse @False_ind cases (not_le_Sn_O O) /2/
qed.
-lemma iter_tape_move_l_nil : ∀T,n,rs.
+lemma iter_tape_move_L_nil : ∀T,n,rs.
iter ? (λt0.tape_move T t0 L) n (mk_tape ? [ ] (None ?) rs) =
mk_tape ? [ ] (None ?) rs.
#T #n #rs elim n // #n0 #IH <IH in ⊢ (???%); cases rs //
qed.
+lemma iter_tape_move_R_nil : ∀T,n,ls.
+ iter ? (λt0.tape_move T t0 R) n (mk_tape ? ls (None ?) [ ]) =
+ mk_tape ? ls (None ?) [ ].
+#T #n #ls elim n // #n0 #IH <IH in ⊢ (???%); cases ls //
+qed.
+
lemma iter_tape_move_L_left : ∀T,n,cs,rs. O < n →
iter ? (λt0.tape_move T t0 L) n
(mk_tape ? [ ] (option_hd ? cs) (tail ? cs@rs)) =
mk_tape ? [ ] (None ?) (cs@rs).
#T #n #cs #rs *
[ cases cs // cases rs //
-| #m #_ whd in ⊢ (??%?); <(iter_tape_move_l_nil ? m) cases cs // cases rs // ]
+| #m #_ whd in ⊢ (??%?); <(iter_tape_move_L_nil ? m) cases cs // cases rs // ]
qed.
lemma iter_tape_move_L : ∀T,n,ls,cs,rs.|cs| = n →
]
qed.
+axiom loop_increase : ∀sig,M,m,n,cfg,cfg'.m < n →
+ loopM sig M m cfg = Some ? cfg' → loopM sig M n cfg = Some ? cfg'.
+
lemma binaryTM_loop :
- ∀sig,M,i,t,q,tf,qf.
- O < FS_crd sig →
- 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 #Hcrd change with (None ?) in ⊢ (??%?→?); #H destruct (H)
-| -i #i #IH #t #q #tf #qf #Hcrd >loopM_unfold
+ ∀sig,M,i,tf,qf. O < FS_crd sig →
+ ∀t,q.∃k.i ≤ k ∧
+ ((loopM sig M i (mk_config ?? q t) = Some ? (mk_config ?? qf tf) →
+ 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))) ∧
+ (loopM sig M i (mk_config ?? q t) = None ? →
+ loopM ? (mk_binaryTM sig M) k
+ (mk_config ?? (state_bin_lift ? M q) (tape_bin_lift ? t)) = None ?)).
+#sig #M #i #tf #qf #Hcrd elim i
+[ #t #q %{O} % // % // change with (None ?) in ⊢ (??%?→?); #H destruct (H)
+| -i #i #IH #t #q >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) [@Hcrd] -IH * #k0 #IH <config_expand in Hloop; #Hloop
- %{(7*(S (FS_crd sig)) + k0)}
- (*** PHASE 0 ***)
- >state_bin_lift_unfold cases (current_None_or_midtape ? t)
- (* 0.1) current = None *)
- [ (* #Hcur >state_bin_lift_unfold in ⊢ (??%?);
- lapply (current_tape_bin_list … Hcur) #Hcur'
- >binaryTM_phase0_None /2 by monotonic_lt_plus_l/
- >(?: FS_crd sig + k0 = S (FS_crd sig + k0 - 1)) [|@daemon]
- >loopM_unfold >loop_S_false // lapply (refl ? t) cases t in ⊢ (???%→?);
- [4: #ls #c #rs normalize in ⊢ (%→?); #H destruct (H) normalize in Hcur; destruct (Hcur)
- | #Ht >Ht >binaryTM_bin4_None // <loopM_unfold *)
- (* 0.2) midtape *)
- | * #ls * #c * #rs #Ht >Ht >tape_bin_lift_unfold
- >left_midtape >current_midtape >right_midtape >opt_bin_char_Some
- >(binaryTM_phase0_midtape … Hhalt ? (refl ??)) [| // ]
- >(?: 7*S (FS_crd sig) + k0 - S (FS_crd sig) = 6*S (FS_crd sig) + k0) [|/2 by plus_minus/]
- (*** PHASE 1 ***)
- >binaryTM_phase1
- [| cases (bin_list ? rs) normalize // #r0 #rs0 #H destruct (H)
- | >length_reverse @daemon
- | // ]
- >(?:6*S (FS_crd sig) + k0 - S (FS_crd sig) = 5*S (FS_crd sig) + k0) [|/2 by plus_minus/]
- >reverse_reverse >opt_cons_hd_tl
+ [ %{(S i)} % //
+ >(loop_S_true ??? (λc.halt ?? (cstate ?? c)) (mk_config ?? q t) Hhalt) %
+ [| #H destruct (H)]
+ #H destruct (H) >loopM_unfold >loop_S_true // ]
+ (* interesting case: more than one step *)
+ >(loop_S_false ??? (λc.halt ?? (cstate ?? c)) (mk_config ?? q t) Hhalt)cases (current_None_or_midtape ? t)
+ (*** current = None ***)
+ [ #Hcur lapply (current_tape_bin_list … Hcur) #Hcur'
+ cut (∃qn,chn,mv.〈qn,chn,mv〉 = trans ? M 〈q,None ?〉)
+ [ cases (trans ? M 〈q,None ?〉) * #qn #chn #mv /4 by ex_intro/ ]
+ * #qn * #chn * #mv cases chn -chn
+ [ #Htrans lapply (binaryTM_phase0_None_None … (None ?) (FS_crd sig) … Hhalt Hcur' Htrans) // [/2 by monotonic_lt_plus_l/]
+ lapply (binaryTM_phase3 ? M qn (None ?) (displ2_of_move sig mv) ? (tape_move FinBool (tape_bin_lift sig t) (mv_tech mv))) [//]
+ cases (IH (tape_move ? t mv) qn) -IH #k0 * #Hk0 * #IH #IHNone
+ #phase3 #phase0 %{(S (S (displ2_of_move sig mv))+k0)} %
+ [ @le_S_S @(le_plus O) // ]
+ >state_bin_lift_unfold >phase0 [|//]
+ >phase3 [|//]
+ >(?: S (S (displ2_of_move sig mv))+k0-1-S (displ2_of_move sig mv) = k0)
+ [| /2 by refl, plus_to_minus/ ]
+ cut (tape_move sig t mv=tape_move sig (tape_write sig t (None sig)) mv) [%] #Hcut
+ >(?: iter ? (λt0.tape_move ? t0 L) (displ2_of_move sig mv) (tape_move ? (tape_bin_lift ? t) (mv_tech mv))
+ =tape_bin_lift ? (tape_move ? t mv))
+ [|cases t in Hcur;
+ [4: #ls #c #rs normalize in ⊢ (%→?); #H destruct (H)
+ | #_ whd in match (tape_bin_lift ??);
+ (* TODO *)
+ (* ∀mv.tape_move ? (niltape ?) mv = niltape ? *)
+ (* ∀n.iter ? (λt.tape_move ? t ?) n (niltape ?) = niltape ? *)
+ @daemon
+ | #r0 #rs0 #_ cases mv
+ [ >tape_bin_lift_unfold whd in match (mv_tech L); whd in match (displ2_of_move sig L);
+ whd in match (rev_bin_list ??); whd in match (option_hd ??);
+ whd in match (right ??); >(?: []@bin_list ? (r0::rs0) = bin_char ? r0@bin_list ? rs0) [|%]
+ (* TODO *)
+ (* tape_move (mk_tape [ ] (None ?) rs R = ... *)
+ (* use iter_tape_move_R *)
+ @daemon
+ | >tape_bin_lift_unfold whd in match (mv_tech R); whd in match (displ2_of_move sig R);
+ whd in match (rev_bin_list ??); whd in match (option_hd ??);
+ whd in match (right ??); >(?: []@bin_list ? (r0::rs0) = bin_char ? r0@bin_list ? rs0) [|%]
+ whd in match (tape_move ? (leftof ???) R);
+ >tape_bin_lift_unfold >left_midtape >opt_bin_char_Some >right_midtape
+ >iter_O
+ (* TODO *)
+ (* tape_move (mk_tape [ ] (None ?) rs R = ... *)
+ @daemon
+ | >tape_bin_lift_unfold % ]
+ | #l0 #ls0 #_ cases mv
+ [ >tape_bin_lift_unfold whd in match (mv_tech L); whd in match (displ2_of_move sig L);
+ whd in match (bin_list ??); >append_nil whd in match (option_hd ??);
+ whd in match (left ??); whd in match (tail ??);
+ whd in match (tape_move ? (rightof ???) L);
+ >(?: rev_bin_list ? (l0::ls0) = reverse ? (bin_char ? l0)@rev_bin_list ? ls0) [|%]
+ (* TODO *)
+ (* tape_move (mk_tape ls (None ?) [ ] R = ... *)
+ (* use iter_tape_move_L *)
+ @daemon
+ | >tape_bin_lift_unfold whd in match (mv_tech R); whd in match (displ2_of_move sig R);
+ whd in match (bin_list ??); >append_nil whd in match (option_hd ??);
+ whd in match (left ??); whd in match (tail ??); >iter_O cases (rev_bin_list ??) //
+ | >tape_bin_lift_unfold % ]
+ ]
+ ]
+ %
+ [ #Hloop @IH <Hloop @eq_f whd in ⊢ (???%); >Hcur <Htrans @eq_f @Hcut
+ | #Hloop @IHNone <Hloop @eq_f whd in ⊢ (???%); >Hcur <Htrans @eq_f @Hcut ]
+ | #chn #Htrans
+ lapply (binaryTM_phase0_None_Some … (None ?) (FS_crd sig) … Hhalt Hcur' Htrans) // [/2 by monotonic_lt_plus_l/]
+ cases t in Hcur;
+ [ 4: #ls #c #rs normalize in ⊢ (%→?); #H destruct (H)
+ | 2: #r0 #rs0 #_ cut (∃b,bs.bin_char ? r0 = b::bs) [ @daemon ] * #b * #bs #Hbs
+ lapply (binaryTM_phase4_extend ???? (tape_move ? (tape_bin_lift ? (leftof ? r0 rs0)) R) b … Htrans)
+ [ >tape_bin_lift_unfold whd in match (option_hd ??); whd in match (tail ??);
+ whd in match (right ??);
+ >(?:bin_list ? (r0::rs0) = bin_char ? r0@bin_list ? rs0) [|%]
+ >Hbs % ]
+ cases (binaryTM_phase5 ? M q (None ?) (FS_crd sig) (bin_list ? (r0::rs0)) ?) [|//]
+ #cs * #Hcs
+ lapply (binaryTM_phase2_Some_ow ?? q (None ?) … [ ] ? (bin_list ? (r0::rs0)) Htrans Hcs)
+ lapply (binaryTM_phase3 ? M qn (None ?) (displ_of_move sig mv) ?
+ (mk_tape FinBool (reverse bool (bin_char sig chn)@[])
+ (option_hd FinBool (bin_list sig (r0::rs0))) (tail FinBool (bin_list sig (r0::rs0))))) [//]
+ cases (IH (tape_move ? (tape_write ? (leftof ? r0 rs0) (Some ? chn)) mv) qn) -IH #k0 * #Hk0 * #IH #IHNone
+ #phase3 #phase2 #phase5 #phase4 #phase0
+ %{(1 + 1 + (S (FS_crd sig)) + (S (FS_crd sig)) + S (displ_of_move sig mv) + k0)} %
+ [ @le_S_S @(le_plus O) // ]
+ >state_bin_lift_unfold >phase0 [|//]
+ >phase4 [|//]
+ >(?: loopM ? (mk_binaryTM ??) ? (mk_config ?? 〈q,bin5,None ?,to_initN ???〉 ?) = ?)
+ [|| @(trans_eq ????? (phase5 ??))
+ [ @eq_f @eq_f
+ >tape_bin_lift_unfold whd in match (rev_bin_list ??);
+ whd in match (right ??); whd in match (bin_list ??);
+ cases (bin_char ? r0) // (* bin_char can't be nil *) @daemon
+ | @le_S_S >associative_plus >associative_plus >commutative_plus @(le_plus O) //
+ |]]
+ >phase2 [| (*arith*) @daemon ]
+ >phase3 [| (*arith*) @daemon ]
+ >(?: 1+1+S (FS_crd sig)+S (FS_crd sig)+S (displ_of_move sig mv)+k0-1-1
+ -S (FS_crd sig)-S (FS_crd sig) -S (displ_of_move sig mv) = k0)
+ [| (*arith*) @daemon ]
+ -phase0 -phase2 -phase3 -phase4 -phase5 <state_bin_lift_unfold
+ >(?: iter ? (λt0.tape_move ? t0 L) (displ_of_move sig mv)
+ (mk_tape ? (reverse ? (bin_char sig chn)@[])
+ (option_hd FinBool (bin_list sig (r0::rs0)))
+ (tail FinBool (bin_list sig (r0::rs0))))
+ = tape_bin_lift ? (tape_move ? (tape_write ? (leftof ? r0 rs0) (Some ? chn)) mv))
+ [ % #Hloop
+ [ @IH <Hloop @eq_f whd in ⊢ (???%); <Htrans %
+ | @IHNone <Hloop @eq_f whd in ⊢ (???%); <Htrans % ]
+ | >(?:bin_list ? (r0::rs0) = bin_char ? r0@bin_list ? rs0) [|%]
+ cases mv
+ [ >(?:displ_of_move sig L = FS_crd sig+FS_crd sig) [|normalize //]
+ >iter_split >iter_tape_move_L [| @daemon ]
+ >hd_tech [|@daemon] >tail_tech [|@daemon] >iter_tape_move_L_left [|//]
+ whd in match (tape_move ???); >tape_bin_lift_unfold %
+ | normalize in match (displ_of_move ??); >iter_O
+ normalize in match (tape_move ???);
+ >tape_bin_lift_unfold >opt_bin_char_Some
+ >hd_tech [|@daemon] >tail_tech [| @daemon ] %
+ | normalize in match (displ_of_move ??);
+ >iter_tape_move_L [|@daemon]
+ normalize in match (tape_move ???); >tape_bin_lift_unfold
+ >opt_bin_char_Some >hd_tech [|@daemon] >tail_tech [|@daemon] % ]
+ ]
+ | #_ lapply (binaryTM_phase4_write ? M q (None ?) (niltape ?) (refl ??))
+ lapply (binaryTM_phase2_Some_of ?? q (None ?) … [ ] Htrans)
+ lapply (binaryTM_phase3 ? M qn (None ?) (displ_of_move sig mv) ?
+ (mk_tape FinBool (reverse bool (bin_char sig chn)@[]) (None ?) [ ])) [//]
+ cases (IH (tape_move ? (midtape ? [ ] chn [ ]) mv) qn) -IH #k0 * #Hk0 * #IH #IHNone
+ #phase3 #phase2 #phase4 #phase0
+ %{(1 + 1 + (S (FS_crd sig)) + S (displ_of_move sig mv) + k0)} %
+ [ @le_S_S @(le_plus O) // ]
+ >state_bin_lift_unfold >phase0 [|//]
+ >phase4 [|//]
+ >phase2 [|(*arith *) @daemon]
+ >phase3 [| (*arith*) @daemon ]
+ >(?: 1+1+S (FS_crd sig) + S (displ_of_move sig mv)+k0-1-1
+ -S (FS_crd sig)-S (displ_of_move sig mv) = k0)
+ [| (*arith*) @daemon ]
+ -phase0 -phase2 -phase3 -phase4 <state_bin_lift_unfold
+ >(?: iter ? (λt0.tape_move ? t0 L) (displ_of_move sig mv)
+ (mk_tape ? (reverse ? (bin_char sig chn)@[]) (None ?) [ ])
+ = tape_bin_lift ? (tape_move ? (tape_write ? (niltape ?) (Some ? chn)) mv))
+ [ % #Hloop
+ [ @IH <Hloop @eq_f whd in ⊢ (???%); <Htrans %
+ | @IHNone <Hloop @eq_f whd in ⊢ (???%); <Htrans % ]
+ | cases mv
+ [ >(?:displ_of_move sig L = FS_crd sig+FS_crd sig) [|normalize //]
+ >iter_split change with (mk_tape ?? (option_hd ? [ ]) (tail ? [ ])) in ⊢ (??(????(????%))?);
+ >iter_tape_move_L [| @daemon ]
+ >append_nil in ⊢ (??(????(???%?))?); >tail_tech [|@daemon]
+ >iter_tape_move_L_left [|//]
+ normalize in match (tape_move ???);
+ >tape_bin_lift_unfold %
+ | normalize in match (displ_of_move ??); >iter_O
+ normalize in match (tape_move ???);
+ >tape_bin_lift_unfold %
+ | normalize in match (displ_of_move ??);
+ change with (mk_tape ?? (option_hd ? [ ]) (tail ? [ ])) in ⊢ (??(????%)?);
+ >iter_tape_move_L [|@daemon]
+ normalize in match (tape_move ???); >tape_bin_lift_unfold
+ >opt_bin_char_Some >hd_tech [|@daemon] >tail_tech [|@daemon] % ]
+ ]
+ | #l0 #ls0 #_ lapply (binaryTM_phase4_write ? M q (None ?) (tape_bin_lift ? (rightof ? l0 ls0)) ?)
+ [ >tape_bin_lift_unfold >current_mk_tape % ]
+ lapply (binaryTM_phase2_Some_of ?? q (None ?) … (rev_bin_list ? (l0::ls0)) Htrans)
+ lapply (binaryTM_phase3 ? M qn (None ?) (displ_of_move sig mv) ?
+ (mk_tape FinBool (reverse bool (bin_char sig chn)@rev_bin_list ? (l0::ls0)) (None ?) [ ])) [//]
+ cases (IH (tape_move ? (midtape ? (l0::ls0) chn [ ]) mv) qn) -IH #k0 * #Hk0 * #IH #IHNone
+ #phase3 #phase2 #phase4 #phase0
+ %{(1 + 1 + (S (FS_crd sig)) + S (displ_of_move sig mv) + k0)} %
+ [ @le_S_S @(le_plus O) // ]
+ >state_bin_lift_unfold >phase0 [|//]
+ >(?:tape_move ? (tape_bin_lift ? (rightof ? l0 ls0)) R = tape_bin_lift ? (rightof ? l0 ls0))
+ [| >tape_bin_lift_unfold normalize in match (option_hd ??); normalize in match (right ??);
+ normalize in match (tail ??); normalize in match (left ??);
+ >(?:rev_bin_list ? (l0::ls0) = reverse ? (bin_char ? l0)@rev_bin_list ? ls0) [|%]
+ cases (reverse ? (bin_char ? l0)) // cases (rev_bin_list ? ls0) // ]
+ >phase4 [|//]
+ >phase2 [|(*arith *) @daemon]
+ >phase3 [| (*arith*) @daemon]
+ >(?: 1+1+S (FS_crd sig) + S (displ_of_move sig mv)+k0-1-1
+ -S (FS_crd sig)-S (displ_of_move sig mv) = k0)
+ [| (*arith*) @daemon ]
+ -phase0 -phase2 -phase3 -phase4 <state_bin_lift_unfold
+ >(?: iter ? (λt0.tape_move ? t0 L) (displ_of_move sig mv)
+ (mk_tape ? (reverse ? (bin_char sig chn)@rev_bin_list ? (l0::ls0)) (None ?) [ ])
+ = tape_bin_lift ? (tape_move ? (tape_write ? (rightof ? l0 ls0) (Some ? chn)) mv))
+ [ % #Hloop
+ [ @IH <Hloop @eq_f whd in ⊢ (???%); <Htrans %
+ | @IHNone <Hloop @eq_f whd in ⊢ (???%); <Htrans % ]
+ | cases mv
+ [ >(?:displ_of_move sig L = FS_crd sig+FS_crd sig) [|normalize //]
+ >iter_split change with (mk_tape ?? (option_hd ? [ ]) (tail ? [ ])) in ⊢ (??(????(????%))?);
+ >iter_tape_move_L [| @daemon ]
+ >append_nil in ⊢ (??(????(???%?))?); >tail_tech [|@daemon]
+ >(?:rev_bin_list ? (l0::ls0) = reverse ? (bin_char ? l0)@rev_bin_list ? ls0) [|%]
+ >append_nil >iter_tape_move_L [|@daemon]
+ normalize in match (tape_move ???);
+ >tape_bin_lift_unfold @eq_f2
+ [ >hd_tech [|@daemon] %
+ | >tail_tech [|@daemon] >opt_bin_char_Some normalize in match (bin_list ??); >append_nil %]
+ | normalize in match (displ_of_move ??); >iter_O
+ normalize in match (tape_move ???);
+ >tape_bin_lift_unfold %
+ | normalize in match (displ_of_move ??);
+ change with (mk_tape ?? (option_hd ? [ ]) (tail ? [ ])) in ⊢ (??(????%)?);
+ >iter_tape_move_L [|@daemon]
+ normalize in match (tape_move ???); >tape_bin_lift_unfold
+ >opt_bin_char_Some >hd_tech [|@daemon] >tail_tech [|@daemon] % ]
+ ]
+ ]
+ ]
+ (*** midtape ***)
+ | * #ls * #c * #rs #Ht >Ht
cut (∃qn,chn,mv.〈qn,chn,mv〉 = trans ? M 〈q,Some ? c〉)
[ cases (trans ? M 〈q,Some ? c〉) * #qn #chn #mv /4 by ex_intro/ ]
- * #qn * #chn * #mv cases chn -chn
- [ (* no write! *) #Htrans >(binaryTM_phase2_None … Htrans) [2,3: //]
+ * #qn * #chn * #mv #Htrans
+ cut (tape_bin_lift ? t = ?) [| >tape_bin_lift_unfold % ]
+ >Ht in ⊢ (???%→?); >opt_bin_char_Some >left_midtape >right_midtape #Ht'
+ lapply (binaryTM_phase0_midtape ?? (tape_bin_lift ? t) q … (None ?) Hhalt Ht')
+ lapply (binaryTM_phase1 ?? q (reverse ? (bin_char ? c)) (rev_bin_list ? ls)
+ (option_hd ? (bin_list ? rs)) (tail ? (bin_list ? rs)) (Some ? c) ??)
+ [ cases (bin_list ? rs) // @daemon | >length_reverse @daemon |]
+ >opt_cons_hd_tl >reverse_reverse
+ cases chn in Htrans; -chn
+ [ #Htrans
+ lapply (binaryTM_phase2_None … Htrans (FS_crd sig) ?
+ (mk_tape FinBool (rev_bin_list sig ls)
+ (option_hd FinBool (bin_char sig c@bin_list sig rs))
+ (tail FinBool (bin_char sig c@bin_list sig rs)))) [//]
+ lapply (binaryTM_phase3 ? M qn (Some ? c) (displ_of_move sig mv) ?
+ (mk_tape FinBool (reverse bool (bin_char sig c)@rev_bin_list ? ls)
+ (option_hd FinBool (bin_list sig rs)) (tail FinBool (bin_list sig rs)))) [//]
+ cases (IH (tape_move ? (tape_write ? (midtape ? ls c rs) (None ?)) mv) qn) -IH #k0 * #Hk0 * #IH #IHNone
+ #phase3 #phase2 #phase1 #phase0
+ %{(S (FS_crd sig) + S (FS_crd sig) + S (FS_crd sig) + S (displ_of_move sig mv) + k0)} %
+ [ @le_S_S @(le_plus O) // ]
+ >state_bin_lift_unfold <Ht >phase0 [|//]
+ >phase1 [|/2 by monotonic_le_minus_l/]
+ >phase2 [|/2 by monotonic_le_minus_l/]
>iter_tape_move_R [|@daemon]
- >(?:5*S (FS_crd sig) + k0 - S (FS_crd sig) = 4*S (FS_crd sig) + k0) [|/2 by plus_minus/]
- >binaryTM_phase3
- [|//| cut (S (displ_of_move sig mv) ≤ 2*(S (FS_crd sig)))
- [ /2 by le_S_S/
- | #H @(transitive_le ??? H) -H -Hcrd @(transitive_le ? (4*S (FS_crd sig))) /2 by le_plus_a/ ]
- ]
- cut (∀sig,M,m,n,cfg,cfg'.m < n → loopM sig M m cfg = Some ? cfg' → loopM sig M n cfg = Some ? cfg') [@daemon]
- #Hcut <(Hcut ??? (4*S (FS_crd sig) + k0 - S (displ_of_move sig mv)) ??? IH)
- [| cases mv
- [ >(?:displ_of_move sig L = 2*FS_crd sig) //
- >eq_minus_S_pred
- @(transitive_le ? (pred (4*FS_crd sig+k0-2*FS_crd sig)))
- [ >(?:4*FS_crd sig+k0-2*FS_crd sig = 2*FS_crd sig + k0)
- [ cases Hcrd /2 by le_minus_to_plus, le_n/
- | <plus_minus [2://]
- >(commutative_times 4) >(commutative_times 2)
- <distributive_times_minus //
- ]
- | @monotonic_pred /2 by monotonic_le_minus_l/ ]
- | whd in match (displ_of_move ??); @(transitive_le ? (4*1+k0-1))
- [ //
- | change with (pred (4*1+k0)) in ⊢ (?%?);
- >eq_minus_S_pred <minus_n_O @monotonic_pred // ]
- | >(?:displ_of_move sig N = FS_crd sig) //
- >eq_minus_S_pred
- @(transitive_le ? (pred (4*FS_crd sig+k0-1*FS_crd sig)))
- [ >(?:4*FS_crd sig+k0-1*FS_crd sig = 3*FS_crd sig + k0)
- [ cases Hcrd /2 by le_minus_to_plus, le_n/
- | <plus_minus [2://]
- >(commutative_times 4) >(commutative_times 1)
- <distributive_times_minus //
- ]
- | @monotonic_pred /2 by transitive_le, le_n/ ] ] ]
- @eq_f @eq_f2
- [ <state_bin_lift_unfold >Ht whd in match (step ???); <Htrans %
- | (* must distinguish mv *)
- cases mv in Htrans; #Htrans
- [ >(?:displ_of_move ? L = FS_crd sig + FS_crd sig) [| normalize // ]
- >iter_split >iter_tape_move_L [|@daemon] >Ht cases ls
- [ normalize in match (rev_bin_list ??);
- >hd_tech [|@daemon] >tail_tech [|@daemon]
- >iter_tape_move_L_left // whd in match (step ???);
- <Htrans whd in match (ctape ???);
+ >phase3 [|/2 by monotonic_le_minus_l/]
+ -phase0 -phase1 -phase2 -phase3
+ >(?: S (FS_crd sig) + S (FS_crd sig) + S (FS_crd sig) + S (displ_of_move sig mv) + k0
+ - S (FS_crd sig) - S (FS_crd sig) - S (FS_crd sig) - S (displ_of_move sig mv)
+ = k0) [| (*arith*) @daemon]
+ <state_bin_lift_unfold
+ >(?: iter ? (λt0.tape_move ? t0 L) (displ_of_move sig mv)
+ (mk_tape ? (reverse ? (bin_char sig c)@rev_bin_list ? ls)
+ (option_hd ? (bin_list ? rs)) (tail ? (bin_list ? rs)))
+ = tape_bin_lift ? (tape_move ? (tape_write ? (midtape ? ls c rs) (None ?)) mv))
+ [ % #Hloop
+ [ @IH <Hloop @eq_f whd in ⊢ (???%); >Ht <Htrans %
+ | @IHNone <Hloop @eq_f whd in ⊢ (???%); >Ht <Htrans % ]
+ | normalize in match (tape_write ???); cases mv in Htrans; #Htrans
+ [ >(?:displ_of_move sig L = FS_crd sig+FS_crd sig) [|normalize //]
+ >iter_split >iter_tape_move_L [| @daemon ]
+ cases ls
+ [ >hd_tech [|@daemon] >tail_tech [|@daemon] >iter_tape_move_L_left [|//]
>tape_bin_lift_unfold %
- | #l0 #ls0 change with (reverse ? (bin_char ? l0)@rev_bin_list ? ls0) in match (rev_bin_list ??);
+ | #l0 #ls0 >(?:rev_bin_list ? (l0::ls0) = reverse ? (bin_char ? l0)@rev_bin_list ? ls0) [|%]
+ normalize in match (tape_move ???);
>iter_tape_move_L [|@daemon]
>hd_tech [|@daemon] >tail_tech [|@daemon]
- whd in match (step ???); <Htrans whd in match (ctape ???);
- >tape_bin_lift_unfold >left_midtape >current_midtape
- >opt_bin_char_Some >right_midtape %
+ >tape_bin_lift_unfold % ]
+ | normalize in match (displ_of_move ??); >iter_O cases rs
+ [ normalize in match (tape_move ???); >tape_bin_lift_unfold %
+ | #r0 #rs0 normalize in match (tape_move ???);
+ >tape_bin_lift_unfold >opt_bin_char_Some
+ >left_midtape >right_midtape
+ >(?:bin_list ? (r0::rs0) = bin_char ? r0@bin_list ? rs0) [|%]
+ >hd_tech [|@daemon] >tail_tech [|@daemon] %
]
- | change with
- (mk_tape ? (reverse ? (bin_char ? c)@rev_bin_list ? ls)
+ | normalize in match (displ_of_move ??); >iter_tape_move_L [|@daemon]
+ >hd_tech [|@daemon] >tail_tech [|@daemon] >tape_bin_lift_unfold %
+ ]
+ ]
+ | #chn #Htrans
+ lapply (binaryTM_phase2_Some_ow ?? q (Some ? c) ??? (rev_bin_list ? ls) (bin_char ? c) (bin_list ? rs) Htrans ?)
+ [@daemon]
+ lapply (binaryTM_phase3 ? M qn (Some ? c) (displ_of_move sig mv) ?
+ (mk_tape FinBool (reverse bool (bin_char sig chn)@rev_bin_list ? ls)
+ (option_hd FinBool (bin_list sig rs)) (tail FinBool (bin_list sig rs)))) [//]
+ cases (IH (tape_move ? (tape_write ? (midtape ? ls c rs) (Some ? chn)) mv) qn) -IH #k0 * #Hk0 * #IH #IHNone
+ #phase3 #phase2 #phase1 #phase0
+ %{(S (FS_crd sig) + S (FS_crd sig) + S (FS_crd sig) + S (displ_of_move sig mv) + k0)} %
+ [ @le_S_S @(le_plus O) // ]
+ >state_bin_lift_unfold <Ht >phase0 [|//]
+ >phase1 [|/2 by monotonic_le_minus_l/]
+ >phase2 [|/2 by monotonic_le_minus_l/]
+ >phase3 [|/2 by monotonic_le_minus_l/]
+ -phase0 -phase1 -phase2 -phase3
+ >(?: S (FS_crd sig) + S (FS_crd sig) + S (FS_crd sig) + S (displ_of_move sig mv) + k0
+ - S (FS_crd sig) - S (FS_crd sig) - S (FS_crd sig) - S (displ_of_move sig mv)
+ = k0) [| (*arith*) @daemon]
+ <state_bin_lift_unfold
+ >(?: iter ? (λt0.tape_move ? t0 L) (displ_of_move sig mv)
+ (mk_tape ? (reverse ? (bin_char sig chn)@rev_bin_list ? ls)
(option_hd ? (bin_list ? rs)) (tail ? (bin_list ? rs)))
- in ⊢ (??%?);
- >reverse_bin_char_list <IH
- >Ht >tape_bin_lift_unfold @eq_f3
- whd in match (step ???); <Htrans cases rs //
- #r0 #rs0 whd in match (ctape ???); >current_midtape >opt_bin_char_Some
- [ <hd_tech in ⊢(???%); // @daemon
- | >right_midtape <tail_tech // @daemon ]
- | whd in match (displ_of_move ? N); >iter_tape_move_L [|@daemon]
- >Ht whd in match (step ???); <Htrans whd in match (ctape ???);
- >tape_bin_lift_unfold >left_midtape >current_midtape >right_midtape
- >opt_bin_char_Some lapply Hcrd >(?:FS_crd sig = |bin_char ? c|) [| @daemon ]
- cases (bin_char ? c) // #H normalize in H; @False_ind
- cases (not_le_Sn_O O) /2/
+ = tape_bin_lift ? (tape_move ? (tape_write ? (midtape ? ls c rs) (Some ? chn)) mv))
+ [ % #Hloop
+ [ @IH <Hloop @eq_f whd in ⊢ (???%); >Ht <Htrans %
+ | @IHNone <Hloop @eq_f whd in ⊢ (???%); >Ht <Htrans % ]
+ | normalize in match (tape_write ???); cases mv in Htrans; #Htrans
+ [ >(?:displ_of_move sig L = FS_crd sig+FS_crd sig) [|normalize //]
+ >iter_split >iter_tape_move_L [| @daemon ]
+ cases ls
+ [ >hd_tech [|@daemon] >tail_tech [|@daemon] >iter_tape_move_L_left [|//]
+ >tape_bin_lift_unfold %
+ | #l0 #ls0 >(?:rev_bin_list ? (l0::ls0) = reverse ? (bin_char ? l0)@rev_bin_list ? ls0) [|%]
+ normalize in match (tape_move ???);
+ >iter_tape_move_L [|@daemon]
+ >hd_tech [|@daemon] >tail_tech [|@daemon]
+ >tape_bin_lift_unfold % ]
+ | normalize in match (displ_of_move ??); >iter_O cases rs
+ [ normalize in match (tape_move ???); >tape_bin_lift_unfold %
+ | #r0 #rs0 normalize in match (tape_move ???);
+ >tape_bin_lift_unfold >opt_bin_char_Some
+ >left_midtape >right_midtape
+ >(?:bin_list ? (r0::rs0) = bin_char ? r0@bin_list ? rs0) [|%]
+ >hd_tech [|@daemon] >tail_tech [|@daemon] %
+ ]
+ | normalize in match (displ_of_move ??); >iter_tape_move_L [|@daemon]
+ >hd_tech [|@daemon] >tail_tech [|@daemon] >tape_bin_lift_unfold %
]
]
-
-
-
-
+ ]
+ ]
+]
+qed.
+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.
+
(*
-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
-
-*)
\ No newline at end of file
+∀sig,M,i,tf,qf. O < FS_crd sig →
+ ∀t,q.∃k.i ≤ k ∧
+ ((loopM sig M i (mk_config ?? q t) = Some ? (mk_config ?? qf tf) →
+ 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))) ∧
+ (loopM sig M i (mk_config ?? q t) = None ? →
+ loopM ? (mk_binaryTM sig M) k
+ (mk_config ?? (state_bin_lift ? M q) (tape_bin_lift ? t)) = None ?)).
+ *)
+axiom loop_incr : ∀sig,M,m,n,cfg,cfg'.m ≤ n →
+ loopM sig M m cfg = Some ? cfg' → loopM sig M n cfg = Some ? cfg'.
+
+theorem sem_binaryTM :
+ ∀sig,M,R.O < FS_crd sig → M ⊫ R → mk_binaryTM sig M ⊫ R_bin_lift ? R.
+#sig #M #R #Hcrd #HM #t #k #outc #Hloopbin #u #Ht
+lapply (refl ? (loopM ? M k (initc ? M u))) cases (loopM ? M k (initc ? M u)) in ⊢ (???%→?);
+[ #H cases (binaryTM_loop ? M k u (start ? M) Hcrd u (start ? M))
+ #k0 * #Hlt * #_ #H1 lapply (H1 H) -H -H1 <Ht
+ whd in match (initc ???) in Hloopbin; whd in match (start ??) in Hloopbin;
+ >state_bin_lift_unfold >(loop_incr … Hlt Hloopbin) #H destruct (H)
+| * #qf #tf #H cases (binaryTM_loop ? M k tf qf Hcrd u (start ? M))
+ #k0 * #Hlt * #H1 #_ lapply (H1 H) -H1 <Ht
+ whd in match (initc ???) in Hloopbin; whd in match (start ??) in Hloopbin;
+ >state_bin_lift_unfold >(loop_incr … Hlt Hloopbin) #Heq destruct (Heq)
+ % [| % [%]] @(HM … H)
+qed.
\ No newline at end of file
projects / helm.git / commitdiff