--- /dev/null
+include "turing/auxiliary_machines1.ma".
+include "turing/multi_to_mono/shift_trace_aux.ma".
+
+(******************************************************************************)
+(* mtiL: complete move L for tape i. We reaching the left border of trace i, *)
+(* add a blank if there is no more tape, then move the i-trace and finally *)
+(* come back to the head position. *)
+(******************************************************************************)
+
+(* we say that a tape is regular if for any trace after the first blank we
+ only have other blanks *)
+
+definition all_blanks_in ≝ λsig,l.
+ ∀x. mem ? x l → x = blank sig.
+
+definition regular_i ≝ λsig,n.λl:list (multi_sig sig n).λi.
+ all_blanks_in ? (after_blank ? (trace sig n i l)).
+
+definition regular_trace ≝ λsig,n,a.λls,rs:list (multi_sig sig n).λi.
+ Or (And (regular_i sig n (a::ls) i) (regular_i sig n rs i))
+ (And (regular_i sig n ls i) (regular_i sig n (a::rs) i)).
+
+axiom regular_tail: ∀sig,n,l,i.
+ regular_i sig n l i → regular_i sig n (tail ? l) i.
+
+axiom regular_extend: ∀sig,n,l,i.
+ regular_i sig n l i → regular_i sig n (l@[all_blank sig n]) i.
+
+axiom all_blank_after_blank: ∀sig,n,l1,b,l2,i.
+ nth i ? (vec … b) (blank ?) = blank ? →
+ regular_i sig n (l1@b::l2) i → all_blanks_in ? (trace sig n i l2).
+
+lemma regular_trace_extl: ∀sig,n,a,ls,rs,i.
+ regular_trace sig n a ls rs i →
+ regular_trace sig n a (ls@[all_blank sig n]) rs i.
+#sig #n #a #ls #rs #i *
+ [* #H1 #H2 % % // @(regular_extend … H1)
+ |* #H1 #H2 %2 % // @(regular_extend … H1)
+ ]
+qed.
+
+lemma regular_cons_hd_rs: ∀sig,n.∀a:multi_sig sig n.∀ls,rs1,rs2,i.
+ regular_trace sig n a ls (rs1@rs2) i →
+ regular_trace sig n a ls (rs1@((hd ? rs2 (all_blank …))::(tail ? rs2))) i.
+#sig #n #a #ls #rs1 #rs2 #i cases rs2 [2: #b #tl #H @H]
+*[* #H1 >append_nil #H2 %1 %
+ [@H1 | whd in match (hd ???); @(regular_extend … rs1) //]
+ |* #H1 >append_nil #H2 %2 %
+ [@H1 | whd in match (hd ???); @(regular_extend … (a::rs1)) //]
+ ]
+qed.
+
+lemma eq_trace_to_regular : ∀sig,n.∀a1,a2:multi_sig sig n.∀ls1,ls2,rs1,rs2,i.
+ nth i ? (vec … a1) (blank ?) = nth i ? (vec … a2) (blank ?) →
+ trace sig n i ls1 = trace sig n i ls2 →
+ trace sig n i rs1 = trace sig n i rs2 →
+ regular_trace sig n a1 ls1 rs1 i →
+ regular_trace sig n a2 ls2 rs2 i.
+#sig #n #a1 #a2 #ls1 #ls2 #rs1 #rs2 #i #H1 #H2 #H3 #H4
+whd in match (regular_trace ??????); whd in match (regular_i ????);
+whd in match (regular_i ?? rs2 ?); whd in match (regular_i ?? ls2 ?);
+whd in match (regular_i ?? (a2::rs2) ?); whd in match (trace ????);
+<trace_def whd in match (trace ??? (a2::rs2)); <trace_def
+<H1 <H2 <H3 @H4
+qed.
+
+(******************************* move_to_blank_L ******************************)
+(* we compose machines together to reduce the number of output cases, and
+ improve semantics *)
+
+definition move_to_blank_L ≝ λsig,n,i.
+ (move_until ? L (no_blank sig n i)) · extend ? (all_blank sig n).
+
+(*
+definition R_move_to_blank_L ≝ λsig,n,i,t1,t2.
+(current ? t1 = None ? →
+ t2 = midtape (multi_sig sig n) (left ? t1) (all_blank …) (right ? t1)) ∧
+∀ls,a,rs.t1 = midtape ? ls a rs →
+ ((no_blank sig n i a = false) ∧ t2 = t1) ∨
+ (∃b,ls1,ls2.
+ (no_blank sig n i b = false) ∧
+ (∀j.j ≤n → to_blank_i ?? j (ls1@b::ls2) = to_blank_i ?? j ls) ∧
+ t2 = midtape ? ls2 b ((reverse ? (a::ls1))@rs)).
+*)
+
+definition R_move_to_blank_L ≝ λsig,n,i,t1,t2.
+(current ? t1 = None ? →
+ t2 = midtape (multi_sig sig n) (left ? t1) (all_blank …) (right ? t1)) ∧
+∀ls,a,rs.
+ t1 = midtape (multi_sig sig n) ls a rs →
+ regular_i sig n (a::ls) i →
+ (∀j. j ≠ i → regular_trace … a ls rs j) →
+ (∃b,ls1,ls2.
+ (regular_i sig n (ls1@b::ls2) i) ∧
+ (∀j. j ≠ i → regular_trace …
+ (hd ? (ls1@b::ls2) (all_blank …)) (tail ? (ls1@b::ls2)) rs j) ∧
+ (no_blank sig n i b = false) ∧
+ (hd (multi_sig sig n) (ls1@[b]) (all_blank …) = a) ∧ (* not implied by the next fact *)
+ (∀j.j ≤n → to_blank_i ?? j (ls1@b::ls2) = to_blank_i ?? j (a::ls)) ∧
+ t2 = midtape ? ls2 b ((reverse ? ls1)@rs)).
+
+theorem sem_move_to_blank_L: ∀sig,n,i.
+ move_to_blank_L sig n i ⊨ R_move_to_blank_L sig n i.
+#sig #n #i
+@(sem_seq_app ??????
+ (ssem_move_until_L ? (no_blank sig n i)) (sem_extend ? (all_blank sig n)))
+#tin #tout * #t1 * * #Ht1a #Ht1b * #Ht2a #Ht2b %
+ [#Hcur >(Ht1a Hcur) in Ht2a; /2 by /
+ |#ls #a #rs #Htin #Hreg #Hreg2 -Ht1a cases (Ht1b … Htin)
+ [* #Hnb #Ht1 -Ht1b -Ht2a >Ht1 in Ht2b; >Htin #H
+ %{a} %{[ ]} %{ls}
+ %[%[%[%[%[@Hreg|@Hreg2]|@Hnb]|//]|//]|@H normalize % #H1 destruct (H1)]
+ |*
+ [(* we find the blank *)
+ * #ls1 * #b * #ls2 * * * #H1 #H2 #H3 #Ht1
+ >Ht1 in Ht2b; #Hout -Ht1b
+ %{b} %{(a::ls1)} %{ls2}
+ %[%[%[%[%[>H1 in Hreg; #H @H
+ |#j #jneqi whd in match (hd ???); whd in match (tail ??);
+ <H1 @(Hreg2 j jneqi)]|@H2] |//]|>H1 //]
+ |@Hout normalize % normalize #H destruct (H)
+ ]
+ |* #b * #lss * * #H1 #H2 #Ht1 -Ht1b >Ht1 in Ht2a;
+ whd in match (left ??); whd in match (right ??); #Hout
+ %{(all_blank …)} %{(lss@[b])} %{[]}
+ %[%[%[%[%[<H2 @regular_extend //
+ |<H2 #j #jneqi whd in match (hd ???); whd in match (tail ??);
+ @regular_trace_extl @Hreg2 //]
+ |whd in match (no_blank ????); >blank_all_blank //]
+ |<H2 //]
+ |#j #lejn <H2 @sym_eq @to_blank_i_ext]
+ |>reverse_append >reverse_single @Hout normalize //
+ ]
+ ]
+ ]
+qed.
+
+(******************************************************************************)
+
+definition shift_i_L ≝ λsig,n,i.
+ ncombf_r (multi_sig …) (shift_i sig n i) (all_blank sig n) ·
+ mti sig n i ·
+ extend ? (all_blank sig n).
+
+definition R_shift_i_L ≝ λsig,n,i,t1,t2.
+ (∀a,ls,rs.
+ t1 = midtape ? ls a rs →
+ ((∃rs1,b,rs2,a1,rss.
+ rs = rs1@b::rs2 ∧
+ nth i ? (vec … b) (blank ?) = (blank ?) ∧
+ (∀x. mem ? x rs1 → nth i ? (vec … x) (blank ?) ≠ (blank ?)) ∧
+ shift_l sig n i (a::rs1) (a1::rss) ∧
+ t2 = midtape (multi_sig sig n) ((reverse ? (a1::rss))@ls) b rs2) ∨
+ (∃b,rss.
+ (∀x. mem ? x rs → nth i ? (vec … x) (blank ?) ≠ (blank ?)) ∧
+ shift_l sig n i (a::rs) (rss@[b]) ∧
+ t2 = midtape (multi_sig sig n)
+ ((reverse ? (rss@[b]))@ls) (all_blank sig n) [ ]))).
+
+definition R_shift_i_L_new ≝ λsig,n,i,t1,t2.
+ (∀a,ls,rs.
+ t1 = midtape ? ls a rs →
+ ∃rs1,b,rs2,rss.
+ b = hd ? rs2 (all_blank sig n) ∧
+ nth i ? (vec … b) (blank ?) = (blank ?) ∧
+ rs = rs1@rs2 ∧
+ (∀x. mem ? x rs1 → nth i ? (vec … x) (blank ?) ≠ (blank ?)) ∧
+ shift_l sig n i (a::rs1) rss ∧
+ t2 = midtape (multi_sig sig n) ((reverse ? rss)@ls) b (tail ? rs2)).
+
+theorem sem_shift_i_L: ∀sig,n,i. shift_i_L sig n i ⊨ R_shift_i_L sig n i.
+#sig #n #i
+@(sem_seq_app ??????
+ (sem_ncombf_r (multi_sig sig n) (shift_i sig n i)(all_blank sig n))
+ (sem_seq ????? (ssem_mti sig n i)
+ (sem_extend ? (all_blank sig n))))
+#tin #tout * #t1 * * #Ht1a #Ht1b * #t2 * * #Ht2a #Ht2b * #Htout1 #Htout2
+#a #ls #rs cases rs
+ [#Htin %2 %{(shift_i sig n i a (all_blank sig n))} %{[ ]}
+ %[%[#x @False_ind | @daemon]
+ |lapply (Ht1a … Htin) -Ht1a -Ht1b #Ht1
+ lapply (Ht2a … Ht1) -Ht2a -Ht2b #Ht2 >Ht2 in Htout1;
+ >Ht1 whd in match (left ??); whd in match (right ??); #Htout @Htout //
+ ]
+ |#a1 #rs1 #Htin
+ lapply (Ht1b … Htin) -Ht1a -Ht1b #Ht1
+ lapply (Ht2b … Ht1) -Ht2a -Ht2b *
+ [(* a1 is blank *) * #H1 #H2 %1
+ %{[ ]} %{a1} %{rs1} %{(shift_i sig n i a a1)} %{[ ]}
+ %[%[%[%[// |//] |#x @False_ind] | @daemon]
+ |>Htout2 [>H2 >reverse_single @Ht1 |>H2 >Ht1 normalize % #H destruct (H)]
+ ]
+ |*
+ [* #rs10 * #b * #rs2 * #rss * * * * #H1 #H2 #H3 #H4
+ #Ht2 %1
+ %{(a1::rs10)} %{b} %{rs2} %{(shift_i sig n i a a1)} %{rss}
+ %[%[%[%[>H1 //|//] |@H3] |@daemon ]
+ |>reverse_cons >associative_append
+ >H2 in Htout2; #Htout >Htout [@Ht2| >Ht2 normalize % #H destruct (H)]
+ ]
+ |* #b * #rss * * #H1 #H2
+ #Ht2 %2
+ %{(shift_i sig n i b (all_blank sig n))} %{(shift_i sig n i a a1::rss)}
+ %[%[@H1 |@daemon ]
+ |>Ht2 in Htout1; #Htout >Htout //
+ whd in match (left ??); whd in match (right ??);
+ >reverse_append >reverse_single >associative_append >reverse_cons
+ >associative_append //
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem sem_shift_i_L_new: ∀sig,n,i.
+ shift_i_L sig n i ⊨ R_shift_i_L_new sig n i.
+#sig #n #i
+@(Realize_to_Realize … (sem_shift_i_L sig n i))
+#t1 #t2 #H #a #ls #rs #Ht1 lapply (H a ls rs Ht1) *
+ [* #rs1 * #b * #rs2 * #a1 * #rss * * * * #H1 #H2 #H3 #H4 #Ht2
+ %{rs1} %{b} %{(b::rs2)} %{(a1::rss)}
+ %[%[%[%[%[//|@H2]|@H1]|@H3]|@H4] | whd in match (tail ??); @Ht2]
+ |* #b * #rss * * #H1 #H2 #Ht2
+ %{rs} %{(all_blank sig n)} %{[]} %{(rss@[b])}
+ %[%[%[%[%[//|@blank_all_blank]|//]|@H1]|@H2] | whd in match (tail ??); @Ht2]
+ ]
+qed.
+
+
+(*******************************************************************************
+The following machine implements a full move of for a trace: we reach the left
+border, shift the i-th trace and come back to the head position. *)
+
+(* this exclude the possibility that traces do not overlap: the head must
+remain inside all traces *)
+
+definition mtiL ≝ λsig,n,i.
+ move_to_blank_L sig n i ·
+ shift_i_L sig n i ·
+ move_until ? L (no_head sig n).
+
+definition Rmtil ≝ λsig,n,i,t1,t2.
+ ∀ls,a,rs.
+ t1 = midtape (multi_sig sig n) ls a rs →
+ nth n ? (vec … a) (blank ?) = head ? →
+ (∀i.regular_trace sig n a ls rs i) →
+ (* next: we cannot be on rightof on trace i *)
+ (nth i ? (vec … a) (blank ?) = (blank ?)
+ → nth i ? (vec … (hd ? rs (all_blank …))) (blank ?) ≠ (blank ?)) →
+ no_head_in … ls →
+ no_head_in … rs →
+ (∃ls1,a1,rs1.
+ t2 = midtape (multi_sig …) ls1 a1 rs1 ∧
+ (∀i.regular_trace … a1 ls1 rs1 i) ∧
+ (∀j. j ≤ n → j ≠ i → to_blank_i ? n j (a1::ls1) = to_blank_i ? n j (a::ls)) ∧
+ (∀j. j ≤ n → j ≠ i → to_blank_i ? n j rs1 = to_blank_i ? n j rs) ∧
+ (to_blank_i ? n i ls1 = to_blank_i ? n i (a::ls)) ∧
+ (to_blank_i ? n i (a1::rs1)) = to_blank_i ? n i rs).
+
+theorem sem_Rmtil: ∀sig,n,i. i < n → mtiL sig n i ⊨ Rmtil sig n i.
+#sig #n #i #lt_in
+@(sem_seq_app ??????
+ (sem_move_to_blank_L … )
+ (sem_seq ????? (sem_shift_i_L_new …)
+ (ssem_move_until_L ? (no_head sig n))))
+#tin #tout * #t1 * * #_ #Ht1 * #t2 * #Ht2 * #_ #Htout
+(* we start looking into Rmitl *)
+#ls #a #rs #Htin (* tin is a midtape *)
+#Hhead #Hreg #no_rightof #Hnohead_ls #Hnohead_rs
+cut (regular_i sig n (a::ls) i)
+ [cases (Hreg i) * //
+ cases (true_or_false (nth i ? (vec … a) (blank ?) == (blank ?))) #Htest
+ [#_ @daemon (* absurd, since hd rs non e' blank *)
+ |#H #_ @daemon]] #Hreg1
+lapply (Ht1 … Htin Hreg1 ?) [#j #_ @Hreg] -Ht1 -Htin
+* #b * #ls1 * #ls2 * * * * * #reg_ls1_i #reg_ls1_j #Hno_blankb #Hhead #Hls1 #Ht1
+lapply (Ht2 … Ht1) -Ht2 -Ht1
+* #rs1 * #b0 * #rs2 * #rss * * * * * #Hb0 #Hb0blank #Hrs1 #Hrs1b #Hrss #Ht2
+(* we need to recover the position of the head of the emulated machine
+ that is the head of ls1. This is somewhere inside rs1 *)
+cut (∃rs11. rs1 = (reverse ? ls1)@rs11)
+ [cut (ls1 = [ ] ∨ ∃aa,tlls1. ls1 = aa::tlls1)
+ [cases ls1 [%1 // | #aa #tlls1 %2 %{aa} %{tlls1} //]] *
+ [#H1ls1 %{rs1} >H1ls1 //
+ |* #aa * #tlls1 #H1ls1 >H1ls1 in Hrs1;
+ cut (aa = a) [>H1ls1 in Hls1; #H @(to_blank_hd … H)] #eqaa >eqaa
+ #Hrs1_aux cases (compare_append … (sym_eq … Hrs1_aux)) #l *
+ [* #H1 #H2 %{l} @H1
+ |(* this is absurd : if l is empty, the case is as before.
+ if l is not empty then it must start with a blank, since it is the
+ first character in rs2. But in this case we would have a blank
+ inside ls1=a::tls1 that is absurd *)
+ @daemon
+ ]]]
+ * #rs11 #H1
+cut (rs = rs11@rs2)
+ [@(injective_append_l … (reverse … ls1)) >Hrs1 <associative_append <H1 //] #H2
+lapply (Htout … Ht2) -Htout -Ht2 *
+ [(* the current character on trace i holds the head-mark.
+ The case is absurd, since b0 is the head of rs2, that is a sublist of rs,
+ and the head-mark is not in rs *)
+ * #H3 @False_ind @(absurd (nth n ? (vec … b0) (blank sig) = head ?))
+ [@(\P ?) @injective_notb @H3 ]
+ @Hnohead_rs >H2 >trace_append @mem_append_l2
+ lapply Hb0 cases rs2
+ [whd in match (hd ???); #H >H in H3; whd in match (no_head ???);
+ >all_blank_n normalize -H #H destruct (H); @False_ind
+ |#c #r #H4 %1 >H4 //
+ ]
+ |*
+ [(* we reach the head position *)
+ (* cut (trace sig n j (a1::ls20)=trace sig n j (ls1@b::ls2)) *)
+ * #ls10 * #a1 * #ls20 * * * #Hls20 #Ha1 #Hnh #Htout
+ cut (∀j.j ≠ i →
+ trace sig n j (reverse (multi_sig sig n) rs1@b::ls2) =
+ trace sig n j (ls10@a1::ls20))
+ [#j #ineqj >append_cons <reverse_cons >trace_def <map_append <reverse_map
+ lapply (trace_shift_neq …lt_in ? (sym_not_eq … ineqj) … Hrss) [//] #Htr
+ <(trace_def … (b::rs1)) <Htr >reverse_map >map_append @eq_f @Hls20 ]
+ #Htracej
+ cut (trace sig n i (reverse (multi_sig sig n) (rs1@[b0])@ls2) =
+ trace sig n i (ls10@a1::ls20))
+ [>trace_def <map_append <reverse_map <map_append <(trace_def … [b0])
+ cut (trace sig n i [b0] = [blank ?]) [@daemon] #Hcut >Hcut
+ lapply (trace_shift … lt_in … Hrss) [//] whd in match (tail ??); #Htr <Htr
+ >reverse_map >map_append <trace_def <Hls20 %
+ ]
+ #Htracei
+ cut (∀j. j ≠ i →
+ (trace sig n j (reverse (multi_sig sig n) rs11) = trace sig n j ls10) ∧
+ (trace sig n j (ls1@b::ls2) = trace sig n j (a1::ls20)))
+ [@daemon (* si fa
+ #j #ineqj @(first_P_to_eq ? (λx. x ≠ head ?))
+ [lapply (Htracej … ineqj) >trace_def in ⊢ (%→?); <map_append
+ >trace_def in ⊢ (%→?); <map_append #H @H
+ | *) ] #H2
+ cut ((trace sig n i (b0::reverse ? rs11) = trace sig n i (ls10@[a1])) ∧
+ (trace sig n i (ls1@ls2) = trace sig n i ls20))
+ [>H1 in Htracei; >reverse_append >reverse_single >reverse_append
+ >reverse_reverse >associative_append >associative_append
+ @daemon
+ ] #H3
+ cut (∀j. j ≠ i →
+ trace sig n j (reverse (multi_sig sig n) ls10@rs2) = trace sig n j rs)
+ [#j #jneqi @(injective_append_l … (trace sig n j (reverse ? ls1)))
+ >map_append >map_append >Hrs1 >H1 >associative_append
+ <map_append <map_append in ⊢ (???%); @eq_f
+ <map_append <map_append @eq_f2 // @sym_eq
+ <(reverse_reverse … rs11) <reverse_map <reverse_map in ⊢ (???%);
+ @eq_f @(proj1 … (H2 j jneqi))] #Hrs_j
+ %{ls20} %{a1} %{(reverse ? (b0::ls10)@tail (multi_sig sig n) rs2)}
+ %[%[%[%[%[@Htout
+ |#j cases (decidable_eq_nat j i)
+ [#eqji >eqji (* by cases wether a1 is blank *)
+ @daemon
+ |#jneqi lapply (reg_ls1_j … jneqi) #H4
+ >reverse_cons >associative_append >Hb0 @regular_cons_hd_rs
+ @(eq_trace_to_regular … H4)
+ [<hd_trace >(proj2 … (H2 … jneqi)) >hd_trace %
+ |<tail_trace >(proj2 … (H2 … jneqi)) >tail_trace %
+ |@sym_eq @Hrs_j //
+ ]
+ ]]
+ |#j #lejn #jneqi <(Hls1 … lejn)
+ >to_blank_i_def >to_blank_i_def @eq_f @sym_eq @(proj2 … (H2 j jneqi))]
+ |#j #lejn #jneqi >reverse_cons >associative_append >Hb0
+ <to_blank_hd_cons >to_blank_i_def >to_blank_i_def @eq_f @Hrs_j //]
+ |<(Hls1 i) [2:@lt_to_le //]
+ lapply (all_blank_after_blank … reg_ls1_i)
+ [@(\P ?) @daemon] #allb_ls2
+ whd in match (to_blank_i ????); <(proj2 … H3)
+ @daemon ]
+ |>reverse_cons >associative_append
+ cut (to_blank_i sig n i rs = to_blank_i sig n i (rs11@[b0])) [@daemon]
+ #Hcut >Hcut >(to_blank_i_chop … b0 (a1::reverse …ls10)) [2: @Hb0blank]
+ >to_blank_i_def >to_blank_i_def @eq_f
+ >trace_def >trace_def @injective_reverse >reverse_map >reverse_cons
+ >reverse_reverse >reverse_map >reverse_append >reverse_single @sym_eq
+ @(proj1 … H3)
+ ]
+ |(*we do not find the head: this is absurd *)
+ * #b1 * #lss * * #H2 @False_ind
+ cut (∀x0. mem ? x0 (trace sig n n (b0::reverse ? rss@ls2)) → x0 ≠ head ?)
+ [@daemon] -H2 #H2
+ lapply (trace_shift_neq sig n i n … lt_in … Hrss)
+ [@lt_to_not_eq @lt_in | // ]
+ #H3 @(absurd
+ (nth n ? (vec … (hd ? (ls1@[b]) (all_blank sig n))) (blank ?) = head ?))
+ [>Hhead //
+ |@H2 >trace_def %2 <map_append @mem_append_l1 <reverse_map <trace_def
+ >H3 >H1 >trace_def >reverse_map >reverse_cons >reverse_append
+ >reverse_reverse >associative_append <map_append @mem_append_l2
+ cases ls1 [%1 % |#x #ll %1 %]
+ ]
+ ]
+ ]
+qed.
--- /dev/null
+include "turing/basic_machines.ma".
+include "turing/if_machine.ma".
+include "turing/multi_to_mono/trace_alphabet.ma".
+
+(* a machine that shift the i trace r starting from the bord of the trace *)
+
+(* vec is a coercion. Why should I insert it? *)
+definition mti_step ≝ λsig:FinSet.λn,i.
+ ifTM (multi_sig sig n) (test_char ? (λc:multi_sig sig n.¬(nth i ? (vec … c) (blank ?))==blank ?))
+ (single_finalTM … (ncombf_r (multi_sig sig n) (shift_i sig n i) (all_blank sig n)))
+ (nop ?) tc_true.
+
+definition Rmti_step_true ≝
+λsig,n,i,t1,t2.
+ ∃b:multi_sig sig n. (nth i ? (vec … b) (blank ?) ≠ blank ?) ∧
+ ((∃ls,a,rs.
+ t1 = midtape (multi_sig sig n) ls b (a::rs) ∧
+ t2 = midtape (multi_sig sig n) ((shift_i sig n i b a)::ls) a rs) ∨
+ (∃ls.
+ t1 = midtape ? ls b [] ∧
+ t2 = rightof ? (shift_i sig n i b (all_blank sig n)) ls)).
+
+(* 〈combf0,all_blank sig n〉 *)
+definition Rmti_step_false ≝
+ λsig,n,i,t1,t2.
+ (∀ls,b,rs. t1 = midtape (multi_sig sig n) ls b rs →
+ (nth i ? (vec … b) (blank ?) = blank ?)) ∧ t2 = t1.
+
+lemma sem_mti_step :
+ ∀sig,n,i.
+ mti_step sig n i ⊨
+ [inr … (inl … (inr … start_nop)): Rmti_step_true sig n i, Rmti_step_false sig n i].
+#sig #n #i
+@(acc_sem_if_app (multi_sig sig n) ??????????
+ (sem_test_char …) (sem_ncombf_r (multi_sig sig n) (shift_i sig n i)(all_blank sig n))
+ (sem_nop (multi_sig sig n)))
+ [#intape #outtape #tapea whd in ⊢ (%→%→%); * * #c *
+ #Hcur cases (current_to_midtape … Hcur) #ls * #rs #Hintape
+ #ctest #Htapea * #Hout1 #Hout2 @(ex_intro … c) %
+ [@(\Pf (injective_notb … )) @ctest]
+ generalize in match Hintape; -Hintape cases rs
+ [#Hintape %2 @(ex_intro …ls) % // @Hout1 >Htapea @Hintape
+ |#a #rs1 #Hintape %1
+ @(ex_intro … ls) @(ex_intro … a) @(ex_intro … rs1) % //
+ @Hout2 >Htapea @Hintape
+ ]
+ |#intape #outtape #tapea whd in ⊢ (%→%→%);
+ * #Htest #tapea #outtape
+ % // #ls #b #rs
+ #intape lapply (Htest b ?) [>intape //] -Htest #Htest
+ lapply (injective_notb ? true Htest) -Htest #Htest @(\P Htest)
+ ]
+qed.
+
+(* move tape i machine *)
+definition mti ≝
+ λsig,n,i.whileTM (multi_sig sig n) (mti_step sig n i) (inr … (inl … (inr … start_nop))).
+
+axiom daemon: ∀P:Prop.P.
+
+definition R_mti ≝
+ λsig,n,i,t1,t2.
+ (∀a,rs. t1 = rightof ? a rs → t2 = t1) ∧
+ (∀a,ls,rs.
+ t1 = midtape (multi_sig sig n) ls a rs →
+ (nth i ? (vec … a) (blank ?) = blank ? ∧ t2 = t1) ∨
+ ((∃rs1,b,rs2,rss.
+ rs = rs1@b::rs2 ∧
+ nth i ? (vec … b) (blank ?) = (blank ?) ∧
+ (∀x. mem ? x (a::rs1) → nth i ? (vec … x) (blank ?) ≠ (blank ?)) ∧
+ shift_l sig n i (a::rs1) rss ∧
+ t2 = midtape (multi_sig sig n) ((reverse ? rss)@ls) b rs2) ∨
+ (∃b,rss.
+ (∀x. mem ? x (a::rs) → nth i ? (vec … x) (blank ?) ≠ (blank ?)) ∧
+ shift_l sig n i (a::rs) (rss@[b]) ∧
+ t2 = rightof (multi_sig sig n) (shift_i sig n i b (all_blank sig n))
+ ((reverse ? rss)@ls)))).
+
+lemma sem_mti:
+ ∀sig,n,i.
+ WRealize (multi_sig sig n) (mti sig n i) (R_mti sig n i).
+#sig #n #i #inc #j #outc #Hloop
+lapply (sem_while … (sem_mti_step sig n i) inc j outc Hloop) [%]
+-Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
+[ whd in ⊢ (% → ?); * #H1 #H2 %
+ [#a #rs #Htape1 @H2
+ |#a #ls #rs #Htapea % % [@(H1 … Htapea) |@H2]
+ ]
+| #tapeb #tapec #Hstar1 #HRtrue #IH #HRfalse
+ lapply (IH HRfalse) -IH -HRfalse whd in ⊢ (%→%);
+ * #IH1 #IH2 %
+ [#b0 #ls #Htapeb cases Hstar1 #b * #_ *
+ [* #ls0 * #a * #rs0 * >Htapeb #H destruct (H)
+ |* #ls0 * >Htapeb #H destruct (H)
+ ]
+ |#b0 #ls #rs #Htapeb cases Hstar1 #b * #btest *
+ [* #ls1 * #a * #rs1 * >Htapeb #H destruct (H) #Htapec
+ %2 cases (IH2 … Htapec)
+ [(* case a = None *)
+ * #testa #Hout %1
+ %{[ ]} %{a} %{rs1} %{[shift_i sig n i b a]} %
+ [%[%[% // |#x #Hb >(mem_single ??? Hb) // ]
+ |@daemon]
+ |>Hout >reverse_single @Htapec
+ ]
+ |*
+ [ (* case a \= None and exists b = None *) -IH1 -IH2 #IH
+ %1 cases IH -IH #rs10 * #b0 * #rs2 * #rss * * * *
+ #H1 #H2 #H3 #H4 #H5
+ %{(a::rs10)} %{b0} %{rs2} %{(shift_i sig n i b a::rss)}
+ %[%[%[%[>H1 //|@H2]
+ |#x * [#eqxa >eqxa (*?? *) @daemon|@H3]]
+ |@daemon]
+ |>H5 >reverse_cons >associative_append //
+ ]
+ | (* case a \= None and we reach the end of the (full) tape *) -IH1 -IH2 #IH
+ %2 cases IH -IH #b0 * #rss * * #H1 #H2 #H3
+ %{b0} %{(shift_i sig n i b a::rss)}
+ %[%[#x * [#eqxb >eqxb @btest|@H1]
+ |@daemon]
+ |>H3 >reverse_cons >associative_append //
+ ]
+ ]
+ ]
+ |(* b \= None but the left tape is empty *)
+ * #ls0 * >Htapeb #H destruct (H) #Htapec
+ %2 %2 %{b} %{[ ]}
+ %[%[#x * [#eqxb >eqxb @btest|@False_ind]
+ |@daemon (*shift of dingle char OK *)]
+ |>(IH1 … Htapec) >Htapec //
+ ]
+ ]
+qed.
+
+lemma WF_mti_niltape:
+ ∀sig,n,i. WF ? (inv ? (Rmti_step_true sig n i)) (niltape ?).
+#sig #n #i @wf #t1 whd in ⊢ (%→?); * #b * #_ *
+ [* #ls * #a * #rs * #H destruct|* #ls * #H destruct ]
+qed.
+
+lemma WF_mti_rightof:
+ ∀sig,n,i,a,ls. WF ? (inv ? (Rmti_step_true sig n i)) (rightof ? a ls).
+#sig #n #i #a #ls @wf #t1 whd in ⊢ (%→?); * #b * #_ *
+ [* #ls * #a * #rs * #H destruct|* #ls * #H destruct ]
+qed.
+
+lemma WF_mti_leftof:
+ ∀sig,n,i,a,ls. WF ? (inv ? (Rmti_step_true sig n i)) (leftof ? a ls).
+#sig #n #i #a #ls @wf #t1 whd in ⊢ (%→?); * #b * #_ *
+ [* #ls * #a * #rs * #H destruct|* #ls * #H destruct ]
+qed.
+
+lemma terminate_mti:
+ ∀sig,n,i.∀t. Terminate ? (mti sig n i) t.
+#sig #n #i #t @(terminate_while … (sem_mti_step sig n i)) [%]
+cases t // #ls #c #rs lapply c -c lapply ls -ls elim rs
+ [#ls #c @wf #t1 whd in ⊢ (%→?); * #b * #_ *
+ [* #ls1 * #a * #rs1 * #H destruct
+ |* #ls1 * #H destruct #Ht1 >Ht1 //
+ ]
+ |#a #rs1 #Hind #ls #c @wf #t1 whd in ⊢ (%→?); * #b * #_ *
+ [* #ls1 * #a2 * #rs2 * #H destruct (H) #Ht1 >Ht1 //
+ |* #ls1 * #H destruct
+ ]
+ ]
+qed.
+
+lemma ssem_mti: ∀sig,n,i.
+ Realize ? (mti sig n i) (R_mti sig n i).
+/2/ qed.
+
+++ /dev/null
-include "turing/basic_machines.ma".
-include "turing/if_machine.ma".
-include "turing/multi_to_mono/trace_alphabet.ma".
-
-(* a machine that shift the i trace r starting from the bord of the trace *)
-
-(* vec is a coercion. Why should I insert it? *)
-definition mti_step ≝ λsig:FinSet.λn,i.
- ifTM (multi_sig sig n) (test_char ? (λc:multi_sig sig n.¬(nth i ? (vec … c) (blank ?))==blank ?))
- (single_finalTM … (ncombf_r (multi_sig sig n) (shift_i sig n i) (all_blank sig n)))
- (nop ?) tc_true.
-
-definition Rmti_step_true ≝
-λsig,n,i,t1,t2.
- ∃b:multi_sig sig n. (nth i ? (vec … b) (blank ?) ≠ blank ?) ∧
- ((∃ls,a,rs.
- t1 = midtape (multi_sig sig n) ls b (a::rs) ∧
- t2 = midtape (multi_sig sig n) ((shift_i sig n i b a)::ls) a rs) ∨
- (∃ls.
- t1 = midtape ? ls b [] ∧
- t2 = rightof ? (shift_i sig n i b (all_blank sig n)) ls)).
-
-(* 〈combf0,all_blank sig n〉 *)
-definition Rmti_step_false ≝
- λsig,n,i,t1,t2.
- (∀ls,b,rs. t1 = midtape (multi_sig sig n) ls b rs →
- (nth i ? (vec … b) (blank ?) = blank ?)) ∧ t2 = t1.
-
-lemma sem_mti_step :
- ∀sig,n,i.
- mti_step sig n i ⊨
- [inr … (inl … (inr … start_nop)): Rmti_step_true sig n i, Rmti_step_false sig n i].
-#sig #n #i
-@(acc_sem_if_app (multi_sig sig n) ??????????
- (sem_test_char …) (sem_ncombf_r (multi_sig sig n) (shift_i sig n i)(all_blank sig n))
- (sem_nop (multi_sig sig n)))
- [#intape #outtape #tapea whd in ⊢ (%→%→%); * * #c *
- #Hcur cases (current_to_midtape … Hcur) #ls * #rs #Hintape
- #ctest #Htapea * #Hout1 #Hout2 @(ex_intro … c) %
- [@(\Pf (injective_notb … )) @ctest]
- generalize in match Hintape; -Hintape cases rs
- [#Hintape %2 @(ex_intro …ls) % // @Hout1 >Htapea @Hintape
- |#a #rs1 #Hintape %1
- @(ex_intro … ls) @(ex_intro … a) @(ex_intro … rs1) % //
- @Hout2 >Htapea @Hintape
- ]
- |#intape #outtape #tapea whd in ⊢ (%→%→%);
- * #Htest #tapea #outtape
- % // #ls #b #rs
- #intape lapply (Htest b ?) [>intape //] -Htest #Htest
- lapply (injective_notb ? true Htest) -Htest #Htest @(\P Htest)
- ]
-qed.
-
-(* move tape i machine *)
-definition mti ≝
- λsig,n,i.whileTM (multi_sig sig n) (mti_step sig n i) (inr … (inl … (inr … start_nop))).
-
-axiom daemon: ∀P:Prop.P.
-
-definition R_mti ≝
- λsig,n,i,t1,t2.
- (∀a,rs. t1 = rightof ? a rs → t2 = t1) ∧
- (∀a,ls,rs.
- t1 = midtape (multi_sig sig n) ls a rs →
- (nth i ? (vec … a) (blank ?) = blank ? ∧ t2 = t1) ∨
- ((∃rs1,b,rs2,rss.
- rs = rs1@b::rs2 ∧
- nth i ? (vec … b) (blank ?) = (blank ?) ∧
- (∀x. mem ? x (a::rs1) → nth i ? (vec … x) (blank ?) ≠ (blank ?)) ∧
- shift_l sig n i (a::rs1) rss ∧
- t2 = midtape (multi_sig sig n) ((reverse ? rss)@ls) b rs2) ∨
- (∃b,rss.
- (∀x. mem ? x (a::rs) → nth i ? (vec … x) (blank ?) ≠ (blank ?)) ∧
- shift_l sig n i (a::rs) (rss@[b]) ∧
- t2 = rightof (multi_sig sig n) (shift_i sig n i b (all_blank sig n))
- ((reverse ? rss)@ls)))).
-
-lemma sem_mti:
- ∀sig,n,i.
- WRealize (multi_sig sig n) (mti sig n i) (R_mti sig n i).
-#sig #n #i #inc #j #outc #Hloop
-lapply (sem_while … (sem_mti_step sig n i) inc j outc Hloop) [%]
--Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
-[ whd in ⊢ (% → ?); * #H1 #H2 %
- [#a #rs #Htape1 @H2
- |#a #ls #rs #Htapea % % [@(H1 … Htapea) |@H2]
- ]
-| #tapeb #tapec #Hstar1 #HRtrue #IH #HRfalse
- lapply (IH HRfalse) -IH -HRfalse whd in ⊢ (%→%);
- * #IH1 #IH2 %
- [#b0 #ls #Htapeb cases Hstar1 #b * #_ *
- [* #ls0 * #a * #rs0 * >Htapeb #H destruct (H)
- |* #ls0 * >Htapeb #H destruct (H)
- ]
- |#b0 #ls #rs #Htapeb cases Hstar1 #b * #btest *
- [* #ls1 * #a * #rs1 * >Htapeb #H destruct (H) #Htapec
- %2 cases (IH2 … Htapec)
- [(* case a = None *)
- * #testa #Hout %1
- %{[ ]} %{a} %{rs1} %{[shift_i sig n i b a]} %
- [%[%[% // |#x #Hb >(mem_single ??? Hb) // ]
- |@daemon]
- |>Hout >reverse_single @Htapec
- ]
- |*
- [ (* case a \= None and exists b = None *) -IH1 -IH2 #IH
- %1 cases IH -IH #rs10 * #b0 * #rs2 * #rss * * * *
- #H1 #H2 #H3 #H4 #H5
- %{(a::rs10)} %{b0} %{rs2} %{(shift_i sig n i b a::rss)}
- %[%[%[%[>H1 //|@H2]
- |#x * [#eqxa >eqxa (*?? *) @daemon|@H3]]
- |@daemon]
- |>H5 >reverse_cons >associative_append //
- ]
- | (* case a \= None and we reach the end of the (full) tape *) -IH1 -IH2 #IH
- %2 cases IH -IH #b0 * #rss * * #H1 #H2 #H3
- %{b0} %{(shift_i sig n i b a::rss)}
- %[%[#x * [#eqxb >eqxb @btest|@H1]
- |@daemon]
- |>H3 >reverse_cons >associative_append //
- ]
- ]
- ]
- |(* b \= None but the left tape is empty *)
- * #ls0 * >Htapeb #H destruct (H) #Htapec
- %2 %2 %{b} %{[ ]}
- %[%[#x * [#eqxb >eqxb @btest|@False_ind]
- |@daemon (*shift of dingle char OK *)]
- |>(IH1 … Htapec) >Htapec //
- ]
- ]
-qed.
-
-lemma WF_mti_niltape:
- ∀sig,n,i. WF ? (inv ? (Rmti_step_true sig n i)) (niltape ?).
-#sig #n #i @wf #t1 whd in ⊢ (%→?); * #b * #_ *
- [* #ls * #a * #rs * #H destruct|* #ls * #H destruct ]
-qed.
-
-lemma WF_mti_rightof:
- ∀sig,n,i,a,ls. WF ? (inv ? (Rmti_step_true sig n i)) (rightof ? a ls).
-#sig #n #i #a #ls @wf #t1 whd in ⊢ (%→?); * #b * #_ *
- [* #ls * #a * #rs * #H destruct|* #ls * #H destruct ]
-qed.
-
-lemma WF_mti_leftof:
- ∀sig,n,i,a,ls. WF ? (inv ? (Rmti_step_true sig n i)) (leftof ? a ls).
-#sig #n #i #a #ls @wf #t1 whd in ⊢ (%→?); * #b * #_ *
- [* #ls * #a * #rs * #H destruct|* #ls * #H destruct ]
-qed.
-
-lemma terminate_mti:
- ∀sig,n,i.∀t. Terminate ? (mti sig n i) t.
-#sig #n #i #t @(terminate_while … (sem_mti_step sig n i)) [%]
-cases t // #ls #c #rs lapply c -c lapply ls -ls elim rs
- [#ls #c @wf #t1 whd in ⊢ (%→?); * #b * #_ *
- [* #ls1 * #a * #rs1 * #H destruct
- |* #ls1 * #H destruct #Ht1 >Ht1 //
- ]
- |#a #rs1 #Hind #ls #c @wf #t1 whd in ⊢ (%→?); * #b * #_ *
- [* #ls1 * #a2 * #rs2 * #H destruct (H) #Ht1 >Ht1 //
- |* #ls1 * #H destruct
- ]
- ]
-qed.
-
-lemma ssem_mti: ∀sig,n,i.
- Realize ? (mti sig n i) (R_mti sig n i).
-/2/ qed.
-
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