+++ /dev/null
-include "turing/auxiliary_machines1.ma".
-include "turing/multi_to_mono/shift_trace_machines.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.
projects / helm.git / blobdiff