X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_to_mono%2Fmulti_to_mono.ma;fp=matita%2Fmatita%2Flib%2Fturing%2Fmulti_to_mono%2Fmulti_to_mono.ma;h=0000000000000000000000000000000000000000;hb=600fba840c748f67593838673a6eb40eab9b68e5;hp=0e9bce53e403f8dde1c6efc62b5660c69c5dff55;hpb=b4f76b0d8fa0e5365fb48e91474febe200b647a7;p=helm.git

diff --git a/matita/matita/lib/turing/multi_to_mono/multi_to_mono.ma b/matita/matita/lib/turing/multi_to_mono/multi_to_mono.ma
deleted file mode 100644
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--- a/matita/matita/lib/turing/multi_to_mono/multi_to_mono.ma
+++ /dev/null
@@ -1,397 +0,0 @@
-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.