-include "turing/basic_machines.ma".
-include "turing/if_machine.ma".
include "turing/auxiliary_machines1.ma".
-include "turing/multi_to_mono/trace_alphabet.ma".
+include "turing/multi_to_mono/shift_trace_machines.ma".
-(* a machine that moves the i trace r: we reach the left margin of the i-trace
-and make a (shifted) copy of the old tape up to the end of the right margin of
-the i-trace. Then come back to the origin. *)
-
-definition shift_i ≝ λsig,n,i.λa,b:Vector (sig_ext sig) n.
- change_vec (sig_ext sig) n a (nth i ? b (blank ?)) i.
-
-(* 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))).
-
-(* v2 is obtained from v1 shifting left the i-th trace *)
-definition shift_l ≝ λsig,n,i,v1,v2. (* multi_sig sig n*)
- |v1| = |v2| ∧
- ∀j.nth j ? v2 (all_blank sig n) =
- change_vec ? n (nth j ? v1 (all_blank sig n))
- (nth i ? (vec … (nth (S j) ? v1 (all_blank sig n))) (blank ?)) i.
-
-lemma trace_shift: ∀sig,n,i,v1,v2. i < n → 0 < |v1| →
- shift_l sig n i v1 v2 → trace ? n i v2 = tail ? (trace ? n i v1)@[blank sig].
-#sig #n #i #v1 #v2 #lein #Hlen * #Hlen1 #Hnth @(nth_to_eq … (blank ?))
- [>length_trace <Hlen1 generalize in match Hlen; cases v1
- [normalize #H @(le_n_O_elim … H) //
- |#b #tl #_ normalize >length_append >length_trace normalize //
- ]
- |#j >nth_trace >Hnth >nth_change_vec // >tail_trace
- cut (trace ? n i [all_blank sig n] = [blank sig])
- [normalize >blank_all_blank //]
- #Hcut <Hcut <trace_append >nth_trace
- <nth_extended //
- ]
-qed.
-
-lemma trace_shift_neq: ∀sig,n,i,j,v1,v2. i < n → 0 < |v1| → i ≠ j →
- shift_l sig n i v1 v2 → trace ? n j v2 = trace ? n j v1.
-#sig #n #i #j #v1 #v2 #lein #Hlen #Hneq * #Hlen1 #Hnth @(nth_to_eq … (blank ?))
- [>length_trace >length_trace @sym_eq @Hlen1
- |#k >nth_trace >Hnth >nth_change_vec_neq // >nth_trace //
- ]
-qed.
-
-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.
-
(******************************************************************************)
(* 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. *)
(******************************************************************************)
-definition no_blank ≝ λsig,n,i.λc:multi_sig sig n.
- ¬(nth i ? (vec … c) (blank ?))==blank ?.
-
-definition no_head ≝ λsig,n.λc:multi_sig sig n.
- ¬((nth n ? (vec … c) (blank ?))==head ?).
-
-let rec to_blank sig l on l ≝
- match l with
- [ nil ⇒ [ ]
- | cons a tl ⇒
- if a == blank sig then [ ] else a::(to_blank sig tl)].
-
-definition to_blank_i ≝ λsig,n,i,l. to_blank ? (trace sig n i l).
-
-lemma to_blank_i_def : ∀sig,n,i,l.
- to_blank_i sig n i l = to_blank ? (trace sig n i l).
-// qed.
-(*
- match l with
- [ nil ⇒ [ ]
- | cons a tl ⇒
- let ai ≝ nth i ? (vec … n a) (blank sig) in
- if ai == blank ? then [ ] else ai::(to_blank sig n i tl)]. *)
-
-lemma to_blank_cons_b : ∀sig,n,i,a,l.
- nth i ? (vec … n a) (blank sig) = blank ? →
- to_blank_i sig n i (a::l) = [ ].
-#sig #n #i #a #l #H whd in match (to_blank_i ????);
->(\b H) //
-qed.
+(* 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.
-lemma to_blank_cons_nb: ∀sig,n,i,a,l.
- nth i ? (vec … n a) (blank sig) ≠ blank ? →
- to_blank_i sig n i (a::l) =
- nth i ? (vec … n a) (blank sig)::(to_blank_i sig n i l).
-#sig #n #i #a #l #H whd in match (to_blank_i ????);
->(\bf H) //
+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.
-axiom to_blank_hd : ∀sig,n,a,b,l1,l2.
- (∀i. i ≤ n → to_blank_i sig n i (a::l1) = to_blank_i sig n i (b::l2)) → a = b.
+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 to_blank_i_ext : ∀sig,n,i,l.
- to_blank_i sig n i l = to_blank_i sig n i (l@[all_blank …]).
-#sig #n #i #l elim l
- [@sym_eq @to_blank_cons_b @blank_all_blank
- |#a #tl #Hind whd in match (to_blank_i ????); >Hind <to_blank_i_def >Hind %
- ]
-qed.
-
-lemma to_blank_hd_cons : ∀sig,n,i,l1,l2.
- to_blank_i sig n i (l1@l2) =
- to_blank_i … i (l1@(hd … l2 (all_blank …))::tail … l2).
-#sig #n #i #l1 #l2 cases l2
- [whd in match (hd ???); whd in match (tail ??); >append_nil @to_blank_i_ext
- |#a #tl %
- ]
+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.
-lemma to_blank_i_chop : ∀sig,n,i,a,l1,l2.
- (nth i ? (vec … a) (blank ?))= blank ? →
- to_blank_i sig n i (l1@a::l2) = to_blank_i sig n i l1.
-#sig #n #i #a #l1 elim l1
- [#l2 #H @to_blank_cons_b //
- |#x #tl #Hind #l2 #H whd in match (to_blank_i ????);
- >(Hind l2 H) <to_blank_i_def >(Hind l2 H) %
- ]
-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)) ∧
(∃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)).
+ t2 = midtape ? ls2 b ((reverse ? (a::ls1))@rs)).
+*)
-definition R_move_to_blank_L_new ≝ λsig,n,i,t1,t2.
+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 →
+∀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 ? (ls1@[b]) (all_blank …) = a) ∧ (* not implied by the next fact *)
+ (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
(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 -Ht1a cases (Ht1b … Htin)
- [* #Hnb #Ht1 -Ht1b -Ht2a >Ht1 in Ht2b; >Htin #H %1
- % [//|@H normalize % #H1 destruct (H1)]
+ |#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 #H4 %2
- %{b} %{ls1} %{ls2}
- %[%[@H2|>H1 //]
- |-Ht1b -Ht2a >H4 in Ht2b; #H5 @H5
- % normalize #H6 destruct (H6)
+ * #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 #H3 %2
- %{(all_blank …)} %{ls} %{[ ]}
- %[%[whd in match (no_blank ????); >blank_all_blank // @daemon
- |@daemon (* make a lemma *)]
- |-Ht1b -Ht2b >H3 in Ht2a;
- whd in match (left ??); whd in match (right ??); #H4
- >H2 >reverse_append >reverse_single @H4 normalize //
+ |* #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.
-theorem sem_move_to_blank_L_new: ∀sig,n,i.
- move_to_blank_L sig n i ⊨ R_move_to_blank_L_new sig n i.
-#sig #n #i
-@(Realize_to_Realize … (sem_move_to_blank_L sig n i))
-#t1 #t2 * #H1 #H2 % [@H1]
-#ls #a #rs #Ht1 lapply (H2 ls a rs Ht1) -H2 *
- [* #Ha #Ht2 >Ht2 %{a} %{[ ]} %{ls}
- %[%[%[@Ha| //]| normalize //] | @Ht1]
- |* #b * #ls1 * #ls2 * * * #Hblank #Ht2
- %{b} %{(a::ls1)} %{ls2}
- %[2:@Ht2|%[%[//|//]|
- #j #lejn whd in match (to_blank_i ????);
- whd in match (to_blank_i ???(a::ls));
- lapply (Hblank j lejn) whd in match (to_blank_i ????);
- whd in match (to_blank_i ???(a::ls)); #H >H %
- ]
-qed.
-
(******************************************************************************)
definition shift_i_L ≝ λsig,n,i.
extend ? (all_blank sig n).
definition R_shift_i_L ≝ λsig,n,i,t1,t2.
- (* ∀b:multi_sig sig n.∀ls.
- (t1 = midtape ? ls b [ ] →
- t2 = midtape (multi_sig sig n)
- ((shift_i sig n i b (all_blank sig n))::ls) (all_blank sig n) [ ]) ∧ *)
(∀a,ls,rs.
t1 = midtape ? ls a rs →
((∃rs1,b,rs2,a1,rss.
(sem_seq ????? (ssem_mti sig n i)
(sem_extend ? (all_blank sig n))))
#tin #tout * #t1 * * #Ht1a #Ht1b * #t2 * * #Ht2a #Ht2b * #Htout1 #Htout2
-(* #b #ls %
- [#Htin 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 // *)
#a #ls #rs cases rs
[#Htin %2 %{(shift_i sig n i a (all_blank sig n))} %{[ ]}
%[%[#x @False_ind | @daemon]
qed.
-(******************************************************************************)
-definition no_head_in ≝ λsig,n,l.
- ∀x. mem ? x (trace sig n n l) → x ≠ head ?.
-
-lemma no_head_true: ∀sig,n,a.
- nth n ? (vec … n a) (blank sig) ≠ head ? → no_head … a = true.
-#sig #n #a normalize cases (nth n ? (vec … n a) (blank sig))
-normalize // * normalize // * #H @False_ind @H //
-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. *)
-lemma no_head_false: ∀sig,n,a.
- nth n ? (vec … n a) (blank sig) = head ? → no_head … a = false.
-#sig #n #a #H normalize >H //
-qed.
+(* 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 ·
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 ? →
+ 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).
-lemma reverse_map: ∀A,B,f,l.
- reverse B (map … f l) = map … f (reverse A l).
-#A #B #f #l elim l //
-#a #l #Hind >reverse_cons >reverse_cons <map_append >Hind //
-qed.
-
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_new … )
+ (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 #Hnohead_ls #Hnohead_rs
-lapply (Ht1 … Htin) -Ht1 -Htin
-* #b * #ls1 * #ls2 * * * #Hno_blankb #Hhead #Hls1 #Ht1
+#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.
+ [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
- frist character in rs2. But in this case we would have a blank
- inside ls1=attls1 that is absurd *)
+ 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 hold the head-mark.
+ [(* 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 *)
- @daemon
- (* something is missing
- * #H @False_ind @(absurd ? H) @eqnot_to_noteq whd in ⊢ (???%);
- cut (rs2 = [ ] ∨ ∃c,rs21. rs2 = c::rs21)
- [cases rs2 [ %1 // | #c #rs22 %2 %{c} %{rs22} //]]
- * (* by cases on rs2 *)
- [#Hrs2 >Hb0 >Hrs2 normalize >all_blank_n //
- |* #c * #rs22 #Hrs2 destruct (Hrs2) @no_head_true @Hnohead_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)) *)
| *) ] #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
+ [>H1 in Htracei; >reverse_append >reverse_single >reverse_append
>reverse_reverse >associative_append >associative_append
@daemon
- ] #H3
- %{ls20} %{a1} %{(reverse ? (b0::ls10)@tail (multi_sig sig n) rs2)}
- %[%[%[%[@Htout
- |#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
- @(injective_append_l … (trace sig n j (reverse ? ls1))) (* >trace_def >trace_def *)
+ ] #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))]
- |<(Hls1 i) [2:@lt_to_le //] (* manca un invariante: dopo un blank
- posso avere solo blank *) @daemon ]
+ @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]
>reverse_reverse >associative_append <map_append @mem_append_l2
cases ls1 [%1 % |#x #ll %1 %]
]
-
-
-
-
-(*
-
- cut (∀j.i ≠ j →
- trace sig n j (reverse (multi_sig sig n) rss@ls2) =
- trace sig n j (ls10@a1::ls20))
- [#j #ineqj >trace_def <map_append in ⊢ (??%?);
- <reverse_map lapply (trace_shift_neq …lt_in ? ineqj … Hrss) [//] #Htr
- <trace_def <trace_def >Htr >reverse_cons *)
-
- %{ls20} %{a1} %{(reverse ? (b0::ls10)@tail (multi_sig sig n) rs2)}
- %[%[%[%[%[@Htout
- |#j #lejn #jneqi <(Hls1 … lejn) -Hls1
- >to_blank_i_def >to_blank_i_def @eq_f
- @(injective_append_l … (trace sig n j (reverse ? rs))) (* >trace_def >trace_def *)
- >map_append >map_append
-
- ]
- |@daemon]
- |@daemon]
- |@daemon]
- |
-
-
- [(* the current character on trace i is blank *)
- * #Hblank #Ht1 <Ht1 in Htin; #Ht1a >Ht1a in Ht2;
- #Ht2 lapply (Ht2 … (refl …)) *
- [(* we reach the margin of the i-th trace *)
- * #rs1 * #b * #rs2 * #a1 * #rss * * * * #H1 #H2 #H3 #H4 #Ht2
- lapply (Htout … Ht2) -Htout *
- [(* the head is on b: this is absurd *)
- * #Hhb @False_ind >Hnohead_rs in Hhb;
- [#H destruct (H) | >H1 @mem_append_l2 %1 //]
- |*
- [(* we reach the head position *)
- * #ls1 * #b0 * #ls2 * * * #H5 #H6 #H7 #Htout
- %{ls2} %{b0} %{(reverse ? (b::ls1)@rs2)}
- cut (ls2 = ls)
- [@daemon (* da finire
- lapply H5 lapply H4 -H5 -H4 cases rss
- [* normalize in ⊢ (%→?); #H destruct (H)
- |#rssa #rsstl #H >reverse_cons >associative_append
- normalize in ⊢ (??(???%)%→?); #H8 @sym_eq
- @(first_P_to_eq (multi_sig sig n)
- (λa.nth n (sig_ext sig) (vec … a) (blank ?) = head ?) ?????? H8)
- *)
- ] #Hls
- %[%[%[%[%[@Htout|>Hls //]
- | #j #lejn #neji >to_blank_i_def >to_blank_i_def
- @eq_f >H1 >trace_def >trace_def >reverse_cons
- >associative_append <map_append <map_append in ⊢ (???%); @eq_f2 //
- @daemon]
- |//]
- |* #H @False_ind @H @daemon
- ]
- |>H1 @daemon
- ]
- |(* we do not find the head: absurd *)
- cut (nth n (sig_ext sig) (vec … a1) (blank sig)=head sig)
- [lapply (trace_shift_neq ??? n … lt_in … H4)
- [@lt_to_not_eq // |//]
- whd in match (trace ????); whd in match (trace ???(a::rs1));
- #H <Hhead @(cons_injective_l … H)]
- #Hcut * #b0 * #lss * * #H @False_ind
- @(absurd (true=false)) [2://] <(H a1)
- [whd in match (no_head ???); >Hcut //
- |%2 @mem_append_l1 >reverse_cons @mem_append_l2 %1 //
- ]
- ]
-
-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 …)
- (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 #Hnohead_ls #Hnohead_rs
-cases (Ht1 … Htin) -Ht1
- [(* the current character on trace i is blank *)
- * #Hblank #Ht1 <Ht1 in Htin; #Ht1a >Ht1a in Ht2;
- #Ht2 lapply (Ht2 … (refl …)) *
- [(* we reach the margin of the i-th trace *)
- * #rs1 * #b * #rs2 * #a1 * #rss * * * * #H1 #H2 #H3 #H4 #Ht2
- lapply (Htout … Ht2) -Htout *
- [(* the head is on b: this is absurd *)
- * #Hhb @False_ind >Hnohead_rs in Hhb;
- [#H destruct (H) | >H1 @mem_append_l2 %1 //]
- |*
- [(* we reach the head position *)
- * #ls1 * #b0 * #ls2 * * * #H5 #H6 #H7 #Htout
- %{ls2} %{b0} %{(reverse ? (b::ls1)@rs2)}
- cut (ls2 = ls)
- [@daemon (* da finire
- lapply H5 lapply H4 -H5 -H4 cases rss
- [* normalize in ⊢ (%→?); #H destruct (H)
- |#rssa #rsstl #H >reverse_cons >associative_append
- normalize in ⊢ (??(???%)%→?); #H8 @sym_eq
- @(first_P_to_eq (multi_sig sig n)
- (λa.nth n (sig_ext sig) (vec … a) (blank ?) = head ?) ?????? H8)
- *)
- ] #Hls
- %[%[%[%[%[@Htout|>Hls //]
- | #j #lejn #neji >to_blank_i_def >to_blank_i_def
- @eq_f >H1 >trace_def >trace_def >reverse_cons
- >associative_append <map_append <map_append in ⊢ (???%); @eq_f2 //
- @daemon]
- |//]
- |* #H @False_ind @H @daemon
- ]
- |>H1 @daemon
- ]
- |(* we do not find the head: absurd *)
- cut (nth n (sig_ext sig) (vec … a1) (blank sig)=head sig)
- [lapply (trace_shift_neq ??? n … lt_in … H4)
- [@lt_to_not_eq // |//]
- whd in match (trace ????); whd in match (trace ???(a::rs1));
- #H <Hhead @(cons_injective_l … H)]
- #Hcut * #b0 * #lss * * #H @False_ind
- @(absurd (true=false)) [2://] <(H a1)
- [whd in match (no_head ???); >Hcut //
- |%2 @mem_append_l1 >reverse_cons @mem_append_l2 %1 //
- ]
- ]
-
-
-theorem sem_Rmtil: ∀sig,n,i. mtiL sig n i ⊨ Rmtil sig n i.
-#sig #n #i
-@(sem_seq_app ??????
- (ssem_move_until_L ? (no_blank sig n i))
- (sem_seq ????? (sem_extend ? (all_blank sig n))
- (sem_seq ????? (sem_ncombf_r (multi_sig sig n) (shift_i sig n i)(all_blank sig n))
- (sem_seq ????? (ssem_mti sig n i)
- (sem_seq ????? (sem_extend ? (all_blank sig n))
- (ssem_move_until_L ? (no_head sig n)))))))
-#tin #tout * #t1 * * #_ #Ht1 * #t2 * * #Ht2a #Ht2b * #t3 * * #Ht3a #Ht3b
-* #t4 * * #Ht4a #Ht4b * #t5 * * #Ht5a #Ht5b * #t6 #Htout
-(* we start looking into Rmitl *)
-#ls #a #rs #Htin (* tin is a midtape *)
-#Hhead #Hnohead_ls #Hnohead_rs
-cases (Ht1 … Htin) -Ht1
- [(* the current character on trace i is blank *)
- -Ht2a * #Hblank #Ht1 <Ht1 in Htin; #Ht1a >Ht1a in Ht2b;
- #Ht2b lapply (Ht2b ?) [% normalize #H destruct] -Ht2b -Ht1 -Ht1a
- lapply Hnohead_rs -Hnohead_rs
- (* by cases on rs *) cases rs
- [#_ (* rs is empty *) #Ht2 -Ht3b lapply (Ht3a … Ht2)
- #Ht3 -Ht4b lapply (Ht4a … Ht3) -Ht4a #Ht4 -Ht5b
- >Ht4 in Ht5a; >Ht3 #Ht5a lapply (Ht5a (refl … )) -Ht5a #Ht5
- cases (Htout … Ht5) -Htout
- [(* the current symbol on trace n is the head: this is absurd *)
- * whd in ⊢ (??%?→?); >all_blank_n whd in ⊢ (??%?→?); #H destruct (H)
- |*
- [(* we return to the head *)
- * #ls1 * #b * #ls2 * * * whd in ⊢ (??%?→?);
- #H1 #H2 #H3
- (* ls1 must be empty *)
- cut (ls1=[])
- [lapply H3 lapply H1 -H3 -H1 cases ls1 // #c #ls3
- whd in ⊢ (???%→?); #H1 destruct (H1)
- >blank_all_blank [|@daemon] #H @False_ind
- lapply (H … (change_vec (sig_ext sig) n a (blank sig) i) ?)
- [%2 %1 // | whd in match (no_head ???); >nth_change_vec_neq [|@daemon]
- >Hhead whd in ⊢ (??%?→?); #H1 destruct (H1)]]
- #Hls1_nil >Hls1_nil in H1; whd in ⊢ (???%→?); #H destruct (H)
- >reverse_single >blank_all_blank [|@daemon]
- whd in match (right ??) ; >append_nil #Htout
- %{ls2} %{(change_vec (sig_ext sig) n a (blank sig) i)} %{[all_blank sig n]}
- %[%[%[%[%[//|//]|@daemon]|//] |(*absurd*)@daemon]
- |normalize >nth_change_vec // @daemon]
- |(* we do not find the head: this is absurd *)
- * #b * #lss * * whd in match (left ??); #HF @False_ind
- lapply (HF … (shift_i sig n i a (all_blank sig n)) ?) [%2 %1 //]
- whd in match (no_head ???); >nth_change_vec_neq [|@daemon]
- >Hhead whd in ⊢ (??%?→?); #H destruct (H)
- ]
- ]
- |(* rs = c::rs1 *)
- #c #rs1 #Hnohead_rs #Ht2 -Ht3a lapply (Ht3b … Ht2) -Ht3b
- #Ht3 -Ht4a lapply (Ht4b … Ht3) -Ht4b *
- [(* the first character is blank *) *
- #Hblank #Ht4 -Ht5a >Ht4 in Ht5b; >Ht3
- normalize in ⊢ ((%→?)→?); #Ht5 lapply (Ht5 ?) [% #H destruct (H)]
- -Ht2 -Ht3 -Ht4 -Ht5 #Ht5 cases (Htout … Ht5) -Htout
- [(* the current symbol on trace n is the head: this is absurd *)
- * >Hnohead_rs [#H destruct (H) |%1 //]
- |*
- [(* we return to the head *)
- * #ls1 * #b * #ls2 * * * whd in ⊢ (??%?→?);
- #H1 #H2 #H3
- (* ls1 must be empty *)
- cut (ls1=[])
- [lapply H3 lapply H1 -H3 -H1 cases ls1 // #x #ls3
- whd in ⊢ (???%→?); #H1 destruct (H1) #H @False_ind
- lapply (H (change_vec (sig_ext sig) n a (nth i (sig_ext sig) (vec … c) (blank sig)) i) ?)
- [%2 %1 // | whd in match (no_head ???); >nth_change_vec_neq [|@daemon]
- >Hhead whd in ⊢ (??%?→?); #H1 destruct (H1)]]
- #Hls1_nil >Hls1_nil in H1; whd in ⊢ (???%→?); #H destruct (H)
- >reverse_single #Htout
- %{ls2} %{(change_vec (sig_ext sig) n a (nth i (sig_ext sig) (vec … c) (blank sig)) i)}
- %{(c::rs1)}
- %[%[%[%[%[@Htout|//]|//]|//] |(*absurd*)@daemon]
- |>to_blank_cons_b
- [>(to_blank_cons_b … Hblank) //| >nth_change_vec [@Hblank |@daemon]]
- ]
- |(* we do not find the head: this is absurd *)
- * #b * #lss * * #HF @False_ind
- lapply (HF … (shift_i sig n i a c) ?) [%2 %1 //]
- whd in match (no_head ???); >nth_change_vec_neq [|@daemon]
- >Hhead whd in ⊢ (??%?→?); #H destruct (H)
- ]
- ]
- |
-
- (* end of the first case !! *)
-
-
-
- #Ht2
-
-cut (∃rs11,rs12. b::rs1 = rs11@a::rs12 ∧ no_head_in ?? rs11)
- [cut (ls1 = [ ] ∨ ∃aa,tlls1. ls1 = aa::tlls1)
- [cases ls1 [%1 // | #aa #tlls1 %2 %{aa} %{tlls1} //]] *
- [#H1ls1 %{[ ]} %{rs1} %
- [ @eq_f2 // >H1ls1 in Hls1; whd in match ([]@b::ls2);
- #Hls1 @(to_blank_hd … Hls1)
- |normalize #x @False_ind
- ]
- |* #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 %{(b::(reverse ? tlls1))} %{l} %
- [@eq_f >H1 >reverse_cons >associative_append //
- |@daemon (* is a sublit of the tail of ls1, and hence of ls *)
- ]
- |(* 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
- frist character in rs2. But in this case we would have a blank
- inside ls1=attls1 that is absurd *)
- @daemon
- ]]]
-
-
+ ]
+ ]
+qed.
projects / helm.git / blobdiff