X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_to_mono%2Fmulti_to_mono.ma;h=0e9bce53e403f8dde1c6efc62b5660c69c5dff55;hb=b4f76b0d8fa0e5365fb48e91474febe200b647a7;hp=181ac7a4d06889937b220ff5515104810bddfadd;hpb=15158ce1b220fb89ba994981add8bcb2c8d6e199;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 index 181ac7a4d..0e9bce53e 100644 --- a/matita/matita/lib/turing/multi_to_mono/multi_to_mono.ma +++ b/matita/matita/lib/turing/multi_to_mono/multi_to_mono.ma @@ -1,427 +1,77 @@ -include "turing/basic_machines.ma". -include "turing/if_machine.ma". -include "basics/vector_finset.ma". include "turing/auxiliary_machines1.ma". +include "turing/multi_to_mono/shift_trace_machines.ma". -(* a multi_sig characheter is composed by n+1 sub-characters: n for each tape -of a multitape machine, and an additional one to mark the position of the head. -We extend the tape alphabet with two new symbols true and false. -false is used to pad shorter tapes, and true to mark the head of the tape *) - -definition sig_ext ≝ λsig. FinSum FinBool sig. -definition blank : ∀sig.sig_ext sig ≝ λsig. inl … false. -definition head : ∀sig.sig_ext sig ≝ λsig. inl … true. - -definition multi_sig ≝ λsig:FinSet.λn.FinVector (sig_ext sig) n. - -let rec all_blank sig n on n : multi_sig sig n ≝ - match n with - [ O ⇒ Vector_of_list ? [] - | S m ⇒ vec_cons ? (blank ?) m (all_blank sig m) - ]. - -lemma blank_all_blank: ∀sig,n,i. - nth i ? (vec … (all_blank sig n)) (blank sig) = blank ?. -#sig #n elim n normalize - [#i >nth_nil // - |#m #Hind #i cases i // #j @Hind - ] -qed. - -lemma all_blank_n: ∀sig,n. - nth n ? (vec … (all_blank sig n)) (blank sig) = blank ?. -#sig #n @blank_all_blank -qed. - -(* compute the i-th trace *) -definition trace ≝ λsig,n,i,l. - map ?? (λa. nth i ? (vec … n a) (blank sig)) l. - -lemma trace_def : ∀sig,n,i,l. - trace sig n i l = map ?? (λa. nth i ? (vec … n a) (blank sig)) l. -// qed. - -(* -let rec trace sig n i l on l ≝ - match l with - [ nil ⇒ nil ? - | cons a tl ⇒ nth i ? (vec … n a) (blank sig)::(trace sig n i tl)]. *) - -lemma tail_trace: ∀sig,n,i,l. - tail ? (trace sig n i l) = trace sig n i (tail ? l). -#sig #n #i #l elim l // -qed. - -lemma trace_append : ∀sig,n,i,l1,l2. - trace sig n i (l1 @ l2) = trace sig n i l1 @ trace sig n i l2. -#sig #n #i #l1 #l2 elim l1 // #a #tl #Hind normalize >Hind // -qed. - -lemma trace_reverse : ∀sig,n,i,l. - trace sig n i (reverse ? l) = reverse (sig_ext sig) (trace sig n i l). -#sig #n #i #l elim l // -#a #tl #Hind whd in match (trace ??? (a::tl)); >reverse_cons >reverse_cons ->trace_append >Hind // -qed. - -lemma nth_trace : ∀sig,n,i,j,l. - nth j ? (trace sig n i l) (blank ?) = - nth i ? (nth j ? l (all_blank sig n)) (blank ?). -#sig #n #i #j elim j - [#l cases l normalize // >blank_all_blank // - |-j #j #Hind #l whd in match (nth ????); whd in match (nth ????); - cases l - [normalize >nth_nil >nth_nil >blank_all_blank // - |#a #tl normalize @Hind] - ] -qed. - -lemma length_trace: ∀sig,n,i,l. - |trace sig n i l| = |l|. -#sig #n #i #l elim l // #a #tl #Hind normalize @eq_f @Hind -qed. - -(* some lemmas over lists *) -lemma injective_append_l: ∀A,l. injective ?? (append A l). -#A #l elim l - [#l1 #l2 normalize // - |#a #tl #Hind #l1 #l2 normalize #H @Hind @(cons_injective_r … H) - ] -qed. - -lemma injective_append_r: ∀A,l. injective ?? (λl1.append A l1 l). -#A #l #l1 #l2 #H -cut (reverse ? (l1@l) = reverse ? (l2@l)) [//] ->reverse_append >reverse_append #Hrev -lapply (injective_append_l … Hrev) -Hrev #Hrev -<(reverse_reverse … l1) <(reverse_reverse … l2) // -qed. - -lemma injective_reverse: ∀A. injective ?? (reverse A). -#A #l1 #l2 #H <(reverse_reverse … l1) <(reverse_reverse … l2) // -qed. - -lemma first_P_to_eq : ∀A:Type[0].∀P:A→Prop.∀l1,l2,l3,l4,a1,a2. - l1@a1::l2 = l3@a2::l4 → (∀x. mem A x l1 → P x) → (∀x. mem A x l2 → P x) → - ¬ P a1 → ¬ P a2 → l1 = l3 ∧ a1::l2 = a2::l4. -#A #P #l1 elim l1 - [#l2 * - [#l4 #a1 #a2 normalize #H destruct /2/ - |#c #tl #l4 #a1 #a2 normalize #H destruct - #_ #H #_ #H1 @False_ind @(absurd ?? H1) @H @mem_append_l2 %1 // - ] - |#b #tl1 #Hind #l2 * - [#l4 #a1 #a2 normalize #H destruct - #H1 #_ #_ #H2 @False_ind @(absurd ?? H2) @H1 %1 // - |#c #tl2 #l4 #a1 #a2 normalize #H1 #H2 #H3 #H4 #H5 - lapply (Hind l2 tl2 l4 … H4 H5) - [destruct @H3 |#x #H6 @H2 %2 // | destruct (H1) // - |* #H6 #H7 % // >H7 in H1; #H1 @(injective_append_r … (a2::l4)) @H1 - ] - ] -qed. - -lemma nth_to_eq: ∀A,l1,l2,a. |l1| = |l2| → - (∀i. nth i A l1 a = nth i A l2 a) → l1 = l2. -#A #l1 elim l1 - [* // #a #tl #d normalize #H destruct (H) - |#a #tl #Hind * - [#d normalize #H destruct (H) - |#b #tl1 #d #Hlen #Hnth @eq_f2 - [@(Hnth 0) | @(Hind tl1 d) [@injective_S @Hlen | #i @(Hnth (S i)) ]] - ] - ] -qed. - -lemma nth_extended: ∀A,i,l,a. - nth i A l a = nth i A (l@[a]) a. -#A #i elim i [* // |#j #Hind * // #b #tl #a normalize @Hind] -qed. - -(* 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 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 nth_trace - 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 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 ????); +(Hind l2 H) (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)) ∧ @@ -430,18 +80,25 @@ definition R_move_to_blank_L ≝ λsig,n,i,t1,t2. (∃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 @@ -449,47 +106,35 @@ theorem sem_move_to_blank_L: ∀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 //] + |@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])} %{[]} + %[%[%[%[%[

blank_all_blank //] + |

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. @@ -498,10 +143,6 @@ 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. @@ -534,10 +175,6 @@ theorem sem_shift_i_L: ∀sig,n,i. shift_i_L sig n i ⊨ R_shift_i_L sig n i. (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] @@ -590,20 +227,12 @@ theorem sem_shift_i_L_new: ∀sig,n,i. 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 · @@ -613,66 +242,71 @@ definition mtiL ≝ λsig,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 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 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)) *) @@ -702,24 +336,40 @@ lapply (Htout … Ht2) -Htout -Ht2 * | *) ] #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_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 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) + [(proj2 … (H2 … jneqi)) >hd_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_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] @@ -742,251 +392,6 @@ lapply (Htout … Ht2) -Htout -Ht2 * >reverse_reverse >associative_append 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 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 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 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 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 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 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 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.