X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=ed4142fa327eeda0f8c9f9bce81a5303df955cf0;hb=b31ab31a99065295b91003a0df95dec817cee5de;hp=60c2c3f9f3e819ab637262bc55798c211456381d;hpb=5876d7c2897e2d1f325fcddf8c243d47a5656e7c;p=helm.git diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma index 60c2c3f9f..ed4142fa3 100644 --- a/matita/matita/lib/turing/multi_universal/match.ma +++ b/matita/matita/lib/turing/multi_universal/match.ma @@ -14,7 +14,7 @@ include "turing/multi_universal/compare.ma". include "turing/multi_universal/par_test.ma". - +include "turing/multi_universal/moves_2.ma". definition Rtc_multi_true ≝ λalpha,test,n,i.λt1,t2:Vector ? (S n). @@ -71,123 +71,289 @@ cases (acc_sem_inject … Hin (sem_test_null alpha) int) #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ] qed. -axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2. - l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧ - ∀a,tla. tl1 = a::tla → is_endc a = true ∨ (∀b,tlb.tl2 = b::tlb → a≠b). - -axiom daemon : ∀X:Prop.X. - -definition match_test ≝ λsrc,dst.λsig:DeqSet.λn,is_endc.λv:Vector ? n. +definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n. match (nth src (option sig) v (None ?)) with [ None ⇒ false - | Some x ⇒ notb ((is_endc x) ∨ (nth dst (DeqOption sig) v (None ?) == None ?))]. + | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ]. + +definition mmove_states ≝ initN 2. + +definition mmove0 : mmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)). +definition mmove1 : mmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)). + +definition trans_mmove ≝ + λi,sig,n,D. + λp:mmove_states × (Vector (option sig) (S n)). + let 〈q,a〉 ≝ p in match (pi1 … q) with + [ O ⇒ 〈mmove1,change_vec ? (S n) (null_action ? n) (〈None ?,D〉) i〉 + | S _ ⇒ 〈mmove1,null_action sig n〉 ]. + +definition mmove ≝ + λi,sig,n,D. + mk_mTM sig n mmove_states (trans_mmove i sig n D) + mmove0 (λq.q == mmove1). + +definition Rm_multi ≝ + λalpha,n,i,D.λt1,t2:Vector ? (S n). + t2 = change_vec ? (S n) t1 (tape_move alpha (nth i ? t1 (niltape ?)) D) i. + +lemma sem_move_multi : + ∀alpha,n,i,D.i ≤ n → + mmove i alpha n D ⊨ Rm_multi alpha n i D. +#alpha #n #i #D #Hin #int %{2} +%{(mk_mconfig ? mmove_states n mmove1 ?)} +[| % + [ whd in ⊢ (??%?); @eq_f whd in ⊢ (??%?); @eq_f % + | whd >tape_move_multi_def + <(change_vec_same … (ctapes …) i (niltape ?)) + >pmap_change tape_move_null_action % ] ] + qed. + +definition rewind ≝ λsrc,dst,sig,n. + parmove src dst sig n L · mmove src sig n R · mmove dst sig n R. -definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc. - compare src dst sig n is_endc · - (ifTM ?? (partest sig n (match_test src dst sig ? is_endc)) +definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). + (∀x,x0,xs,rs. + nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs → + ∀ls0,y,y0,target,rs0.|xs| = |target| → + nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 → + outt = change_vec ?? + (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src) + (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst). + +theorem accRealize_to_Realize : + ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc. + M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse. +#sig #n #M #Rtrue #Rfalse #acc #HR #t +cases (HR t) #k * #outc * * #Hloop +#Htrue #Hfalse %{k} %{outc} % // +cases (true_or_false (cstate sig (states sig n M) n outc == acc)) #Hcase +[ % @Htrue @(\P Hcase) | %2 @Hfalse @(\Pf Hcase) ] +qed. + +lemma sem_rewind : ∀src,dst,sig,n. + src ≠ dst → src < S n → dst < S n → + rewind src dst sig n ⊨ R_rewind src dst sig n. +#src #dst #sig #n #Hneq #Hsrc #Hdst +@(sem_seq_app sig n ????? (sem_parmoveL src dst sig n Hneq Hsrc Hdst) ?) +[| @(sem_seq_app sig n ????? (sem_move_r_multi …) (sem_move_r_multi …)) // + @le_S_S_to_le // ] +#ta #tb * #tc * * #Htc #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb +#x #x0 #xs #rs #Hmidta_src #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_dst +>(Htc ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in Htd; +[|>Hmidta_dst // +|>length_append >length_append >Hlen % ] * #_ +[ whd in ⊢ (%→?); * #x1 * #x2 * * + >change_vec_commute in ⊢ (%→?); // >nth_change_vec // + cases (reverse sig (xs@[x0])@x::rs) + [|#z #zs] normalize in ⊢ (%→?); #H destruct (H) +| whd in ⊢ (%→?); * #_ #Htb >Htb -Htb FAIL + + normalize in ⊢ (%→?); + (sem_parmove_step src dst sig n R Hneq Hsrc Hdst)) + (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?)) + (sem_seq … + (sem_parmoveL ???? Hneq Hsrc Hdst) + (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? ))) + (sem_nop …))) + + +definition match_step ≝ λsrc,dst,sig,n. + compare src dst sig n · + (ifTM ?? (partest sig n (match_test src dst sig ?)) (single_finalTM ?? - (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst))) + (rewind src dst sig n · (inject_TM ? (move_r ?) n dst))) (nop …) partest1). definition R_match_step_false ≝ - λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n). - ∀ls,x,xs,end,rs. - nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) → - (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true → - ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨ - (current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨ - (∃ls0,rs0. - nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧ - ∀rsj,c. - rs0 = c::rsj → - outt = change_vec ?? - (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src) - (midtape sig (reverse ? xs@x::ls0) c rsj) dst). + λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). + ∀ls,x,xs. + nth src ? int (niltape ?) = midtape sig ls x xs → + ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨ + (∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧ + xs = rs0@xs0 ∧ + current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨ + (∃ls0,rs0. + nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧ + (* ∀rsj,c. + rs0 = c::rsj → *) + outt = change_vec ?? + (change_vec ?? int (mk_tape sig (reverse ? xs@x::ls) (None ?) [ ]) src) + (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst). -definition R_match_step_true ≝ - λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n). - ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s → - is_startc s = true → - (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) → +(*definition R_match_step_true ≝ + λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). + ∀s,rs.current sig (nth src (tape sig) int (niltape sig)) = Some ? s → current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧ (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 → outt = change_vec ?? int - (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧ - (∀ls,x,xs,ci,cj,rs,ls0,rs0. + (tape_move_mono … (nth dst ? int (niltape ?)) (〈Some ? s1,R〉)) dst) ∧ + (∀ls,x,xs,ci,rs,ls0,rs0. nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → - nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → - (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → - ci ≠ cj → - (outt = change_vec ?? int - (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false)). -(* ∧ - (rs0 = [ ] → - outt = change_vec ?? - (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src) - (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)). *) - + nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) → + rs0 ≠ [] ∧ + ∀cj,rs1.rs0 = cj::rs1 → + ci ≠ cj → + (outt = change_vec ?? int + (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst)). +*) +definition R_match_step_true ≝ + λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). + ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s → + ∃s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 ∧ + (left ? (nth src ? int (niltape ?)) = [ ] → + (s ≠ s1 → + outt = change_vec ?? int + (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst) ∧ + (∀xs,ci,rs,ls0,rs0. + nth src ? int (niltape ?) = midtape sig [] s (xs@ci::rs) → + nth dst ? int (niltape ?) = midtape sig ls0 s (xs@rs0) → + rs0 ≠ [] ∧ + ∀cj,rs1.rs0 = cj::rs1 → + ci ≠ cj → + (outt = change_vec ?? int + (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst))). + lemma sem_match_step : - ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n → - match_step src dst sig n is_startc is_endc ⊨ + ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → + match_step src dst sig n ⊨ [ inr ?? (inr ?? (inl … (inr ?? start_nop))) : - R_match_step_true src dst sig n is_startc is_endc, - R_match_step_false src dst sig n is_endc ]. -#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst -@(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst) - (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ? is_endc)) + R_match_step_true src dst sig n, + R_match_step_false src dst sig n ]. +#src #dst #sig #n #Hneq #Hsrc #Hdst +@(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst) + (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?)) (sem_seq … - (sem_parmoveL ???? is_startc Hneq Hsrc Hdst) + (sem_parmoveL ???? Hneq Hsrc Hdst) (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? ))) (sem_nop …))) -[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd >Htd -Htd - * #te * #Hte #Htb whd - #s #Hcurta_src #Hstart #Hnotstart % [ % - [ @daemon - | #s1 #Hcurta_dst #Hneqss1 -Hcomp2 - cut (tc = ta) - [@Hcomp1 %2 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //] - #H destruct (H) -Hcomp1 cases Hte #_ -Hte #Hte - cut (te = ta) [@Hte %1 %1 %{s} % //] -Hte #H destruct (H) % - [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … )) - #i #Hi cases (decidable_eq_nat i dst) #Hidst - [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst) - #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs // - | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ] - | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src - >Hcurta_src in Htest; whd in ⊢ (??%?→?); - cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq // - ]] - |#ls #x #xs #ci #cj #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc #Hcicj - cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ] - * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2 - lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc % - [ cases Hte -Hte #Hte #_ whd in Hte; - >Htasrc_mid in Hcurta_src; whd in ⊢ (??%?→?); #H destruct (H) - lapply (Hte ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??) ?) // - [ >Htc >nth_change_vec // - | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid cases (orb_true_l … Hc0) -Hc0 #Hc0 - [@memb_append_l2 >(\P Hc0) @memb_hd - |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse // +[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * #Htest + * #te * #Hte #Htb #s #Hcurta_src whd + cut (∃s1.current sig (nth dst (tape sig) ta (niltape sig))=Some sig s1) + [ lapply Hcomp1 -Hcomp1 + lapply (refl ? (current ? (nth dst ? ta (niltape ?)))) + cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%); + [ #Hcurta_dst #Hcomp1 >Hcomp1 in Htest; // * + change with (vec_map ?????) in match (current_chars ???); whd in ⊢ (??%?→?); + <(nth_vec_map ?? (current ?) src ? ta (niltape ?)) + <(nth_vec_map ?? (current ?) dst ? ta (niltape ?)) + >Hcurta_src >Hcurta_dst whd in ⊢ (??%?→?); #H destruct (H) + | #s1 #_ #_ %{s1} % ] ] + * #s1 #Hcurta_dst %{s1} % // #Hleftta % + [ #Hneqss1 -Hcomp2 cut (tc = ta) + [@Hcomp1 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //] + #H destruct (H) -Hcomp1 cut (td = ta) + [ cases Htest -Htest // ] #Htdta destruct (Htdta) + cases Hte -Hte #Hte #_ + cases (current_to_midtape … Hcurta_src) #ls * #rs #Hmidta_src + cases (current_to_midtape … Hcurta_dst) #ls0 * #rs0 #Hmidta_dst + >Hmidta_src in Hleftta; normalize in ⊢ (%→?); #Hls destruct (Hls) + >(Hte s [ ] rs Hmidta_src ls0 s1 [ ] rs0 (refl ??) Hmidta_dst) in Htb; + * whd in ⊢ (%→?); + mid + + in Htb; + cut (te = ta) + [ cases Htest -Htest #Htest #Htdta Htdta @Hcurta_src %{s} % //] + -Hte #H destruct (H) % + [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … )) + #i #Hi cases (decidable_eq_nat i dst) #Hidst + [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst) + #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs // + | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ] + | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src + >Hcurta_src in Htest; whd in ⊢ (??%?→?); + cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq // + ] + <(nth_vec_map ?? (current ?) dst ? tc (niltape ?)) + >Hcurta_src normalize + lapply (refl ? (current ? (nth dst ? ta (niltape ?)))) + cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%); + [| #s1 #Hcurta_dst % + [ % #Hfalse destruct (Hfalse) + | #s1' #Hs1 destruct (Hs1) #Hneqss1 -Hcomp2 + cut (tc = ta) + [@Hcomp1 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //] + #H destruct (H) -Hcomp1 cases Hte -Hte #_ #Hte + cut (te = ta) [ cases Htest -Htest #Htest #Htdta Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst) + #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs // + | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ] + | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src + >Hcurta_src in Htest; whd in ⊢ (??%?→?); + cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq // + ] + ] - | >Htc >change_vec_commute // >nth_change_vec // ] -Hte - >Htc >change_vec_commute // >change_vec_change_vec - >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte - >Hte in Htb; * * #_ >reverse_reverse #Htbdst1 #Htbdst2 -Hte @(eq_vec … (niltape ?)) - #i #Hi cases (decidable_eq_nat i dst) #Hidst - [ >Hidst >nth_change_vec // >(Htbdst1 ls0 s (xs@cj'::rs0')) - [| >nth_change_vec // ] - >Htadst_mid cases xs // - | >nth_change_vec_neq [|@sym_not_eq // ] - nth_change_vec_neq [| @sym_not_eq // ] - change_vec_same % ] - | >Hcurta_src in Htest; whd in ⊢(??%?→?); - >Htc >change_vec_commute // - change with (current ? (niltape ?)) in match (None ?); - nth_change_vec // whd in ⊢ (??%?→?); - cases (is_endc ci) whd in ⊢ (??%?→?); #H destruct (H) % + #Hcurta_dst >Hcomp1 in Htest; [| %2 %2 //] + whd in ⊢ (??%?→?); change with (current ? (niltape ?)) in match (None ?); + Hcurta_src whd in ⊢ (??%?→?); Hcurta_dst cases (is_endc s) normalize in ⊢ (%→?); #H destruct (H) + | #Hstart #Hnotstart % + [ #s1 #Hcurta_dst #Hneqss1 -Hcomp2 + cut (tc = ta) + [@Hcomp1 %2 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //] + #H destruct (H) -Hcomp1 cases Hte #_ -Hte #Hte + cut (te = ta) [@Hte %1 %1 %{s} % //] -Hte #H destruct (H) % + [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … )) + #i #Hi cases (decidable_eq_nat i dst) #Hidst + [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst) + #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs // + | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ] + | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src + >Hcurta_src in Htest; whd in ⊢ (??%?→?); + cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq // + ] + |#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc + cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) + [ * #Hrs00 #Htc >Htc in Htest; whd in ⊢ (??%?→?); + <(nth_vec_map ?? (current sig) ??? (niltape ?)) + >change_vec_commute // >nth_change_vec // whd in ⊢ (??%?→?); + cases (is_endc ci) + [ whd in ⊢ (??%?→?); #H destruct (H) + | <(nth_vec_map ?? (current sig) ??? (niltape ?)) + >change_vec_commute [| @sym_not_eq // ] >nth_change_vec // + >(?:current ? (mk_tape ?? (None ?) ?) = None ?) + [ whd in ⊢ (??%?→?); #H destruct (H) + | cases (reverse sig xs@x::ls0) normalize // ] ] ] + * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2 % [ % + [ cases (true_or_false (is_endc ci)) // + #Hendci >(Hcomp2 (or_introl … Hendci)) in Htest; + whd in ⊢ (??%?→?); <(nth_vec_map ?? (current sig) ??? (niltape ?)) + >change_vec_commute // >nth_change_vec // whd in ⊢ (??%?→?); + >Hendci normalize // + | % #H destruct (H) ] ] #cj #rs1 #H destruct (H) #Hcicj + lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc % + [ cases Hte -Hte #Hte #_ whd in Hte; + >Htasrc_mid in Hcurta_src; whd in ⊢ (??%?→?); #H destruct (H) + lapply (Hte ls ci (reverse ? xs) rs s ??? ls0 cj (reverse ? xs) s rs1 (refl ??) ?) // + [ >Htc >nth_change_vec // + | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid cases (orb_true_l … Hc0) -Hc0 #Hc0 + [@memb_append_l2 >(\P Hc0) @memb_hd + |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse // + ] + | >Htc >change_vec_commute // >nth_change_vec // ] -Hte + >Htc >change_vec_commute // >change_vec_change_vec + >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte + >Hte in Htb; * * #_ >reverse_reverse #Htbdst1 #Htbdst2 -Hte @(eq_vec … (niltape ?)) + #i #Hi cases (decidable_eq_nat i dst) #Hidst + [ >Hidst >nth_change_vec // >(Htbdst1 ls0 s (xs@cj::rs1)) + [| >nth_change_vec // ] + >Htadst_mid cases xs // + | >nth_change_vec_neq [|@sym_not_eq // ] + nth_change_vec_neq [| @sym_not_eq // ] + change_vec_same % ] + | >Hcurta_src in Htest; whd in ⊢(??%?→?); + >Htc >change_vec_commute // + change with (current ? (niltape ?)) in match (None ?); + nth_change_vec // whd in ⊢ (??%?→?); + cases (is_endc ci) whd in ⊢ (??%?→?); #H destruct (H) % + ] + ] ] - ] |#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb whd in ⊢ (%→?); #Hout >Hout >Htb whd #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend @@ -199,15 +365,37 @@ lemma sem_match_step : #ls_dst * #rs_dst #Hmid_dst cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * * #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq >Hrs_dst in Hmid_dst; #Hmid_dst - cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src; + cut (∃r1,rs1.rsi = r1::rs1) + [cases rsi in Hrs_src; + [ >append_nil #H (Hnotendxs1 end) in Hend; [ #H1 destruct (H1) ] + @memb_append_l2 @memb_hd + | #r1 #rs1 #_ %{r1} %{rs1} % ] ] + * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src; #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?) [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0 [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //] ] * - [ * #Hrsj >Hrsj #Hta % %2 >Hta >nth_change_vec // cases (reverse ? xs1) // + [ * #Hrsj >Hrsj #Hta % %2 >Hta >nth_change_vec // + %{ls_dst} %{xs1} cut (∃xs0.xs = xs1@xs0) + [lapply Hnotendxs1 -Hnotendxs1 lapply Hrs_src lapply xs elim xs1 + [ #l #_ #_ %{l} % + | #x2 #xs2 #IH * + [ whd in ⊢ (??%%→?); #H destruct (H) #Hnotendxs2 + >Hnotendxs2 in Hend; [ #H destruct (H) |@memb_hd ] + | #x2' #xs2' whd in ⊢ (??%%→?); #H destruct (H) + #Hnotendxs2 cases (IH xs2' e0 ?) + [ #xs0 #Hxs2 %{xs0} @eq_f // + |#c #Hc @Hnotendxs2 @memb_cons // ] + ] + ] + ] * #xs0 #Hxs0 %{xs0} % [ % + [ >Hmid_dst >Hrsj >append_nil % + | @Hxs0 ] + | cases (reverse ? xs1) // ] | * #cj * #rs2 * #Hrsj #Hta lapply (Hta ?) - [ cases (Hneq ?? Hrs1) /2/ #Hr1 %2 @(Hr1 ?? Hrsj) ] -Hta #Hta + [ cases (Hneq ?? Hrs1) /2/ * #_ #Hr1 %2 @(Hr1 ?? Hrsj) ] -Hta #Hta %2 >Hta in Hc; whd in ⊢ (??%?→?); change with (current ? (niltape ?)) in match (None ?); nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // @@ -251,7 +439,7 @@ lemma sem_match_step : ] ] ] -qed. +qed. definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc. whileTM … (match_step src dst sig n is_startc is_endc) @@ -259,10 +447,10 @@ definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc. definition R_match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n). -(* (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ *) ∀ls,x,xs,end,rs. nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) → (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true → + (∀c0. memb ? c0 (xs@end::rs) = true → is_startc c0 = false) → (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ (is_startc x = true → (∀ls0,x0,rs0. @@ -274,30 +462,19 @@ definition R_match_m ≝ (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨ ∀l,l1.x0::rs0 ≠ l@x::xs@l1)). -(* -definition R_match_m ≝ - λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n). - (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨ - current ? (nth i ? int (niltape ?)) = None ? ∨ - current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧ - (∀ls,x,xs,ci,rs,ls0,x0,rs0. - (∀x. is_startc x ≠ is_endc x) → - is_startc x = true → is_endc ci = true → - (∀z. memb ? z (x::xs) = true → is_endc x = false) → - nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → - nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 → - (∃l,l1.x0::rs0 = l@x::xs@l1 ∧ - ∀cj,l2.l1=cj::l2 → - outt = change_vec ?? - (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i) - (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨ - ∀l,l1.x0::rs0 ≠ l@x::xs@l1). -*) +lemma not_sub_list_merge : + ∀T.∀a,b:list T. (∀l1.a ≠ b@l1) → (∀t,l,l1.a ≠ t::l@b@l1) → ∀l,l1.a ≠ l@b@l1. +#T #a #b #H1 #H2 #l elim l normalize // +qed. + +lemma not_sub_list_merge_2 : + ∀T:DeqSet.∀a,b:list T.∀t. (∀l1.t::a ≠ b@l1) → (∀l,l1.a ≠ l@b@l1) → ∀l,l1.t::a ≠ l@b@l1. +#T #a #b #t #H1 #H2 #l elim l // +#t0 #l1 #IH #l2 cases (true_or_false (t == t0)) #Htt0 +[ >(\P Htt0) % normalize #H destruct (H) cases (H2 l1 l2) /2/ +| normalize % #H destruct (H) cases (\Pf Htt0) /2/ ] +qed. -(* -axiom sub_list_dec: ∀A.∀l,ls:list A. - ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2. -*) lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc. src ≠ dst → src < S n → dst < S n → @@ -305,12 +482,29 @@ src ≠ dst → src < S n → dst < S n → #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) // -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend +[ #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse - [(* current dest = None *) * #Hcur_dst #Houtc % - [#_ >Houtc // - |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst; - normalize in ⊢ (%→?); #H destruct (H) + [(* current dest = None *) * + [ * #Hcur_dst #Houtc % + [#_ >Houtc // + |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst; + normalize in ⊢ (%→?); #H destruct (H) + ] + | * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone % + [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H) + | #Hstart #ls1 #x1 #rs1 >Htc_dst #H destruct (H) + >Hrs0 cases xs0 + [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %] + #cj #ls2 #H destruct (H) + | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs) + >length_append whd in ⊢ (??%(??%)→?); >length_append + >length_append normalize >commutative_plus whd in ⊢ (???%→?); + #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?); + >associative_plus >associative_plus + #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?); + #e2 destruct (e2) + ] + ] ] |* #ls0 * #rs0 * #Hmid_dst #HFalse % [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H) @@ -319,57 +513,288 @@ lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc >reverse_cons >associative_append @(HFalse ?? Hnotnil) ] ] -|#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd - #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend +|-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd + #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart lapply (refl ? (current ? (nth dst ? ta (niltape ?)))) cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?); [#Hmid_dst % [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue - cases (Htrue x (refl … ) Hstart ?) -Htrue [2: @daemon] - * #Htb #_ #_ >Htb in IH; // #IH - cases (IH ls x xs end rs Hmid_src Hstart Hnotend Hend) - #Hcur_outc #_ @Hcur_outc // - |#ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?); + cases (Htrue x (refl … )) -Htrue * #Htaneq #_ + @False_ind >Hmid_dst in Htaneq; /2/ + |#Hstart #ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?); #H destruct (H) ] | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ] - #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?); + #Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?); #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue - cases (Htrue x (refl …) Hstart ?) -Htrue - [2: #z #membz @daemon (*aggiungere l'ipotesi*)] + cases (Htrue x (refl …)) -Htrue #_ #Htrue cases (Htrue Hstart Hnotstart) -Htrue cases (true_or_false (x==c)) #eqx - [ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc) + [ lapply (\P eqx) -eqx #eqx destruct (eqx) + #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc) #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1 cases tl1 in Hxs; - [>append_nil #Hx1 @daemon (* absurd by Hx1 e notendx1 *)] - #ci -tl1 #tl1 #Hxs #H cases (H … (refl … )) - [(* this is absurd, since Htrue conlcudes is_endc ci =false *) - #Hend_ci @daemon (* lapply(Htrue … (refl …)) -Htrue *) - |#Hcomp lapply (Htrue ls x x1 ci tl1 ls0 tl2 ???) - [ #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0 - [ @Hnotend >(\P Hc0) @memb_hd - | @Hnotendx1 // ] - | >Hmid_dst >Hrs0 >(\P eqx) % - | >Hxs % - | * cases tl2 in Hrs0; - [ >append_nil #Hrs0 #_ #Htb whd in IH; - lapply (IH ls x x1 ci tl1 ? Hstart ??) - [ - | - | >Htb // >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // - - >Hrs0 in Hmid_dst; #Hmid_dst - cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx - whd in IH; - cases(IH ls x xs end rs ? Hstart Hnotend Hend) - [* #H1 #H2 >Htb in H1; >nth_change_vec // - >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)] - #_ %2 @daemon (* si dimostra *) - |@daemon - |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src + [>append_nil #Hx1 Hend #H destruct (H) ] + #ci -tl1 #tl1 #Hxs #H cases (H … (refl … )) -H + [ #Hendci % >Hrs0 in Hmid_dst; cut (ci = end ∧ x1 = xs) + [ lapply Hxs lapply Hnotendx1 lapply x1 elim xs in Hnotend; + [ #_ * + [ #_ normalize #H destruct (H) /2/ + | #x2 #xs2 #Hnotendx2 normalize #H destruct (H) + >(Hnotendx2 ? (memb_hd …)) in Hend; #H destruct (H) ] + | #x2 #xs2 #IH #Hnotendx2 * + [ #_ normalize #H destruct (H) >(Hnotendx2 ci ?) in Hendci; + [ #H destruct (H) + | @memb_cons @memb_hd ] + | #x3 #xs3 #Hnotendx3 normalize #H destruct (H) + cases (IH … e0) + [ #H1 #H2 /2/ + | #c0 #Hc0 @Hnotendx2 cases (orb_true_l … Hc0) -Hc0 #Hc0 + [ >(\P Hc0) @memb_hd + | @memb_cons @memb_cons @Hc0 ] + | #c0 #Hc0 @Hnotendx3 @memb_cons @Hc0 ] + ] + ] + | * #Hcieq #Hx1eq >Hx1eq #Hmid_dst + cases (Htrue ??????? (refl ??) Hmid_dst Hnotend) + Hendci * #H destruct (H) ] + |cases tl2 in Hrs0; + [ >append_nil #Hrs0 destruct (Hrs0) * #Hcifalse#_ %2 + cut (∃l.xs = x1@ci::l) + [lapply Hxs lapply Hnotendx1 lapply Hnotend lapply xs + -Hxs -xs -Hnotendx1 elim x1 + [ * + [ #_ #_ normalize #H1 destruct (H1) >Hend in Hcifalse; + #H1 destruct (H1) + | #x2 #xs2 #_ #_ normalize #H >(cons_injective_l ????? H) %{xs2} % ] + | #x2 #xs2 #IHin * + [ #_ #Hnotendxs2 normalize #H destruct (H) + >(Hnotendxs2 ? (memb_hd …)) in Hend; #H destruct (H) + | #x3 #xs3 #Hnotendxs3 #Hnotendxs2 normalize #H destruct (H) + cases (IHin ??? e0) + [ #xs4 #Hxs4 >Hxs4 %{xs4} % + | #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0 + [ >(\P Hc0) @Hnotendxs3 @memb_hd + | @Hnotendxs3 @memb_cons @memb_cons @Hc0 ] + | #c0 #Hc0 @Hnotendxs2 @memb_cons @Hc0 ] + ] + ] + ] * #l #Hxs' >Hxs' + #l0 #l1 % #H lapply (eq_f ?? (length ?) ?? H) -H + >length_append normalize >length_append >length_append + normalize >commutative_plus normalize #H destruct (H) -H + >associative_plus in e0; >associative_plus + >(plus_n_O (|x1|)) in ⊢(??%?→?); #H lapply (injective_plus_r … H) + -H normalize #H destruct (H) + | #cj #tl2' #Hrs0 * #Hcifalse #Hcomp + lapply (Htrue ls c x1 ci tl1 ls0 (cj::tl2') ???) + [ #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0 + [ @Hnotend >(\P Hc0) @memb_hd + | @Hnotendx1 // ] + | >Hmid_dst >Hrs0 % + | >Hxs % + | * * #_ #_ -Htrue #Htrue lapply (Htrue ?? (refl ??) ?) [ @(Hcomp ?? (refl ??)) ] + * #Htb >Htb #Hendci >Hrs0 >Hxs + cases (IH ls c xs end rs ? Hnotend Hend Hnotstart) -IH + [| >Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src ] + #_ #IH lapply Hxs lapply Hnotendx1 -Hxs -Hnotendx1 cases x1 in Hrs0; + [ #Hrs0 #_ whd in ⊢ (???%→?); #Hxs + cases (IH Hstart (c::ls0) cj tl2' ?) + [ -IH * #l * #l1 * #Hll1 #IH % %{(c::l)} %{l1} + % [ @eq_f @Hll1 ] + #cj0 #l2 #Hcj0 >(IH … Hcj0) >Htb + >change_vec_commute // >change_vec_change_vec + >change_vec_commute [|@sym_not_eq // ] @eq_f3 // + >reverse_cons >associative_append % + | #IH %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%] + @not_sub_list_merge + [ #l2 cut (∃xs'.xs = ci::xs') + [ cases xs in Hxs; + [ normalize #H destruct (H) >Hend in Hendci; #H destruct (H) + | #ci' #xs' normalize #H lapply (cons_injective_l ????? H) + #H1 >H1 %{xs'} % ] + ] + * #xs' #Hxs' >Hxs' normalize % #H destruct (H) + lapply (Hcomp … (refl ??)) * /2/ + |#t #l2 #l3 % normalize #H lapply (cons_injective_r ????? H) + -H #H >H in IH; #IH cases (IH l2 l3) -IH #IH @IH % ] + | >Htb >nth_change_vec // >Hmid_dst >Hrs0 % ] + | #x2 #xs2 normalize in ⊢ (%→?); #Hrs0 #Hnotendxs2 normalize in ⊢ (%→?); + #Hxs cases (IH Hstart (c::ls0) x2 (xs2@cj::tl2') ?) + [ -IH * #l * #l1 * #Hll1 #IH % %{(c::l)} %{l1} + % [ @eq_f @Hll1 ] + #cj0 #l2 #Hcj0 >(IH … Hcj0) >Htb + >change_vec_commute // >change_vec_change_vec + >change_vec_commute [|@sym_not_eq // ] @eq_f3 // + >reverse_cons >associative_append % + | -IH #IH %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%] + @not_sub_list_merge_2 [| @IH] + cut (∃l2.xs = (x2::xs2)@ci::l2) + [lapply Hnotendxs2 + lapply Hnotend -Hnotend lapply Hxs + >(?:x2::xs2@ci::tl1 = (x2::xs2)@ci::tl1) [|%] + lapply (x2::xs2) elim xs + [ * + [ normalize in ⊢ (%→?); #H1 destruct (H1) + >Hendci in Hend; #Hend destruct (Hend) + | #x3 #xs3 normalize in ⊢ (%→?); #H1 destruct (H1) + #_ #Hnotendx3 >(Hnotendx3 ? (memb_hd …)) in Hend; + #Hend destruct (Hend) + ] + | #x3 #xs3 #IHin * + [ normalize in ⊢ (%→?); #Hxs3 destruct (Hxs3) #_ #_ + %{xs3} % + | #x4 #xs4 normalize in ⊢ (%→?); #Hxs3xs4 #Hnotend + #Hnotendxs4 destruct (Hxs3xs4) cases (IHin ? e0 ??) + [ #l0 #Hxs3 >Hxs3 %{l0} % + | #c0 #Hc0 @Hnotend cases (orb_true_l … Hc0) -Hc0 #Hc0 + [ >(\P Hc0) @memb_hd + | @memb_cons @memb_cons @Hc0 ] + | #c0 #Hc0 @Hnotendxs4 @memb_cons // + ] + ] + ] + ] * #l2 #Hxs' + >Hxs' #l3 normalize >associative_append normalize % #H + destruct (H) lapply (append_l2_injective ?????? e1) // + #H1 destruct (H1) cases (Hcomp ?? (refl ??)) /2/ + | >Htb >nth_change_vec // >Hmid_dst >Hrs0 % ] + ] + ] + ] + ] + |lapply (\Pf eqx) -eqx #eqx >Hmid_dst #Htrue + cases (Htrue ? (refl ??) eqx) -Htrue #Htb #Hendcx #_ + cases rs0 in Htb; + [ #_ %2 #l #l1 cases l + [ normalize cases xs + [ cases l1 + [ normalize % #H destruct (H) cases eqx /2/ + | #tmp1 #l2 normalize % #H destruct (H) ] + | #tmp1 #l2 normalize % #H destruct (H) ] + | #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0; + [ normalize #H1 destruct (H1) + | #tmp2 #l3 normalize #H1 destruct (H1) ] + ] + | #r1 #rs1 normalize in ⊢ (???(????%?)→?); #Htb >Htb in IH; #IH + cases (IH ls x xs end rs ? Hnotend Hend Hnotstart) + [| >Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src ] -IH + #_ #IH cases (IH Hstart (c::ls0) r1 rs1 ?) + [|| >nth_change_vec // ] -IH + [ * #l * #l1 * #Hll1 #Hout % %{(c::l)} %{l1} % >Hll1 // + >reverse_cons >associative_append #cj0 #ls #Hl1 >(Hout ?? Hl1) + >change_vec_commute in ⊢ (??(???%??)?); // @sym_not_eq // + | #IH %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@IH] + #l1 % #H destruct (H) cases eqx /2/ ] ] ] ] qed. +definition Pre_match_m ≝ + λsrc,sig,n,is_startc,is_endc.λt: Vector (tape sig) (S n). + ∃start,xs,end. + nth src (tape sig) t (niltape sig) = midtape ? [] start (xs@[end]) ∧ + is_startc start = true ∧ + (∀c.c ∈ (xs@[end]) = true → is_startc c = false) ∧ + (∀c.c ∈ (start::xs) = true → is_endc c = false) ∧ + is_endc end = true. + +lemma terminate_match_m : + ∀src,dst,sig,n,is_startc,is_endc,t. + src ≠ dst → src < S n → dst < S n → + Pre_match_m src sig n is_startc is_endc t → + match_m src dst sig n is_startc is_endc ↓ t. +#src #dst #sig #n #is_startc #is_endc #t #Hneq #Hsrc #Hdst * #start * #xs * #end +* * * * #Hmid_src #Hstart #Hnotstart #Hnotend #Hend +@(terminate_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst)) // +<(change_vec_same … t dst (niltape ?)) +lapply (refl ? (nth dst (tape sig) t (niltape ?))) +cases (nth dst (tape sig) t (niltape ?)) in ⊢ (???%→?); +[ #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //] + >Hmid_src #HR cases (HR ? (refl ??)) -HR + >nth_change_vec // >Htape_dst normalize in ⊢ (%→?); + * #H @False_ind @H % +| #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //] + >Hmid_src #HR cases (HR ? (refl ??)) -HR + >nth_change_vec // >Htape_dst normalize in ⊢ (%→?); + * #H @False_ind @H % +| #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //] + >Hmid_src #HR cases (HR ? (refl ??)) -HR + >nth_change_vec // >Htape_dst normalize in ⊢ (%→?); + * #H @False_ind @H % +| #ls #s #rs lapply s -s lapply ls -ls lapply Hmid_src lapply t -t elim rs + [#t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //] + >Hmid_src >nth_change_vec // >Hmid_dst #HR cases (HR ? (refl ??)) -HR #_ + #HR cases (HR Hstart Hnotstart) + cases (true_or_false (start == s)) #Hs + [ lapply (\P Hs) -Hs #Hs Hxs in Htrue; #Htrue + cases (Htrue [ ] start [ ] ? xs1 ? [ ] (refl ??) (refl ??) ?) + [ * #_ * #H @False_ind @H % ] + #c0 #Hc0 @Hnotend >(memb_single … Hc0) @memb_hd + | lapply (\Pf Hs) -Hs #Hs #Htrue #_ + cases (Htrue ? (refl ??) Hs) -Htrue #Ht1 #_ % + #t2 whd in ⊢ (%→?); #HR cases (HR start ?) + [ >Ht1 >nth_change_vec // normalize in ⊢ (%→?); * #H @False_ind @H % + | >Ht1 >nth_change_vec_neq [|@sym_not_eq //] + >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src % ] + ] + |#r0 #rs0 #IH #t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?); + >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src + #Htrue cases (Htrue ? (refl ??)) -Htrue #_ #Htrue + <(change_vec_same … t1 dst (niltape ?)) + cases (Htrue Hstart Hnotstart) -Htrue + cases (true_or_false (start == s)) #Hs + [ lapply (\P Hs) -Hs #Hs Hrs + cut (∃y,ys. xs1 = y::ys) + [ lapply Hxs0notend lapply Hxs lapply xs0 elim xs + [ * + [ normalize #Hxs1 Hend #H1 destruct (H1) + ] + | #y #ys #IH0 * + [ normalize in ⊢ (%→?); #Hxs1 Hxs1 in Hxs; #Hxs >Hmid_src >Hmid_dst >Hxs >Hrs + %{ls} %{xs0} %{y} %{ys} %{xs2} + % [ % // | @Hcomp // ] ] + * #ls0 * #xs0 * #ci * #rs * #rs0 * * #Hmid_src' #Hmid_dst' #Hcomp + nth_change_vec // >Hs #Htrue destruct (Hs) + lapply (Htrue ??????? Hmid_src' Hmid_dst' ?) -Htrue + [ #c0 #Hc0 @Hnotend cases (orb_true_l … Hc0) -Hc0 #Hc0 + [ whd in ⊢ (??%?); >Hc0 % + | @memb_cons >Hmid_src in Hmid_src'; #Hmid_src' destruct (Hmid_src') + lapply e0 -e0 @(list_elim_left … rs) + [ #e0 destruct (e0) lapply (append_l1_injective_r ?????? e0) // + | #x1 #xs1 #_ >append_cons in ⊢ (???%→?); + e1 @memb_append_l1 @memb_append_l1 // ] ] + | * * #Hciendc cases rs0 in Hcomp; + [ #_ * #H @False_ind @H % + | #r1 #rs1 * [ >Hciendc #H destruct (H) ] + * #_ #Hcomp lapply (Hcomp ?? (refl ??)) -Hcomp #Hcomp #_ #Htrue + cases (Htrue ?? (refl ??) Hcomp) #Ht1 #_ >Ht1 @(IH ?? (s::ls) r0) + [ >nth_change_vec_neq [|@sym_not_eq //] + >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src + | >nth_change_vec // >Hmid_dst % ] ] ] + | >Hmid_dst >nth_change_vec // lapply (\Pf Hs) -Hs #Hs #Htrue #_ + cases (Htrue ? (refl ??) Hs) #Ht1 #_ >Ht1 @(IH ?? (s::ls) r0) + [ >nth_change_vec_neq [|@sym_not_eq //] + >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src + | >nth_change_vec // ] ] ] ] +qed. \ No newline at end of file