X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=c771222e3bd73c03373a1f4fc6789c49cc0a9715;hb=38c81062ae1aedf89d426d5dcd9a27824c4b0fb0;hp=857477e430ced162b9c36dd11d90145eaee583ef;hpb=292ea72d5160e31b8516fe7dfd92bb9e487db9fc;p=helm.git diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma index 857477e43..c771222e3 100644 --- a/matita/matita/lib/turing/multi_universal/match.ma +++ b/matita/matita/lib/turing/multi_universal/match.ma @@ -12,106 +12,99 @@ (* *) (**************************************************************************) +include "turing/simple_machines.ma". 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). - (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1. - -definition Rtc_multi_false ≝ - λalpha,test,n,i.λt1,t2:Vector ? (S n). - (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1. +lemma eq_vec_change_vec : ∀sig,n.∀v1,v2:Vector sig n.∀i,t,d. + nth i ? v2 d = t → + (∀j.i ≠ j → nth j ? v1 d = nth j ? v2 d) → + v2 = change_vec ?? v1 t i. +#sig #n #v1 #v2 #i #t #d #H1 #H2 @(eq_vec … d) +#i0 #Hlt cases (decidable_eq_nat i0 i) #Hii0 +[ >Hii0 >nth_change_vec // +| >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ] +qed. -lemma sem_test_char_multi : - ∀alpha,test,n,i.i ≤ n → - inject_TM ? (test_char ? test) n i ⊨ - [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ]. -#alpha #test #n #i #Hin #int -cases (acc_sem_inject … Hin (sem_test_char alpha test) int) -#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ % -[ @Hloop -| #Hqtrue lapply (Htrue Hqtrue) * * * #c * - #Hcur #Htestc #Hnth_i #Hnth_j % - [ %{c} % // - | @(eq_vec … (niltape ?)) #i0 #Hi0 - cases (decidable_eq_nat i0 i) #Hi0i - [ >Hi0i @Hnth_i - | @sym_eq @Hnth_j @sym_not_eq // ] ] ] -| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j % - [ @Htestc - | @(eq_vec … (niltape ?)) #i0 #Hi0 - cases (decidable_eq_nat i0 i) #Hi0i - [ >Hi0i @Hnth_i - | @sym_eq @Hnth_j @sym_not_eq // ] ] ] +lemma right_mk_tape : + ∀sig,ls,c,rs.(c = None ? → ls = [ ] ∨ rs = [ ]) → right ? (mk_tape sig ls c rs) = rs. +#sig #ls #c #rs cases c // cases ls +[ cases rs // +| #l0 #ls0 #H normalize cases (H (refl ??)) #H1 [ destruct (H1) | >H1 % ] ] qed. -definition Rm_test_null_true ≝ - λalpha,n,i.λt1,t2:Vector ? (S n). - current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1. - -definition Rm_test_null_false ≝ - λalpha,n,i.λt1,t2:Vector ? (S n). - current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1. +lemma left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls. +#sig #ls #c #rs cases c // cases ls // cases rs // +qed. -lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n → - inject_TM ? (test_null ?) n i ⊨ - [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ]. -#alpha #n #i #Hin #int -cases (acc_sem_inject … Hin (sem_test_null alpha) int) -#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ % -[ @Hloop -| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % // - @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i - [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ] -| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j % - [ @Hcur - | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) // - #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ] +lemma current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c. +#sig #ls #c #rs cases c // cases ls // cases rs // qed. -definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n. - match (nth src (option sig) v (None ?)) with - [ None ⇒ false - | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ]. +lemma length_tail : ∀A,l.0 < |l| → |tail A l| < |l|. +#A * normalize // +qed. + +(* +[ ] → [ ], l2, 1 +a::al → + [ ] → [ ], l1, 2 + b::bl → match rec(al,bl) + x, y, 1 → b::x, y, 1 + x, y, 2 → a::x, y, 2 +*) -definition mmove_states ≝ initN 2. +lemma lists_length_split : + ∀A.∀l1,l2:list A.(∃la,lb.(|la| = |l1| ∧ l2 = la@lb) ∨ (|la| = |l2| ∧ l1 = la@lb)). +#A #l1 elim l1 +[ #l2 %{[ ]} %{l2} % % % +| #hd1 #tl1 #IH * + [ %{[ ]} %{(hd1::tl1)} %2 % % + | #hd2 #tl2 cases (IH tl2) #x * #y * + [ * #IH1 #IH2 %{(hd2::x)} %{y} % normalize % // + | * #IH1 #IH2 %{(hd1::x)} %{y} %2 normalize % // ] + ] +] +qed. -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 option_cons ≝ λsig.λc:option sig.λl. + match c with [ None ⇒ l | Some c0 ⇒ c0::l ]. -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〉 ]. +lemma opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l). +#A * // +qed. -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 match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n. + match (nth src (option sig) v (None ?)) with + [ None ⇒ false + | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ]. definition rewind ≝ λsrc,dst,sig,n. parmove src dst sig n L · mmove src sig n R · mmove dst sig n R. +definition R_rewind_strong ≝ 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) ∧ + (∀x,x0,xs,rs. + nth dst ? int (niltape ?) = midtape sig (xs@[x0]) x rs → + ∀ls0,y,y0,target,rs0.|xs| = |target| → + nth src ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 → + outt = change_vec ?? + (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) dst) + (midtape sig ls0 y0 (reverse ? target@y::rs0)) src) ∧ + (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs → + ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 → + outt = int) ∧ + (∀x,rs.nth dst ? int (niltape ?) = midtape sig [] x rs → + ∀ls0,y,rs0.nth src ? int (niltape ?) = midtape sig ls0 y rs0 → + outt = int). + 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 → @@ -136,16 +129,16 @@ cases (true_or_false (cstate sig (states sig n M) n outc == acc)) #Hcase qed. *) -lemma sem_rewind : ∀src,dst,sig,n. +lemma sem_rewind_strong : ∀src,dst,sig,n. src ≠ dst → src < S n → dst < S n → - rewind src dst sig n ⊨ R_rewind src dst sig n. + rewind src dst sig n ⊨ R_rewind_strong 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_multi … R ?) (sem_move_multi … R ?)) // @le_S_S_to_le // ] -#ta #tb * #tc * * #Htc #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb % +#ta #tb * #tc * * * #Htc1 #Htc2 #_ * #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; + >(Htc1 ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in Htd; [|>Hmidta_dst // |>length_append >length_append >Hlen % ] >change_vec_commute [|@sym_not_eq //] @@ -156,8 +149,20 @@ lemma sem_rewind : ∀src,dst,sig,n. >rev_append_def >append_nil #Htd >Htd in Htb; >change_vec_change_vec >nth_change_vec // cases ls0 [|#l1 #ls1] normalize in match (tape_move ???); // +| #x #x0 #xs #rs #Hmidta_dst #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_src + >(Htc2 ??? Hmidta_dst ls0 y (target@[y0]) rs0 ??) in Htd; + [|>Hmidta_src // + |>length_append >length_append >Hlen % ] + >change_vec_change_vec + >change_vec_commute [|@sym_not_eq //] + >nth_change_vec // + >reverse_append >reverse_single + >reverse_append >reverse_single + cases ls0 [|#l1 #ls1] normalize in match (tape_move ???); + #Htd >Htd in Htb; >change_vec_change_vec >nth_change_vec // + >rev_append_def >change_vec_commute // normalize in match (tape_move ???); // ] | #x #rs #Hmidta_src #ls0 #y #rs0 #Hmidta_dst - lapply (Htc … Hmidta_src … (refl ??) Hmidta_dst) -Htc #Htc >Htc in Htd; + lapply (Htc1 … Hmidta_src … (refl ??) Hmidta_dst) -Htc1 #Htc >Htc in Htd; >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%); @@ -169,7 +174,28 @@ lemma sem_rewind : ∀src,dst,sig,n. >nth_change_vec // >change_vec_change_vec whd in match (tape_move ???);whd in match (tape_move ???); change_vec_same >change_vec_same // -]] + ] ] +| #x #rs #Hmidta_dst #ls0 #y #rs0 #Hmidta_src + lapply (Htc2 … Hmidta_dst … (refl ??) Hmidta_src) -Htc2 #Htc >Htc in Htd; + >change_vec_change_vec >change_vec_commute [|@sym_not_eq //] + >nth_change_vec // lapply (refl ? ls0) cases ls0 in ⊢ (???%→%); + [ #Hls0 destruct (Hls0) #Htd >Htd in Htb; + >nth_change_vec // >change_vec_change_vec + whd in match (tape_move ???);whd in match (tape_move ???); + change_vec_same >change_vec_same // + | #l1 #ls1 #Hls0 destruct (Hls0) #Htd >Htd in Htb; + >nth_change_vec // >change_vec_change_vec + whd in match (tape_move ???); whd in match (tape_move ???); change_vec_same >change_vec_same // + ] +] +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 @(Realize_to_Realize … (sem_rewind_strong …)) // +#ta #tb * * * #H1 #H2 #H3 #H4 % /2 by / qed. definition match_step ≝ λsrc,dst,sig,n. @@ -219,14 +245,13 @@ definition R_match_step_true ≝ ∀s,rs.nth src ? int (niltape ?) = midtape ? [ ] s rs → outt = change_vec ?? int (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst ∧ - (current sig (nth dst (tape sig) int (niltape sig)) = Some ? s → + (∃s0.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s0 ∧ + (s0 = s → ∃xs,ci,rs',ls0,cj,rs0. rs = xs@ci::rs' ∧ nth dst ? int (niltape ?) = midtape sig ls0 s (xs@cj::rs0) ∧ - ci ≠ cj). + ci ≠ cj)). -axiom daemon : ∀X:Prop.X. - lemma sem_match_step : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → match_step src dst sig n ⊨ @@ -288,8 +313,8 @@ lemma sem_match_step : | nth_change_vec_neq [| @(\Pf Hdsti)] >Hrs0 >reverse_reverse >nth_change_vec_neq in ⊢ (???%); change_vec_same % ] - | #_ >Hmidta_dst >Hrs0 - %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % // + | >Hmidta_dst %{s'} % [%] #_ + >Hrs0 %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % // ] ] | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta) @@ -303,7 +328,7 @@ lemma sem_match_step : [ <(\P Hdsti) >(Htb1 … Hmidta_dst) >nth_change_vec // >Hmidta_dst cases rs0 [|#r2 #rs2] % | nth_change_vec_neq [| @(\Pf Hdsti)] % ] - | >Hs0 #H destruct (H) @False_ind cases (Hss0) /2/ ] + | >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ] ] ] ] @@ -344,7 +369,8 @@ definition R_match_m ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). ∀x,rs. nth src ? int (niltape ?) = midtape sig [ ] x rs → - (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ + (current sig (nth dst (tape sig) int (niltape sig)) = None ? → + right ? (nth dst (tape sig) int (niltape sig)) = [ ] → outt = int) ∧ (∀ls0,x0,rs0. nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 → (∃l,l1.x0::rs0 = l@x::rs@l1 ∧ @@ -413,10 +439,18 @@ lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) / lapply (refl ? (current ? (nth dst ? ta (niltape ?)))) cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?); [#Hcurta_dst % - [#_ whd in Htrue; >Hmidta_src in Htrue; #Htrue - cases (Htrue ?? (refl ??)) -Htrue >Hcurta_dst - (* dovremmo sapere che ta.dst è sul margine destro, da cui la move non - ha effetto *) #_ cut (tc = ta) [@daemon] #Htc destruct (Htc) #_ + [#Hcurta_dst #Hrightta_dst whd in Htrue; >Hmidta_src in Htrue; #Htrue + cases (Htrue ?? (refl ??)) -Htrue #Htc + cut (tc = ta) + [ >Htc whd in match (tape_move_mono ???); whd in match (tape_write ???); + <(change_vec_same … ta dst (niltape ?)) in ⊢ (???%); + lapply Hrightta_dst lapply Hcurta_dst -Hrightta_dst -Hcurta_dst + cases (nth dst ? ta (niltape ?)) + [ #_ #_ % + | #r0 #rs0 #_ normalize in ⊢ (%→?); #H destruct (H) + | #l0 #ls0 #_ #_ % + | #ls #x0 #rs normalize in ⊢ (%→?); #H destruct (H) ] ] + -Htc #Htc destruct (Htc) #_ cases (IH … Hmidta_src) #Houtc #_ @Houtc // |#ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst; normalize in ⊢ (%→?); #H destruct (H) @@ -426,8 +460,8 @@ lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) / #H destruct (H) whd in Htrue; >Hmidta_src in Htrue; #Htrue cases (Htrue ?? (refl …)) -Htrue >Hmidta_dst #Htc cases (true_or_false (x==c)) #eqx - [ lapply (\P eqx) -eqx #eqx destruct (eqx) - #Htrue cases (Htrue (refl ??)) -Htrue + [ lapply (\P eqx) -eqx #eqx destruct (eqx) * #s0 * whd in ⊢ (??%?→?); #Hs0 + destruct (Hs0) #Htrue cases (Htrue (refl ??)) -Htrue #xs0 * #ci * #rs' * #ls1 * #cj * #rs1 * * #Hxs #H destruct (H) #Hcicj >Htc in IH; whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //] #IH cases (IH … Hmidta_src) -IH #_ >nth_change_vec // @@ -436,10 +470,10 @@ lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) / #Hxs1 >Hxs1 #IH cases (IH … (refl ??)) -IH [ * #l * #l1 * #Hxs1' >change_vec_commute // >change_vec_change_vec - #Houtc % %{(c::l)} %{l1} % + #Houtc % %{(s0::l)} %{l1} % [ normalize reverse_cons >associative_append >change_vec_commute // @Houtc ] - | #H %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%] + | #H %2 #l #l1 >(?:l@s0::xs@l1 = l@(s0::xs)@l1) [|%] @not_sub_list_merge [ #l2 >Hxs associative_append #H2 lapply (append_l2_injective ????? (refl ??) H2) @@ -474,103 +508,310 @@ lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) / ] qed. -definition Pre_match_m ≝ - λsrc,sig,n.λt: Vector (tape sig) (S n). - ∃x,xs. - nth src (tape sig) t (niltape sig) = midtape ? [] x xs. - +definition R_match_step_true_naive ≝ + λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). + |left ? (nth src ? outt (niltape ?))| + + |option_cons ? (current ? (nth dst ? outt (niltape ?))) (right ? (nth dst ? outt (niltape ?)))| < + |left ? (nth src ? int (niltape ?))| + + |option_cons ? (current ? (nth dst ? int (niltape ?))) (right ? (nth dst ? int (niltape ?)))|. + +lemma sem_match_step_termination : + ∀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_naive 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_rewind_strong ???? Hneq Hsrc Hdst) + (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? ))) + (sem_nop …))) +[ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?)))) + cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%); + [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %] + whd in ⊢ (%→?); * whd in ⊢ (??%?→?); + >nth_current_chars >Hcurta_src normalize in ⊢ (%→?); #H destruct (H) + | #s #Hs lapply (refl ? (current ? (nth dst ? ta (niltape ?)))) + cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%); + [ #Hcurta_dst #Hcomp #_ * #td * >Hcomp [| %2 %] + whd in ⊢ (%→?); * whd in ⊢ (??%?→?); + >nth_current_chars >nth_current_chars >Hs >Hcurta_dst + normalize in ⊢ (%→?); #H destruct (H) + | #s0 #Hs0 + cases (current_to_midtape … Hs) #ls * #rs #Hmidta_src >Hmidta_src + cases (current_to_midtape … Hs0) #ls0 * #rs0 #Hmidta_dst >Hmidta_dst + cases (true_or_false (s == s0)) #Hss0 + [ lapply (\P Hss0) -Hss0 #Hss0 destruct (Hss0) + #_ #Hcomp cases (Hcomp ????? (refl ??) (refl ??)) -Hcomp [ * + [ * #rs' * #_ #Hcurtc_dst * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?); + >nth_current_chars >nth_current_chars >Hcurtc_dst + cases (current ? (nth src …)) + [normalize in ⊢ (%→?); #H destruct (H) + | #x >nth_change_vec // cases (reverse ? rs0) + [ normalize in ⊢ (%→?); #H destruct (H) + | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ] + | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?); + >(?:nth src ? (current_chars ?? tc) (None ?) = None ?) + [|>nth_current_chars >Hcurtc_src >nth_change_vec_neq + [>nth_change_vec [cases (append ???) // | @Hsrc] + |@(not_to_not … Hneq) // + ]] + normalize in ⊢ (%→?); #H destruct (H) ] + | * #xs * #ci * #cj * #rs'' * #rs0' * * * #Hcicj #Hrs #Hrs0 + #Htc * #td * * #Hmatch #Htd destruct (Htd) * #te * * * + >Htc >change_vec_commute // >nth_change_vec // + >change_vec_commute [|@sym_not_eq //] >nth_change_vec // + cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb + destruct (Hlsalsb) * + [ #Hte #_ #_ <(reverse_reverse … ls) in Hte; <(reverse_reverse … lsa) + cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen' + @(list_cases2 … Hlen') + [ #H1 #H2 >H1 >H2 -H1 -H2 normalize in match (reverse ? [ ]); #Hte #_ + lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte + >change_vec_commute // >change_vec_change_vec + >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte + >Hte * * #_ >nth_change_vec // >reverse_reverse + #H lapply (H … (refl ??)) -H #Htb1 #Htb2 + cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (midtape sig [ ] s0 (xs@ci::rs'')) src) (mk_tape sig (s0::lsb) (option_hd sig (xs@cj::rs0')) (tail sig (xs@cj::rs0'))) dst) + [ @(eq_vec_change_vec … (niltape ?)) + [@Htb1| #j #Hj (nth_change_vec_neq ??????? Hj) % ] ] + -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec // + >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // + >right_mk_tape [|cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H)] + normalize in match (left ??); + >Hmidta_src >Hmidta_dst >current_mk_tape Hrs0 + normalize in ⊢ (?(?%)%); // + | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2 + >reverse_cons >reverse_cons #Hte + lapply (Hte ci hdb (reverse ? xs@s0::reverse ? tlb) rs'' ? + lsb cj hda (reverse ? xs@s0::reverse ? tla) rs0' ??) + [ /2 by cons_injective_l, nil/ + | >length_append >length_append @eq_f @(eq_f ?? S) + >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen' + >length_reverse >length_reverse destruct (Hlen') // + | /2 by refl, trans_eq/ ] -Hte + #Hte #_ * * #_ >Hte >nth_change_vec // #Htb1 #Htb2 + cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta + (mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0')) (tail ? (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0'))) dst) + (midtape ? [ ] hdb (reverse sig (reverse sig xs@s0::reverse sig tlb)@ci::rs'')) src) + [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?)) + [ @Htb1 % + | #j #Hj change_vec_commute // >change_vec_change_vec + >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec + >nth_change_vec_neq in ⊢ (???%); // ] ] + -Htb1 -Htb2 #Htb >Htb whd + >nth_change_vec // >nth_change_vec_neq // >nth_change_vec // + >right_mk_tape + [| cases (reverse sig (reverse sig xs@s0::reverse sig tla)) + [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ] + >Hmidta_src >Hmidta_dst + whd in match (left ??); whd in match (left ??); whd in match (right ??); + >current_mk_tape Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse + >length_append >commutative_plus in match (|reverse ??| + ?); + whd in match (|?::?|); >length_reverse >length_reverse + <(length_reverse ? ls) H1 normalize // ] + | #_ #Hte #_ <(reverse_reverse … ls0) in Hte; <(reverse_reverse … lsa) + cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen' + @(list_cases2 … Hlen') + [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #Hte + lapply (Hte … (refl ??) … (refl ??) (refl ??)) -Hte + >change_vec_change_vec >change_vec_commute [|@sym_not_eq //] + >change_vec_change_vec #Hte #_ + >Hte * * #_ >nth_change_vec // >reverse_reverse + #H lapply (H … (refl ??)) -H #Htb1 #Htb2 + cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta (mk_tape ? [s0] (option_hd ? (xs@cj::rs0')) (tail ? (xs@cj::rs0'))) dst) + (midtape ? lsb s0 (xs@ci::rs'')) src) + [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?)) + [ @Htb1 + | #j #Hj nth_change_vec_neq in ⊢ (???%); // ] ] + -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec // + >nth_change_vec_neq // >nth_change_vec // + >right_mk_tape + [| cases xs [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ] + normalize in match (left ??); + >Hmidta_src >Hmidta_dst >current_mk_tape Hrs0 + >length_append normalize >length_append >length_append + <(reverse_reverse ? lsa) >H1 normalize // + | #hda #hdb #tla #tlb #H1 #H2 >H1 >H2 + >reverse_cons >reverse_cons #Hte + lapply (Hte cj hdb (reverse ? xs@s0::reverse ? tlb) rs0' ? + lsb ci hda (reverse ? xs@s0::reverse ? tla) rs'' ??) + [ /2 by cons_injective_l, nil/ + | >length_append >length_append @eq_f @(eq_f ?? S) + >H1 in Hlen'; >H2 whd in ⊢ (??%%→?); #Hlen' + >length_reverse >length_reverse destruct (Hlen') // + | /2 by refl, trans_eq/ ] -Hte + #Hte #_ * * #_ >Hte >nth_change_vec_neq // >nth_change_vec // #Htb1 #Htb2 + cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta + (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0')) (tail ? (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0'))) dst) + (midtape ? lsb hda (reverse sig (reverse sig xs@s0::reverse sig tla)@ci::rs'')) src) + [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?)) + [ @Htb1 % + | #j #Hj change_vec_change_vec + >change_vec_commute [|@sym_not_eq //] + >change_vec_change_vec >nth_change_vec_neq in ⊢ (???%); // ] ] + -Htb1 -Htb2 #Htb >Htb whd + >nth_change_vec // >nth_change_vec_neq // >nth_change_vec // + >right_mk_tape + [| cases (reverse sig (reverse sig xs@s0::reverse sig tlb)) + [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ] + >Hmidta_src >Hmidta_dst + whd in match (left ??); whd in match (left ??); whd in match (right ??); + >current_mk_tape Hrs0 >length_append whd in ⊢ (??(??%)); >length_append >length_reverse + >length_append >commutative_plus in match (|reverse ??| + ?); + whd in match (|?::?|); >length_reverse >length_reverse + <(length_reverse ? lsa) >Hlen' >H2 >length_append + normalize // + ] + ] + ] + | lapply (\Pf Hss0) -Hss0 #Hss0 #Htc cut (tc = ta) + [@Htc % % @(not_to_not ??? Hss0) #H destruct (H) %] + -Htc #Htc destruct (Htc) #_ * #td * whd in ⊢ (%→?); * #_ + #Htd destruct (Htd) * #te * * * * >Hmidta_src >Hmidta_dst + cases (lists_length_split ? ls ls0) #lsa * #lsb * * #Hlen #Hlsalsb + destruct (Hlsalsb) + [ <(reverse_reverse … ls) <(reverse_reverse … lsa) + cut (|reverse ? lsa| = |reverse ? ls|) [ // ] #Hlen' + @(list_cases2 … Hlen') + [ #H1 #H2 >H1 >H2 -H1 -H2 #_ #_ normalize in match (reverse ? [ ]); #Hte #_ + lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte) * * #_ + >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2 + cut (tb = change_vec ?? ta (mk_tape ? (s0::lsa@lsb) (option_hd ? rs0) (tail ? rs0)) dst) + [@(eq_vec_change_vec … (niltape ?)) [@Htb1|@Htb2] ] + -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec // + >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst + >right_mk_tape + [| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H)] ] + normalize in match (left ??); normalize in match (right ??); + >Hmidta_src >Hmidta_dst >current_mk_tape H1 >H2 + >reverse_cons >reverse_cons >associative_append #Hte + lapply (Hte ???? (refl ??) ? s0 ? (reverse ? tla) ?? (refl ??)) + [ >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) // + normalize #H destruct (H) // ] #Hte #_ #_ #_ + * * #_ >Hte >nth_change_vec // #Htb1 #Htb2 + cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta + (mk_tape sig (hda::lsb) (option_hd ? (reverse sig (reverse sig tla)@s0::rs0)) (tail ? (reverse sig (reverse sig tla)@s0::rs0))) dst) + (midtape ? [ ] hdb (reverse sig (reverse sig tlb)@s::rs)) src) + [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?)) + [ @Htb1 // + | #j #Hj nth_change_vec_neq in ⊢ (???%); // ]] + -Htb1 -Htb2 #Htb >Htb whd + >nth_change_vec // >nth_change_vec_neq // >nth_change_vec // + >right_mk_tape + [| cases (reverse sig (reverse sig tla)) + [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ] + >Hmidta_src >Hmidta_dst + whd in match (left ??); whd in match (left ??); whd in match (right ??); + >current_mk_tape length_append + >length_reverse >length_reverse <(length_reverse ? ls) H1 normalize // ] + | #_ <(reverse_reverse … ls0) <(reverse_reverse … lsa) + cut (|reverse ? lsa| = |reverse ? ls0|) [ // ] #Hlen' + @(list_cases2 … Hlen') + [ #H1 #H2 >H1 >H2 normalize in match (reverse ? [ ]); #_ #_ #Hte + lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte) + * * #_ >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2 + cut (tb = change_vec (tape sig) (S n) ta (mk_tape ? (s0::ls0) (option_hd ? rs0) (tail ? rs0)) dst) + [ @(eq_vec_change_vec … (niltape ?)) // @Htb2 ] + -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec // + >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst + >current_mk_tape >right_mk_tape + [| cases rs0 [ #_ %2 % | #x0 #xs0 normalize in ⊢ (??%?→?); #H destruct (H) ]] + normalize in ⊢ (??%); H1 >H2 + >reverse_cons >reverse_cons #Hte #_ #_ + lapply (Hte s0 hdb (reverse ? tlb) rs0 ? + lsb s hda (reverse ? tla) rs ??) + [ /2 by cons_injective_l, nil/ + | >length_reverse >length_reverse cut (|hda::tla| = |hdb::tlb|) // + normalize #H destruct (H) // + | /2 by refl, trans_eq/ ] -Hte + #Hte * * #_ >Hte >nth_change_vec_neq // >nth_change_vec // #Htb1 #Htb2 + cut (tb = change_vec ?? (change_vec (tape sig) (S n) ta + (mk_tape sig [hdb] (option_hd ? (reverse sig (reverse sig tlb)@s0::rs0)) (tail ? (reverse sig (reverse sig tlb)@s0::rs0))) dst) + (midtape ? lsb hda (reverse sig (reverse sig tla)@s::rs)) src) + [ >change_vec_commute [|@sym_not_eq //] @(eq_vec_change_vec … (niltape ?)) + [ @Htb1 % + | #j #Hj change_vec_commute [|@sym_not_eq //] + >nth_change_vec_neq in ⊢ (???%); // ]] + -Htb1 -Htb2 #Htb >Htb whd + >nth_change_vec // >nth_change_vec_neq // >nth_change_vec // + >right_mk_tape + [| cases (reverse ? (reverse ? tlb)) [|#x0 #xs0] normalize in ⊢ (??%?→?); #H destruct (H) ] + >Hmidta_src >Hmidta_dst + whd in match (left ??); whd in match (left ??); whd in match (right ??); + >current_mk_tape length_append + normalize in ⊢ (??%); >length_append >reverse_reverse + <(length_reverse ? lsa) >Hlen' >H2 normalize // + ] + ] + ] + ] + ] +| #ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd destruct (Htd) + whd in ⊢ (%→?); #Htb destruct (Htb) #ls #x #xs #Hta_src + lapply (refl ? (current ? (nth dst ? ta (niltape ?)))) + cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?); + [ #Hcurta_dst % % % // @Hcomp1 %2 // + | #x0 #Hcurta_dst cases (current_to_midtape … Hcurta_dst) -Hcurta_dst + #ls0 * #rs0 #Hta_dst cases (true_or_false (x == x0)) #Hxx0 + [ lapply (\P Hxx0) -Hxx0 #Hxx0 destruct (Hxx0) + | >(?:tc=ta) in Htest; + [|@Hcomp1 % % >Hta_src >Hta_dst @(not_to_not ??? (\Pf Hxx0)) normalize + #Hxx0' destruct (Hxx0') % ] + whd in ⊢ (??%?→?); + >nth_current_chars >Hta_src >nth_current_chars >Hta_dst + whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1 + cases (Hcomp2 … Hta_src Hta_dst) [ * + [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} % + [ % // | >Hcurtc % ] + | * #rs0' * #Hxs #Htc %2 >Htc %{ls0} %{rs0'} % // ] + | * #xs0 * #ci * #cj * #rs' * #rs0' * * * + #Hci #Hxs #Hrs0 #Htc @False_ind + whd in Htest:(??%?); + >(?:nth src ? (current_chars ?? tc) (None ?) = Some ? ci) in Htest; + [|>nth_current_chars >Htc >nth_change_vec_neq [|@(not_to_not … Hneq) //] + >nth_change_vec //] + >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj) + [|>nth_current_chars >Htc >nth_change_vec //] + normalize #H destruct (H) ] ] ] +qed. + +(* lemma WF_to_WF_f : ∀A,B,R,f,b. WF A R (f b) → WF B (λx,y.R (f x) (f y)) b. *) +let rec WF_to_WF_f A B R f b (Hwf: WF A R (f b)) on Hwf: WF B (λx,y.R (f x) (f y)) b ≝ + match Hwf return (λa0,r.f b = a0 → WF B (λx,y:B. R (f x) (f y)) b) with + [ wf a Hwfa ⇒ λHeq.? ] (refl ??). +% #b1 #HRb @WF_to_WF_f @Hwfa nth_change_vec_neq [|@sym_not_eq //] - >Hmid_src #HR cases (HR ?? (refl ??)) -HR - >nth_change_vec // >Htape_dst -| #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 // ] ] ] ] +#src #dst #sig #n #ta #Hneq #Hsrc #Hdst +@(terminate_while … (sem_match_step_termination src dst sig n Hneq Hsrc Hdst)) // +letin f ≝ (λt0:Vector (tape sig) (S n).|left ? (nth src (tape ?) t0 (niltape ?))| + +|option_cons ? (current ? (nth dst (tape ?) t0 (niltape ?))) + (right ? (nth dst (tape ?) t0 (niltape ?)))|) +change with (λx,y.f x < f y) in ⊢ (??%?); @WF_to_WF_f @lt_WF +qed. + +lemma sem_match_m : ∀src,dst,sig,n. +src ≠ dst → src < S n → dst < S n → + match_m src dst sig n \vDash R_match_m src dst sig n. +#src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize [/2/| @wsem_match_m // ] qed. \ No newline at end of file