X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=93e6b340df1d5fb0a35c72f247dff132796979be;hb=637ff9311e16f1d58e03d873f84c354e1cf1e716;hp=74fd89cdf93b862f609a7da3cf40f5aedf7727d9;hpb=95ea23408caad83226b7a9206f3e020accf3f9ce;p=helm.git diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma index 74fd89cdf..93e6b340d 100644 --- a/matita/matita/lib/turing/multi_universal/match.ma +++ b/matita/matita/lib/turing/multi_universal/match.ma @@ -12,102 +12,73 @@ (* *) (**************************************************************************) +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. -definition mmove_states ≝ initN 2. +(* +[ ] → [ ], 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 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 …)). +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 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 option_cons ≝ λsig.λc:option sig.λl. + match c with [ None ⇒ l | Some c0 ⇒ c0::l ]. -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. +lemma opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l). +#A * // +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. @@ -224,14 +195,14 @@ 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/ +#ta #tb * * * #H1 #H2 #H3 #H4 % /2 by / qed. definition match_step ≝ λsrc,dst,sig,n. compare src dst sig n · (ifTM ?? (partest sig n (match_test src dst sig ?)) (single_finalTM ?? - (rewind src dst sig n · (inject_TM ? (move_r ?) n dst))) + (rewind src dst sig n · mmove dst ?? R)) (nop …) partest1). @@ -292,8 +263,8 @@ lemma sem_match_step : (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?)) (sem_seq … (sem_rewind ???? Hneq Hsrc Hdst) - (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? ))) - (sem_nop …))) + (sem_move_multi … R ?)) + (sem_nop …))) /2/ [ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?)))) cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%); [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %] @@ -333,15 +304,10 @@ lemma sem_match_step : 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 in Htb; * * #_ >nth_change_vec // #Htb1 - lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2 % - [ @(eq_vec … (niltape ?)) #i #Hi - cases (true_or_false ((dst : DeqNat) == i)) #Hdsti - [ <(\P Hdsti) >Htb1 >nth_change_vec // >Hmidta_dst - >Hrs0 >reverse_reverse cases xs [|#r1 #rs1] % - | nth_change_vec_neq [| @(\Pf Hdsti)] - >Hrs0 >reverse_reverse >nth_change_vec_neq in ⊢ (???%); - change_vec_same % ] + >Hte in Htb; whd in ⊢ (%→?); #Htb >Htb % + [ >change_vec_change_vec >nth_change_vec // + >reverse_reverse change_vec_same Hmidta_dst %{s'} % [%] #_ >Hrs0 %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % // ] @@ -349,14 +315,10 @@ lemma sem_match_step : | 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 * * #_ #Hte * * #_ #Htb1 #Htb2 + #Htd destruct (Htd) * #te * * #_ #Hte whd in ⊢ (%→?); #Htb #s1 #rs1 >Hmidta_src #H destruct (H) lapply (Hte … Hmidta_src … Hmidta_dst) -Hte #Hte destruct (Hte) % - [ @(eq_vec … (niltape ?)) #i #Hi - cases (true_or_false ((dst : DeqNat) == i)) #Hdsti - [ <(\P Hdsti) >(Htb1 … Hmidta_dst) >nth_change_vec // >Hmidta_dst - cases rs0 [|#r2 #rs2] % - | nth_change_vec_neq [| @(\Pf Hdsti)] % ] + [ >Htb % | >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ] ] ] @@ -537,12 +499,6 @@ lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) / ] qed. -axiom daemon : ∀P:Prop.P. - -(* XXX: move to turing (or mono) *) -definition option_cons ≝ λsig.λc:option sig.λl. - match c with [ None ⇒ l | Some c0 ⇒ c0::l ]. - definition R_match_step_true_naive ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). |left ? (nth src ? outt (niltape ?))| + @@ -550,14 +506,6 @@ definition R_match_step_true_naive ≝ |left ? (nth src ? int (niltape ?))| + |option_cons ? (current ? (nth dst ? int (niltape ?))) (right ? (nth dst ? int (niltape ?)))|. -axiom right_mk_tape : ∀sig,ls,c,rs.right ? (mk_tape sig ls c rs) = rs. -axiom left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls. -axiom current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c. -axiom length_tail : ∀A,l.0 < |l| → |tail A l| < |l|. -axiom lists_length_split : - ∀A.∀l1,l2:list A.(∃la,lb.(|la| = |l1| ∧ l2 = la@lb) ∨ (|la| = |l2| ∧ l1 = la@lb)). -axiom opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l). - lemma sem_match_step_termination : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → match_step src dst sig n ⊨ @@ -569,8 +517,8 @@ lemma sem_match_step_termination : (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 …))) + (sem_move_multi … R ?)) + (sem_nop …))) [/2/] [ #ta #tb #tc * lapply (refl ? (current ? (nth src ? ta (niltape ?)))) cases (current ? (nth src ? ta (niltape ?))) in ⊢ (???%→%); [ #Hcurta_src #Hcomp #_ * #td * >Hcomp [| % %2 %] @@ -592,7 +540,7 @@ lemma sem_match_step_termination : >nth_current_chars >nth_current_chars >Hcurtc_dst cases (current ? (nth src …)) [normalize in ⊢ (%→?); #H destruct (H) - | #x >nth_change_vec // cases (reverse ? rs0) + | #x >nth_change_vec [|@Hdst] cases (reverse ? rs0) [ normalize in ⊢ (%→?); #H destruct (H) | #r1 #rs1 normalize in ⊢ (%→?); #H destruct (H) ] ] | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?); @@ -604,8 +552,8 @@ lemma sem_match_step_termination : 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 // + >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) @@ -613,14 +561,16 @@ lemma sem_match_step_termination : @(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 [|//] >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) - [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec // - >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // - >right_mk_tape normalize in match (left ??); + >Hte whd in ⊢ (%→?); >change_vec_change_vec >nth_change_vec [|//] + >reverse_reverse #Htb + 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) + [ >Htb @eq_f3 // cases (xs@cj::rs0') // ] + -Htb #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 ⊢ (?(?%)%); // @@ -633,13 +583,20 @@ lemma sem_match_step_termination : >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 + #Hte #_ whd in ⊢ (%→?); #Htb 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) - [@daemon] -Htb1 -Htb2 #Htb >Htb whd - >nth_change_vec // >nth_change_vec_neq // >nth_change_vec // - >right_mk_tape >Hmidta_src >Hmidta_dst + [ >Htb >Hte >nth_change_vec // >change_vec_change_vec >change_vec_commute [|//] + >change_vec_change_vec >change_vec_commute [|@sym_not_eq //] + >change_vec_change_vec >change_vec_commute [|//] + @eq_f3 // cases (reverse sig (reverse sig xs@s0::reverse sig tla)@cj::rs0') // ] + -Htb #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 @@ -653,13 +610,16 @@ lemma sem_match_step_termination : 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 + >Hte whd in ⊢ (%→?); >nth_change_vec [|//] >reverse_reverse #Htb 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) - [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec // - >nth_change_vec_neq // >nth_change_vec // - >right_mk_tape normalize in match (left ??); + [ >Htb >change_vec_change_vec >change_vec_commute [|//] + @eq_f3 // 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 // @@ -672,13 +632,20 @@ lemma sem_match_step_termination : >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 + #Hte #_ whd in ⊢ (%→?); >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb 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) - [@daemon] -Htb1 -Htb2 #Htb >Htb whd - >nth_change_vec // >nth_change_vec_neq // >nth_change_vec // - >right_mk_tape >Hmidta_src >Hmidta_dst + [ >Htb >change_vec_change_vec >change_vec_commute [|//] + >change_vec_change_vec >change_vec_commute [|@sym_not_eq //] + >change_vec_change_vec >change_vec_commute [|//] + @eq_f3 // cases (reverse sig (reverse sig xs@s0::reverse sig tlb)@cj::rs0') // ] + -Htb #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 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 + lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte) + whd in ⊢ (%→?); >Hmidta_dst #Htb cut (tb = change_vec ?? ta (mk_tape ? (s0::lsa@lsb) (option_hd ? rs0) (tail ? rs0)) dst) - [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec // + [ >Htb cases rs0 // ] + -Htb #Htb >Htb whd >nth_change_vec [|//] >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst - >right_mk_tape normalize in match (left ??); normalize in match (right ??); + >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 @@ -713,13 +683,17 @@ lemma sem_match_step_termination : 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 + whd in ⊢ (%→?); >Hte >change_vec_change_vec >nth_change_vec // #Htb 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) - [@daemon] -Htb1 -Htb2 #Htb >Htb whd - >nth_change_vec // >nth_change_vec_neq // >nth_change_vec // - >right_mk_tape >Hmidta_src >Hmidta_dst + [ >Htb >change_vec_commute [|//] @eq_f3 // cases (reverse sig (reverse sig tla)@s0::rs0) // ] + -Htb #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 >H2 normalize in match (reverse ? [ ]); #_ #_ #Hte lapply (Hte … (refl ??) … (refl ??)) -Hte #Hte destruct (Hte) - * * #_ >Hmidta_dst #Htb1 lapply (Htb1 … (refl ??)) -Htb1 #Htb1 #Htb2 + whd in ⊢ (%→?); #Htb whd >Hmidta_dst cut (tb = change_vec (tape sig) (S n) ta (mk_tape ? (s0::ls0) (option_hd ? rs0) (tail ? rs0)) dst) - [@daemon] -Htb1 -Htb2 #Htb >Htb whd >nth_change_vec // + [ >Htb >Hmidta_dst cases rs0 // ] + -Htb #Htb >Htb whd >nth_change_vec [|//] >nth_change_vec_neq [|@sym_not_eq //] >Hmidta_src >Hmidta_dst - >current_mk_tape >right_mk_tape normalize in ⊢ (??%); 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 #_ #_ @@ -743,13 +720,17 @@ lemma sem_match_step_termination : | >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 + #Hte whd in ⊢ (%→?); >Hte >nth_change_vec_neq [|//] >nth_change_vec [|//] #Htb 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) - [@daemon] -Htb1 -Htb2 #Htb >Htb whd - >nth_change_vec // >nth_change_vec_neq // >nth_change_vec // - >right_mk_tape >Hmidta_src >Hmidta_dst + [ >Htb >change_vec_commute [|//] >change_vec_change_vec + @eq_f3 // cases (reverse sig (reverse sig tlb)@s0::rs0) // ] + -Htb #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 @@ -788,50 +769,31 @@ lemma sem_match_step_termination : 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 #_ * #s0 * normalize in ⊢ (%→?); #H destruct (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 ⊢ (%→?); - * #s0 * #H destruct (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 ⊢ (%→?); - * #s0 * #H destruct (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 - >change_vec_change_vec #Ht1 #_ % #t2 whd in ⊢ (%→?); - >Ht1 >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src #HR - cases (HR ?? (refl ??)) -HR #_ - >nth_change_vec // * #s1 * normalize in ⊢ (%→?); #H destruct (H) - |#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 >change_vec_change_vec - >nth_change_vec // >Hmid_dst whd in match (tape_move_mono ???); #Ht1 - * #s0 * whd in ⊢ (??%?→?); #H destruct (H) #_ >Ht1 - lapply (IH t1 ? (s0::ls) r0 ?) - [ >Ht1 >nth_change_vec // - | >Ht1 >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src - | >Ht1 >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