From 30f12b94fb7f9f201fb092a1b25a1c7e2f9b4564 Mon Sep 17 00:00:00 2001 From: Wilmer Ricciotti Date: Fri, 11 Jan 2013 23:28:57 +0000 Subject: [PATCH] match almost finished --- .../lib/turing/multi_universal/compare.ma | 103 ++- .../lib/turing/multi_universal/match.ma | 659 ++++++------------ 2 files changed, 266 insertions(+), 496 deletions(-) diff --git a/matita/matita/lib/turing/multi_universal/compare.ma b/matita/matita/lib/turing/multi_universal/compare.ma index 688c88638..423a5af33 100644 --- a/matita/matita/lib/turing/multi_universal/compare.ma +++ b/matita/matita/lib/turing/multi_universal/compare.ma @@ -197,18 +197,21 @@ definition R_compare ≝ ((current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨ current ? (nth i ? int (niltape ?)) = None ? ∨ current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧ - (∀ls,x,xs,ci,rs,ls0,rs0. - nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → - nth j ? int (niltape ?) = midtape sig ls0 x (xs@rs0) → - (rs0 = [ ] ∧ + (∀ls,x,rs,ls0,rs0. +(* nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → *) + nth i ? int (niltape ?) = midtape sig ls x rs → + nth j ? int (niltape ?) = midtape sig ls0 x rs0 → + (∃rs'.rs = rs0@rs' ∧ current ? (nth j ? outt (niltape ?)) = None ?) ∨ + (∃rs0'.rs0 = rs@rs0' ∧ outt = change_vec ?? - (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i) - (mk_tape sig (reverse ? xs@x::ls0) (None ?) []) j) ∨ - ∃cj,rs1.rs0 = cj::rs1 ∧ - (ci ≠ cj → - outt = change_vec ?? - (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i) - (midtape sig (reverse ? xs@x::ls0) cj rs1) j)). + (change_vec ?? int + (mk_tape sig (reverse sig rs@x::ls) (None sig) []) i) + (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs0') + (tail sig rs0')) j) ∨ + (∃xs,ci,cj,rs',rs0'.ci ≠ cj ∧ rs = xs@ci::rs' ∧ rs0 = xs@cj::rs0' ∧ + outt = change_vec ?? + (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs') i) + (midtape sig (reverse ? xs@x::ls0) cj rs0') j)). lemma wsem_compare : ∀i,j,sig,n.i ≠ j → i < S n → j < S n → compare i j sig n ⊫ R_compare i j sig n. @@ -218,61 +221,53 @@ lapply (sem_while … (sem_comp_step i j sig n Hneq Hi Hj) … Hloop) // [ whd in ⊢ (%→?); * * [ * [ #Hcicj #Houtc % [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj + | #ls #x #rs #ls0 #rs0 #Hnthi #Hnthj >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H % ] | #Hci #Houtc % [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi >Hnthi in Hci; + | #ls #x #rs #ls0 #rs0 #Hnthi >Hnthi in Hci; normalize in ⊢ (%→?); #H destruct (H) ] ] | #Hcj #Houtc % [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #rs0 #_ #Hnthj >Hnthj in Hcj; + | #ls #x #rs #ls0 #rs0 #_ #Hnthj >Hnthj in Hcj; normalize in ⊢ (%→?); #H destruct (H) ] ] | #td #te * #x * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH * #IH1 #IH2 % [ >Hci >Hcj * [ * [ * #H @False_ind @H % | #H destruct (H)] | #H destruct (H)] - | #ls #c0 #xs #ci #rs #ls0 #rs0 cases xs - [ #Hnthi #Hnthj cases rs0 in Hnthj; - [ #Hnthj % % // >IH1 - [ >Hd @eq_f3 // - [ @eq_f3 // >Hnthi % - | >Hnthj % ] - | >Hd %2 >nth_change_vec // >Hnthj % ] - | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // #Hcir1 >IH1 - [ >Hd @eq_f3 // - [ @eq_f3 // >Hnthi % - | >Hnthj % ] - | >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // >Hnthi >Hnthj normalize % % - @(not_to_not … Hcir1) #H destruct (H) % - ] - ] - | #x0 #xs0 #Hnthi #Hnthj - cut (c0 = x) [ >Hnthi in Hci; normalize #H destruct (H) // ] - #Hcut destruct (Hcut) cases rs0 in Hnthj; - [ #Hnthj % % // - cases (IH2 (x::ls) x0 xs0 ci rs (x::ls0) [ ] ??) -IH2 - [ * #_ #IH2 >IH2 >Hd >change_vec_commute in ⊢ (??%?); // - >change_vec_change_vec >change_vec_commute in ⊢ (??%?); // - @sym_not_eq // - | * #cj * #rs1 * #H destruct (H) - | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // - >Hnthi % - | >Hd >nth_change_vec // >Hnthj % ] - | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // #Hcir1 - cases(IH2 (x::ls) x0 xs0 ci rs (x::ls0) (r1::rs1) ??) - [ * #H destruct (H) - | * #r1' * #rs1' * #H destruct (H) #Hc1r1 >Hc1r1 // - >Hd >change_vec_commute in ⊢ (??%?); // - >change_vec_change_vec >change_vec_commute in ⊢ (??%?); // - @sym_not_eq // - | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // - >Hnthi // - | >Hd >nth_change_vec // >Hnthi >Hnthj % -]]]]] -qed. + | #ls #c0 #rs #ls0 #rs0 cases rs + [ -IH2 #Hnthi #Hnthj % %2 %{rs0} % [%] + >Hnthi in Hd; #Hd >Hd in IH1; #IH1 >IH1 + [| % %2 >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // % ] + >Hnthj cases rs0 [| #r1 #rs1 ] % + | #r1 #rs1 #Hnthi cases rs0 + [ -IH2 #Hnthj % % %{(r1::rs1)} % [%] + >Hnthj in Hd; #Hd >Hd in IH1; #IH1 >IH1 + [| %2 >nth_change_vec // ] + >nth_change_vec // + | #r2 #rs2 #Hnthj lapply IH2; >Hd in IH1; >Hnthi >Hnthj + >nth_change_vec // + >nth_change_vec_neq [| @sym_not_eq // ] >nth_change_vec // + cases (true_or_false (r1 == r2)) #Hr1r2 + [ >(\P Hr1r2) #_ #IH2 cases (IH2 … (refl ??) (refl ??)) [ * + [ * #rs' * #Hrs1 #Hcurout_j % % %{rs'} + >Hrs1 >Hcurout_j normalize % // + | * #rs0' * #Hrs2 #Hcurout_i % %2 %{rs0'} + >Hrs2 >Hcurout_i % // + >change_vec_commute // >change_vec_change_vec + >change_vec_commute [|@sym_not_eq//] >change_vec_change_vec + >reverse_cons >associative_append >associative_append % ] + | * #xs * #ci * #cj * #rs' * #rs0' * * * #Hcicj #Hrs1 #Hrs2 + >change_vec_commute // >change_vec_change_vec + >change_vec_commute [| @sym_not_eq ] // >change_vec_change_vec + #Houtc %2 %{(r2::xs)} %{ci} %{cj} %{rs'} %{rs0'} + % [ % [ % [ // | >Hrs1 // ] | >Hrs2 // ] + | >reverse_cons >associative_append >associative_append >Houtc % ] ] + | lapply (\Pf Hr1r2) -Hr1r2 #Hr1r2 #IH1 #_ %2 + >IH1 [| % % normalize @(not_to_not … Hr1r2) #H destruct (H) % ] + %{[]} %{r1} %{r2} %{rs1} %{rs2} % [ % [ % /2/ | % ] | % ] ]]]]] +qed. lemma terminate_compare : ∀i,j,sig,n,t. i ≠ j → i < S n → j < S n → diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma index ed4142fa3..38dba8d72 100644 --- a/matita/matita/lib/turing/multi_universal/match.ma +++ b/matita/matita/lib/turing/multi_universal/match.ma @@ -119,8 +119,11 @@ definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). 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). - + (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧ + (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs → + ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 → + outt = int). + theorem accRealize_to_Realize : ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc. M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse. @@ -129,34 +132,43 @@ 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. - +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 …)) // +[| @(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 -#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 …))) - +#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 % ] + >change_vec_commute [|@sym_not_eq //] + >change_vec_change_vec + >nth_change_vec_neq [|@sym_not_eq //] + >nth_change_vec // >reverse_append >reverse_single + >reverse_append >reverse_single normalize in match (tape_move ???); + >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 #rs #Hmidta_src #ls0 #y #rs0 #Hmidta_dst + lapply (Htc … Hmidta_src … (refl ??) Hmidta_dst) -Htc #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 ⊢ (???%→%); + [ #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 #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. definition match_step ≝ λsrc,dst,sig,n. compare src dst sig n · @@ -165,7 +177,12 @@ definition match_step ≝ λsrc,dst,sig,n. (rewind src dst sig n · (inject_TM ? (move_r ?) n dst))) (nop …) partest1). - + +(* we assume the src is a midtape + we stop + if the dst is out of bounds (outt = int) + or dst.right is shorter than src.right (outt.current → None) + or src.right is a prefix of dst.right (out = just right of the common prefix) *) definition R_match_step_false ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). ∀ls,x,xs. @@ -182,39 +199,30 @@ definition R_match_step_false ≝ (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.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_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@rs0) → - rs0 ≠ [] ∧ - ∀cj,rs1.rs0 = cj::rs1 → - ci ≠ cj → - (outt = change_vec ?? int - (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst)). -*) +(* + we assume the src is a midtape [ ] s rs + if we iterate + then dst.current = Some ? s1 + and if s ≠ s1 then outt = int.dst.move_right() + and if s = s1 + then int.src.right and int.dst.right have a common prefix + and the heads of their suffixes are different + and outt = int.dst.move_right(). + + *) 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))). - + ∀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 → + ∃xs,ci,rs',ls0,cj,rs0. + rs = xs@ci::rs' ∧ + nth dst ? int (niltape ?) = midtape sig ls0 s (xs@cj::rs0) ∧ + 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 ⊨ @@ -225,242 +233,115 @@ lemma sem_match_step : @(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 ???? Hneq Hsrc Hdst) + (sem_rewind ???? Hneq Hsrc Hdst) (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? ))) (sem_nop …))) -[#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 // - ] - - ] - #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 // +[ #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 src ? (current_chars ?? ta) (None ?) = None ?) + [ normalize in ⊢ (%→?); #H destruct (H) + | @daemon ] + | #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 src ? (current_chars ?? ta) (None ?) = Some ? s) [|@daemon] + >(?:nth dst ? (current_chars ?? ta) (None ?) = None ?) [|@daemon] + 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 dst ? (current_chars ?? tc) (None ?) = None ?) [|@daemon] + cases (nth src ? (current_chars ?? tc) (None ?)) + [| #x ] normalize in ⊢ (%→?); #H destruct (H) + | * #rs0' * #_ #Hcurtc_src * #td * whd in ⊢ (%→?); * whd in ⊢ (??%?→?); + >(?:nth src ? (current_chars ?? tc) (None ?) = None ?) [|@daemon] + 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 // #Hte #_ #Htb + #s' #rs' >Hmidta_src #H destruct (H) + 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 % ] + | #_ >Hmidta_dst >Hrs0 + %{xs} %{ci} %{rs''} %{ls0} %{cj} %{rs0'} % // % // + ] ] - | >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) % + | 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 + #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)] % ] + | >Hs0 #H destruct (H) @False_ind cases (Hss0) /2/ ] + ] ] - ] ] -|#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb - whd in ⊢ (%→?); #Hout >Hout >Htb whd - #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend - lapply (current_to_midtape sig (nth dst ? intape (niltape ?))) - cases (current … (nth dst ? intape (niltape ?))) in Hcomp1; - [#Hcomp1 #_ %1 % % [% | @Hcomp1 %2 %2 % ] - |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq - [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst - #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) - [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 // - %{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 - %2 >Hta in Hc; whd in ⊢ (??%?→?); - change with (current ? (niltape ?)) in match (None ?); - nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // - whd in ⊢ (??%?→?); #Hc cut (is_endc r1 = true) - [ cases (is_endc r1) in Hc; whd in ⊢ (??%?→?); // - change with (current ? (niltape ?)) in match (None ?); - nth_change_vec // normalize #H destruct (H) ] - #Hendr1 cut (xs = xs1) - [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1 - -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs - [ * normalize in ⊢ (%→?); // - #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1 - lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H) - | #x2 #xs2 #IH * - [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc - >Hnotendc in Hendr1; [| @memb_cons @memb_hd ] - normalize in ⊢ (%→?); #H destruct (H) - | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq) - #Hnotendc #Hnotendcxs1 @eq_f @IH - [ @(cons_injective_r … Heq) - | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0 - [ >(\P Hc0) @memb_hd - | @memb_cons @memb_cons // ] - | #c #Hc @Hnotendcxs1 @memb_cons // ] - ] - ] - | #Hxsxs1 destruct (Hxsxs1) >Hmid_dst %{ls_dst} %{rsj} % // - #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) - lapply (append_l2_injective … Hrs_src) // #Hrs' destruct (Hrs') % - ] - ] - |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst - @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize - @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape >Hintape in Hc; - whd in ⊢(??%?→?); >Hmid_src - change with (current ? (niltape ?)) in match (None ?); - Hmid_src whd in ⊢ (??%?→?); - >(Hnotend c_src) [|@memb_hd] - change with (current ? (niltape ?)) in match (None ?); - Hmid_src whd in ⊢ (??%?→?); >Hdst normalize #H destruct (H) - ] - ] -] +| #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 src ? (current_chars ?? ta) (None ?) = Some ? x) + [| @daemon ] + >(?:nth dst ? (current_chars ?? ta) (None ?) = Some ? x0) [|@daemon] + whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ] -Hcomp1 + cases (Hcomp2 … Hta_src Hta_dst) [ * + [ * #rs' * #Hxs #Hcurtc % %2 %{ls0} %{rs0} %{rs'} % // % // + | * #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; [|@daemon] + >(?:nth dst ? (current_chars ?? tc) (None ?) = Some ? cj) [|@daemon] + normalize #H destruct (H) ] ] ] qed. -definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc. - whileTM … (match_step src dst sig n is_startc is_endc) +definition match_m ≝ λsrc,dst,sig,n. + whileTM … (match_step src dst sig n) (inr ?? (inr ?? (inl … (inr ?? start_nop)))). definition R_match_m ≝ - λsrc,dst,sig,n,is_startc,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 → - (∀c0. memb ? c0 (xs@end::rs) = true → is_startc c0 = false) → + 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) ∧ - (is_startc x = true → - (∀ls0,x0,rs0. - nth dst ? 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) end rs) src) - (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨ - ∀l,l1.x0::rs0 ≠ l@x::xs@l1)). + (∀ls0,x0,rs0. + nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 → + (∃l,l1.x0::rs0 = l@x::rs@l1 ∧ + outt = change_vec ?? + (change_vec ?? int + (mk_tape sig (reverse ? rs@[x]) (None ?) [ ]) src) + (mk_tape sig ((reverse ? (l@x::rs))@ls0) (option_hd ? l1) (tail ? l1)) dst) ∨ + ∀l,l1.x0::rs0 ≠ l@x::rs@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. @@ -476,26 +357,27 @@ lemma not_sub_list_merge_2 : qed. -lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc. +lemma wsem_match_m : ∀src,dst,sig,n. src ≠ dst → src < S n → dst < S n → - match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc. -#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) // + match_m src dst sig n ⊫ R_match_m src dst sig n. +#src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop +lapply (sem_while … (sem_match_step src dst sig n Hneq Hsrc Hdst) … Hloop) // -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart - cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse +[ #Hfalse #x #xs #Hmid_src + cases (Hfalse … Hmid_src) -Hfalse [(* current dest = None *) * [ * #Hcur_dst #Houtc % [#_ >Houtc // - |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst; - normalize in ⊢ (%→?); #H destruct (H) + | #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) + | #ls1 #x1 #rs1 >Htc_dst #H destruct (H) >Hrs0 cases xs0 [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %] - #cj #ls2 #H destruct (H) + (* change false case + #cj #ls2 #H destruct (H) *) @daemon | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs) >length_append whd in ⊢ (??%(??%)→?); >length_append >length_append normalize >commutative_plus whd in ⊢ (???%→?); @@ -506,168 +388,61 @@ lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc ] ] ] - |* #ls0 * #rs0 * #Hmid_dst #HFalse % + |* #ls0 * #rs0 * #Hmid_dst #Houtc % [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H) - | #Hstart #ls1 #x1 #rs1 >Hmid_dst #H destruct (H) - %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil - >reverse_cons >associative_append @(HFalse ?? Hnotnil) + |#ls1 #x1 #rs1 >Hmid_dst #H destruct (H) + %1 %{[ ]} %{rs0} % [%] + >reverse_cons >associative_append >Houtc % ] ] |-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd - #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart + #x #xs #Hmidta_src 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 … )) -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) + [#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) #_ + cases (IH … Hmidta_src) #Houtc #_ @Houtc // + |#ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst; + normalize in ⊢ (%→?); #H destruct (H) ] | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ] - #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 …)) -Htrue #_ #Htrue cases (Htrue Hstart Hnotstart) -Htrue + #ls0 #x0 #rs0 #Hmidta_dst >Hmidta_dst in Hcurta_dst; normalize in ⊢ (%→?); + #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 (comp_list ? (xs@end::rs) rs0 is_endc) - #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1 - cases tl1 in Hxs; - [>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 % ] - ] - ] + #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 // + cut (∃x1,xs1.xs0@cj::rs1 = x1::xs1) + [ cases xs0 [ %{cj} %{rs1} % | #x1 #xs1 %{x1} %{(xs1@cj::rs1)} % ] ] * #x1 * #xs1 + #Hxs1 >Hxs1 #IH cases (IH … (refl ??)) -IH + [ * #l * #l1 * #Hxs1' + >change_vec_commute // >change_vec_change_vec + #Houtc % %{(c::l)} %{l1} % + [ normalize reverse_cons >associative_append >change_vec_commute // @Houtc ] + | #H %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%] + @not_sub_list_merge + [ #l2 >Hxs associative_append #H2 lapply (append_l2_injective ????? (refl ??) H2) + #H3 lapply (cons_injective_l ????? H3) #H3 >H3 in Hcicj; * /2/ + |#t #l2 #l3 % normalize #H1 lapply (cons_injective_r ????? H1) + -H1 #H1 cases (H l2 l3) #H2 @H2 @H1 ] ] - |lapply (\Pf eqx) -eqx #eqx >Hmid_dst #Htrue - cases (Htrue ? (refl ??) eqx) -Htrue #Htb #Hendcx #_ - cases rs0 in Htb; - [ #_ %2 #l #l1 cases l + | (* in match_step_true manca il caso di fallimento immediato + (con i due current diversi) *) + @daemon + (* + #_ lapply (\Pf eqx) -eqx #eqx >Hmidta_dst + 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/ @@ -687,7 +462,7 @@ lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc >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/ - ] + ] *) ] ] ] -- 2.39.2