X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=9cd11f61007dce904310557b583ee95ba990e41c;hb=d6b8021e8c83eb19033cad0aeaeebf95b327e78a;hp=bf5e3d42b4fa33fbce9f37708f5faf6f9dc343d4;hpb=06efa0bfaf379b7d3e2f93a41ce5e6f0f01e76a5;p=helm.git diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma index bf5e3d42b..9cd11f610 100644 --- a/matita/matita/lib/turing/multi_universal/match.ma +++ b/matita/matita/lib/turing/multi_universal/match.ma @@ -12,530 +12,718 @@ (* *) (**************************************************************************) -include "turing/multi_universal/moves.ma". -include "turing/if_multi.ma". -include "turing/inject.ma". -include "turing/basic_machines.ma". +include "turing/auxiliary_multi_machines.ma". -definition compare_states ≝ initN 3. +(* rewind *) +definition rewind ≝ λsrc,dst,sig,n. + parmove src dst sig n L · mmove src sig n R · mmove dst sig n R. -definition comp0 : compare_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)). -definition comp1 : compare_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)). -definition comp2 : compare_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)). - -(* +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). -0) (x,x) → (x,x)(R,R) → 1 - (x,y≠x) → None 2 -1) (_,_) → None 1 -2) (_,_) → None 2 +definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). + (∀x,x0,xs,rs. + nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs → + ∀ls0,y,y0,target,rs0.|xs| = |target| → + nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 → + outt = change_vec ?? + (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src) + (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst) ∧ + (∀x,rs.nth src ? int (niltape ?) = midtape sig [] x rs → + ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 → + outt = int). -*) +lemma sem_rewind_strong : ∀src,dst,sig,n. + src ≠ dst → src < S n → dst < S 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 * * * #Htc1 #Htc2 #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb % [ % [ % +[ #x #x0 #xs #rs #Hmidta_src #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_dst + >(Htc1 ??? 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 #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 (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 ⊢ (???%→%); + [ #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 // + ] ] +| #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. -definition trans_compare_step ≝ - λi,j.λsig:FinSet.λn.λis_endc. - λp:compare_states × (Vector (option sig) (S n)). - let 〈q,a〉 ≝ p in - match pi1 … q with - [ O ⇒ match nth i ? a (None ?) with - [ None ⇒ 〈comp2,null_action ? n〉 - | Some ai ⇒ match nth j ? a (None ?) with - [ None ⇒ 〈comp2,null_action ? n〉 - | Some aj ⇒ if notb (is_endc ai) ∧ ai == aj - then 〈comp1,change_vec ? (S n) - (change_vec ? (S n) (null_action ? n) (Some ? 〈ai,R〉) i) - (Some ? 〈aj,R〉) j〉 - else 〈comp2,null_action ? n〉 ] - ] - | S q ⇒ match q with - [ O ⇒ (* 1 *) 〈comp1,null_action ? n〉 - | S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ]. +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 compare_step ≝ - λi,j,sig,n,is_endc. - mk_mTM sig n compare_states (trans_compare_step i j sig n is_endc) - comp0 (λq.q == comp1 ∨ q == comp2). +(* match step *) -definition R_comp_step_true ≝ - λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n). - ∃x. - is_endc x = false ∧ - current ? (nth i ? int (niltape ?)) = Some ? x ∧ - current ? (nth j ? int (niltape ?)) = Some ? x ∧ - outt = change_vec ?? - (change_vec ?? int - (tape_move ? (nth i ? int (niltape ?)) (Some ? 〈x,R〉)) i) - (tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) j. +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 R_comp_step_false ≝ - λi,j:nat.λsig,n,is_endc.λint,outt: Vector (tape sig) (S n). - ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨ - current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨ - current ? (nth i ? int (niltape ?)) = None ? ∨ - current ? (nth j ? int (niltape ?)) = None ?) ∧ outt = int. +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 · mmove dst ?? R)) + (nop …) + partest1). -lemma comp_q0_q2_null : - ∀i,j,sig,n,is_endc,v.i < S n → j < S n → - (nth i ? (current_chars ?? v) (None ?) = None ? ∨ - nth j ? (current_chars ?? v) (None ?) = None ?) → - step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) - = mk_mconfig ??? comp2 v. -#i #j #sig #n #is_endc #v #Hi #Hj -whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?); -* #Hcurrent -[ @eq_f2 - [ whd in ⊢ (??(???%)?); >Hcurrent % - | whd in ⊢ (??(???????(???%))?); >Hcurrent @tape_move_null_action ] -| @eq_f2 - [ whd in ⊢ (??(???%)?); >Hcurrent cases (nth i ?? (None sig)) // - | whd in ⊢ (??(???????(???%))?); >Hcurrent - cases (nth i ?? (None sig)) [|#x] @tape_move_null_action ] ] -qed. +(* 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. + nth src ? int (niltape ?) = midtape sig ls x xs → + ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨ + (∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧ + xs = rs0@xs0 ∧ + outt = change_vec ?? + (change_vec ?? int (mk_tape sig (reverse ? rs0@x::ls) (option_hd ? xs0) (tail ? xs0)) src) + (mk_tape ? (reverse ? rs0@x::ls0) (None ?) [ ]) dst) ∨ + (∃ls0,rs0. + nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧ + (* ∀rsj,c. + rs0 = c::rsj → *) + outt = change_vec ?? + (change_vec ?? int (mk_tape sig (reverse ? xs@x::ls) (None ?) [ ]) src) + (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst). -lemma comp_q0_q2_neq : - ∀i,j,sig,n,is_endc,v.i < S n → j < S n → - ((∃x.nth i ? (current_chars ?? v) (None ?) = Some ? x ∧ is_endc x = true) ∨ - nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?)) → - step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) - = mk_mconfig ??? comp2 v. -#i #j #sig #n #is_endc #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?))) -cases (nth i ?? (None ?)) in ⊢ (???%→?); -[ #Hnth #_ @comp_q0_q2_null // % // -| #ai #Hai lapply (refl ? (nth j ?(current_chars ?? v)(None ?))) - cases (nth j ?? (None ?)) in ⊢ (???%→?); - [ #Hnth #_ @comp_q0_q2_null // %2 // - | #aj #Haj * - [ * #c * >Hai #Heq #Hendc whd in ⊢ (??%?); - >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 - [ whd in match (trans ????); >Hai >Haj destruct (Heq) - whd in ⊢ (??(???%)?); >Hendc // - | whd in match (trans ????); >Hai >Haj destruct (Heq) - whd in ⊢ (??(???????(???%))?); >Hendc @tape_move_null_action - ] - | #Hneq - whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 - [ whd in match (trans ????); >Hai >Haj - whd in ⊢ (??(???%)?); cut ((¬is_endc ai∧ai==aj)=false) - [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) // |#Hcut >Hcut //] - | whd in match (trans ????); >Hai >Haj - whd in ⊢ (??(???????(???%))?); cut ((¬is_endc ai∧ai==aj)=false) - [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) // - |#Hcut >Hcut @tape_move_null_action +(* + 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,rs.nth src ? int (niltape ?) = midtape ? [ ] s rs → + outt = change_vec ?? int + (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst ∧ + (∃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)). + +lemma sem_match_step : + ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → + match_step src dst sig n ⊨ + [ inr ?? (inr ?? (inl … (inr ?? start_nop))) : + R_match_step_true src dst sig n, + 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 ???? Hneq Hsrc Hdst) + (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 %] + 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 // #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; 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'} % // % // ] ] + | 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 whd in ⊢ (%→?); #Htb + #s1 #rs1 >Hmidta_src #H destruct (H) + lapply (Hte … Hmidta_src … Hmidta_dst) -Hte #Hte destruct (Hte) % + [ >Htb % + | >Hs0 %{s0} % // #H destruct (H) @False_ind cases (Hss0) /2/ ] ] ] -] + ] +| #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 comp_q0_q1 : - ∀i,j,sig,n,is_endc,v,a.i ≠ j → i < S n → j < S n → - nth i ? (current_chars ?? v) (None ?) = Some ? a → is_endc a = false → - nth j ? (current_chars ?? v) (None ?) = Some ? a → - step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) = - mk_mconfig ??? comp1 - (change_vec ? (S n) - (change_vec ?? v - (tape_move ? (nth i ? v (niltape ?)) (Some ? 〈a,R〉)) i) - (tape_move ? (nth j ? v (niltape ?)) (Some ? 〈a,R〉)) j). -#i #j #sig #n #is_endc #v #a #Heq #Hi #Hj #Ha1 #Hnotendc #Ha2 -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 -[ whd in match (trans ????); - >Ha1 >Ha2 whd in ⊢ (??(???%)?); >Hnotendc >(\b ?) // -| whd in match (trans ????); - >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >Hnotendc >(\b ?) // - change with (change_vec ?????) in ⊢ (??(???????%)?); - <(change_vec_same … v j (niltape ?)) in ⊢ (??%?); - <(change_vec_same … v i (niltape ?)) in ⊢ (??%?); - >pmap_change >pmap_change >tape_move_null_action - @eq_f2 // @eq_f2 // >nth_change_vec_neq // -] +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.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 ? → + 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 ∧ + 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. +#T #a #b #H1 #H2 #l elim l normalize // qed. -lemma sem_comp_step : - ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n → - compare_step i j sig n is_endc ⊨ - [ comp1: R_comp_step_true i j sig n is_endc, - R_comp_step_false i j sig n is_endc ]. -#i #j #sig #n #is_endc #Hneq #Hi #Hj #int -lapply (refl ? (current ? (nth i ? int (niltape ?)))) -cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?); -[ #Hcuri %{2} % - [| % [ % - [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ % comp_q0_q2_null /2/ %2 Ha >Hcurj % % %2 % #H destruct (H) ] ] - | #b #Hb %{2} - cases (true_or_false (is_endc a)) #Haendc - [ % - [| % [ % - [whd in ⊢ (??%?); >comp_q0_q2_neq // - % %{a} % // Ha %{a} % // ] - ] - |cases (true_or_false (a == b)) #Hab - [ % - [| % [ % - [whd in ⊢ (??%?); >(comp_q0_q1 … a Hneq Hi Hj) // - [>(\P Hab) (\P Hab) %{b} % // % // <(\P Hab) % // ] - | * #H @False_ind @H % - ] ] - | % - [| % [ % - [whd in ⊢ (??%?); >comp_q0_q2_neq // - <(nth_vec_map ?? (current …) i ? int (niltape ?)) - <(nth_vec_map ?? (current …) j ? int (niltape ?)) %2 >Ha >Hb - @(not_to_not ??? (\Pf Hab)) #H destruct (H) % - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // % % %2 >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ] - ] - ] - ] -] +lemma not_sub_list_merge_2 : + ∀T:DeqSet.∀a,b:list T.∀t. (∀l1.t::a ≠ b@l1) → (∀l,l1.a ≠ l@b@l1) → ∀l,l1.t::a ≠ l@b@l1. +#T #a #b #t #H1 #H2 #l elim l // +#t0 #l1 #IH #l2 cases (true_or_false (t == t0)) #Htt0 +[ >(\P Htt0) % normalize #H destruct (H) cases (H2 l1 l2) /2/ +| normalize % #H destruct (H) cases (\Pf Htt0) /2/ ] qed. -definition compare ≝ λi,j,sig,n,is_endc. - whileTM … (compare_step i j sig n is_endc) comp1. -definition R_compare ≝ - λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n). - ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨ - (current ? (nth i ? int (niltape ?)) ≠ 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,cj,rs0. - nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → - nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → - (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → - (is_endc ci = true ∨ ci ≠ cj) → - 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,is_endc.i ≠ j → i < S n → j < S n → - compare i j sig n is_endc ⊫ R_compare i j sig n is_endc. -#i #j #sig #n #is_endc #Hneq #Hi #Hj #ta #k #outc #Hloop -lapply (sem_while … (sem_comp_step i j sig n is_endc Hneq Hi Hj) … Hloop) // +lemma wsem_match_m : ∀src,dst,sig,n. +src ≠ dst → src < S n → dst < S n → + 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 -[ #tc whd in ⊢ (%→?); * * [ * [ * - [* #curi * #Hcuri #Hendi #Houtc % - [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj #Hnotendc - @False_ind - >Hnthi in Hcuri; normalize in ⊢ (%→?); #H destruct (H) - >(Hnotendc ? (memb_hd … )) in Hendi; #H destruct (H) +[ #Hfalse #x #xs #Hmid_src + cases (Hfalse … Hmid_src) -Hfalse + [(* current dest = None *) * + [ * #Hcur_dst #Houtc % + [#_ >Houtc // + | #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) + | #ls1 #x1 #rs1 >Htc_dst #H destruct (H) + >Hrs0 >HNone cases xs0 + [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %] + @eq_f3 // + [ >reverse_append % + | >reverse_append >reverse_cons >reverse_append + >associative_append >associative_append % ] + | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs) + >length_append whd in ⊢ (??%(??%)→?); >length_append + >length_append normalize >commutative_plus whd in ⊢ (???%→?); + #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?); + >associative_plus >associative_plus + #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?); + #e2 destruct (e2) + ] + ] + ] + |* #ls0 * #rs0 * #Hmid_dst #Houtc % + [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H) + |#ls1 #x1 #rs1 >Hmid_dst #H destruct (H) + %1 %{[ ]} %{rs0} % [%] + >reverse_cons >associative_append >Houtc % ] - |#Hcicj #Houtc % - [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj - >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H % - ]] - | #Hci #Houtc % - [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci; - normalize in ⊢ (%→?); #H destruct (H) ] ] - | #Hcj #Houtc % - [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj; - normalize in ⊢ (%→?); #H destruct (H) ] ] - | #tc #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH * - #IH1 #IH2 % - [ >Hci >Hcj * [* #x0 * #H destruct (H) >Hendcx #H destruct (H) - |* [* #H @False_ind [cases H -H #H @H % | destruct (H)] | #H destruct (H)]] - | #ls #c0 #xs #ci #rs #ls0 #cj #rs0 cases xs - [ #Hnthi #Hnthj #Hnotendc #Hcicj >IH1 - [ >Hd @eq_f3 // - [ @eq_f3 // >(?:c0=x) [ >Hnthi % ] - >Hnthi in Hci;normalize #H destruct (H) % - | >(?:c0=x) [ >Hnthj % ] - >Hnthi in Hci;normalize #H destruct (H) % ] - | >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // >Hnthi >Hnthj normalize - cases Hcicj #Hcase - [%1 %{ci} % // | %2 %1 %1 @(not_to_not ??? Hcase) #H destruct (H) % ] + ] +|-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd + #x #xs #Hmidta_src + lapply (refl ? (current ? (nth dst ? ta (niltape ?)))) + cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?); + [#Hcurta_dst % + [#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) + ] + | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ] + #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) * #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 // + 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 % %{(s0::l)} %{l1} % + [ normalize reverse_cons >associative_append >change_vec_commute // @Houtc ] + | #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) + #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 ] - | #x0 #xs0 #Hnthi #Hnthj #Hnotendc #Hcicj - >(IH2 (c0::ls) x0 xs0 ci rs (c0::ls0) cj rs0 … Hcicj) - [ >Hd >change_vec_commute in ⊢ (??%?); // - >change_vec_change_vec >change_vec_commute in ⊢ (??%?); // - @sym_not_eq // - | #c1 #Hc1 @Hnotendc @memb_cons @Hc1 - | >Hd >nth_change_vec // >Hnthj normalize - >Hnthi in Hci;normalize #H destruct (H) % - | >Hd >nth_change_vec_neq [|@sym_not_eq //] >Hnthi - >nth_change_vec // normalize - >Hnthi in Hci;normalize #H destruct (H) % + ] + | #_ cases (IH x xs ?) -IH + [| >Htc >nth_change_vec_neq [|@sym_not_eq //] @Hmidta_src ] + >Htc >nth_change_vec // cases rs0 + [ #_ #_ %2 #l #l1 cases l + [ normalize cases xs + [ cases l1 + [ normalize % #H destruct (H) cases (\Pf eqx) /2/ + | #tmp1 #l2 normalize % #H destruct (H) ] + | #tmp1 #l2 normalize % #H destruct (H) ] + | #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0; + [ normalize #H1 destruct (H1) + | #tmp2 #l3 normalize #H1 destruct (H1) ] ] + | #r1 #rs1 #_ #IH cases (IH … (refl ??)) -IH + [ * #l * #l1 * #Hll1 #Houtc % %{(c::l)} %{l1} % [ >Hll1 % ] + >Houtc >change_vec_commute // >change_vec_change_vec + >change_vec_commute [|@sym_not_eq //] + >reverse_cons >associative_append % + | #Hll1 %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@Hll1] + #l1 % #H destruct (H) cases (\Pf eqx) /2/ ] -]]] -qed. - -lemma terminate_compare : ∀i,j,sig,n,is_endc,t. - i ≠ j → i < S n → j < S n → - compare i j sig n is_endc ↓ t. -#i #j #sig #n #is_endc #t #Hneq #Hi #Hj -@(terminate_while … (sem_comp_step …)) // -<(change_vec_same … t i (niltape ?)) -cases (nth i (tape sig) t (niltape ?)) -[ % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct -|2,3: #a0 #al0 % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct -| #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs - [#t #ls #c % #t1 * #x * * * #Hendcx >nth_change_vec // normalize in ⊢ (%→?); - #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 % - #t2 * #x0 * * * #Hendcx0 >Ht1 >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) - |#r0 #rs0 #IH #t #ls #c % #t1 * #x * * >nth_change_vec // - normalize in ⊢ (%→?); #H destruct (H) #Hcur - >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH + ] + ] ] ] qed. -lemma sem_compare : ∀i,j,sig,n,is_endc. - i ≠ j → i < S n → j < S n → - compare i j sig n is_endc ⊨ R_compare i j sig n is_endc. -#i #j #sig #n #is_endc #Hneq #Hi #Hj @WRealize_to_Realize /2/ -qed. +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 ?)))|. -(* - |conf1 $ - |confin 0/1 confout move - - match machine step ≝ - compare; - if (cur(src) != $) - then - parmoveL; - moveR(dst); - else nop - *) - -definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc. - compare src dst sig n · - (ifTM ?? (inject_TM ? (test_char ? (λa.is_endc a == false)) n src) - (single_finalTM ?? - (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst))) - (nop …) - tc_true). - -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. - -definition R_match_step_false ≝ - λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n). - (((∃x.current ? (nth src ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨ - (* current ? (nth src ? int (niltape ?)) ≠ current ? (nth dst ? int (niltape ?)) ∨ *) - current sig (nth src (tape sig) int (niltape sig)) = None ? ∨ - current sig (nth dst (tape sig) int (niltape sig)) = None ? ) ∧ outt = int) ∨ - ∃ls,ls0,rs,rs0,x,xs. ∀rsi,rsj,end,c. - rs = end::rsi → rs0 = c::rsj → - is_endc x = false ∧ is_endc end = true ∧ - nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧ - nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧ - outt = change_vec ?? - (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rsi) src) - (midtape sig (reverse ? xs@x::ls0) c rsj) dst. - -definition R_match_step_true ≝ - λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n). - ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s → - is_startc s = true → - (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) → - (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → - s ≠ s1 → - outt = change_vec ?? int - (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧ - (∀ls,x,xs,ci,rs,ls0,cj,rs0. - nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → - nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → ci ≠ cj → - outt = change_vec ?? int - (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false). - -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 // ] ] ] -qed. - -axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S. ∃l,tl1,tl2. - l1 = l@tl1 ∧ l2 = l@tl2 ∧ ∀a,b,tla,tlb. tl1 = a::tla → tl2 = b::tlb → a≠b. - -axiom daemon : ∀X:Prop.X. - -lemma sem_match_step : - ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n → - match_step src dst sig n is_startc is_endc ⊨ +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 src dst sig n is_startc is_endc, - R_match_step_false src dst sig n is_endc ]. -#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst + 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_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc)) + (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?)) (sem_seq … - (sem_parmoveL ???? is_startc Hneq Hsrc Hdst) - (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? ))) - (sem_nop …))) -[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * * #c * #Hcurtc #Hcend #Htd >Htd -Htd - #Htb #s #Hcurta_src #Hstart #Hnotstart % - [ #s1 #Hcurta_dst #Hneqss1 - lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta) - [|@Hcomp1 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ] - #Hcurtc * #te * * #_ #Hte >Hte // whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse % - [ @(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 >(Htbdst … Hta_mid) >Hta_mid cases rs // - | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Htbelse @sym_not_eq // ] - | >Hcurtc in Hcurta_src; #H destruct (H) cases (is_endc s) in Hcend; - normalize #H destruct (H) // ] - |#ls #x #xs #ci #rs #ls0 #cj #rs0 #Htasrc_mid #Htadst_mid #Hcicj - lapply (Hcomp2 … Htasrc_mid Htadst_mid Hcicj) -Hcomp2 #Hcomp2 - cases Htb #td * * #Htd #_ >Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?); - #H destruct (H) - >(Htd ls ci (reverse ? xs) rs s ??? ls0 cj (reverse ? xs) s rs0 (refl ??)) // - [| >Hcomp2 >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 // - ] - | >Hcomp2 >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ] - * * #_ #Htbdst #Htbelse % - [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst - [ >Hidst >nth_change_vec // >Htadst_mid >(Htbdst ls0 s (xs@cj::rs0)) - [ cases xs // - | >nth_change_vec // ] - | >nth_change_vec_neq [|@sym_not_eq //] - nth_change_vec_neq [|@sym_not_eq //] - (* STOP. *) - cases (decidable_eq_nat i src) #Hisrc - [ >Hisrc >nth_change_vec // >Htasrc_mid // - | >nth_change_vec_neq [|@sym_not_eq //] - <(Htbelse i) [|@sym_not_eq // ] - >Hcomp2 >nth_change_vec_neq [|@sym_not_eq // ] - >nth_change_vec_neq [|@sym_not_eq // ] // + (sem_rewind_strong ???? Hneq Hsrc Hdst) + (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 %] + 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 [|@Hdst] 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 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 ⊢ (?(?%)%); // + | #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 #_ 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) + [ >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 + >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 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) + [ >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 // + | #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 #_ 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) + [ >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 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) + whd in ⊢ (%→?); >Hmidta_dst #Htb + cut (tb = change_vec ?? ta (mk_tape ? (s0::lsa@lsb) (option_hd ? rs0) (tail ? rs0)) dst) + [ >Htb cases rs0 // ] + -Htb #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 #_ #_ #_ + 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) + [ >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 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) + 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) + [ >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 + [| 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 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) + [ >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 + <(length_reverse ? lsa) >Hlen' >H2 normalize // + ] ] ] - | >Hcomp2 in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // whd in ⊢ (??%?→?); - #H destruct (H) cases (is_endc c) in Hcend; - normalize #H destruct (H) // ] - ] -|#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb - whd in ⊢ (%→?); #Hout >Hout >Htb whd - lapply (current_to_midtape sig (nth src ? intape (niltape ?))) - cases (current … (nth src ? intape (niltape ?))) in Hcomp1; - [#Hcomp1 #_ %1 % [%1 %2 // | @Hcomp1 %1 %2 %] - |#c_src lapply (current_to_midtape sig (nth dst ? intape (niltape ?))) - cases (current … (nth dst ? intape (niltape ?))) - [#_ #Hcomp1 #_ %1 % [%2 % | @Hcomp1 %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 #_ - #Hmid_src cases (Hmid_src c_src (refl …)) -Hmid_src - #ls_src * #rs_src #Hmid_src %2 - cases (comp_list … rs_src rs_dst) #xs * #rsi * #rsj * * - #Hrs_src #Hrs_dst #Hneq - %{ls_src} %{ls_dst} %{rsi} %{rsj} %{c_src} %{xs} - #rsi0 #rsj0 #end #c #Hend #Hc_dst - >Hrs_src in Hmid_src; >Hend #Hmid_src - >Hrs_dst in Hmid_dst; >Hc_dst <(\P Hceq) #Hmid_dst - lapply(Hcomp2 … Hmid_src Hmid_dst ?) - [@(Hneq … Hend Hc_dst)] - -Hcomp2 #Hcomp2 Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // #H lapply (H ? (refl …)) - cases (is_endc end) normalize // - |@Hmid_src] - |@Hmid_dst] - |#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc - %1 % - [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1 - [| % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ] - cases (is_endc c_src) // - >Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H) - |@Hcomp1 %1 %1 @(not_to_not ??? (\Pf Hceq)) #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_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. -definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc. - whileTM … (match_step src dst sig n is_startc is_endc) - (inr ?? (inr ?? (inl … (inr ?? start_nop)))). - -definition R_match_m ≝ - λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n). - (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨ - current ? (nth i ? int (niltape ?)) = None ? ∨ - current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧ - (∀ls,x,xs,ci,rs,ls0,x0,rs0. - is_startc x = true → is_endc ci = true → - nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → - nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 → - ∃l,cj,l1.x0::rs0 = l@x::xs@cj::l1 ∧ - outt = change_vec ?? - (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i) - (midtape sig ((reverse ? (l@x::xs))@ls0) cj l1) j). - -lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc. -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) // --Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ #tc whd in ⊢ (%→?); * - [ * * [ * - [ * #cur_src * #H1 #H2 #Houtc % - [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #Hnthi #Hnthj - >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H % - ] - | #Hci #Houtc % - [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci; - normalize in ⊢ (%→?); #H destruct (H) ] ] - | #Hcj #Houtc % - [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj; - normalize in ⊢ (%→?); #H destruct (H) ] ] - +(* 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