X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=f16190491a202810abf3f2e5fb192b2d477a2884;hb=302d83e414e27210fbfbc1de4d213786d9580e23;hp=d986ac4e3779147c532e095a144814b13ccd1d4a;hpb=26aa4427b7075c6587faf3ead1dc44910ec86a5c;p=helm.git diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma index d986ac4e3..f16190491 100644 --- a/matita/matita/lib/turing/multi_universal/match.ma +++ b/matita/matita/lib/turing/multi_universal/match.ma @@ -215,14 +215,19 @@ 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,cj,rs0. + (∀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@cj::rs0) → + nth j ? int (niltape ?) = midtape sig ls0 x (xs@rs0) → (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → - (is_endc ci = true ∨ ci ≠ cj) → + (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 ∧ + ((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). + (midtape sig (reverse ? xs@x::ls0) cj rs1) 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. @@ -232,53 +237,84 @@ lapply (sem_while … (sem_comp_step i j sig n is_endc Hneq Hi Hj) … Hloop) // [ #tc whd in ⊢ (%→?); * * [ * [ * [* #curi * #Hcuri #Hendi #Houtc % [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj #Hnotendc + | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj #Hnotendc @False_ind >Hnthi in Hcuri; normalize in ⊢ (%→?); #H destruct (H) >(Hnotendc ? (memb_hd … )) in Hendi; #H destruct (H) ] |#Hcicj #Houtc % [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj + | #ls #x #xs #ci #rs #ls0 #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; + | #ls #x #xs #ci #rs #ls0 #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; + | #ls #x #xs #ci #rs #ls0 #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 // + | #ls #c0 #xs #ci #rs #ls0 #rs0 cases xs + [ #Hnthi #Hnthj #Hnotendc cases rs0 in Hnthj; + [ #Hnthj % % // >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 %2 %2 >nth_change_vec // >Hnthj % ] + | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // * + [ #Hendci >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 % %{ci} % // + ] + |#Hcir1 >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) % ] - ] - | #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 ⊢ (??%?); // + | >Hd %2 % % >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 #Hnotendc + 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 % + | #c0 #Hc0 @Hnotendc @memb_cons @Hc0 ] + | #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 // - | #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) % - ] -]]] + | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // + >Hnthi // + | >Hd >nth_change_vec // >Hnthi >Hnthj % + | #c0 #Hc0 @Hnotendc @memb_cons @Hc0 +]]]]] qed. lemma terminate_compare : ∀i,j,sig,n,is_endc,t. @@ -321,14 +357,6 @@ qed. else nop *) -definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc. - compare src dst sig n is_endc · - (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. @@ -336,37 +364,7 @@ definition Rtc_multi_true ≝ 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 → - (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = 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 → - (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → - 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 ⊨ @@ -396,6 +394,42 @@ axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2. axiom daemon : ∀X:Prop.X. +(* +definition R_match_step_false ≝ + λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n). + ∀ls,x,xs,end,rs. + nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) → + (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true → + ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨ + (∃ls0,rs0. + nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧ + ∀rsj,c. + rs0 = c::rsj → + outt = change_vec ?? + (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src) + (midtape sig (reverse ? xs@x::ls0) c rsj) dst). + +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) → + (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ + (∀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,rs0. + nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → + nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) → + (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → + (∀cj,rs1.rs0 = cj::rs1 → ci ≠ cj → + (outt = change_vec ?? int + (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false)) ∧ + (rs0 = [ ] → + outt = change_vec ?? + (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src) + (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)). + 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 ⊨ @@ -410,119 +444,404 @@ lemma sem_match_step : (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 + #Htb #s #Hcurta_src #Hstart #Hnotstart % [ % + [#Hdst_none @daemon + | #s1 #Hcurta_dst #Hneqss1 lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta) [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ] - #Hcurtc * #te * * #_ #Hte >Hte // whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse % + #Hcurtc * #te * * #_ #Hte >Hte [2: %1 %1 %{s} % //] + 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 #Hnotendc - lapply (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc (or_intror ?? 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 // + ] + |#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc + cases rs00 in Htadst_mid; + [(* case rs empty *) #Htadst_mid % [ #cj #rs1 #H destruct (H) ] + #_ cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2 + [2: * #x0 * #rs1 * #H destruct (H) ] + * #_ #Htc cases Htb #td * * #_ #Htd >Htasrc_mid in Hcurta_src; + normalize in ⊢ (%→?); #H destruct (H) + >Htd [2: %2 >Htc >nth_change_vec // cases (reverse sig ?) //] + >Htc * * >nth_change_vec // #Htbdst #_ #Htbelse + @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst + [ >Hidst >nth_change_vec // Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?); + #H destruct (H) + >(Htd ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??)) // + [| >Htc >nth_change_vec // + | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid + cases (orb_true_l … Hc0) -Hc0 #Hc0 + [@memb_append_l2 >(\P Hc0) @memb_hd + |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse // + ] + | >Htc >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 //] + 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 // ] + >Htc >nth_change_vec_neq [|@sym_not_eq // ] + >nth_change_vec_neq [|@sym_not_eq // ] // + ] + ] + | >Htc 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) // ] ] - | >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 // ] // + ] +|#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 %2 + cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * * + #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >Hrs_dst in Hmid_dst; #Hmid_dst + cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #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 #Hta % + [ >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // + #Hc lapply (Hc ? (refl ??)) #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 >Hmid_dst >Hxsxs1 % ] + | #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) ] + | * #cj * #rs2 * #Hrs2 #Hta lapply (Hta ?) + [ cases (Hneq … Hrs1) /2/ #H %2 @(H ?? Hrs2) ] + -Hta #Hta >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] + >nth_change_vec // #Hc lapply (Hc ? (refl ??)) #Hendr1 + (* lemmatize this proof *) 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 >Hmid_dst >Hxsxs1 % // + #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 // + @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) // + #Hendr1 destruct (Hendr1) % ] ] ] - | >Hcomp2 in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // whd in ⊢ (??%?→?); - #H destruct (H) cases (is_endc c) in Hcend; + (* STOP *) + |#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; >Hmid_src #Hc lapply (Hc ? (refl …)) -Hc + >(Hnotend c_src) // normalize #H destruct (H) + ] + ] +] +qed. +*) + +definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc. + compare src dst sig n is_endc · + (ifTM ?? (inject_TM ? (test_char ? (λa.is_endc a == false)) n src) + (ifTM ?? (inject_TM ? (test_null ?) n src) + (single_finalTM ?? + (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst))) + (nop …) tc_true) + (nop …) + tc_true). + +definition R_match_step_false ≝ + λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n). + ∀ls,x,xs,end,rs. + nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) → + (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true → + ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨ + (∃ls0,rs0. + nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧ + ∀rsj,c. + rs0 = c::rsj → + outt = change_vec ?? + (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src) + (midtape sig (reverse ? xs@x::ls0) c rsj) dst). + +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) → + current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧ + (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 → + outt = change_vec ?? int + (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧ + (∀ls,x,xs,ci,rs,ls0,rs0. + nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → + nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) → + (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → + (∀cj,rs1.rs0 = cj::rs1 → ci ≠ cj → + (outt = change_vec ?? int + (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false)) ∧ + (rs0 = [ ] → + outt = change_vec ?? + (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src) + (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) dst)). + +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 ⊨ + [ inr ?? (inr ?? (inl … (inr ?? (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 +(* test_null versione multi? *) +@(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc 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_test_null sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc)) + + 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 % [ % + [#Hdst_none @daemon + | #s1 #Hcurta_dst #Hneqss1 + lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta) + [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ] + #Hcurtc * #te * * #_ #Hte >Hte [2: %1 %1 %{s} % //] + 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 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc + cases rs00 in Htadst_mid; + [(* case rs empty *) #Htadst_mid % [ #cj #rs1 #H destruct (H) ] + #_ cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2 + [2: * #x0 * #rs1 * #H destruct (H) ] + * #_ #Htc cases Htb #td * * #_ #Htd >Htasrc_mid in Hcurta_src; + normalize in ⊢ (%→?); #H destruct (H) + >Htd [2: %2 >Htc >nth_change_vec // cases (reverse sig ?) //] + >Htc * * >nth_change_vec // #Htbdst #_ #Htbelse + @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst + [ >Hidst >nth_change_vec // Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?); + #H destruct (H) + >(Htd ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s rs0' (refl ??)) // + [| >Htc >nth_change_vec // + | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid + cases (orb_true_l … Hc0) -Hc0 #Hc0 + [@memb_append_l2 >(\P Hc0) @memb_hd + |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse // + ] + | >Htc >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 //] + 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 // ] + >Htc >nth_change_vec_neq [|@sym_not_eq // ] + >nth_change_vec_neq [|@sym_not_eq // ] // + ] + ] + | >Htc 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 %2 %1 %2 %] - |#c_src lapply (current_to_midtape sig (nth dst ? intape (niltape ?))) - cases (current … (nth dst ? intape (niltape ?))) - [#_ #Hcomp1 #_ %1 % [%2 % | @Hcomp1 %2 % % % #H destruct (H)] - |#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 #Hcomp1 - #Hmid_src cases (Hmid_src c_src (refl …)) -Hmid_src - #ls_src * #rs_src #Hmid_src - cases (true_or_false (is_endc c_src)) #Hc_src - [ % % [ % % %{c_src} % // | @Hcomp1 % %{c_src} % // ] - | %2 cases (comp_list … rs_src rs_dst is_endc) #xs * #rsi * #rsj * * * - #Hrs_src #Hrs_dst #Hnotendc #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 - cut (is_endc end = true ∨ end ≠ c) - [cases (Hneq … Hend) /2/ -Hneq #Hneq %2 @(Hneq … Hc_dst) ] #Hneq - lapply (Hcomp2 … Hmid_src Hmid_dst ? Hneq) - [#c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0 - [ >(\P Hc0) // - | @Hnotendc // ] - ] - -Hcomp2 #Hcomp2 Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // #H lapply (H ? (refl …)) - cases (is_endc end) [|normalize #H destruct (H) ] - #_ % // #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0 - [ >(\P Hc0) // | @Hnotendc // ] - |@Hmid_src] - |@Hmid_dst] ] - |#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc - %1 % - [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1 - [| %2 % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ] - cases (is_endc c_src) // - >Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H) - |@Hcomp1 %2 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // - ] - ] - ] - ] -qed. + #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 %2 + cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * * + #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >Hrs_dst in Hmid_dst; #Hmid_dst + cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #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 #Hta % + [ >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // + #Hc lapply (Hc ? (refl ??)) #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 >Hmid_dst >Hxsxs1 % ] + | #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) ] + | * #cj * #rs2 * #Hrs2 #Hta lapply (Hta ?) + [ cases (Hneq … Hrs1) /2/ #H %2 @(H ?? Hrs2) ] + -Hta #Hta >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] + >nth_change_vec // #Hc lapply (Hc ? (refl ??)) #Hendr1 + (* lemmatize this proof *) 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 >Hmid_dst >Hxsxs1 % // + #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 // + @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) // + #Hendr1 destruct (Hendr1) % ] + ] + ] + (* STOP *) + |#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; >Hmid_src #Hc lapply (Hc ? (refl …)) -Hc + >(Hnotend c_src) // 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 ≝ + λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n). +(* (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ *) + ∀ls,x,xs,end,rs. + nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) → + (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true → + (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)). + +(* 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. + (∀ls,x,xs,ci,rs,ls0,x0,rs0. + (∀x. is_startc x ≠ is_endc x) → is_startc x = true → is_endc ci = true → + (∀z. memb ? z (x::xs) = true → is_endc x = false) → nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 → - ∃l,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). + (∃l,l1.x0::rs0 = l@x::xs@l1 ∧ + ∀cj,l2.l1=cj::l2 → + outt = change_vec ?? + (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i) + (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨ + ∀l,l1.x0::rs0 ≠ l@x::xs@l1). +*) + +(* +axiom sub_list_dec: ∀A.∀l,ls:list A. + ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2. +*) lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc. src ≠ dst → src < S n → dst < S n → @@ -530,23 +849,71 @@ src ≠ dst → src < S n → dst < S n → #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) // -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ #tc 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) ] ] - - - -[ #tc whd in ⊢ (%→?); * * [ * +[ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend + cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse + [(* current dest = None *) * #Hcur_dst #Houtc % + [#_ >Houtc // + |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst; + normalize in ⊢ (%→?); #H destruct (H) + ] + |* #ls0 * #rs0 * #Hmid_dst #HFalse % + [ >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) + ] + ] +|#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd + #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend + lapply (refl ? (current ? (nth dst ? ta (niltape ?)))) + cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?); + [#Hmid_dst % + [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue + cases (Htrue x (refl … ) Hstart ?) -Htrue [2: @daemon] + * #Htb #_ #_ >Htb in IH; // #IH + cases (IH ls x xs end rs Hmid_src Hstart Hnotend Hend) + #Hcur_outc #_ @Hcur_outc // + |#ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?); + #H destruct (H) + ] + | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ] + #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?); + #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue + cases (Htrue x (refl …) Hstart ?) -Htrue + [2: #z #membz @daemon (*aggiungere l'ipotesi*)] + cases (true_or_false (x==c)) #eqx + [ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc) + #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1 + cases tl1 in Hxs; + [>append_nil #Hx1 @daemon (* absurd by Hx1 e notendx1 *)] + #ci -tl1 #tl1 #Hxs #H cases (H … (refl … )) + [(* this is absurd, since Htrue conlcudes is_endc ci =false *) + #Hend_ci @daemon (* lapply(Htrue … (refl …)) -Htrue *) + |#Hcomp lapply (Htrue ls x x1 ci tl1 ls0 tl2 ???) + [ #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0 + [ @Hnotend >(\P Hc0) @memb_hd + | @Hnotendx1 // ] + | >Hmid_dst >Hrs0 >(\P eqx) % + | >Hxs % + | * cases tl2 in Hrs0; + [ >append_nil #Hrs0 #_ #Htb whd in IH; + lapply (IH ls x x1 ci tl1 ? Hstart ??) + [ + | + | >Htb // >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // + + >Hrs0 in Hmid_dst; #Hmid_dst + cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx + whd in IH; + cases(IH ls x xs end rs ? Hstart Hnotend Hend) + [* #H1 #H2 >Htb in H1; >nth_change_vec // + >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)] + #_ %2 @daemon (* si dimostra *) + |@daemon + |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src + ] + ] + ] +] +qed.