X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=8f525c9f4f7188df65acfeb91114d4d1b072b18c;hb=afd1e4522f61a72711ed822267c5ca86a3eb6d63;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..8f525c9f4 100644 --- a/matita/matita/lib/turing/multi_universal/match.ma +++ b/matita/matita/lib/turing/multi_universal/match.ma @@ -215,10 +215,15 @@ 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) → + (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) @@ -322,7 +327,7 @@ qed. *) definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc. - compare src dst sig n · + 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))) @@ -336,34 +341,50 @@ 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). + ∀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,end,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_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) ∧ + (∃ls,ls0,rs,rs0,x,xs. + nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧ is_endc x = false ∧ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧ + ∀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 dst ? int (niltape ?) = midtape sig ls0 x (xs@c::rsj) ∧ outt = change_vec ?? (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rsi) src) - (midtape sig (reverse ? xs@x::ls0) c rsj) dst. + (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 → + (∀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). @@ -390,8 +411,9 @@ cases (acc_sem_inject … Hin (sem_test_char alpha test) int) | @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 comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2. + l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧ + ∀a,tla. tl1 = a::tla → is_endc a = true ∨ (∀b,tlb.tl2 = b::tlb → a≠b). axiom daemon : ∀X:Prop.X. @@ -402,7 +424,7 @@ lemma sem_match_step : 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 -@(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst) +@(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)) (sem_seq … (sem_parmoveL ???? is_startc Hneq Hsrc Hdst) @@ -412,7 +434,7 @@ lemma sem_match_step : #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) % ] + [|@Hcomp1 %2 % % >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) @@ -420,8 +442,9 @@ lemma sem_match_step : | >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 + |#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 ??)) // @@ -455,40 +478,95 @@ lemma sem_match_step : normalize #H destruct (H) // ] ] |#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 #Hnotendc #Hneq + %{ls_dst} %{rsj} % + [(\P Hceq) // ]] + #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_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. + +#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 %] + [#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 %] + [#_ #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 #_ + #ls_dst * #rs_dst #Hmid_dst #Hcomp1 #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] + #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} % + [% [% // (\P Hceq) // ]] + #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_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) // ] + [| %2 % % @(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) // + |@Hcomp1 %2 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ] ] ] @@ -504,14 +582,23 @@ definition R_match_m ≝ (((∃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 → @@ -519,23 +606,48 @@ 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 ⊢ (%→?); * +[ #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 % + | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #Hnotend #Hnthi + @False_ind + >Hnthi in H1; whd in ⊢ (??%?→?); #H destruct (H) cases (Hdiff cur_src) + #Habs @Habs // ] - | #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 ⊢ (%→?); * * [ * + | #Hci #Houtc % + [ #_ @Houtc + | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #Hnotend + #Hnthi >Hnthi in Hci; normalize in ⊢ (%→?); #H destruct (H) ] ] + | #Hcj #Houtc % + [ #_ @Houtc + | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #_ #_ #Hnthj >Hnthj in Hcj; + normalize in ⊢ (%→?); #H destruct (H) ] + ] + |* #ls * #ls0 * #rs * #rs0 * #x0 * #xs * * * #Hsrc #Hx0 #Hdst #H % + [>Hsrc * + [* [* #x * whd in ⊢ (??%?→?); #Habs destruct (Habs) >Hx0 #Habs destruct (Habs) + |whd in ⊢ (??%?→?); #Habs destruct (Habs) ] + |>Hdst whd in ⊢ (??%?→?); #Habs destruct (Habs) ] + |#ls1 #x1 #xs1 #ci #rsi #ls2 #x2 #rs2 + #Hdiff #Hstart #Hend #Hnotend + >Hsrc #Hsrc1 destruct (Hsrc1) >Hdst #Hdst1 destruct (Hdst1) + %1 %{[ ]} %{rs0} normalize in ⊢ (%→?); #Heq #cj #l2 #Hl1 + cut (xs=xs1) + [@(append_l1_injective_r … rs0 rs0 (refl …)) @(cons_injective_r …Heq)] + #eqxs reverse_cons >associative_append + normalize in match (append ? [x2] ls2); + cases (H rsi l2 ci cj ? Hl1) + [* #_ #_ #H3 @H3 + |>eqxs in e0; #e0 @(append_l2_injective … e0) // + ] + ] + ] +|#tc #td #te #Hd #Hstar #IH #He lapply (IH He) -IH * + #IH1 #IH2 % [@IH1] + + + cases (comp_list ? (x1::xs1@ci::rsi) (x2::rs2) is_endc) + #l * #tl1 * #tl2 * * * #H1 #H2 #H3 #H4