From 99865bbdbb8b4694c85085abb0e98b4d3be7ea9f Mon Sep 17 00:00:00 2001 From: Wilmer Ricciotti Date: Mon, 26 Nov 2012 17:55:19 +0000 Subject: [PATCH] test null, match --- matita/matita/lib/turing/basic_machines.ma | 21 + .../lib/turing/multi_universal/match.ma | 439 +++++++++++++----- 2 files changed, 340 insertions(+), 120 deletions(-) diff --git a/matita/matita/lib/turing/basic_machines.ma b/matita/matita/lib/turing/basic_machines.ma index 82d4758c2..a23576829 100644 --- a/matita/matita/lib/turing/basic_machines.ma +++ b/matita/matita/lib/turing/basic_machines.ma @@ -197,6 +197,27 @@ lemma test_char_inv : #sig #P #f #t #t0 #HPt * #_ // qed. +definition test_null ≝ λalpha.test_char alpha (λ_.true). + +definition R_test_null_true ≝ λalpha,t1,t2. + current alpha t1 ≠ None ? ∧ t1 = t2. + +definition R_test_null_false ≝ λalpha,t1,t2. + current alpha t1 = None ? ∧ t1 = t2. + +lemma sem_test_null : ∀alpha. + test_null alpha ⊨ [ tc_true : R_test_null_true alpha, R_test_null_false alpha]. +#alpha #t1 cases (sem_test_char alpha (λ_.true) t1) #k * #outc * * #Hloop #Htrue +#Hfalse %{k} %{outc} % [ % +[ @Hloop +| #Houtc cases (Htrue ?) [| @Houtc] * #c * #Hcurt1 #_ #Houtc1 % + [ >Hcurt1 % #H destruct (H) | 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.∀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. + +(* definition R_match_step_false ≝ λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n). ∀ls,x,xs,end,rs. @@ -386,23 +408,6 @@ definition R_match_step_false ≝ 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 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. - 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). -*) definition R_match_step_true ≝ λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n). @@ -417,43 +422,14 @@ definition R_match_step_true ≝ 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 → + (∀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)) ∨ + (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_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.∀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. - 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 ⊨ @@ -484,8 +460,8 @@ lemma sem_match_step : ] |#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc cases rs00 in Htadst_mid; - [(* case rs empty *) #Htadst_mid %2 #_ - cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2 + [(* 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) @@ -495,15 +471,16 @@ lemma sem_match_step : [ >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 + |#cj0 #rs0 #Htadst_mid % [| #H destruct (H) ] + #cj #rs1 #H destruct (H) #Hcicj + cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ] + * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2 + lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc + 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 ??)) // + [| >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 // @@ -511,24 +488,232 @@ lemma sem_match_step : | >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 // ] // - ] + [ >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 + #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_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) // ] + >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 @@ -618,13 +803,12 @@ definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc. 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) → - is_startc x = true → (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true → - ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) → - current sig (nth dst (tape sig) outt (niltape sig)) = None ?) - (* outt = int) *) ∧ + (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 ∧ @@ -632,7 +816,7 @@ definition R_match_m ≝ 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). + ∀l,l1.x0::rs0 ≠ l@x::xs@l1)). (* definition R_match_m ≝ @@ -665,54 +849,69 @@ 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 #Hfalse #ls #x #xs #end #rs #Hmid_src #Hstart #Hnotend #Hend +[ #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 // - |#ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst; + |#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) - |#ls1 #x1 #rs1 >Hmid_dst #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 #Hstart #Hnotend #Hend + #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) - [#H @H // - | - - |#cur_dst #Hcur_dst %2 #ls0 #x0 #rs0 #Hmid_dst - whd in Htrue; >Hmid_src in Htrue; #Htrue - cases (Htrue x (refl …) Hstart ?) -Htrue + [#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==cur_dst)) #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 Hxs e notendx1 *)] - #ci -tl1 #tl1 #Hxs #H cases (H … (refl … )) + 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 *) - |#Htrue #_ cases(Htrue cur_dst Hcur_dst (\Pf eqx)) -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 - ] + #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 + ] ] ] ] -- 2.39.2