X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Flib%2Fturing%2Fmulti_universal%2Fmatch.ma;h=76a1fc30d0a6dbd270d7c926a79b02de98239910;hb=c16905138e385d30856d587f07c396a3cab301ed;hp=f16190491a202810abf3f2e5fb192b2d477a2884;hpb=99865bbdbb8b4694c85085abb0e98b4d3be7ea9f;p=helm.git diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma index f16190491..76a1fc30d 100644 --- a/matita/matita/lib/turing/multi_universal/match.ma +++ b/matita/matita/lib/turing/multi_universal/match.ma @@ -12,350 +12,9 @@ (* *) (**************************************************************************) -include "turing/multi_universal/moves.ma". -include "turing/if_multi.ma". -include "turing/inject.ma". -include "turing/basic_machines.ma". +include "turing/multi_universal/compare.ma". +include "turing/multi_universal/par_test.ma". -definition compare_states ≝ initN 3. - -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 …)). - -(* - -0) (x,x) → (x,x)(R,R) → 1 - (x,y≠x) → None 2 -1) (_,_) → None 1 -2) (_,_) → None 2 - -*) - -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〉 ] ]. - -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). - -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 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. - -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. - -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 - ] - ] - ] - ] -] -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 // -] -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) % ] ] - ] - ] - ] -] -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,rs0. - nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → - 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) - (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. -#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) // --Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ #tc whd in ⊢ (%→?); * * [ * [ * - [* #curi * #Hcuri #Hendi #Houtc % - [ #_ @Houtc - | #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 #rs0 #Hnthi #Hnthj - >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H % - ]] - | #Hci #Houtc % - [ #_ @Houtc - | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi >Hnthi in Hci; - normalize in ⊢ (%→?); #H destruct (H) ] ] - | #Hcj #Houtc % - [ #_ @Houtc - | #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 #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 %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 // - | >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. - 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. - -(* - |conf1 $ - |confin 0/1 confout move - - match machine step ≝ - compare; - if (cur(src) != $) - then - parmoveL; - moveR(dst); - else nop - *) definition Rtc_multi_true ≝ λalpha,test,n,i.λt1,t2:Vector ? (S n). @@ -388,217 +47,48 @@ cases (acc_sem_inject … Hin (sem_test_char alpha test) int) | @sym_eq @Hnth_j @sym_not_eq // ] ] ] qed. +definition Rm_test_null_true ≝ + λalpha,n,i.λt1,t2:Vector ? (S n). + current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1. + +definition Rm_test_null_false ≝ + λalpha,n,i.λt1,t2:Vector ? (S n). + current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1. + +lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n → + inject_TM ? (test_null ?) n i ⊨ + [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ]. +#alpha #n #i #Hin #int +cases (acc_sem_inject … Hin (sem_test_null alpha) int) +#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ % +[ @Hloop +| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % // + @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i + [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ] +| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j % + [ @Hcur + | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) // + #Hi0i @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. - 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 ⊨ - [ 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 -@(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) - (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 - #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_test ≝ λsrc,dst.λsig:DeqSet.λn,is_endc.λv:Vector ? n. + match (nth src (option sig) v (None ?)) with + [ None ⇒ false + | Some x ⇒ notb ((is_endc x) ∨ (nth dst (DeqOption sig) v (None ?) == None ?))]. 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) + (ifTM ?? (partest sig n (match_test src dst sig ? is_endc)) + (single_finalTM ?? + (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst))) (nop …) - tc_true). + partest1). definition R_match_step_false ≝ λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n). @@ -606,196 +96,188 @@ definition R_match_step_false ≝ 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). + (∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧ + xs = rs0@xs0 ∧ + current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨ + (∃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 + (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,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)) ∧ + (∀ls,x,xs,ci,cj,rs,ls0,rs0. + nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → + nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → + (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → + 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)). + (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)))) : + [ 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 -(* 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 … + (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ? is_endc)) + (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 // ] // +[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * #Htest #Htd >Htd -Htd + * #te * #Hte #Htb whd + #s #Hcurta_src % + [ lapply (refl ? (current ? (nth dst ? ta (niltape ?)))) + cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%); + [| #c #_ % #Hfalse destruct (Hfalse) ] + #Hcurta_dst >Hcomp1 in Htest; [| %2 %2 //] + whd in ⊢ (??%?→?); change with (current ? (niltape ?)) in match (None ?); + Hcurta_src whd in ⊢ (??%?→?); Hcurta_dst cases (is_endc s) normalize in ⊢ (%→?); #H destruct (H) + | #Hstart #Hnotstart % + [ #s1 #Hcurta_dst #Hneqss1 -Hcomp2 + cut (tc = ta) + [@Hcomp1 %2 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //] + #H destruct (H) -Hcomp1 cases Hte #_ -Hte #Hte + cut (te = ta) [@Hte %1 %1 %{s} % //] -Hte #H destruct (H) % + [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … )) + #i #Hi cases (decidable_eq_nat i dst) #Hidst + [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst) + #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs // + | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ] + | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src + >Hcurta_src in Htest; whd in ⊢ (??%?→?); + cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq // + ] + |#ls #x #xs #ci #cj #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc #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 Hte -Hte #Hte #_ whd in Hte; + >Htasrc_mid in Hcurta_src; whd in ⊢ (??%?→?); #H destruct (H) + lapply (Hte ls ci (reverse ? xs) rs s ??? ls0 cj' (reverse ? xs) s 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 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) // ] + | >Htc >change_vec_commute // >nth_change_vec // ] -Hte + >Htc >change_vec_commute // >change_vec_change_vec + >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte + >Hte in Htb; * * #_ >reverse_reverse #Htbdst1 #Htbdst2 -Hte @(eq_vec … (niltape ?)) + #i #Hi cases (decidable_eq_nat i dst) #Hidst + [ >Hidst >nth_change_vec // >(Htbdst1 ls0 s (xs@cj'::rs0')) + [| >nth_change_vec // ] + >Htadst_mid cases xs // + | >nth_change_vec_neq [|@sym_not_eq // ] + nth_change_vec_neq [| @sym_not_eq // ] + change_vec_same % ] + | >Hcurta_src in Htest; whd in ⊢(??%?→?); + >Htc >change_vec_commute // + change with (current ? (niltape ?)) in match (None ?); + nth_change_vec // whd in ⊢ (??%?→?); + cases (is_endc ci) whd in ⊢ (??%?→?); #H destruct (H) % ] + ] ] |#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 % ] + [#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 + #ls_dst * #rs_dst #Hmid_dst cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * * - #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >Hrs_dst in Hmid_dst; #Hmid_dst + #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq >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 // ] - ] + [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //] ] + * + [ * #Hrsj >Hrsj #Hta % %2 >Hta >nth_change_vec // + %{ls_dst} %{xs1} cut (∃xs0.xs = xs1@xs0) + [lapply Hnotendxs1 -Hnotendxs1 lapply Hrs_src lapply xs elim xs1 + [ #l #_ #_ %{l} % + | #x2 #xs2 #IH * + [ whd in ⊢ (??%%→?); #H destruct (H) #Hnotendxs2 + >Hnotendxs2 in Hend; [ #H destruct (H) |@memb_hd ] + | #x2' #xs2' whd in ⊢ (??%%→?); #H destruct (H) + #Hnotendxs2 cases (IH xs2' e0 ?) + [ #xs0 #Hxs2 %{xs0} @eq_f // + |#c #Hc @Hnotendxs2 @memb_cons // ] + ] + ] + ] * #xs0 #Hxs0 %{xs0} % [ % + [ >Hmid_dst >Hrsj >append_nil % + | @Hxs0 ] + | cases (reverse ? xs1) // ] + | * #cj * #rs2 * #Hrsj #Hta lapply (Hta ?) + [ cases (Hneq ?? Hrs1) /2/ #Hr1 %2 @(Hr1 ?? Hrsj) ] -Hta #Hta + %2 >Hta in Hc; whd in ⊢ (??%?→?); + change with (current ? (niltape ?)) in match (None ?); + nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // + whd in ⊢ (??%?→?); #Hc cut (is_endc r1 = true) + [ cases (is_endc r1) in Hc; whd in ⊢ (??%?→?); // + change with (current ? (niltape ?)) in match (None ?); + nth_change_vec // normalize #H destruct (H) ] + #Hendr1 cut (xs = xs1) + [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1 + -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs + [ * normalize in ⊢ (%→?); // + #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1 + lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H) + | #x2 #xs2 #IH * + [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc + >Hnotendc in Hendr1; [| @memb_cons @memb_hd ] + normalize in ⊢ (%→?); #H destruct (H) + | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq) + #Hnotendc #Hnotendcxs1 @eq_f @IH + [ @(cons_injective_r … Heq) + | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0 + [ >(\P Hc0) @memb_hd + | @memb_cons @memb_cons // ] + | #c #Hc @Hnotendcxs1 @memb_cons // ] ] - | #Hxsxs1 >Hmid_dst >Hxsxs1 % // - #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 // - @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) // - #Hendr1 destruct (Hendr1) % ] + ] + | #Hxsxs1 destruct (Hxsxs1) >Hmid_dst %{ls_dst} %{rsj} % // + #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) + lapply (append_l2_injective … Hrs_src) // #Hrs' destruct (Hrs') % ] ] - (* 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) + |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst + @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize + @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape >Hintape in Hc; + whd in ⊢(??%?→?); >Hmid_src + change with (current ? (niltape ?)) in match (None ?); + Hmid_src whd in ⊢ (??%?→?); + >(Hnotend c_src) [|@memb_hd] + change with (current ? (niltape ?)) in match (None ?); + Hmid_src whd in ⊢ (??%?→?); >Hdst normalize #H destruct (H) ] ] ] -qed. +qed. definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc. whileTM … (match_step src dst sig n is_startc is_endc) @@ -851,10 +333,27 @@ lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar [ #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) + [(* 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 * #xs0 * * #Htc_dst #Hrs0 #HNone % + [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H) + | #Hstart #ls1 #x1 #rs1 >Htc_dst #H destruct (H) + >Hrs0 cases xs0 + [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %] + #cj #ls2 #H destruct (H) + | #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 #HFalse % [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H) @@ -869,17 +368,15 @@ lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc 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 ⊢ (%→?); + cases (Htrue x (refl … )) -Htrue * #Htaneq #_ + @False_ind >Hmid_dst in Htaneq; /2/ + |#Hstart #ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?); #H destruct (H) ] | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ] - #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?); + #Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?); #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue - cases (Htrue x (refl …) Hstart ?) -Htrue + cases (Htrue x (refl …)) -Htrue #_ #Htrue cases (Htrue 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) @@ -889,18 +386,22 @@ lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc #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 // + |cases tl2 in Hrs0; + [ + | #cj #tl2' #Hrs0 #Hcomp lapply (Htrue ls x x1 ci cj tl1 ls0 tl2' ????) + [ @(Hcomp ?? (refl ??)) + | #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0 + [ @Hnotend >(\P Hc0) @memb_hd + | @Hnotendx1 // ] + | >Hmid_dst >Hrs0 >(\P eqx) % + | >Hxs % + | * #Htb >Htb #Hendci %2 >Hrs0 >Hxs + cases (IH ls x xs end rs ? Hnotend Hend) [| + STOP + + + + >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // >Hrs0 in Hmid_dst; #Hmid_dst cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx