From 06efa0bfaf379b7d3e2f93a41ce5e6f0f01e76a5 Mon Sep 17 00:00:00 2001 From: Andrea Asperti Date: Wed, 21 Nov 2012 15:38:05 +0000 Subject: [PATCH] compare con terminatore --- .../lib/turing/multi_universal/match.ma | 259 ++++++++++++------ 1 file changed, 175 insertions(+), 84 deletions(-) diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma index d054a9c83..bf5e3d42b 100644 --- a/matita/matita/lib/turing/multi_universal/match.ma +++ b/matita/matita/lib/turing/multi_universal/match.ma @@ -33,7 +33,7 @@ definition comp2 : compare_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …) *) definition trans_compare_step ≝ - λi,j.λsig:FinSet.λn. + λi,j.λsig:FinSet.λn.λis_endc. λp:compare_states × (Vector (option sig) (S n)). let 〈q,a〉 ≝ p in match pi1 … q with @@ -41,7 +41,7 @@ definition trans_compare_step ≝ [ None ⇒ 〈comp2,null_action ? n〉 | Some ai ⇒ match nth j ? a (None ?) with [ None ⇒ 〈comp2,null_action ? n〉 - | Some aj ⇒ if ai == aj + | 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〉 @@ -52,13 +52,14 @@ definition trans_compare_step ≝ | S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ]. definition compare_step ≝ - λi,j,sig,n. - mk_mTM sig n compare_states (trans_compare_step i j sig n) + 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.λint,outt: Vector (tape sig) (S n). + 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 ?? @@ -67,18 +68,19 @@ definition R_comp_step_true ≝ (tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) j. definition R_comp_step_false ≝ - λi,j:nat.λsig,n.λint,outt: Vector (tape sig) (S n). - (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨ + 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,v.i < S n → j < S n → + ∀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) (mk_mconfig ??? comp0 v) + step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) = mk_mconfig ??? comp2 v. -#i #j #sig #n #v #Hi #Hj +#i #j #sig #n #is_endc #v #Hi #Hj whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?); * #Hcurrent [ @eq_f2 @@ -91,42 +93,57 @@ whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??% qed. lemma comp_q0_q2_neq : - ∀i,j,sig,n,v.i < S n → j < S n → - nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?) → - step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) + ∀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 #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?))) +#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 #Hneq - whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 - [ whd in match (trans ????); >Hai >Haj - whd in ⊢ (??(???%)?); >(\bf ?) // @(not_to_not … Hneq) // - | whd in match (trans ????); >Hai >Haj - whd in ⊢ (??(???????(???%))?); >(\bf ?) /2 by not_to_not/ - @tape_move_null_action -] ] + | #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,v,a.i ≠ j → i < S n → j < S n → - nth i ? (current_chars ?? v) (None ?) = Some ? a → + ∀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) (mk_mconfig ??? comp0 v) = + 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 #v #a #Heq #Hi #Hj #Ha1 #Ha2 +#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 ⊢ (??(???%)?); >(\b ?) // + >Ha1 >Ha2 whd in ⊢ (??(???%)?); >Hnotendc >(\b ?) // | whd in match (trans ????); - >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >(\b ?) // + >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 ⊢ (??%?); @@ -136,11 +153,11 @@ whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 qed. lemma sem_comp_step : - ∀i,j,sig,n.i ≠ j → i < S n → j < S n → - compare_step i j sig n ⊨ - [ comp1: R_comp_step_true i j sig n, - R_comp_step_false i j sig n ]. -#i #j #sig #n #Hneq #Hi #Hj #int + ∀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} % @@ -156,56 +173,75 @@ cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?); [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2 Ha >Hcurj % % % #H destruct (H) ] ] - | #b #Hb %{2} cases (true_or_false (a == b)) #Hab + | #_ % // >Ha >Hcurj % % %2 % #H destruct (H) ] ] + | #b #Hb %{2} + cases (true_or_false (is_endc a)) #Haendc [ % - [| % [ % - [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 ?)) >Ha >Hb - @(not_to_not ??? (\Pf Hab)) #H destruct (H) % + % %{a} % // Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ] + | #_ % // % % % >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. - whileTM … (compare_step i j sig n) comp1. +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.λint,outt: Vector (tape sig) (S n). - ((current ? (nth i ? int (niltape ?)) - ≠ current ? (nth j ? int (niltape ?)) ∨ + λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n). + ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨ + (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨ current ? (nth i ? int (niltape ?)) = None ? ∨ current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧ (∀ls,x,xs,ci,rs,ls0,cj,rs0. nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → - nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → ci ≠ cj → + nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → + (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → + (is_endc ci = true ∨ ci ≠ cj) → outt = change_vec ?? (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i) (midtape sig (reverse ? xs@x::ls0) cj rs0) j). -lemma wsem_compare : ∀i,j,sig,n.i ≠ j → i < S n → j < S n → - compare i j sig n ⊫ R_compare i j sig n. -#i #j #sig #n #Hneq #Hi #Hj #ta #k #outc #Hloop -lapply (sem_while … (sem_comp_step i j sig n Hneq Hi Hj) … Hloop) // +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 ⊢ (%→?); * * [ * - [ #Hcicj #Houtc % +[ #tc whd in ⊢ (%→?); * * [ * [ * + [* #curi * #Hcuri #Hendi #Houtc % + [ #_ @Houtc + | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj #Hnotendc + @False_ind + >Hnthi in Hcuri; normalize in ⊢ (%→?); #H destruct (H) + >(Hnotendc ? (memb_hd … )) in Hendi; #H destruct (H) + ] + |#Hcicj #Houtc % [ #_ @Houtc | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H % - ] + ]] | #Hci #Houtc % [ #_ @Houtc | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci; @@ -214,24 +250,28 @@ lapply (sem_while … (sem_comp_step i j sig n Hneq Hi Hj) … Hloop) // [ #_ @Houtc | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj; normalize in ⊢ (%→?); #H destruct (H) ] ] - | #tc #td #te * #x * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH * + | #tc #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH * #IH1 #IH2 % - [ >Hci >Hcj * [* [* #H @False_ind @H % | #H destruct (H)] | #H destruct (H)] + [ >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 #Hcicj >IH1 + [ #Hnthi #Hnthj #Hnotendc #Hcicj >IH1 [ >Hd @eq_f3 // [ @eq_f3 // >(?:c0=x) [ >Hnthi % ] >Hnthi in Hci;normalize #H destruct (H) % | >(?:c0=x) [ >Hnthj % ] >Hnthi in Hci;normalize #H destruct (H) % ] | >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // >Hnthi >Hnthj normalize %1 %1 @(not_to_not ??? Hcicj) - #H destruct (H) % ] - | #x0 #xs0 #Hnthi #Hnthj #Hcicj + >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 ⊢ (??%?); // @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 @@ -241,19 +281,19 @@ lapply (sem_while … (sem_comp_step i j sig n Hneq Hi Hj) … Hloop) // ]]] qed. -lemma terminate_compare : ∀i,j,sig,n,t. +lemma terminate_compare : ∀i,j,sig,n,is_endc,t. i ≠ j → i < S n → j < S n → - compare i j sig n ↓ t. -#i #j #sig #n #t #Hneq #Hi #Hj + 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 +[ % #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 * * >nth_change_vec // normalize in ⊢ (%→?); + [#t #ls #c % #t1 * #x * * * #Hendcx >nth_change_vec // normalize in ⊢ (%→?); #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 % - #t2 * #x0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //] + #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 @@ -262,10 +302,10 @@ cases (nth i (tape sig) t (niltape ?)) ] qed. -lemma sem_compare : ∀i,j,sig,n. +lemma sem_compare : ∀i,j,sig,n,is_endc. i ≠ j → i < S n → j < S n → - compare i j sig n ⊨ R_compare i j sig n. -#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize /2/ + 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. (* @@ -299,12 +339,13 @@ definition Rtc_multi_false ≝ definition R_match_step_false ≝ λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n). - ((current ? (nth src ? int (niltape ?)) ≠ current ? (nth dst ? int (niltape ?)) ∨ + (((∃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 end = true ∧ + is_endc x = false ∧ is_endc end = true ∧ nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧ outt = change_vec ?? @@ -314,7 +355,8 @@ definition R_match_step_false ≝ 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 → + 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 @@ -352,7 +394,7 @@ axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S. ∃l,tl1,tl2. l1 = l@tl1 ∧ l2 = l@tl2 ∧ ∀a,b,tla,tlb. tl1 = a::tla → tl2 = b::tlb → a≠b. axiom daemon : ∀X:Prop.X. - + lemma sem_match_step : ∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n → match_step src dst sig n is_startc is_endc ⊨ @@ -367,7 +409,7 @@ 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 % + #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) % ] @@ -384,7 +426,11 @@ lemma sem_match_step : #H destruct (H) >(Htd ls ci (reverse ? xs) rs s ??? ls0 cj (reverse ? xs) s rs0 (refl ??)) // [| >Hcomp2 >nth_change_vec // - | @daemon + | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid + cases (orb_true_l … Hc0) -Hc0 #Hc0 + [@memb_append_l2 >(\P Hc0) @memb_hd + |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse // + ] | >Hcomp2 >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ] * * #_ #Htbdst #Htbelse % [ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst @@ -436,8 +482,12 @@ lemma sem_match_step : cases (is_endc end) normalize // |@Hmid_src] |@Hmid_dst] - |#_ #Hcomp1 #_ %1 % - [% % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // + |#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc + %1 % + [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1 + [| % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ] + cases (is_endc c_src) // + >Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H) |@Hcomp1 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ] ] @@ -445,6 +495,47 @@ lemma sem_match_step : ] qed. +definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc. + whileTM … (match_step src dst sig n is_startc is_endc) + (inr ?? (inr ?? (inl … (inr ?? start_nop)))). + +definition R_match_m ≝ + λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n). + (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨ + current ? (nth i ? int (niltape ?)) = None ? ∨ + current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧ + (∀ls,x,xs,ci,rs,ls0,x0,rs0. + is_startc x = true → is_endc ci = true → + nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → + nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 → + ∃l,cj,l1.x0::rs0 = l@x::xs@cj::l1 ∧ + outt = change_vec ?? + (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i) + (midtape sig ((reverse ? (l@x::xs))@ls0) cj l1) j). + +lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc. +src ≠ dst → src < S n → dst < S n → + match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc. +#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop +lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) // +-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar +[ #tc whd in ⊢ (%→?); * + [ * * [ * + [ * #cur_src * #H1 #H2 #Houtc % + [ #_ @Houtc + | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #_ #Hnthi #Hnthj + >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H % + ] + | #Hci #Houtc % + [ #_ @Houtc + | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci; + normalize in ⊢ (%→?); #H destruct (H) ] ] + | #Hcj #Houtc % + [ #_ @Houtc + | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj; + normalize in ⊢ (%→?); #H destruct (H) ] ] +[ #tc whd in ⊢ (%→?); * * [ * + -- 2.39.2