From: Andrea Asperti Date: Thu, 7 Feb 2013 09:10:31 +0000 (+0000) Subject: restructuring X-Git-Tag: make_still_working~1267 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=69d5ccfff20e1001735c102239fb40912eb8360e;hp=baba23b670cb20eb478975fa9cb419c7ae58f7bc;p=helm.git restructuring --- diff --git a/matita/matita/lib/turing/multi_universal/compare.ma b/matita/matita/lib/turing/multi_universal/compare.ma deleted file mode 100644 index 8ad92d1e8..000000000 --- a/matita/matita/lib/turing/multi_universal/compare.ma +++ /dev/null @@ -1,313 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "turing/multi_universal/moves.ma". -include "turing/if_multi.ma". -include "turing/inject.ma". -include "turing/basic_machines.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. - 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 sig n〉 - | Some ai ⇒ match nth j ? a (None ?) with - [ None ⇒ 〈comp2,null_action ? n〉 - | Some aj ⇒ if ai == aj - then 〈comp1,change_vec ? (S n) - (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) i) - (〈None ?,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. - mk_mTM sig n compare_states (trans_compare_step i j sig n) - comp0 (λq.q == comp1 ∨ q == comp2). - -definition R_comp_step_true ≝ - λi,j,sig,n.λint,outt: Vector (tape sig) (S n). - ∃x. - current ? (nth i ? int (niltape ?)) = Some ? x ∧ - current ? (nth j ? int (niltape ?)) = Some ? x ∧ - outt = change_vec ?? - (change_vec ?? int - (tape_move_right ? (nth i ? int (niltape ?))) i) - (tape_move_right ? (nth j ? int (niltape ?))) 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 ?)) ∨ - 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 → - (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) - = mk_mconfig ??? comp2 v. -#i #j #sig #n #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,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) - = mk_mconfig ??? comp2 v. -#i #j #sig #n #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 ⊢ (??(???%)?); cut ((ai==aj)=false) - [>(\bf ?) /2 by not_to_not/ % #Haiaj @Hneq - >Hai >Haj // - | #Haiaj >Haiaj % ] - | whd in match (trans ????); >Hai >Haj - whd in ⊢ (??(????(???%))?); cut ((ai==aj)=false) - [>(\bf ?) /2 by not_to_not/ % #Haiaj @Hneq - >Hai >Haj // - |#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 → - nth j ? (current_chars ?? v) (None ?) = Some ? a → - step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) = - mk_mconfig ??? comp1 - (change_vec ? (S n) - (change_vec ?? v - (tape_move_right ? (nth i ? v (niltape ?))) i) - (tape_move_right ? (nth j ? v (niltape ?))) j). -#i #j #sig #n #v #a #Heq #Hi #Hj #Ha1 #Ha2 -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 -[ whd in match (trans ????); - >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) // -| whd in match (trans ????); - >Ha1 >Ha2 whd in ⊢ (??(????(???%))?); >(\b ?) // - change with (change_vec ?????) in ⊢ (??(????%)?); - <(change_vec_same … v j (niltape ?)) in ⊢ (??%?); - <(change_vec_same … v i (niltape ?)) in ⊢ (??%?); - >tape_move_multi_def - >pmap_change >pmap_change tape_move_null_action - @eq_f2 // >nth_change_vec_neq // -] -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 -lapply (refl ? (current ? (nth i ? int (niltape ?)))) -cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?); -[ #Hcuri %{2} % - [| % [ % - [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // % %2 // ] ] -| #a #Ha lapply (refl ? (current ? (nth j ? int (niltape ?)))) - cases (current ? (nth j ? int (niltape ?))) in ⊢ (???%→?); - [ #Hcurj %{2} % - [| % [ % - [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2 - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // >Ha >Hcurj % % % #H destruct (H) ] ] - | #b #Hb %{2} 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 ?)) >Ha >Hb - @(not_to_not ??? (\Pf Hab)) #H destruct (H) % - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // % % >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. - -(* (∃rs'.rs = rs0@rs' ∧ current ? (nth j ? outt (niltape ?)) = None ?) ∨ - (∃rs0'.rs0 = rs@rs0' ∧ - outt = change_vec ?? - (change_vec ?? int - (mk_tape sig (reverse sig rs@x::ls) (None sig) []) i) - (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs0') - (tail sig rs0')) j) ∨ - (∃xs,ci,cj,rs',rs0'.ci ≠ cj ∧ rs = xs@ci::rs' ∧ rs0 = xs@cj::rs0' ∧ - outt = change_vec ?? - (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs') i) - (midtape sig (reverse ? xs@x::ls0) cj rs0') j)).*) -definition R_compare ≝ - λi,j,sig,n.λint,outt: Vector (tape sig) (S n). - ((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,rs,ls0,rs0. -(* nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → *) - nth i ? int (niltape ?) = midtape sig ls x rs → - nth j ? int (niltape ?) = midtape sig ls0 x rs0 → - (∃rs'.rs = rs0@rs' ∧ - outt = change_vec ?? - (change_vec ?? int - (mk_tape sig (reverse sig rs0@x::ls) (option_hd sig rs') (tail ? rs')) i) - (mk_tape sig (reverse sig rs0@x::ls0) (None ?) [ ]) j) ∨ - (∃rs0'.rs0 = rs@rs0' ∧ - outt = change_vec ?? - (change_vec ?? int - (mk_tape sig (reverse sig rs@x::ls) (None sig) []) i) - (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs0') - (tail sig rs0')) j) ∨ - (∃xs,ci,cj,rs',rs0'.ci ≠ cj ∧ rs = xs@ci::rs' ∧ rs0 = xs@cj::rs0' ∧ - 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) // --Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ whd in ⊢ (%→?); * * [ * - [ #Hcicj #Houtc % - [ #_ @Houtc - | #ls #x #rs #ls0 #rs0 #Hnthi #Hnthj - >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H % - ] - | #Hci #Houtc % - [ #_ @Houtc - | #ls #x #rs #ls0 #rs0 #Hnthi >Hnthi in Hci; - normalize in ⊢ (%→?); #H destruct (H) ] ] - | #Hcj #Houtc % - [ #_ @Houtc - | #ls #x #rs #ls0 #rs0 #_ #Hnthj >Hnthj in Hcj; - normalize in ⊢ (%→?); #H destruct (H) ] ] -| #td #te * #x * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH * - #IH1 #IH2 % - [ >Hci >Hcj * [ * - [ * #H @False_ind @H % | #H destruct (H)] | #H destruct (H)] - | #ls #c0 #rs #ls0 #rs0 cases rs - [ -IH2 #Hnthi #Hnthj % %2 %{rs0} % [%] - >Hnthi in Hd; #Hd >Hd in IH1; #IH1 >IH1 - [| % %2 >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // % ] - >Hnthj cases rs0 [| #r1 #rs1 ] % - | #r1 #rs1 #Hnthi cases rs0 - [ -IH2 #Hnthj % % %{(r1::rs1)} % [%] - >Hnthj in Hd; #Hd >Hd in IH1; #IH1 >IH1 - [| %2 >nth_change_vec // ] - >Hnthi >Hnthj % - | #r2 #rs2 #Hnthj lapply IH2; >Hd in IH1; >Hnthi >Hnthj - >nth_change_vec // - >nth_change_vec_neq [| @sym_not_eq // ] >nth_change_vec // - cases (true_or_false (r1 == r2)) #Hr1r2 - [ >(\P Hr1r2) #_ #IH2 cases (IH2 … (refl ??) (refl ??)) [ * - [ * #rs' * #Hrs1 #Hcurout_j % % %{rs'} - >Hrs1 % - [ % - | >Hcurout_j >change_vec_commute // >change_vec_change_vec - >change_vec_commute // @sym_not_eq // ] - | * #rs0' * #Hrs2 #Hcurout_i % %2 %{rs0'} - >Hrs2 >Hcurout_i % // - >change_vec_commute // >change_vec_change_vec - >change_vec_commute [|@sym_not_eq//] >change_vec_change_vec - >reverse_cons >associative_append >associative_append % ] - | * #xs * #ci * #cj * #rs' * #rs0' * * * #Hcicj #Hrs1 #Hrs2 - >change_vec_commute // >change_vec_change_vec - >change_vec_commute [| @sym_not_eq ] // >change_vec_change_vec - #Houtc %2 %{(r2::xs)} %{ci} %{cj} %{rs'} %{rs0'} - % [ % [ % [ // | >Hrs1 // ] | >Hrs2 // ] - | >reverse_cons >associative_append >associative_append >Houtc % ] ] - | lapply (\Pf Hr1r2) -Hr1r2 #Hr1r2 #IH1 #_ %2 - >IH1 [| % % normalize @(not_to_not … Hr1r2) #H destruct (H) % ] - %{[]} %{r1} %{r2} %{rs1} %{rs2} % [ % [ % /2/ | % ] | % ] ]]]]] -qed. - -lemma terminate_compare : ∀i,j,sig,n,t. - i ≠ j → i < S n → j < S n → - compare i j sig n ↓ t. -#i #j #sig #n #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 * * >nth_change_vec // normalize in ⊢ (%→?); - #H1 destruct (H1) #_ >change_vec_change_vec #Ht1 % - #t2 * #x0 * * >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. - 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/| @wsem_compare // ] -qed. diff --git a/matita/matita/lib/turing/multi_universal/copy.ma b/matita/matita/lib/turing/multi_universal/copy.ma deleted file mode 100644 index e06ef0a42..000000000 --- a/matita/matita/lib/turing/multi_universal/copy.ma +++ /dev/null @@ -1,244 +0,0 @@ -(* - ||M|| This file is part of HELM, an Hypertextual, Electronic - ||A|| Library of Mathematics, developed at the Computer Science - ||T|| Department of the University of Bologna, Italy. - ||I|| - ||T|| - ||A|| - \ / This file is distributed under the terms of the - \ / GNU General Public License Version 2 - V_____________________________________________________________*) - -include "turing/multi_universal/moves.ma". -include "turing/if_multi.ma". -include "turing/inject.ma". -include "turing/basic_machines.ma". - -definition copy_states ≝ initN 3. - -definition copy0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)). -definition copy1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)). -definition copy2 : copy_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)). - - -definition trans_copy_step ≝ - λsrc,dst.λsig:FinSet.λn. - λp:copy_states × (Vector (option sig) (S n)). - let 〈q,a〉 ≝ p in - match pi1 … q with - [ O ⇒ match nth src ? a (None ?) with - [ None ⇒ 〈copy2,null_action sig n〉 - | Some ai ⇒ match nth dst ? a (None ?) with - [ None ⇒ 〈copy2,null_action ? n〉 - | Some aj ⇒ - 〈copy1,change_vec ? (S n) - (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src) - (〈Some ? ai,R〉) dst〉 - ] - ] - | S q ⇒ match q with - [ O ⇒ (* 1 *) 〈copy1,null_action ? n〉 - | S _ ⇒ (* 2 *) 〈copy2,null_action ? n〉 ] ]. - -definition copy_step ≝ - λsrc,dst,sig,n. - mk_mTM sig n copy_states (trans_copy_step src dst sig n) - copy0 (λq.q == copy1 ∨ q == copy2). - -definition R_copy_step_true ≝ - λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). - ∃x,y. - current ? (nth src ? int (niltape ?)) = Some ? x ∧ - current ? (nth dst ? int (niltape ?)) = Some ? y ∧ - outt = change_vec ?? - (change_vec ?? int - (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src) - (tape_move_mono ? (nth dst ? int (niltape ?)) 〈Some ? x, R〉) dst. - -definition R_copy_step_false ≝ - λsrc,dst:nat.λsig,n.λint,outt: Vector (tape sig) (S n). - (current ? (nth src ? int (niltape ?)) = None ? ∨ - current ? (nth dst ? int (niltape ?)) = None ?) ∧ outt = int. - -lemma copy_q0_q2_null : - ∀src,dst,sig,n,v.src < S n → dst < S n → - (nth src ? (current_chars ?? v) (None ?) = None ? ∨ - nth dst ? (current_chars ?? v) (None ?) = None ?) → - step sig n (copy_step src dst sig n) (mk_mconfig ??? copy0 v) - = mk_mconfig ??? copy2 v. -#src #dst #sig #n #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 src ?? (None sig)) // - | whd in ⊢ (??(????(???%))?); >Hcurrent - cases (nth src ?? (None sig)) [|#x] @tape_move_null_action ] ] -qed. - -lemma copy_q0_q1 : - ∀src,dst,sig,n,v,a,b.src ≠ dst → src < S n → dst < S n → - nth src ? (current_chars ?? v) (None ?) = Some ? a → - nth dst ? (current_chars ?? v) (None ?) = Some ? b → - step sig n (copy_step src dst sig n) (mk_mconfig ??? copy0 v) = - mk_mconfig ??? copy1 - (change_vec ? (S n) - (change_vec ?? v - (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src) - (tape_move_mono ? (nth dst ? v (niltape ?)) 〈Some ? a, R〉) dst). -#src #dst #sig #n #v #a #b #Heq #Hsrc #Hdst #Ha1 #Ha2 -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 -[ whd in match (trans ????); - >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) // -| whd in match (trans ????); - >Ha1 >Ha2 whd in ⊢ (??(????(???%))?); >(\b ?) // - change with (change_vec ?????) in ⊢ (??(????%)?); - <(change_vec_same … v dst (niltape ?)) in ⊢ (??%?); - <(change_vec_same … v src (niltape ?)) in ⊢ (??%?); - >tape_move_multi_def - >pmap_change >pmap_change tape_move_null_action - @eq_f2 // >nth_change_vec_neq // -] -qed. - -lemma sem_copy_step : - ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → - copy_step src dst sig n ⊨ - [ copy1: R_copy_step_true src dst sig n, - R_copy_step_false src dst sig n ]. -#src #dst #sig #n #Hneq #Hsrc #Hdst #int -lapply (refl ? (current ? (nth src ? int (niltape ?)))) -cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?); -[ #Hcur_src %{2} % - [| % [ % - [ whd in ⊢ (??%?); >copy_q0_q2_null /2/ - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // % // ] ] -| #a #Ha lapply (refl ? (current ? (nth dst ? int (niltape ?)))) - cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?); - [ #Hcur_dst %{2} % - [| % [ % - [ whd in ⊢ (??%?); >copy_q0_q2_null /2/ - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // %2 >Hcur_dst % ] ] - | #b #Hb %{2} % - [| % [ % - [whd in ⊢ (??%?); >(copy_q0_q1 … a b Hneq Hsrc Hdst) // - | #_ %{a} %{b} % // % //] - | * #H @False_ind @H % - ] - ] - ] -] -qed. - -definition copy ≝ λsrc,dst,sig,n. - whileTM … (copy_step src dst sig n) copy1. - -definition R_copy ≝ - λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). - ((current ? (nth src ? int (niltape ?)) = None ? ∨ - current ? (nth dst ? int (niltape ?)) = None ?) → outt = int) ∧ - (∀ls,x,x0,rs,ls0,rs0. - nth src ? int (niltape ?) = midtape sig ls x rs → - nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 → - (∃rs01,rs02.rs0 = rs01@rs02 ∧ |rs01| = |rs| ∧ - outt = change_vec ?? - (change_vec ?? int - (mk_tape sig (reverse sig rs@x::ls) (None sig) []) src) - (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs02) - (tail sig rs02)) dst) ∨ - (∃rs1,rs2.rs = rs1@rs2 ∧ |rs1| = |rs0| ∧ - outt = change_vec ?? - (change_vec ?? int - (mk_tape sig (reverse sig rs1@x::ls) (option_hd sig rs2) - (tail sig rs2)) src) - (mk_tape sig (reverse sig rs1@x::ls0) (None sig) []) dst)). - -lemma wsem_copy : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → - copy src dst sig n ⊫ R_copy src dst sig n. -#src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop -lapply (sem_while … (sem_copy_step src dst sig n Hneq Hsrc Hdst) … Hloop) // --Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ whd in ⊢ (%→?); * #Hnone #Hout % - [#_ @Hout - |#ls #x #x0 #rs #ls0 #rs0 #Hsrc1 #Hdst1 @False_ind cases Hnone - [>Hsrc1 normalize #H destruct (H) | >Hdst1 normalize #H destruct (H)] - ] -|#tc #td * #x * #y * * #Hcx #Hcy #Htd #Hstar #IH #He lapply (IH He) -IH * - #IH1 #IH2 % - [* [>Hcx #H destruct (H) | >Hcy #H destruct (H)] - |#ls #x' #y' #rs #ls0 #rs0 #Hnth_src #Hnth_dst - >Hnth_src in Hcx; whd in ⊢ (??%?→?); #H destruct (H) - >Hnth_dst in Hcy; whd in ⊢ (??%?→?); #H destruct (H) - >Hnth_src in Htd; >Hnth_dst -Hnth_src -Hnth_dst - cases rs - [(* the source tape is empty after the move *) - #Htd lapply (IH1 ?) - [%1 >Htd >nth_change_vec_neq [2:@(not_to_not … Hneq) //] >nth_change_vec //] - #Hout (* whd in match (tape_move ???); *) %1 %{([])} %{rs0} % - [% [// | // ] - |whd in match (reverse ??); whd in match (reverse ??); - >Hout >Htd @eq_f2 // cases rs0 // - ] - |#c1 #tl1 cases rs0 - [(* the dst tape is empty after the move *) - #Htd lapply (IH1 ?) [%2 >Htd >nth_change_vec //] - #Hout (* whd in match (tape_move ???); *) %2 %{[ ]} %{(c1::tl1)} % - [% [// | // ] - |whd in match (reverse ??); whd in match (reverse ??); - >Hout >Htd @eq_f2 // - ] - |#c2 #tl2 whd in match (tape_move_mono ???); whd in match (tape_move_mono ???); - #Htd - cut (nth src (tape sig) td (niltape sig)=midtape sig (x::ls) c1 tl1) - [>Htd >nth_change_vec_neq [2:@(not_to_not … Hneq) //] @nth_change_vec //] - #Hsrc_td - cut (nth dst (tape sig) td (niltape sig)=midtape sig (x::ls0) c2 tl2) - [>Htd @nth_change_vec //] - #Hdst_td cases (IH2 … Hsrc_td Hdst_td) -Hsrc_td -Hdst_td - [* #rs01 * #rs02 * * #H1 #H2 #H3 %1 - %{(c2::rs01)} %{rs02} % [% [@eq_f //|normalize @eq_f @H2]] - >Htd in H3; >change_vec_commute // >change_vec_change_vec - >change_vec_commute [2:@(not_to_not … Hneq) //] >change_vec_change_vec - #H >reverse_cons >associative_append >associative_append @H - |* #rs11 * #rs12 * * #H1 #H2 #H3 %2 - %{(c1::rs11)} %{rs12} % [% [@eq_f //|normalize @eq_f @H2]] - >Htd in H3; >change_vec_commute // >change_vec_change_vec - >change_vec_commute [2:@(not_to_not … Hneq) //] >change_vec_change_vec - #H >reverse_cons >associative_append >associative_append @H - ] - ] - ] - ] -qed. - - -lemma terminate_copy : ∀src,dst,sig,n,t. - src ≠ dst → src < S n → dst < S n → copy src dst sig n ↓ t. -#src #dst #sig #n #t #Hneq #Hsrc #Hdts -@(terminate_while … (sem_copy_step …)) // -<(change_vec_same … t src (niltape ?)) -cases (nth src (tape sig) t (niltape ?)) -[ % #t1 * #x * #y * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct -|2,3: #a0 #al0 % #t1 * #x * #y * * >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 * #y * * >nth_change_vec // normalize in ⊢ (%→?); - #H1 destruct (H1) #_ >change_vec_change_vec #Ht1 % - #t2 * #x0 * #y0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) - |#r0 #rs0 #IH #t #ls #c % #t1 * #x * #y * * >nth_change_vec // - normalize in ⊢ (%→?); #H destruct (H) #Hcur - >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH - ] -] -qed. - -lemma sem_copy : ∀src,dst,sig,n. - src ≠ dst → src < S n → dst < S n → - copy src dst sig n ⊨ R_copy src dst sig n. -#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize [/2/| @wsem_copy // ] -qed. diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma index 93e6b340d..9cd11f610 100644 --- a/matita/matita/lib/turing/multi_universal/match.ma +++ b/matita/matita/lib/turing/multi_universal/match.ma @@ -12,74 +12,9 @@ (* *) (**************************************************************************) -include "turing/simple_machines.ma". -include "turing/multi_universal/compare.ma". -include "turing/multi_universal/par_test.ma". -include "turing/multi_universal/moves_2.ma". +include "turing/auxiliary_multi_machines.ma". -lemma eq_vec_change_vec : ∀sig,n.∀v1,v2:Vector sig n.∀i,t,d. - nth i ? v2 d = t → - (∀j.i ≠ j → nth j ? v1 d = nth j ? v2 d) → - v2 = change_vec ?? v1 t i. -#sig #n #v1 #v2 #i #t #d #H1 #H2 @(eq_vec … d) -#i0 #Hlt cases (decidable_eq_nat i0 i) #Hii0 -[ >Hii0 >nth_change_vec // -| >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ] -qed. - -lemma right_mk_tape : - ∀sig,ls,c,rs.(c = None ? → ls = [ ] ∨ rs = [ ]) → right ? (mk_tape sig ls c rs) = rs. -#sig #ls #c #rs cases c // cases ls -[ cases rs // -| #l0 #ls0 #H normalize cases (H (refl ??)) #H1 [ destruct (H1) | >H1 % ] ] -qed. - -lemma left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls. -#sig #ls #c #rs cases c // cases ls // cases rs // -qed. - -lemma current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c. -#sig #ls #c #rs cases c // cases ls // cases rs // -qed. - -lemma length_tail : ∀A,l.0 < |l| → |tail A l| < |l|. -#A * normalize // -qed. - -(* -[ ] → [ ], l2, 1 -a::al → - [ ] → [ ], l1, 2 - b::bl → match rec(al,bl) - x, y, 1 → b::x, y, 1 - x, y, 2 → a::x, y, 2 -*) - -lemma lists_length_split : - ∀A.∀l1,l2:list A.(∃la,lb.(|la| = |l1| ∧ l2 = la@lb) ∨ (|la| = |l2| ∧ l1 = la@lb)). -#A #l1 elim l1 -[ #l2 %{[ ]} %{l2} % % % -| #hd1 #tl1 #IH * - [ %{[ ]} %{(hd1::tl1)} %2 % % - | #hd2 #tl2 cases (IH tl2) #x * #y * - [ * #IH1 #IH2 %{(hd2::x)} %{y} % normalize % // - | * #IH1 #IH2 %{(hd1::x)} %{y} %2 normalize % // ] - ] -] -qed. - -definition option_cons ≝ λsig.λc:option sig.λl. - match c with [ None ⇒ l | Some c0 ⇒ c0::l ]. - -lemma opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l). -#A * // -qed. - -definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n. - match (nth src (option sig) v (None ?)) with - [ None ⇒ false - | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ]. - +(* rewind *) definition rewind ≝ λsrc,dst,sig,n. parmove src dst sig n L · mmove src sig n R · mmove dst sig n R. @@ -117,18 +52,6 @@ definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). ∀ls0,y,rs0.nth dst ? int (niltape ?) = midtape sig ls0 y rs0 → outt = int). -(* -theorem accRealize_to_Realize : - ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc. - M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse. -#sig #n #M #Rtrue #Rfalse #acc #HR #t -cases (HR t) #k * #outc * * #Hloop -#Htrue #Hfalse %{k} %{outc} % // -cases (true_or_false (cstate sig (states sig n M) n outc == acc)) #Hcase -[ % @Htrue @(\P Hcase) | %2 @Hfalse @(\Pf Hcase) ] -qed. -*) - lemma sem_rewind_strong : ∀src,dst,sig,n. src ≠ dst → src < S n → dst < S n → rewind src dst sig n ⊨ R_rewind_strong src dst sig n. @@ -198,6 +121,13 @@ lemma sem_rewind : ∀src,dst,sig,n. #ta #tb * * * #H1 #H2 #H3 #H4 % /2 by / qed. +(* match step *) + +definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n. + match (nth src (option sig) v (None ?)) with + [ None ⇒ false + | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ]. + definition match_step ≝ λsrc,dst,sig,n. compare src dst sig n · (ifTM ?? (partest sig n (match_test src dst sig ?)) diff --git a/matita/matita/lib/turing/multi_universal/moves.ma b/matita/matita/lib/turing/multi_universal/moves.ma deleted file mode 100644 index f725a1940..000000000 --- a/matita/matita/lib/turing/multi_universal/moves.ma +++ /dev/null @@ -1,280 +0,0 @@ -(* - ||M|| This file is part of HELM, an Hypertextual, Electronic - ||A|| Library of Mathematics, developed at the Computer Science - ||T|| Department of the University of Bologna, Italy. - ||I|| - ||T|| - ||A|| - \ / This file is distributed under the terms of the - \ / GNU General Public License Version 2 - V_____________________________________________________________*) - -include "turing/turing.ma". -include "turing/inject.ma". -include "turing/while_multi.ma". - -definition parmove_states ≝ initN 3. - -definition parmove0 : parmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)). -definition parmove1 : parmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)). -definition parmove2 : parmove_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)). - -(* - -src: a b c ... z # ---→ a b c ... z # - ^ ^ - -dst: _ _ _ ... _ d ---→ a b c ... z d - ^ ^ - -0) (x ≠ sep,_) → (x,x)(R,R) → 1 - (sep,d) → None 2 -1) (_,_) → None 1 -2) (_,_) → None 2 - -*) - -definition trans_parmove_step ≝ - λsrc,dst,sig,n,D,is_sep. - λp:parmove_states × (Vector (option sig) (S n)). - let 〈q,a〉 ≝ p in - match pi1 … q with - [ O ⇒ match nth src ? a (None ?) with - [ None ⇒ 〈parmove2,null_action ? n〉 - | Some a0 ⇒ - if is_sep a0 then 〈parmove2,null_action ? n〉 - else match nth dst ? a (None ?) with - [ None ⇒ 〈parmove2,null_action ? n〉 - | Some a1 ⇒ 〈parmove1,change_vec ? (S n) - (change_vec ?(S n) - (null_action ? n) (〈None sig,D〉) src) - (〈None ?,D〉) dst〉 ] ] - | S q ⇒ match q with - [ O ⇒ (* 1 *) 〈parmove1,null_action ? n〉 - | S _ ⇒ (* 2 *) 〈parmove2,null_action ? n〉 ] ]. - -definition parmove_step ≝ - λsrc,dst,sig,n,D,is_sep. - mk_mTM sig n parmove_states (trans_parmove_step src dst sig n D is_sep) - parmove0 (λq.q == parmove1 ∨ q == parmove2). - -definition R_parmove_step_true ≝ - λsrc,dst,sig,n,D,is_sep.λint,outt: Vector (tape sig) (S n). - ∃x1,x2. - current ? (nth src ? int (niltape ?)) = Some ? x1 ∧ - current ? (nth dst ? int (niltape ?)) = Some ? x2 ∧ - is_sep x1 = false ∧ - outt = change_vec ?? - (change_vec ?? int - (tape_move_mono ? (nth src ? int (niltape ?)) (〈None ?,D〉)) src) - (tape_move_mono ? (nth dst ? int (niltape ?)) (〈None ?,D〉)) dst. - -definition R_parmove_step_false ≝ - λsrc,dst:nat.λsig,n,is_sep.λint,outt: Vector (tape sig) (S n). - ((∃x1. - current ? (nth src ? int (niltape ?)) = Some ? x1 ∧ - is_sep x1 = true) ∨ - current ? (nth src ? int (niltape ?)) = None ? ∨ - current ? (nth dst ? int (niltape ?)) = None ?) ∧ - outt = int. - -lemma parmove_q0_q2_null_src : - ∀src,dst,sig,n,D,is_sep,v.src < S n → dst < S n → - nth src ? (current_chars ?? v) (None ?) = None ? → - step sig n (parmove_step src dst sig n D is_sep) - (mk_mconfig ??? parmove0 v) = - mk_mconfig ??? parmove2 v. -#src #dst #sig #n #D #is_sep #v #Hsrc #Hdst #Hcurrent -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); -@eq_f2 -[ whd in ⊢ (??(???%)?); >Hcurrent % -| whd in ⊢ (??(????(???%))?); >Hcurrent @tape_move_null_action ] -qed. - -lemma parmove_q0_q2_sep : - ∀src,dst,sig,n,D,is_sep,v,s.src < S n → dst < S n → - nth src ? (current_chars ?? v) (None ?) = Some ? s → is_sep s = true → - step sig n (parmove_step src dst sig n D is_sep) - (mk_mconfig ??? parmove0 v) = - mk_mconfig ??? parmove2 v. -#src #dst #sig #n #D #is_sep #v #s #Hsrc #Hdst #Hcurrent #Hsep -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); -@eq_f2 -[ whd in ⊢ (??(???%)?); >Hcurrent whd in ⊢ (??(???%)?); >Hsep % -| whd in ⊢ (??(????(???%))?); >Hcurrent - whd in ⊢ (??(????(???%))?); >Hsep @tape_move_null_action ] -qed. - -lemma parmove_q0_q2_null_dst : - ∀src,dst,sig,n,D,is_sep,v,s.src < S n → dst < S n → - nth src ? (current_chars ?? v) (None ?) = Some ? s → is_sep s = false → - nth dst ? (current_chars ?? v) (None ?) = None ? → - step sig n (parmove_step src dst sig n D is_sep) - (mk_mconfig ??? parmove0 v) = - mk_mconfig ??? parmove2 v. -#src #dst #sig #n #D #is_sep #v #s #Hsrc #Hdst #Hcursrc #Hsep #Hcurdst -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); -@eq_f2 -[ whd in ⊢ (??(???%)?); >Hcursrc whd in ⊢ (??(???%)?); >Hsep >Hcurdst % -| whd in ⊢ (??(????(???%))?); >Hcursrc - whd in ⊢ (??(????(???%))?); >Hsep >Hcurdst @tape_move_null_action ] -qed. - -lemma parmove_q0_q1 : - ∀src,dst,sig,n,D,is_sep,v.src ≠ dst → src < S n → dst < S n → - ∀a1,a2. - nth src ? (current_chars ?? v) (None ?) = Some ? a1 → - nth dst ? (current_chars ?? v) (None ?) = Some ? a2 → - is_sep a1 = false → - step sig n (parmove_step src dst sig n D is_sep) - (mk_mconfig ??? parmove0 v) = - mk_mconfig ??? parmove1 - (change_vec ? (S n) - (change_vec ?? v - (tape_move_mono ? (nth src ? v (niltape ?)) (〈None ?, D〉)) src) - (tape_move_mono ? (nth dst ? v (niltape ?)) (〈None ?, D〉)) dst). -#src #dst #sig #n #D #is_sep #v #Hneq #Hsrc #Hdst -#a1 #a2 #Hcursrc #Hcurdst #Hsep -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 -[ whd in match (trans ????); - >Hcursrc >Hcurdst whd in ⊢ (??(???%)?); >Hsep // -| whd in match (trans ????); - >Hcursrc >Hcurdst whd in ⊢ (??(????(???%))?); >Hsep whd in ⊢ (??(????(???%))?); - <(change_vec_same ?? v dst (niltape ?)) in ⊢ (??%?); - >tape_move_multi_def >pmap_change - <(change_vec_same ?? v src (niltape ?)) in ⊢ (??%?); - >pmap_change tape_move_null_action - @eq_f2 // @eq_f2 // >nth_change_vec_neq // -] -qed. - -lemma sem_parmove_step : - ∀src,dst,sig,n,D,is_sep.src ≠ dst → src < S n → dst < S n → - parmove_step src dst sig n D is_sep ⊨ - [ parmove1: R_parmove_step_true src dst sig n D is_sep, - R_parmove_step_false src dst sig n is_sep ]. -#src #dst #sig #n #D #is_sep #Hneq #Hsrc #Hdst #int -lapply (refl ? (current ? (nth src ? int (niltape ?)))) -cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?); -[ #Hcursrc %{2} % - [| % [ % - [ whd in ⊢ (??%?); >parmove_q0_q2_null_src /2/ - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // % %2 // ] ] -| #a #Ha cases (true_or_false (is_sep a)) #Hsep - [ %{2} % - [| % [ % - [ whd in ⊢ (??%?); >(parmove_q0_q2_sep … Hsep) /2/ - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // % % %{a} % // ] ] - | lapply (refl ? (current ? (nth dst ? int (niltape ?)))) - cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?); - [ #Hcurdst %{2} % - [| % [ % - [ whd in ⊢ (??%?); >(parmove_q0_q2_null_dst … Hsep) /2/ - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // %2 // ] ] - | #b #Hb %{2} % - [| % [ % - [whd in ⊢ (??%?); >(parmove_q0_q1 … Hneq Hsrc Hdst ? b ?? Hsep) // - | #_ %{a} %{b} % // % // % // ] - | * #H @False_ind @H % ] -]]]] -qed. - -definition parmove ≝ λsrc,dst,sig,n,D,is_sep. - whileTM … (parmove_step src dst sig n D is_sep) parmove1. - -definition R_parmoveL ≝ - λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n). - (∀ls,x,xs,rs,sep. - nth src ? int (niltape ?) = midtape sig (xs@sep::ls) x rs → - (∀c.memb ? c (x::xs) = true → is_sep c = false) → is_sep sep = true → - ∀ls0,x0,target,c,rs0.|xs| = |target| → - nth dst ? int (niltape ?) = midtape sig (target@c::ls0) x0 rs0 → - outt = change_vec ?? - (change_vec ?? int (midtape sig ls sep (reverse ? xs@x::rs)) src) - (midtape sig ls0 c (reverse ? target@x0::rs0)) dst) ∧ - (((∃s.current ? (nth src ? int (niltape ?)) = Some ? s ∧ is_sep s = true) ∨ - current ? (nth src ? int (niltape ?)) = None ? ∨ - current ? (nth dst ? int (niltape ?)) = None ?) → - outt = int). - -lemma wsem_parmoveL : ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst < S n → - parmove src dst sig n L is_sep ⊫ R_parmoveL src dst sig n is_sep. -#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #ta #k #outc #Hloop -lapply (sem_while … (sem_parmove_step src dst sig n L is_sep Hneq Hsrc Hdst) … Hloop) // --Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ whd in ⊢ (%→?); * #H #Houtc % [2: #_ @Houtc ] cases H - [ * [ * #x * #Hx #Hsep #ls #x0 #xs #rs #sep #Hsrctc #Hnosep >Hsrctc in Hx; normalize in ⊢ (%→?); - #Hx0 destruct (Hx0) lapply (Hnosep ? (memb_hd …)) >Hsep - #Hfalse destruct (Hfalse) - | #Hcur_src #ls #x0 #xs #rs #sep #Hsrctc >Hsrctc in Hcur_src; - normalize in ⊢ (%→?); #H destruct (H)] - |#Hcur_dst #ls #x0 #xs #rs #sep #Hsrctc #Hnosep #Hsep #ls0 #x1 #target - #c #rs0 #Hlen #Hdsttc >Hdsttc in Hcur_dst; normalize in ⊢ (%→?); #H destruct (H) - ] -| #td #te * #c0 * #c1 * * * #Hc0 #Hc1 #Hc0nosep #Hd #Hstar #IH #He - lapply (IH He) -IH * #IH1 #IH2 % - [ #ls #x #xs #rs #sep #Hsrc_tc #Hnosep #Hsep #ls0 #x0 #target - #c #rs0 #Hlen #Hdst_tc - >Hsrc_tc in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0) - >Hdst_tc in Hd; >Hsrc_tc @(list_cases2 … Hlen) - [ #Hxsnil #Htargetnil >Hxsnil >Htargetnil #Hd >IH2 - [2: %1 %1 %{sep} % // >Hd >nth_change_vec_neq [|@(sym_not_eq … Hneq)] - >nth_change_vec //] - >Hd -Hd @(eq_vec … (niltape ?)) - #i #Hi cases (decidable_eq_nat i src) #Hisrc - [ >Hisrc >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)] - >nth_change_vec // - | cases (decidable_eq_nat i dst) #Hidst - [ >Hidst >nth_change_vec // - | >nth_change_vec_neq [|@(sym_not_eq … Hidst)] - >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] % - ] - ] - | #hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd - >(IH1 ls hd1 tl1 (c0::rs) sep ?? Hsep ls0 hd2 tl2 c (x0::rs0)) - [ >Hd >(change_vec_commute … ?? td ?? src dst) // - >change_vec_change_vec - >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //] - >change_vec_change_vec - >reverse_cons >associative_append - >reverse_cons >associative_append % - | >Hd >nth_change_vec // - | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) // - | Hd >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // ] - ] - | >Hc0 >Hc1 * [* [ * #c * #Hc destruct (Hc) >Hc0nosep]] #Habs destruct (Habs) - ] ] -qed. - -lemma terminate_parmoveL : ∀src,dst,sig,n,is_sep,t. - src ≠ dst → src < S n → dst < S n → - parmove src dst sig n L is_sep ↓ t. -#src #dst #sig #n #is_sep #t #Hneq #Hsrc #Hdst -@(terminate_while … (sem_parmove_step …)) // -<(change_vec_same … t src (niltape ?)) -cases (nth src (tape sig) t (niltape ?)) -[ % #t1 * #x1 * #x2 * * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct -|2,3: #a0 #al0 % #t1 * #x1 * #x2 * * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct -| #ls lapply t -t elim ls - [#t #c #rs % #t1 * #x1 * #x2 * * * >nth_change_vec // normalize in ⊢ (%→?); - #H1 destruct (H1) #Hcurdst #Hxsep >change_vec_change_vec #Ht1 % - #t2 * #y1 * #y2 * * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) - |#l0 #ls0 #IH #t #c #rs % #t1 * #x1 * #x2 * * * >nth_change_vec // - normalize in ⊢ (%→?); #H destruct (H) #Hcurdst #Hxsep - >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH - ] -] -qed. - -lemma sem_parmoveL : ∀src,dst,sig,n,is_sep. - src ≠ dst → src < S n → dst < S n → - parmove src dst sig n L is_sep ⊨ R_parmoveL src dst sig n is_sep. -#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst @WRealize_to_Realize -[/2/ | @wsem_parmoveL //] -qed. \ No newline at end of file diff --git a/matita/matita/lib/turing/multi_universal/moves_2.ma b/matita/matita/lib/turing/multi_universal/moves_2.ma deleted file mode 100644 index a3af6106c..000000000 --- a/matita/matita/lib/turing/multi_universal/moves_2.ma +++ /dev/null @@ -1,421 +0,0 @@ -(* - ||M|| This file is part of HELM, an Hypertextual, Electronic - ||A|| Library of Mathematics, developed at the Computer Science - ||T|| Department of the University of Bologna, Italy. - ||I|| - ||T|| - ||A|| - \ / This file is distributed under the terms of the - \ / GNU General Public License Version 2 - V_____________________________________________________________*) - -include "turing/turing.ma". -include "turing/inject.ma". -include "turing/while_multi.ma". -include "turing/while_machine.ma". -include "turing/simple_machines.ma". -include "turing/if_machine.ma". - -definition parmove_states ≝ initN 3. - -definition parmove0 : parmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)). -definition parmove1 : parmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)). -definition parmove2 : parmove_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)). - -(* - -src: a b c ... z ---→ a b c ... z - ^ ^ - -dst: _ _ _ ... _ ---→ a b c ... z - ^ ^ - -0) (x,_) → (x,_)(R,R) → 1 - (None,_) → None 2 -1) (_,_) → None 1 -2) (_,_) → None 2 - -*) - -definition trans_parmove_step ≝ - λsrc,dst,sig,n,D. - λp:parmove_states × (Vector (option sig) (S n)). - let 〈q,a〉 ≝ p in - match pi1 … q with - [ O ⇒ match nth src ? a (None ?) with - [ None ⇒ 〈parmove2,null_action sig n〉 - | Some a0 ⇒ match nth dst ? a (None ?) with - [ None ⇒ 〈parmove2,null_action ? n〉 - | Some a1 ⇒ 〈parmove1,change_vec ? (S n) - (change_vec ?(S n) - (null_action ? n) (〈None ?,D〉) src) - (〈None ?,D〉) dst〉 ] ] - | S q ⇒ match q with - [ O ⇒ (* 1 *) 〈parmove1,null_action ? n〉 - | S _ ⇒ (* 2 *) 〈parmove2,null_action ? n〉 ] ]. - -definition parmove_step ≝ - λsrc,dst,sig,n,D. - mk_mTM sig n parmove_states (trans_parmove_step src dst sig n D) - parmove0 (λq.q == parmove1 ∨ q == parmove2). - -definition R_parmove_step_true ≝ - λsrc,dst,sig,n,D.λint,outt: Vector (tape sig) (S n). - ∃x1,x2. - current ? (nth src ? int (niltape ?)) = Some ? x1 ∧ - current ? (nth dst ? int (niltape ?)) = Some ? x2 ∧ - outt = change_vec ?? - (change_vec ?? int - (tape_move ? (nth src ? int (niltape ?)) D) src) - (tape_move ? (nth dst ? int (niltape ?)) D) dst. - -definition R_parmove_step_false ≝ - λsrc,dst:nat.λsig,n.λint,outt: Vector (tape sig) (S n). - (current ? (nth src ? int (niltape ?)) = None ? ∨ - current ? (nth dst ? int (niltape ?)) = None ?) ∧ - outt = int. - -lemma parmove_q0_q2_null_src : - ∀src,dst,sig,n,D,v.src < S n → dst < S n → - nth src ? (current_chars ?? v) (None ?) = None ? → - step sig n (parmove_step src dst sig n D) - (mk_mconfig ??? parmove0 v) = - mk_mconfig ??? parmove2 v. -#src #dst #sig #n #D #v #Hsrc #Hdst #Hcurrent -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); -@eq_f2 -[ whd in ⊢ (??(???%)?); >Hcurrent % -| whd in ⊢ (??(????(???%))?); >Hcurrent @tape_move_null_action ] -qed. - -lemma parmove_q0_q2_null_dst : - ∀src,dst,sig,n,D,v,s.src < S n → dst < S n → - nth src ? (current_chars ?? v) (None ?) = Some ? s → - nth dst ? (current_chars ?? v) (None ?) = None ? → - step sig n (parmove_step src dst sig n D) - (mk_mconfig ??? parmove0 v) = - mk_mconfig ??? parmove2 v. -#src #dst #sig #n #D #v #s #Hsrc #Hdst #Hcursrc #Hcurdst -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); -@eq_f2 -[ whd in ⊢ (??(???%)?); >Hcursrc whd in ⊢ (??(???%)?); >Hcurdst % -| whd in ⊢ (??(????(???%))?); >Hcursrc - whd in ⊢ (??(????(???%))?); >Hcurdst @tape_move_null_action ] -qed. - -lemma parmove_q0_q1 : - ∀src,dst,sig,n,D,v.src ≠ dst → src < S n → dst < S n → - ∀a1,a2. - nth src ? (current_chars ?? v) (None ?) = Some ? a1 → - nth dst ? (current_chars ?? v) (None ?) = Some ? a2 → - step sig n (parmove_step src dst sig n D) - (mk_mconfig ??? parmove0 v) = - mk_mconfig ??? parmove1 - (change_vec ? (S n) - (change_vec ?? v - (tape_move ? (nth src ? v (niltape ?)) D) src) - (tape_move ? (nth dst ? v (niltape ?)) D) dst). -#src #dst #sig #n #D #v #Hneq #Hsrc #Hdst -#a1 #a2 #Hcursrc #Hcurdst -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2 -[ whd in match (trans ????); - >Hcursrc >Hcurdst % -| whd in match (trans ????); - >Hcursrc >Hcurdst whd in ⊢ (??(????(???%))?); - >tape_move_multi_def <(change_vec_same ?? v dst (niltape ?)) in ⊢ (??%?); - >pmap_change <(change_vec_same ?? v src (niltape ?)) in ⊢(??%?); - >pmap_change tape_move_null_action - @eq_f2 // >nth_change_vec_neq // -] -qed. - -lemma sem_parmove_step : - ∀src,dst,sig,n,D.src ≠ dst → src < S n → dst < S n → - parmove_step src dst sig n D ⊨ - [ parmove1: R_parmove_step_true src dst sig n D, - R_parmove_step_false src dst sig n ]. -#src #dst #sig #n #D #Hneq #Hsrc #Hdst #int -lapply (refl ? (current ? (nth src ? int (niltape ?)))) -cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?); -[ #Hcursrc %{2} % - [| % [ % - [ whd in ⊢ (??%?); >parmove_q0_q2_null_src /2/ - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // % // ] ] -| #a #Ha lapply (refl ? (current ? (nth dst ? int (niltape ?)))) - cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?); - [ #Hcurdst %{2} % - [| % [ % - [ whd in ⊢ (??%?); >(parmove_q0_q2_null_dst …) /2/ - | normalize in ⊢ (%→?); #H destruct (H) ] - | #_ % // %2 // ] ] - | #b #Hb %{2} % - [| % [ % - [whd in ⊢ (??%?); >(parmove_q0_q1 … Hneq Hsrc Hdst ? b ??) - [2: <(nth_vec_map ?? (current …) dst ? int (niltape ?)) // - |3: <(nth_vec_map ?? (current …) src ? int (niltape ?)) // - | // ] - | #_ %{a} %{b} % // % // ] - | * #H @False_ind @H % ] -]]] -qed. - -definition parmove ≝ λsrc,dst,sig,n,D. - whileTM … (parmove_step src dst sig n D) parmove1. - -definition R_parmoveL ≝ - λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n). - (∀x,xs,rs. - nth src ? int (niltape ?) = midtape sig xs x rs → - ∀ls0,x0,target,rs0.|xs| = |target| → - nth dst ? int (niltape ?) = midtape sig (target@ls0) x0 rs0 → - outt = change_vec ?? - (change_vec ?? int (mk_tape sig [] (None ?) (reverse ? xs@x::rs)) src) - (mk_tape sig (tail ? ls0) (option_hd ? ls0) (reverse ? target@x0::rs0)) dst) ∧ - (∀x,xs,rs. - nth dst ? int (niltape ?) = midtape sig xs x rs → - ∀ls0,x0,target,rs0.|xs| = |target| → - nth src ? int (niltape ?) = midtape sig (target@ls0) x0 rs0 → - outt = change_vec ?? - (change_vec ?? int (mk_tape sig [] (None ?) (reverse ? xs@x::rs)) dst) - (mk_tape sig (tail ? ls0) (option_hd ? ls0) (reverse ? target@x0::rs0)) src) ∧ - ((current ? (nth src ? int (niltape ?)) = None ? ∨ - current ? (nth dst ? int (niltape ?)) = None ?) → - outt = int). - -lemma wsem_parmoveL : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → - parmove src dst sig n L ⊫ R_parmoveL src dst sig n. -#src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop -lapply (sem_while … (sem_parmove_step src dst sig n L Hneq Hsrc Hdst) … Hloop) // --Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ whd in ⊢ (%→?); * #H #Houtc % [2: #_ @Houtc ] cases H #Hcurtb - [ % - [ #x #xs #rs #Hsrctb >Hsrctb in Hcurtb; normalize in ⊢ (%→?); - #Hfalse destruct (Hfalse) - | #x #xs #rs #Hdsttb #ls0 #x0 #target #rs0 #Hlen #Hsrctb >Hsrctb in Hcurtb; - normalize in ⊢ (%→?); #H destruct (H) - ] - | % - [ #x #xs #rs #Hsrctb #ls0 #x0 #target - #rs0 #Hlen #Hdsttb >Hdsttb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H) - | #x #xs #rs #Hdsttb >Hdsttb in Hcurtb; normalize in ⊢ (%→?); - #Hfalse destruct (Hfalse) - ] - ] -| #td #te * #c0 * #c1 * * #Hc0 #Hc1 #Hd #Hstar #IH #He - lapply (IH He) -IH * * #IH1a #IH1b #IH2 % [ % - [ #x #xs #rs #Hsrc_td #ls0 #x0 #target - #rs0 #Hlen #Hdst_td - >Hsrc_td in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0) - >Hdst_td in Hd; >Hsrc_td @(list_cases2 … Hlen) - [ #Hxsnil #Htargetnil >Hxsnil >Htargetnil #Hd >IH2 - [2: %1 >Hd >nth_change_vec_neq [|@(sym_not_eq … Hneq)] - >nth_change_vec //] - >Hd -Hd @(eq_vec … (niltape ?)) - #i #Hi cases (decidable_eq_nat i src) #Hisrc - [ >Hisrc >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)] - >nth_change_vec // - >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)] - >nth_change_vec // - | cases (decidable_eq_nat i dst) #Hidst - [ >Hidst >nth_change_vec // >nth_change_vec // - >Hdst_td in Hc1; >Htargetnil - normalize in ⊢ (%→?); #Hc1 destruct (Hc1) cases ls0 // - | >nth_change_vec_neq [|@(sym_not_eq … Hidst)] - >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] - >nth_change_vec_neq [|@(sym_not_eq … Hidst)] - >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] % - ] - ] - | #hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd - >(IH1a hd1 tl1 (c0::rs) ? ls0 hd2 tl2 (x0::rs0)) - [ >Hd >(change_vec_commute … ?? td ?? src dst) // - >change_vec_change_vec - >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //] - >change_vec_change_vec - >reverse_cons >associative_append - >reverse_cons >associative_append % - | >Hd >nth_change_vec // - | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) // - | >Hd >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // ] - ] - | #x #xs #rs #Hdst_td #ls0 #x0 #target - #rs0 #Hlen #Hsrc_td - >Hdst_td in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0) - >Hsrc_td in Hd; >Hdst_td @(list_cases2 … Hlen) - [ #Hxsnil #Htargetnil >Hxsnil >Htargetnil #Hd >IH2 - [2: %2 >Hd >nth_change_vec //] - >Hd -Hd @(eq_vec … (niltape ?)) - #i #Hi cases (decidable_eq_nat i dst) #Hidst - [ >Hidst >(nth_change_vec_neq … dst src) // - >nth_change_vec // >nth_change_vec // - | cases (decidable_eq_nat i src) #Hisrc - [ >Hisrc >nth_change_vec // >(nth_change_vec_neq …) [|@sym_not_eq //] - >Hsrc_td in Hc1; >Htargetnil - normalize in ⊢ (%→?); #Hc1 destruct (Hc1) >nth_change_vec // - cases ls0 // - | >nth_change_vec_neq [|@(sym_not_eq … Hidst)] - >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] - >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] - >nth_change_vec_neq [|@(sym_not_eq … Hidst)] % - ] - ] - | #hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd - >(IH1b hd1 tl1 (x::rs) ? ls0 hd2 tl2 (x0::rs0)) - [ >Hd >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //] - >change_vec_change_vec - >(change_vec_commute … ?? td ?? src dst) // - >change_vec_change_vec - >reverse_cons >associative_append - >reverse_cons >associative_append - >change_vec_commute [|@sym_not_eq //] % - | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // - | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) // - | >Hd >nth_change_vec // ] - ] - ] -| >Hc0 >Hc1 * [ #Hc0 destruct (Hc0) | #Hc1 destruct (Hc1) ] -] ] -qed. - -lemma terminate_parmoveL : ∀src,dst,sig,n,t. - src ≠ dst → src < S n → dst < S n → - parmove src dst sig n L ↓ t. -#src #dst #sig #n #t #Hneq #Hsrc #Hdst -@(terminate_while … (sem_parmove_step …)) // -<(change_vec_same … t src (niltape ?)) -cases (nth src (tape sig) t (niltape ?)) -[ % #t1 * #x1 * #x2 * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct -|2,3: #a0 #al0 % #t1 * #x1 * #x2 * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct -| #ls lapply t -t elim ls - [#t #c #rs % #t1 * #x1 * #x2 * * >nth_change_vec // normalize in ⊢ (%→?); - #H1 destruct (H1) #Hcurdst >change_vec_change_vec #Ht1 % - #t2 * #y1 * #y2 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //] - >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) - |#l0 #ls0 #IH #t #c #rs % #t1 * #x1 * #x2 * * >nth_change_vec // - normalize in ⊢ (%→?); #H destruct (H) #Hcurdst - >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH - ] -] -qed. - -lemma sem_parmoveL : ∀src,dst,sig,n. - src ≠ dst → src < S n → dst < S n → - parmove src dst sig n L ⊨ R_parmoveL src dst sig n. -#src #dst #sig #n #Hneq #Hsrc #Hdst @WRealize_to_Realize -[/2/ | @wsem_parmoveL //] -qed. - -(* while { - if current != null - then move_r - else nop - } - *) - -definition mte_step ≝ λalpha,D. -ifTM ? (test_null alpha) (single_finalTM ? (move alpha D)) (nop ?) tc_true. - -definition R_mte_step_true ≝ λalpha,D,t1,t2. - ∃ls,c,rs. - t1 = midtape alpha ls c rs ∧ t2 = tape_move ? t1 D. - -definition R_mte_step_false ≝ λalpha.λt1,t2:tape alpha. - current ? t1 = None ? ∧ t1 = t2. - -definition mte_acc : ∀alpha,D.states ? (mte_step alpha D) ≝ -λalpha,D.(inr … (inl … (inr … start_nop))). - -lemma sem_mte_step : - ∀alpha,D.mte_step alpha D ⊨ - [ mte_acc … : R_mte_step_true alpha D, R_mte_step_false alpha ] . -#alpha #D #ta -@(acc_sem_if_app ??????????? (sem_test_null …) - (sem_move_single …) (sem_nop alpha) ??) -[ #tb #tc #td * #Hcurtb - lapply (refl ? (current ? tb)) cases (current ? tb) in ⊢ (???%→?); - [ #H @False_ind >H in Hcurtb; * /2/ ] - -Hcurtb #c #Hcurtb #Htb whd in ⊢ (%→?); #Htc whd - cases (current_to_midtape … Hcurtb) #ls * #rs #Hmidtb - %{ls} %{c} %{rs} % // -| #tb #tc #td * #Hcurtb #Htb whd in ⊢ (%→?); #Htc whd % // ] -qed. - -definition move_to_end ≝ λsig,D.whileTM sig (mte_step sig D) (mte_acc …). - -definition R_move_to_end_r ≝ - λsig,int,outt. - (current ? int = None ? → outt = int) ∧ - ∀ls,c,rs.int = midtape sig ls c rs → outt = mk_tape ? (reverse ? rs@c::ls) (None ?) [ ]. - -lemma wsem_move_to_end_r : ∀sig. move_to_end sig R ⊫ R_move_to_end_r sig. -#sig #ta #k #outc #Hloop -lapply (sem_while … (sem_mte_step sig R) … Hloop) // --Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ * #Hcurtb #Houtc % /2/ #ls #c #rs #Htb >Htb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H) -| #tc #td * #ls * #c * #rs * #Htc >Htc cases rs - [ normalize in ⊢ (%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse - lapply (IH Hfalse) -IH * #Htd1 #_ % - [ normalize in ⊢ (%→?); #H destruct (H) - | #ls0 #c0 #rs0 #H destruct (H) >Htd1 // ] - | #r0 #rs0 whd in ⊢ (???%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse - lapply (IH Hfalse) -IH * #_ #IH % - [ normalize in ⊢ (%→?); #H destruct (H) - | #ls1 #c1 #rs1 #H destruct (H) >reverse_cons >associative_append @IH % ] ] ] -qed. - -lemma terminate_move_to_end_r : ∀sig,t.move_to_end sig R ↓ t. -#sig #t @(terminate_while … (sem_mte_step sig R …)) // -cases t -[ % #t1 * #ls * #c * #rs * #H destruct -|2,3: #a0 #al0 % #t1 * #ls * #c * #rs * #H destruct -| #ls #c #rs lapply c -c lapply ls -ls elim rs - [ #ls #c % #t1 * #ls0 * #c0 * #rs0 * #Hmid #Ht1 destruct % - #t2 * #ls1 * #c1 * #rs1 * normalize in ⊢ (%→?); #H destruct - | #r0 #rs0 #IH #ls #c % #t1 * #ls1 * #c1 * #rs1 * #Hmid #Ht1 destruct @IH - ] -] -qed. - -lemma sem_move_to_end_r : ∀sig. move_to_end sig R ⊨ R_move_to_end_r sig. -#sig @WRealize_to_Realize // -qed. - -definition R_move_to_end_l ≝ - λsig,int,outt. - (current ? int = None ? → outt = int) ∧ - ∀ls,c,rs.int = midtape sig ls c rs → outt = mk_tape ? [ ] (None ?) (reverse ? ls@c::rs). - -lemma wsem_move_to_end_l : ∀sig. move_to_end sig L ⊫ R_move_to_end_l sig. -#sig #ta #k #outc #Hloop -lapply (sem_while … (sem_mte_step sig L) … Hloop) // --Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -[ * #Hcurtb #Houtc % /2/ #ls #c #rs #Htb >Htb in Hcurtb; normalize in ⊢ (%→?); #H destruct (H) -| #tc #td * #ls * #c * #rs * #Htc >Htc cases ls - [ normalize in ⊢ (%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse - lapply (IH Hfalse) -IH * #Htd1 #_ % - [ normalize in ⊢ (%→?); #H destruct (H) - | #ls0 #c0 #rs0 #H destruct (H) >Htd1 // ] - | #l0 #ls0 whd in ⊢ (???%→?); #Htd >Htd #Hstar #IH whd in ⊢ (%→?); #Hfalse - lapply (IH Hfalse) -IH * #_ #IH % - [ normalize in ⊢ (%→?); #H destruct (H) - | #ls1 #c1 #rs1 #H destruct (H) >reverse_cons >associative_append @IH % ] ] ] -qed. - -lemma terminate_move_to_end_l : ∀sig,t.move_to_end sig L ↓ t. -#sig #t @(terminate_while … (sem_mte_step sig L …)) // -cases t -[ % #t1 * #ls * #c * #rs * #H destruct -|2,3: #a0 #al0 % #t1 * #ls * #c * #rs * #H destruct -| #ls elim ls - [ #c #rs % #t1 * #ls0 * #c0 * #rs0 * #Hmid #Ht1 destruct % - #t2 * #ls1 * #c1 * #rs1 * normalize in ⊢ (%→?); #H destruct - | #l0 #ls0 #IH #c #rs % #t1 * #ls1 * #c1 * #rs1 * #Hmid #Ht1 destruct @IH - ] -] -qed. - -lemma sem_move_to_end_l : ∀sig. move_to_end sig L ⊨ R_move_to_end_l sig. -#sig @WRealize_to_Realize // -qed. diff --git a/matita/matita/lib/turing/multi_universal/par_test.ma b/matita/matita/lib/turing/multi_universal/par_test.ma deleted file mode 100644 index 1ab0e34a4..000000000 --- a/matita/matita/lib/turing/multi_universal/par_test.ma +++ /dev/null @@ -1,81 +0,0 @@ -(* - ||M|| This file is part of HELM, an Hypertextual, Electronic - ||A|| Library of Mathematics, developed at the Computer Science - ||T|| Department of the University of Bologna, Italy. - ||I|| - ||T|| - ||A|| - \ / This file is distributed under the terms of the - \ / GNU General Public License Version 2 - V_____________________________________________________________*) - -include "turing/turing.ma". -include "turing/inject.ma". -include "turing/while_multi.ma". - -definition partest_states ≝ initN 3. - -definition partest0 : partest_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)). -definition partest1 : partest_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)). -definition partest2 : partest_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)). - -definition trans_partest ≝ - λsig,n,test. - λp:partest_states × (Vector (option sig) (S n)). - let 〈q,a〉 ≝ p in - if test a then 〈partest1,null_action sig n〉 - else 〈partest2,null_action ? n〉. - -definition partest ≝ - λsig,n,test. - mk_mTM sig n partest_states (trans_partest sig n test) - partest0 (λq.q == partest1 ∨ q == partest2). - -definition R_partest_true ≝ - λsig,n,test.λint,outt: Vector (tape sig) (S n). - test (current_chars ?? int) = true ∧ outt = int. - -definition R_partest_false ≝ - λsig,n,test.λint,outt: Vector (tape sig) (S n). - test (current_chars ?? int) = false ∧ outt = int. - -lemma partest_q0_q1: - ∀sig,n,test,v. - test (current_chars ?? v) = true → - step sig n (partest sig n test)(mk_mconfig ??? partest0 v) - = mk_mconfig ??? partest1 v. -#sig #n #test #v #Htest -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); -@eq_f2 -[ whd in ⊢ (??(???%)?); >Htest % -| whd in ⊢ (??(????(???%))?); >Htest @tape_move_null_action ] -qed. - -lemma partest_q0_q2: - ∀sig,n,test,v. - test (current_chars ?? v) = false → - step sig n (partest sig n test)(mk_mconfig ??? partest0 v) - = mk_mconfig ??? partest2 v. -#sig #n #test #v #Htest -whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); -@eq_f2 -[ whd in ⊢ (??(???%)?); >Htest % -| whd in ⊢ (??(????(???%))?); >Htest @tape_move_null_action ] -qed. - -lemma sem_partest: - ∀sig,n,test. - partest sig n test ⊨ - [ partest1: R_partest_true sig n test, R_partest_false sig n test ]. -#sig #n #test #int -cases (true_or_false (test (current_chars ?? int))) #Htest -[ %{2} %{(mk_mconfig ? partest_states n partest1 int)} % - [ % [ whd in ⊢ (??%?); >partest_q0_q1 /2/ | #_ % // ] - | * #H @False_ind @H % - ] -| %{2} %{(mk_mconfig ? partest_states n partest2 int)} % - [ % [ whd in ⊢ (??%?); >partest_q0_q2 /2/ - | whd in ⊢ (??%%→?); #H destruct (H)] - | #_ % //] -] -qed. \ No newline at end of file diff --git a/matita/matita/lib/turing/multi_universal/tuples.ma b/matita/matita/lib/turing/multi_universal/tuples.ma index 425257ea5..4a7f2b5f2 100644 --- a/matita/matita/lib/turing/multi_universal/tuples.ma +++ b/matita/matita/lib/turing/multi_universal/tuples.ma @@ -227,59 +227,3 @@ cut (〈q11,a11,m1〉=〈q21,a21,m2〉) #Heqout tuplet1 @append_l2_injective % qed. - -lemma cfg_in_table_to_tuple: ∀n,l,h,c. is_config n c → - ∀ll,lr.table_TM n l h = ll@c@lr → - ∃out,m,lr0. lr = out@m::lr0 ∧ is_config n (bar::out) ∧ m ≠ bar. -#n #l #h #c * #qin * #cin * * * #H1 #H2 #H3 #H4 -#ll #lr lapply ll -ll elim l - [>H4 #ll cases ll normalize [|#hd #tl ] #Habs destruct - |#t1 #othert #Hind #ll >table_TM_cons #Htuple - cut (S n < |ll@c@lr|) - [length_append >(length_of_tuple … (is_tuple … )) - /2 by transitive_lt, le_n/] #Hsplit lapply Htuple -Htuple - cases (is_tuple … n h t1) #q1 * #c1 * #q2 * #c2 * #m - * * * * * * * #Hq1 #Hq2 #Hc1 #Hc2 #Hm #Hlen1 #Hlen2 - whd in ⊢ (???%→?); #Ht1 - (* if ll is empty we match the first tuple t1, otherwise - we match inside othert *) - cases ll - [>H4 >Ht1 normalize in ⊢ (???%→?); - >associative_append whd in ⊢ (??%?→?); #Heq destruct (Heq) -Heq - >associative_append in e0; #e0 - lapply (append_l1_injective … e0) [>H3 @Hlen1] #Heq1 - lapply (append_l2_injective … e0) [>H3 @Hlen1] - normalize in ⊢ (???%→?); whd in ⊢ (??%?→?); #Htemp - lapply (cons_injective_l ????? Htemp) #Hc1 - lapply (cons_injective_r ????? Htemp) -Htemp #Heq2 - %{(q2@[c2])} %{m} %{(table_TM n othert h)} % // % - [ associative_append >associative_append % | %{q2} %{c2} % // % // % // ] - |(* ll not nil *) - #b #tl >Ht1 normalize in ⊢ (???%→?); - whd in ⊢ (??%?→?); #Heq destruct (Heq) - cases (compare_append … e0) #l * - [* cases l - [#_ #Htab cases (Hind [ ] (sym_eq … Htab)) #out * #m0 * #lr0 * * #Hlr #Hcfg #Hm0 - %{out} %{m0} %{lr0} % // % // - |(* this case is absurd *) - #al #tll #Heq1 >H4 #Heq2 @False_ind - lapply (cons_injective_l ? bar … Heq2) #Hbar Heq1 @mem_append_l2 %1 // - |% #Hmembar cases (mem_append ???? Hmembar) -Hmembar - [#Hmembar lapply(Hq1 bar Hmembar) normalize #Habs destruct (Habs) - |* [#Habs @absurd //] - #Hmembar cases (mem_append ???? Hmembar) -Hmembar - [#Hmembar lapply(Hq2 bar Hmembar) normalize #Habs destruct (Habs) - |* [#Habs @absurd //] #Hmembar @(absurd ?? Hm) @sym_eq @mem_single // - ] - ] - ] - ] - |* #Htl #Htab cases (Hind … Htab) #out * #m0 * #lr0 * * #Hlr #Hcfg #Hm0 - %{out} %{m0} %{lr0} % // % // - ] - ] - ] -qed. - diff --git a/matita/matita/lib/turing/multi_universal/unistep.ma b/matita/matita/lib/turing/multi_universal/unistep.ma index 76ec137ff..ee5948141 100644 --- a/matita/matita/lib/turing/multi_universal/unistep.ma +++ b/matita/matita/lib/turing/multi_universal/unistep.ma @@ -11,6 +11,7 @@ include "turing/multi_universal/unistep_aux.ma". +include "turing/multi_universal/match.ma". definition exec_move ≝ cfg_to_obj · tape_move_obj · restart_tape prg 2 · obj_to_cfg. diff --git a/matita/matita/lib/turing/multi_universal/unistep_aux.ma b/matita/matita/lib/turing/multi_universal/unistep_aux.ma index c4a5dd128..8033c2226 100644 --- a/matita/matita/lib/turing/multi_universal/unistep_aux.ma +++ b/matita/matita/lib/turing/multi_universal/unistep_aux.ma @@ -9,9 +9,8 @@ \ / GNU General Public License Version 2 V_____________________________________________________________*) -include "turing/multi_universal/moves_2.ma". -include "turing/multi_universal/match.ma". -include "turing/multi_universal/copy.ma". +include "turing/auxiliary_machines.ma". +include "turing/auxiliary_multi_machines.ma". include "turing/multi_universal/alphabet.ma". include "turing/multi_universal/tuples.ma". diff --git a/matita/matita/lib/turing/multi_universal/universal.ma b/matita/matita/lib/turing/multi_universal/universal.ma index 9b245cde9..ce8f329c8 100644 --- a/matita/matita/lib/turing/multi_universal/universal.ma +++ b/matita/matita/lib/turing/multi_universal/universal.ma @@ -9,7 +9,7 @@ \ / GNU General Public License Version 2 V_____________________________________________________________*) -include "turing/simple_machines.ma". +include "turing/auxiliary_multi_machines.ma". include "turing/multi_universal/unistep.ma". definition stop ≝ λc.