From 302d83e414e27210fbfbc1de4d213786d9580e23 Mon Sep 17 00:00:00 2001 From: Andrea Asperti Date: Tue, 27 Nov 2012 07:25:16 +0000 Subject: [PATCH] splitting files --- .../lib/turing/multi_universal/compare.ma | 345 ++++++++++++++++++ 1 file changed, 345 insertions(+) create mode 100644 matita/matita/lib/turing/multi_universal/compare.ma diff --git a/matita/matita/lib/turing/multi_universal/compare.ma b/matita/matita/lib/turing/multi_universal/compare.ma new file mode 100644 index 000000000..232a4962b --- /dev/null +++ b/matita/matita/lib/turing/multi_universal/compare.ma @@ -0,0 +1,345 @@ +(**************************************************************************) +(* ___ *) +(* ||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.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. -- 2.39.2