-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
+(*
+ ||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/multi_universal/moves.ma".
+include "turing/if_multi.ma".
include "turing/inject.ma".
+include "turing/basic_machines.ma".
definition copy_states ≝ initN 3.
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 …)).
-(*
-
-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_copy_step ≝
- λsrc,dst,sig,n,is_sep.
+ 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 ? n〉
- | Some a0 ⇒ if is_sep a0 then 〈copy2,null_action ? n〉
- else 〈copy1,change_vec ? (S n)
- (change_vec ?(S n)
- (null_action ? n) (Some ? 〈a0,R〉) src)
- (Some ? 〈a0,R〉) dst〉 ]
+ [ 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,is_sep.
- mk_mTM sig n copy_states (trans_copy_step src dst sig n is_sep)
+ 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 ≝
- λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n).
- (∀x1,x2,xls,xrs.
- nth src ? int (niltape ?) = midtape sig xls x1 (x2::xrs) →
- (is_sep x1 = true ∧ outt = int) ∨
- (is_sep x1 = false ∧
- outt = change_vec ??
- (change_vec ?? int (midtape sig (x1::xls) x2 xrs) src)
- (tape_move ? (nth dst ? int (niltape ?)) (Some ? 〈x1,R〉)) dst)) ∧
- (current ? (nth src ? int (niltape ?)) = None ? →
- outt = int).
+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.
-lemma copy_q0_q2_null :
- ∀src,dst,sig,n,is_sep,v,t.src < S n → dst < S n →
- current ? t = None ? →
- step sig n (copy_step src dst sig n is_sep)
- (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) =
- mk_mconfig ??? copy2 (change_vec ? (S n) v t src).
-#src #dst #sig #n #is_sep #v #t #Hsrc #Hdst #Hcurrent
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent %
-| >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent @tape_move_null_action
-]
-qed.
+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_sep :
- ∀src,dst,sig,n,is_sep,v,t.src < S n → dst < S n →
- ∀s.current ? t = Some ? s → is_sep s = true →
- step sig n (copy_step src dst sig n is_sep)
- (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) =
- mk_mconfig ??? copy2 (change_vec ? (S n) v t src).
-#src #dst #sig #n #is_sep #v #t #Hsrc #Hdst #s #Hcurrent #Hsep
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent whd in ⊢ (??(???%)?); >Hsep %
-| >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent whd in ⊢ (??(???????(???%))?);
- >Hsep @tape_move_null_action
-]
+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,is_sep,v,t.src < S n → dst < S n →
- ∀s.current ? t = Some ? s → is_sep s = false →
- step sig n (copy_step src dst sig n is_sep)
- (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) =
+ ∀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 ? (nth src ? v (niltape ?)) (Some ? 〈s,R〉)) src)
- (tape_move ? (nth dst ? v (niltape ?)) (Some ? 〈s,R〉)) dst).
-#src #dst #sig #n #is_sep #v #t #Hsrc #Hdst #s #Hcurrent #Hsep
+ (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
-[ >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent whd in ⊢ (??(???%)?); >Hsep %
-| >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent whd in ⊢ (??(???????(???%))?);
- >Hsep whd in ⊢ (??(???????(???%))?); >pmap_change
- (* le due change commutano *)
+[ 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_multi_def
+ >tape_move_null_action
+ @eq_f2 // >nth_change_vec_neq //
]
+qed.
lemma sem_copy_step :
- ∀src,dst,sig,n,is_sep.src < S n → dst < S n →
- copy_step src dst sig n is_sep ⊨ R_copy_step src dst sig n is_sep.
-#src #dst #sig #n #is_sep #Hsrc #Hdst #int
-<(change_vec_same ?? int src (niltape ?)) cases (nth src ? int (niltape ?))
-[ %{2} % [| %
- [ whd in ⊢ (??%?); >copy_q0_q2 //
- | % // #x1 #x2 #xls #xrs >nth_change_vec // #Hfalse destruct ] ]
-| #a #al %{2} % [| %
- [ whd in ⊢ (??%?); >copy_q0_q2 //
- | % // #x1 #x2 #xls #xrs >nth_change_vec // #Hfalse destruct ] ]
-| #a #al %{2} % [| %
- [ whd in ⊢ (??%?); >copy_q0_q2 //
- | % // #x1 #x2 #xls #xrs >nth_change_vec // #Hfalse destruct ] ]
-| #ls #c #rs %{2} % [| %
- [
\ No newline at end of file
+ ∀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.
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