include "turing/turing.ma".
include "turing/inject.ma".
include "turing/while_multi.ma".
+include "turing/while_machine.ma".
definition parmove_states ≝ initN 3.
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_states ≝ initN 3.
+definition mte0 : mte_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition mte1 : mte_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition mte2 : mte_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
+definition mte_step ≝
+ λalpha:FinSet.λD.mk_TM alpha mte_states
+ (λp.let 〈q,a〉 ≝ p in
+ match a with
+ [ None ⇒ 〈mte1,None ?,N〉
+ | Some a' ⇒ match (pi1 … q) with
+ [ O ⇒ 〈mte2,Some ? a',D〉
+ | S q ⇒ 〈mte2,None ?,N〉 ] ])
+ mte0 (λq.q == mte1 ∨ q == mte2).
+
+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.
+
+lemma sem_mte_step :
+ ∀alpha,D.mte_step alpha D ⊨ [ mte2 : R_mte_step_true alpha D, R_mte_step_false alpha ] .
+#alpha #D #intape @(ex_intro ?? 2) cases intape
+[ @ex_intro
+ [| % [ % [ % | normalize #H destruct ] | #_ % // ] ]
+|#a #al @ex_intro
+ [| % [ % [ % | normalize #H destruct ] | #_ % // ] ]
+|#a #al @ex_intro
+ [| % [ % [ % | normalize #H destruct ] | #_ % // ] ]
+| #ls #c #rs
+ @ex_intro [| % [ % [ % | #_ %{ls} %{c} %{rs} % // ]
+ | normalize in ⊢ (?(??%?)→?); * #H @False_ind /2/ ] ] ]
+qed.
+
+definition move_to_end ≝ λsig,D.whileTM sig (mte_step sig D) mte2.
+
+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.
+
+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.
\ No newline at end of file
--- /dev/null
+(*
+ ||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_2.ma".
+
+(*
+
+ in.obj : ... x ...
+ ^
+ in.cfg : ... ? ? ...
+ ^
+
+ out.cfg : ... 1 x ...
+ ^
+
+ ---------------------
+ current (in.obj) = None
+
+ in.cfg : ... ? ? ...
+ ^
+
+ out.cfg : ... 0 0 ...
+ ^
+
+ obj_to_cfg ≝
+ move_l(cfg);
+ move_l(cfg);
+ (if (current(in.obj)) == None
+ then write(0,cfg);
+ move_r(cfg);
+ write(0,cfg);
+ else write(1,cfg);
+ move_r(cfg);
+ copy_step(obj,cfg);
+ move_l(obj);)
+ move_to_end_l(cfg);
+ move_r(cfg);
+
+
+ cfg_to_obj
+*)
+
+definition obj_to_cfg ≝
+ mmove cfg unialpha 3 L ·
+ mmove cfg unialpha 3 L ·
+ if_TM ?? (inject_TM ? (test_null ?) 3 obj)
+ (
+
+
+
+
+definition o2c_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_comp_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_comp_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_multi_def
+ >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_comp_step_true src dst sig n,
+ R_comp_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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