--- /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/basic_machines.ma".
+include "turing/if_machine.ma".
+
+(* 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.
+
--- /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/basic_machines.ma".
+include "turing/if_multi.ma".
+include "turing/while_multi.ma".
+include "turing/inject.ma".
+include "turing/basic_multi_machines.ma".
+
+(**************************** injected machines *******************************)
+
+definition Rtc_multi_true ≝
+ λalpha,test,n,i.λt1,t2:Vector ? (S n).
+ (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
+
+definition Rtc_multi_false ≝
+ λalpha,test,n,i.λt1,t2:Vector ? (S n).
+ (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
+
+lemma sem_test_char_multi :
+ ∀alpha,test,n,i.i ≤ n →
+ inject_TM ? (test_char ? test) n i ⊨
+ [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
+#alpha #test #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
+ #Hcur #Htestc #Hnth_i #Hnth_j %
+ [ %{c} % //
+ | @(eq_vec … (niltape ?)) #i0 #Hi0
+ cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @Hnth_i
+ | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
+ [ @Htestc
+ | @(eq_vec … (niltape ?)) #i0 #Hi0
+ cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @Hnth_i
+ | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+qed.
+
+definition Rm_test_null_true ≝
+ λalpha,n,i.λt1,t2:Vector ? (S n).
+ current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
+
+definition Rm_test_null_false ≝
+ λalpha,n,i.λt1,t2:Vector ? (S n).
+ current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
+
+lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n →
+ inject_TM ? (test_null ?) n i ⊨
+ [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
+#alpha #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_null alpha) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
+ @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
+ [ @Hcur
+ | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) //
+ #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ]
+qed.
+
+(* move a single tape *)
+definition mmove ≝ λi,sig,n,D.inject_TM sig (move sig D) n i.
+
+definition Rm_multi ≝
+ λalpha,n,i,D.λt1,t2:Vector ? (S n).
+ t2 = change_vec ? (S n) t1 (tape_move alpha (nth i ? t1 (niltape ?)) D) i.
+
+lemma sem_move_multi :
+ ∀alpha,n,i,D.i ≤ n →
+ mmove i alpha n D ⊨ Rm_multi alpha n i D.
+#alpha #n #i #D #Hin #ta cases (sem_inject … Hin (sem_move_single alpha D) ta)
+#k * #outc * #Hloop * whd in ⊢ (%→?); #Htb1 #Htb2 %{k} %{outc} % [ @Hloop ]
+whd @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
+[ >Hi0i >Htb1 >nth_change_vec //
+| >nth_change_vec_neq [|@sym_not_eq //] <Htb2 // @sym_not_eq // ]
+qed.
+
+(********************** look_ahead test *************************)
+
+definition mtestR ≝ λsig,i,n,test.
+ (mmove i sig n R) ·
+ (ifTM ?? (inject_TM ? (test_char ? test) n i)
+ (single_finalTM ?? (mmove i sig n L))
+ (mmove i sig n L) tc_true).
+
+
+(* underspecified *)
+definition RmtestR_true ≝ λsig,i,n,test.λt1,t2:Vector ? n.
+ ∀ls,c,c1,rs.
+ nth i ? t1 (niltape ?) = midtape sig ls c (c1::rs) →
+ t1 = t2 ∧ (test c1 = true).
+
+definition RmtestR_false ≝ λsig,i,n,test.λt1,t2:Vector ? n.
+ ∀ls,c,c1,rs.
+ nth i ? t1 (niltape ?) = midtape sig ls c (c1::rs) →
+ t1 = t2 ∧ (test c1 = false).
+
+definition mtestR_acc: ∀sig,i,n,test.states ?? (mtestR sig i n test).
+#sig #i #n #test @inr @inr @inl @inr @start_nop
+qed.
+
+lemma sem_mtestR: ∀sig,i,n,test. i ≤ n →
+ mtestR sig i n test ⊨
+ [mtestR_acc sig i n test:
+ RmtestR_true sig i (S n) test, RmtestR_false sig i (S n) test].
+#sig #i #n #test #Hlein
+@(acc_sem_seq_app sig n … (sem_move_multi … R Hlein)
+ (acc_sem_if sig n ????????
+ (sem_test_char_multi sig test n i Hlein)
+ (sem_move_multi … L Hlein)
+ (sem_move_multi … L Hlein)))
+ [#t1 #t2 #t3 whd in ⊢ (%→?); #Hmove * #tx * whd in ⊢ (%→?); * *
+ #cx #Hcx #Htx #Ht2 #ls #c #c1 #rs #Ht1
+ >Ht1 in Hmove; whd in match (tape_move ???); #Ht3
+ >Ht3 in Hcx; >nth_change_vec [|@le_S_S //] * whd in ⊢ (??%?→?);
+ #Hsome destruct (Hsome) #Htest % [2:@Htest]
+ >Htx in Ht2; whd in ⊢ (%→?); #Ht2 @(eq_vec … (niltape ?))
+ #j #lejn cases (true_or_false (eqb i j)) #Hij
+ [lapply lejn <(eqb_true_to_eq … Hij) #lein >Ht2 >nth_change_vec [2://]
+ >Ht3 >nth_change_vec >Ht1 //
+ |lapply (eqb_false_to_not_eq … Hij) #Hneq >Ht2 >nth_change_vec_neq [2://]
+ >Ht3 >nth_change_vec_neq //
+ ]
+ |#t1 #t2 #t3 whd in ⊢ (%→?); #Hmove * #tx * whd in ⊢ (%→?); *
+ #Hcx #Heqtx #Htx #ls #c #c1 #rs #Ht1
+ >Ht1 in Hmove; whd in match (tape_move ???); #Ht3
+ >Ht3 in Hcx; >nth_change_vec [2:@le_S_S //] #Hcx lapply (Hcx ? (refl ??))
+ #Htest % // @(eq_vec … (niltape ?))
+ #j #lejn cases (true_or_false (eqb i j)) #Hij
+ [lapply lejn <(eqb_true_to_eq … Hij) #lein >Htx >nth_change_vec [2://]
+ >Heqtx >Ht3 >nth_change_vec >Ht1 //
+ |lapply (eqb_false_to_not_eq … Hij) #Hneq >Htx >nth_change_vec_neq [2://]
+ >Heqtx >Ht3 >nth_change_vec_neq //
+ ]
+ ]
+qed.
+(* advance in parallel along two tapes src and dst until we reach the end
+ of one tape *)
+
+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.
+
+(* compare *)
+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.
+
+(* copy *)
+
+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.
#ls1 #c1 #rs1 #H destruct cases ls1 // ] ] ]
qed.
+(* a slightly different move machine. *)
+definition smove_states ≝ initN 2.
+
+definition smove0 : smove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
+definition smove1 : smove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
+
+definition trans_smove ≝
+ λsig,D.
+ λp:smove_states × (option sig).
+ let 〈q,a〉 ≝ p in match (pi1 … q) with
+ [ O ⇒ 〈smove1,None sig, D〉
+ | S _ ⇒ 〈smove1,None sig, N〉 ].
+
+definition move ≝
+ λsig,D.mk_TM sig smove_states (trans_smove sig D) smove0 (λq.q == smove1).
+
+definition Rmove ≝
+ λalpha,D,t1,t2. t2 = tape_move alpha t1 D.
+
+lemma sem_move_single :
+ ∀alpha,D.move alpha D ⊨ Rmove alpha D.
+#alpha #D #int %{2} %{(mk_config ? smove_states smove1 ?)} [| % % ]
+qed.
+
(********************************* test char **********************************)
(* the test_char machine ends up in two different states q1 and q2 wether or not
--- /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/turing.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 …)).
+
+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_multi_def
+ >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) <Hb @sym_eq @nth_vec_map
+ | #_ whd >(\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.
+(* copy a character from src tape to dst tape without moving them *)
+
+definition copy_states ≝ initN 3.
+
+definition cc0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition cc1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+
+definition trans_copy_char_N ≝
+ λsrc,dst.λsig:FinSet.λn.
+ λp:copy_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ 〈cc1,change_vec ? (S n)
+ (change_vec ? (S n) (null_action ? n) (〈None ?,N〉) src)
+ (〈nth src ? a (None ?),N〉) dst〉
+ | S _ ⇒ 〈cc1,null_action ? n〉 ].
+
+definition copy_char_N ≝
+ λsrc,dst,sig,n.
+ mk_mTM sig n copy_states (trans_copy_char_N src dst sig n)
+ cc0 (λq.q == cc1).
+
+definition R_copy_char_N ≝
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ outt = change_vec ?? int
+ (tape_write ? (nth dst ? int (niltape ?))
+ (current ? (nth src ? int (niltape ?)))) dst.
+
+lemma copy_char_N_q0_q1 :
+ ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n →
+ step sig n (copy_char_N src dst sig n) (mk_mconfig ??? cc0 v) =
+ mk_mconfig ??? cc1
+ (change_vec ?? v
+ (tape_write ? (nth dst ? v (niltape ?))
+ (current ? (nth src ? v (niltape ?)))) dst).
+#src #dst #sig #n #v #Heq #Hsrc #Hdst
+whd in ⊢ (??%?); @eq_f
+<(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_f3 //
+[ >change_vec_same %
+| >change_vec_same >change_vec_same >nth_current_chars // ]
+qed.
+
+lemma sem_copy_char_N:
+ ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
+ copy_char_N src dst sig n ⊨ R_copy_char_N src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst #int
+%{2} % [| % [ % | whd >copy_char_N_q0_q1 // ]]
+qed.
+
+(* copy a character from src tape to dst tape and advance both tape to
+ the right - useful for copying stings
+
+definition copy_char_states ≝ initN 3.
+
+definition trans_copy_char ≝
+ λsrc,dst.λsig:FinSet.λn.
+ λp:copy_char_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ 〈cc1,change_vec ? (S n)
+ (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
+ (〈nth src ? a (None ?),R〉) dst〉
+ | S _ ⇒ 〈cc1,null_action ? n〉 ].
+
+definition copy_char ≝
+ λsrc,dst,sig,n.
+ mk_mTM sig n copy_char_states (trans_copy_char src dst sig n)
+ cc0 (λq.q == cc1).
+
+definition R_copy_char ≝
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ outt = change_vec ??
+ (change_vec ?? int
+ (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
+ (tape_move_mono ? (nth dst ? int (niltape ?))
+ 〈current ? (nth src ? int (niltape ?)), R〉) dst.
+
+lemma copy_char_q0_q1 :
+ ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n →
+ step sig n (copy_char src dst sig n) (mk_mconfig ??? cc0 v) =
+ mk_mconfig ??? cc1
+ (change_vec ? (S n)
+ (change_vec ?? v
+ (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
+ (tape_move_mono ? (nth dst ? v (niltape ?)) 〈current ? (nth src ? v (niltape ?)), R〉) dst).
+#src #dst #sig #n #v #Heq #Hsrc #Hdst
+whd in ⊢ (??%?);
+<(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
+<(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
+>tape_move_multi_def @eq_f2 //
+>pmap_change >pmap_change <tape_move_multi_def
+>tape_move_null_action @eq_f2 // @eq_f2
+[ >change_vec_same %
+| >change_vec_same >change_vec_same // ]
+qed.
+
+lemma sem_copy_char:
+ ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
+ copy_char src dst sig n ⊨ R_copy_char src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst #int
+%{2} % [| % [ % | whd >copy_char_q0_q1 // ]]
+qed.*)
+
+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_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_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.
+
+
+(* advance in parallel on tapes src and dst; stops if one of the
+ two tapes is in oveflow *)
+
+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_multi_def >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.
+
+(* perform a symultaneous test on all tapes; ends up in state partest1 if
+ the test is succesfull and partest2 otherwise *)
+
+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
V_____________________________________________________________*)
include "turing/turing.ma".
+(* include "turing/basic_machines.ma". *)
(******************* inject a mono machine into a multi tape one **********************)
+
definition inject_trans ≝ λsig,states:FinSet.λn,i:nat.
λtrans:states × (option sig) → states × (option sig × move).
λp:states × (Vector (option sig) (S n)).
]
qed.
-(*
-lemma cstate_inject: ∀sig,n,M,i,x. *)
-
definition inject_R ≝ λsig.λR:relation (tape sig).λn,i:nat.
λv1,v2: (Vector (tape sig) (S n)).
R (nth i ? v1 (niltape ?)) (nth i ? v2 (niltape ?)) ∧
∀j. i ≠ j → nth j ? v1 (niltape ?) = nth j ? v2 (niltape ?).
-(*
-lemma nth_make : ∀A,i,n,j,a,d. i < n → nth i ? (make_veci A a n j) d = a (j+i).
-#A #i elim i
- [#n #j #a #d #ltOn @(lt_O_n_elim … ltOn) <plus_n_O //
- |#m #Hind #n #j #a #d #Hlt lapply Hlt @(lt_O_n_elim … (ltn_to_ltO … Hlt))
- #p <plus_n_Sm #ltmp @Hind @le_S_S_to_le //
- ]
-qed. *)
-
-(*
-lemma mk_config_eq_s: ∀S,sig,s1,s2,t1,t2.
- mk_config S sig s1 t1 = mk_config S sig s2 t2 → s1=s2.
-#S #sig #s1 #s2 #t1 #t2 #H destruct //
-qed.
-
-lemma mk_config_eq_t: ∀S,sig,s1,s2,t1,t2.
- mk_config S sig s1 t1 = mk_config S sig s2 t2 → s1=s2.
-#S #sig #s1 #s2 #t1 #t2 #H destruct //
-qed.
-*)
-
theorem sem_inject: ∀sig.∀M:TM sig.∀R.∀n,i.
i≤n → M ⊨ R → inject_TM sig M n i ⊨ inject_R sig R n i.
#sig #M #R #n #i #lein #HR #vt cases (HR (nth i ? vt (niltape ?)))
| cons r0 rs0 ⇒ leftof ? r0 rs0 ]
| cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
+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 current_to_midtape: ∀sig,t,c. current sig t = Some ? c →
∃ls,rs. t = midtape ? ls c rs.
#sig *
+++ /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/if_multi.ma".
-include "turing/inject.ma".
-include "turing/basic_machines.ma".
-
-definition Rtc_multi_true ≝
- λalpha,test,n,i.λt1,t2:Vector ? (S n).
- (∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
-
-definition Rtc_multi_false ≝
- λalpha,test,n,i.λt1,t2:Vector ? (S n).
- (∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
-
-lemma sem_test_char_multi :
- ∀alpha,test,n,i.i ≤ n →
- inject_TM ? (test_char ? test) n i ⊨
- [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
-#alpha #test #n #i #Hin #int
-cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
-#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
-[ @Hloop
-| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
- #Hcur #Htestc #Hnth_i #Hnth_j %
- [ %{c} % //
- | @(eq_vec … (niltape ?)) #i0 #Hi0
- cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @Hnth_i
- | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
-| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
- [ @Htestc
- | @(eq_vec … (niltape ?)) #i0 #Hi0
- cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @Hnth_i
- | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
-qed.
-
-definition Rm_test_null_true ≝
- λalpha,n,i.λt1,t2:Vector ? (S n).
- current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
-
-definition Rm_test_null_false ≝
- λalpha,n,i.λt1,t2:Vector ? (S n).
- current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
-
-lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n →
- inject_TM ? (test_null ?) n i ⊨
- [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
-#alpha #n #i #Hin #int
-cases (acc_sem_inject … Hin (sem_test_null alpha) int)
-#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
-[ @Hloop
-| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
- @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
- [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ]
-| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
- [ @Hcur
- | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) //
- #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ]
-qed.
-
-(* move a single tape *)
-definition smove_states ≝ initN 2.
-
-definition smove0 : smove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
-definition smove1 : smove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
-
-definition trans_smove ≝
- λsig,D.
- λp:smove_states × (option sig).
- let 〈q,a〉 ≝ p in match (pi1 … q) with
- [ O ⇒ 〈smove1,None sig, D〉
- | S _ ⇒ 〈smove1,None sig, N〉 ].
-
-definition move ≝
- λsig,D.mk_TM sig smove_states (trans_smove sig D) smove0 (λq.q == smove1).
-
-definition mmove ≝ λi,sig,n,D.inject_TM sig (move sig D) n i.
-
-definition Rmove ≝
- λalpha,D,t1,t2. t2 = tape_move alpha t1 D.
-
-lemma sem_move_single :
- ∀alpha,D.move alpha D ⊨ Rmove alpha D.
-#alpha #D #int %{2} %{(mk_config ? smove_states smove1 ?)} [| % % ]
-qed.
-
-definition Rm_multi ≝
- λalpha,n,i,D.λt1,t2:Vector ? (S n).
- t2 = change_vec ? (S n) t1 (tape_move alpha (nth i ? t1 (niltape ?)) D) i.
-
-lemma sem_move_multi :
- ∀alpha,n,i,D.i ≤ n →
- mmove i alpha n D ⊨ Rm_multi alpha n i D.
-#alpha #n #i #D #Hin #ta cases (sem_inject … Hin (sem_move_single alpha D) ta)
-#k * #outc * #Hloop * whd in ⊢ (%→?); #Htb1 #Htb2 %{k} %{outc} % [ @Hloop ]
-whd @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
-[ >Hi0i >Htb1 >nth_change_vec //
-| >nth_change_vec_neq [|@sym_not_eq //] <Htb2 // @sym_not_eq // ]
-qed.
-
-(* simple copy with no move *)
-definition copy_states ≝ initN 3.
-
-definition cc0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
-definition cc1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
-
-definition trans_copy_char_N ≝
- λsrc,dst.λsig:FinSet.λn.
- λp:copy_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in
- match pi1 … q with
- [ O ⇒ 〈cc1,change_vec ? (S n)
- (change_vec ? (S n) (null_action ? n) (〈None ?,N〉) src)
- (〈nth src ? a (None ?),N〉) dst〉
- | S _ ⇒ 〈cc1,null_action ? n〉 ].
-
-definition copy_char_N ≝
- λsrc,dst,sig,n.
- mk_mTM sig n copy_states (trans_copy_char_N src dst sig n)
- cc0 (λq.q == cc1).
-
-definition R_copy_char_N ≝
- λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
- outt = change_vec ?? int
- (tape_write ? (nth dst ? int (niltape ?))
- (current ? (nth src ? int (niltape ?)))) dst.
-
-lemma copy_char_N_q0_q1 :
- ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n →
- step sig n (copy_char_N src dst sig n) (mk_mconfig ??? cc0 v) =
- mk_mconfig ??? cc1
- (change_vec ?? v
- (tape_write ? (nth dst ? v (niltape ?))
- (current ? (nth src ? v (niltape ?)))) dst).
-#src #dst #sig #n #v #Heq #Hsrc #Hdst
-whd in ⊢ (??%?); @eq_f
-<(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_f3 //
-[ >change_vec_same %
-| >change_vec_same >change_vec_same >nth_current_chars // ]
-qed.
-
-lemma sem_copy_char_N:
- ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
- copy_char_N src dst sig n ⊨ R_copy_char_N src dst sig n.
-#src #dst #sig #n #Hneq #Hsrc #Hdst #int
-%{2} % [| % [ % | whd >copy_char_N_q0_q1 // ]]
-qed.
-
-(**************** copy and advance ***********************)
-definition copy_char_states ≝ initN 3.
-
-definition trans_copy_char ≝
- λsrc,dst.λsig:FinSet.λn.
- λp:copy_char_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in
- match pi1 … q with
- [ O ⇒ 〈cc1,change_vec ? (S n)
- (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
- (〈nth src ? a (None ?),R〉) dst〉
- | S _ ⇒ 〈cc1,null_action ? n〉 ].
-
-definition copy_char ≝
- λsrc,dst,sig,n.
- mk_mTM sig n copy_char_states (trans_copy_char src dst sig n)
- cc0 (λq.q == cc1).
-
-definition R_copy_char ≝
- λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
- outt = change_vec ??
- (change_vec ?? int
- (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
- (tape_move_mono ? (nth dst ? int (niltape ?))
- 〈current ? (nth src ? int (niltape ?)), R〉) dst.
-
-lemma copy_char_q0_q1 :
- ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n →
- step sig n (copy_char src dst sig n) (mk_mconfig ??? cc0 v) =
- mk_mconfig ??? cc1
- (change_vec ? (S n)
- (change_vec ?? v
- (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
- (tape_move_mono ? (nth dst ? v (niltape ?)) 〈current ? (nth src ? v (niltape ?)), R〉) dst).
-#src #dst #sig #n #v #Heq #Hsrc #Hdst
-whd in ⊢ (??%?);
-<(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
-<(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
->tape_move_multi_def @eq_f2 //
->pmap_change >pmap_change <tape_move_multi_def
->tape_move_null_action @eq_f2 // @eq_f2
-[ >change_vec_same %
-| >change_vec_same >change_vec_same // ]
-qed.
-
-lemma sem_copy_char:
- ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
- copy_char src dst sig n ⊨ R_copy_char src dst sig n.
-#src #dst #sig #n #Hneq #Hsrc #Hdst #int
-%{2} % [| % [ % | whd >copy_char_q0_q1 // ]]
-qed.
-
-
-
-
-
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