--- /dev/null
+(* include "turing/auxiliary_machines.ma". *)
+
+include "turing/multi_to_mono/shift_trace.ma".
+include "turing/multi_universal/normalTM.ma". (* for DeqMove *)
+
+(******************************************************************************)
+
+(* exec_move_R : before shifting the trace left - to simulate a right move of
+the head - we must be sure we are not in rightoverflow, that is that the symbol
+at the right of the head, if any, is not blank *)
+
+(* a simple look-ahead machine *)
+definition mtestR ≝ λsig,test.
+ (move sig R) ·
+ (ifTM ? (test_char ? test)
+ (single_finalTM ? (move sig L))
+ (move sig L) tc_true).
+
+(* underspecified *)
+definition RmtestR_true ≝ λsig,test.λt1,t2.
+ ∀ls,c,rs.
+ t1 = midtape sig ls c rs →
+ ∃c1,rs1. rs = c1::rs1 ∧ t1 = t2 ∧ (test c1 = true).
+
+definition RmtestR_false ≝ λsig,test.λt1,t2.
+ (∀ls,c,c1,rs.
+ t1 = midtape sig ls c (c1::rs) →
+ t1 = t2 ∧ (test c1 = false)) ∧
+ (∀ls,c.
+ t1 = midtape sig ls c [] → t1 = t2).
+
+definition mtestR_acc: ∀sig,test.states ? (mtestR sig test).
+#sig #test @inr @inr @inl @inr @start_nop
+qed.
+
+lemma sem_mtestR: ∀sig,test.
+ mtestR sig test ⊨
+ [mtestR_acc sig test:
+ RmtestR_true sig test, RmtestR_false sig test].
+#sig #test
+@(acc_sem_seq_app sig … (sem_move_single … )
+ (acc_sem_if sig …
+ (sem_test_char sig test)
+ (sem_move_single … )
+ (sem_move_single … )))
+ [#t1 #t2 #t3 whd in ⊢ (%→?); #Hmove * #tx * whd in ⊢ (%→?); * *
+ #cx * #Hcx #H1cx #Htx #Ht2 #ls #c #rs #Ht1
+ >Ht1 in Hmove; cases rs
+ [#H >H in Hcx; whd in ⊢ (??%?→?); #H1 destruct (H1)
+ |#c1 #rs1 #Ht3 %{c1} %{rs1} %
+ [% [//|>Htx in Ht2; >Ht3 whd in ⊢ (%→?); #H @sym_eq @H ]
+ |>Ht3 in Hcx; whd in ⊢ (??%?→?); #H1 destruct (H1) //
+ ]
+ ]
+ |#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; #Hcx lapply (Hcx ? (refl ??))
+ #Htest % // >Heqtx in Htx; >Ht3 whd in ⊢ (%→?); #H @sym_eq @H
+ |#ls0 #c0 #Ht1 >Ht1 in Hmove; whd in match (tape_move ???);
+ <Heqtx #H1tx >H1tx in Htx; #H @sym_eq @H
+ ]
+ ]
+qed.
+
+definition guarded_M ≝ λsig,n,i,M.
+ (ifTM ? (test_char ? (no_blank sig n i))
+ M
+ (ifTM ? (mtestR ? (no_blank sig n i))
+ M
+ (nop ?) (mtestR_acc …)) tc_true).
+
+definition R_guarded_M ≝ λsig,n,i,RM,t1,t2.
+ ∀ls,a,rs. t1 = midtape ? ls a rs →
+ (no_blank sig n i a = false →
+ no_blank sig n i (hd ? rs (all_blank …)) = false → t1=t2) ∧
+ (no_blank sig n i a = true ∨
+ no_blank sig n i (hd ? rs (all_blank …)) = true → RM t1 t2).
+
+lemma sem_R_guarded: ∀sig,n,i,M,RM. M ⊨ RM →
+ guarded_M sig n i M ⊨ R_guarded_M sig n i RM.
+#sig #n #i #M #RM #HM
+@(sem_if_app … (sem_test_char … ) HM
+ (sem_if … (sem_mtestR … ) HM (sem_nop ?)))
+#tin #tout #t1 *
+ [* * * #c * #Hc #H1c #Ht1 #Htout #ls #a #rs #Htin
+ >Htin in Hc; normalize in ⊢ (%→?); #H1 destruct (H1) %
+ [>H1c #H2 @False_ind destruct (H2)
+ |#H1 <Htin <Ht1 @Htout
+ ]
+ |* * #Hc #Ht1 #H #ls #a #rs lapply (Hc a) <Ht1 -Ht1 #Ha #Ht1
+ cases H
+ [* #t2 * #Ht2 lapply (Ht2 … Ht1)
+ * #c1 * #rs1 * * #H1 #H2 #H3 #H4 % [2: //]
+ #Ha >H1 whd in match (hd ???); >H3 #H destruct (H)
+ |lapply Ht1 -Ht1 cases rs
+ [#Ht1 * #t2 * * #_ #Ht2 lapply (Ht2 … Ht1) -Ht2 #Ht2
+ whd in ⊢ (%→?); #Htout % [//] *
+ [>Ha [#H destruct (H)| >Ht1 %]
+ |whd in ⊢ (??%?→?); >blank_all_blank whd in ⊢ (??%?→?);
+ #H destruct (H)
+ ]
+ |#c #rs1 #Ht1 * #t2 * * #Ht2 #_ lapply (Ht2 … Ht1) -Ht2 *
+ #Ht2 whd in ⊢ (??%?→?); #Hnb whd in ⊢ (%→?); #Htout % [//] *
+ [>Ha [#H destruct (H)| >Ht1 %]
+ |whd in ⊢ (??%?→?); whd in match (hd ???); >Hnb whd in ⊢ (??%?→?);
+ #H destruct (H)
+ ]
+ ]
+ ]
+ ]
+qed.
+
+definition move_R_i ≝ λsig,n,i. guarded_M sig n i (mtiL sig n i).
+
+definition Rmove_R_i ≝ λsig,n,i. R_guarded_M sig n i (Rmtil sig n i).
+
+
+(*************************** readback of the tape *****************************)
+
+definition opt_cur ≝ λsig,a.
+ if a == blank sig then None ? else Some ? a.
+
+definition rb_trace ≝ λsig,ls,a,rs.
+ mk_tape ? (to_blank ? ls) (opt_cur sig a) (to_blank ? rs).
+
+lemma rb_trace_def: ∀sig,ls,a,rs.
+ rb_trace sig ls a rs =
+ mk_tape ? (to_blank ? ls) (opt_cur sig a) (to_blank ? rs).
+// qed.
+
+definition rb_trace_i ≝ λsig,n,ls,a,rs,i.
+ rb_trace sig (trace ? n i ls) (nth i ? a (blank ?)) (trace ? n i rs).
+
+lemma rb_trace_i_def: ∀sig,n,ls,a,rs,i.
+ rb_trace_i sig n ls a rs i =
+ rb_trace sig (trace ? n i ls) (nth i ? a (blank ?)) (trace ? n i rs).
+// qed.
+
+let rec readback sig n ls a rs i on i : Vector (tape (sig_ext sig)) i ≝
+ match i with
+ [ O ⇒ mk_Vector ? 0 (nil ?) (refl ??)
+ | S j ⇒ vec_cons ? (rb_trace_i sig n ls a rs j) j (readback sig n ls a rs j)
+ ].
+
+lemma orb_false_l: ∀b1,b2:bool.
+ (b1 ∨ b2) = false → (b1 = false) ∧ (b2 = false).
+* * normalize /2/ qed.
+
+lemma no_blank_true_to_not_blank: ∀sig,n,a,i.
+ (no_blank sig n i a = true) → nth i ? (vec … n a) (blank sig) ≠ blank ?.
+#sig #n #a #i #H @(not_to_not … (eqnot_to_noteq … false H))
+-H #H normalize >H %
+qed.
+
+lemma rb_move_R : ∀sig,n,ls,a,rs,outt,i.
+ (∀i.regular_trace sig n a ls rs i) →
+ nth n ? (vec … a) (blank ?) = head ? →
+ no_head_in … ls →
+ no_head_in … rs →
+ Rmove_R_i … i (midtape ? ls a rs) outt →
+ ∃ls1,a1,rs1.
+ outt = midtape ? ls1 a1 rs1 ∧
+ (∀i.regular_trace sig n a1 ls1 rs1 i) ∧
+ rb_trace_i sig n ls1 (vec … a1) rs1 i =
+ tape_move ? (rb_trace_i sig n ls (vec … a) rs i) R ∧
+ ∀j. j ≤n → j ≠ i →
+ rb_trace_i sig n ls1 (vec … a1) rs1 j =
+ rb_trace_i sig n ls (vec … a) rs j.
+#sig #n #ls #a #rs #outt #i #Hreg #Hha #Hhls #Hhrs #Hmove
+lapply (Hmove … (refl …)) -Hmove * #HMove1 #HMove2
+cases (true_or_false (no_blank sig n i a ∨
+ no_blank sig n i (hd (multi_sig sig n) rs (all_blank sig n))))
+ [2: #Hcase cases (orb_false_l … Hcase) -Hcase #Hb1 #Hb2
+ lapply (HMove1 … Hb1 Hb2) #Hout %{ls} %{a} %{rs}
+ %[%[%[@sym_eq @Hout |@Hreg]
+ |>rb_trace_i_def
+ cut (nth i ? (vec … a) (blank ?) = blank sig)
+ [@(\P ?) @injective_notb @Hb1] #Ha >Ha
+ >rb_trace_def whd in match (opt_cur ??);
+ cut (to_blank sig (trace sig n i rs) = [])
+ [@daemon] #Hrs >Hrs
+ cases (to_blank sig (trace sig n i ls)) //
+ ]
+ |//]
+ |-HMove1 #Hb
+ lapply(HMove2 (orb_true_l … Hb) … (refl …) Hha Hreg ? Hhls Hhrs) -HMove2
+ [#Hb1 lapply Hb -Hb whd in match (no_blank sig n i a);
+ >Hb1 #H @no_blank_true_to_not_blank @H]
+ * #ls1 * #a1 * #rs1 * * * * * #H1 #H2 #H3 #H4 #H5 #H6
+ %{ls1} %{a1} %{rs1}
+ %[%[%[@H1|@H2]
+ |(* either a is blank or not *)
+ cases (true_or_false (no_blank sig n i a)) #Hba
+ [(* a not blank *)
+ >rb_trace_i_def >rb_trace_def <to_blank_i_def >H5 >to_blank_cons_nb
+ [2: @no_blank_true_to_not_blank //]
+ lapply H6 -H6 #Hrs >(rb_trace_i_def … rs i) >rb_trace_def
+ <(to_blank_i_def … rs) <Hrs
+ cut (opt_cur sig (nth i ? (vec … a) (blank sig)) =
+ Some ? (nth i ? (vec … a) (blank sig))) [@daemon] #Ha >Ha
+ (* either a1 is blank or not *)
+ cases (true_or_false (no_blank sig n i a1)) #Hba1
+ [cut (opt_cur sig (nth i ? (vec … a1) (blank sig)) =
+ Some ? (nth i ? (vec … a1) (blank sig))) [@daemon] #Ha1 >Ha1
+ >to_blank_cons_nb [%|@(\Pf ?) @injective_notb @Hba1]
+ |cut (opt_cur sig (nth i ? (vec … a1) (blank sig)) = None ?) [@daemon]
+ #Ha1 >Ha1
+ cut (to_blank_i … i (a1::rs1) = [ ]) [@daemon] #Ha1rs1 >Ha1rs1
+ cut (to_blank_i … i rs1 = [ ]) [@daemon] #Hrs1 <to_blank_i_def >Hrs1 %
+ ]
+ |>rb_trace_i_def >rb_trace_def <to_blank_i_def (* >H5 >to_blank_cons_nb *)
+ >rb_trace_i_def >rb_trace_def <(to_blank_i_def … rs) <H6 >H5
+ cut (to_blank_i … i (a::ls) = [ ]) [@daemon] #Hals >Hals
+ cut (to_blank_i … i ls = [ ]) [@daemon] #Hls <(to_blank_i_def … ls) >Hls
+ cut (opt_cur sig (nth i ? (vec … a) (blank sig)) = None ?) [@daemon] #Ha >Ha
+ cut (nth i ? (vec … a1) (blank ?) ≠ blank ?) [@daemon] #Ha1
+ >(to_blank_cons_nb … Ha1)
+ cut (opt_cur sig (nth i ? (vec … a1) (blank sig)) =
+ Some ? (nth i ? (vec … a1) (blank sig))) [@daemon] -Ha1 #Ha1 >Ha1 %
+ ]
+ ]
+ |(* case j ≠ i *)
+ #j #Hle #Hneq >rb_trace_i_def >rb_trace_i_def >rb_trace_def >rb_trace_def
+ <(to_blank_i_def … rs) <(to_blank_i_def … rs1) >(H4 j Hle Hneq)
+ cut ((to_blank_i sig n j ls1 = to_blank_i sig n j ls) ∧
+ (opt_cur sig (nth j (sig_ext sig) (vec … a1) (blank sig)) =
+ opt_cur sig (nth j (sig_ext sig) (vec … a) (blank sig))))
+ [@daemon] * #H7 #H8 >H7 >H8 //
+ ]
+ ]
+qed.
+
+definition Rmove_R_i_rb ≝ λsig,n,i,t1,t2.
+∀ls,a,rs.
+ t1 = midtape ? ls a rs →
+ (∀i.regular_trace sig n a ls rs i) →
+ nth n ? (vec … a) (blank ?) = head ? →
+ no_head_in … ls →
+ no_head_in … rs →
+ ∃ls1,a1,rs1.
+ t2 = midtape (multi_sig sig n) ls1 a1 rs1 ∧
+ (∀i.regular_trace sig n a1 ls1 rs1 i) ∧
+ rb_trace_i sig n ls1 (vec … a1) rs1 i =
+ tape_move ? (rb_trace_i sig n ls (vec … a) rs i) R ∧
+ ∀j. j ≤n → j ≠ i →
+ rb_trace_i sig n ls1 (vec … a1) rs1 j =
+ rb_trace_i sig n ls (vec … a) rs j.
+
+lemma sem_move_R_i : ∀sig,n,i.i < n →
+ move_R_i sig n i ⊨ Rmove_R_i_rb sig n i.
+#sig #n #i #ltin @(Realize_to_Realize … (Rmove_R_i sig n i))
+ [#t1 #t2 #H #ls #a #rs #H1 #H2 #H3 #H4 #H5 @rb_move_R // <H1 //
+ |@sem_R_guarded @sem_Rmtil //
+ ]
+qed.
+
+(* move_L_i axiomatized *)
+
+axiom move_L_i : ∀sig.∀n,i:nat.TM (multi_sig sig n).
+
+definition Rmove_L_i_rb ≝ λsig,n,i,t1,t2.
+∀ls,a,rs.
+ t1 = midtape ? ls a rs →
+ (∀i.regular_trace sig n a ls rs i) →
+ nth n ? (vec … a) (blank ?) = head ? →
+ no_head_in … ls →
+ no_head_in … rs →
+ ∃ls1,a1,rs1.
+ t2 = midtape (multi_sig sig n) ls1 a1 rs1 ∧
+ (∀i.regular_trace sig n a1 ls1 rs1 i) ∧
+ rb_trace_i sig n ls1 (vec … a1) rs1 i =
+ tape_move ? (rb_trace_i sig n ls (vec … a) rs i) L ∧
+ ∀j. j ≤n → j ≠ i →
+ rb_trace_i sig n ls1 (vec … a1) rs1 j =
+ rb_trace_i sig n ls (vec … a) rs j.
+
+axiom sem_move_L_i : ∀sig,n,i.i < n →
+ move_L_i sig n i ⊨ Rmove_L_i_rb sig n i.
+
+(*
+lemma rb_move_L : ∀sig,n,ls,a,rs,outt,i.
+ (∀i.regular_trace sig n a ls rs i) →
+ nth n ? (vec … a) (blank ?) = head ? →
+ no_head_in … ls →
+ no_head_in … rs →
+ Rmove_L_i … i (midtape ? ls a rs) outt →
+ ∃ls1,a1,rs1.
+ outt = midtape ? ls1 a1 rs1 ∧
+ rb_trace_i sig n ls1 (vec … a1) rs1 i =
+ tape_move ? (rb_trace_i sig n ls (vec … a) rs i) L ∧
+ ∀j. j ≤n → j ≠ i →
+ rb_trace_i sig n ls1 (vec … a1) rs1 j =
+ rb_trace_i sig n ls (vec … a) rs j. *)
+
+(* The following function read a move from a vector of moves and executes it *)
+
+axiom get_move : ∀sig.∀n.∀a:multi_sig sig n.nat → DeqMove.
+
+definition exec_move_i ≝ λsig,n,i.
+ (ifTM ? (test_char ? (λa. (get_move sig n a i == R)))
+ (move_R_i sig n i)
+ (ifTM ? (test_char ? (λa. (get_move sig n a i == L)))
+ (move_L_i sig n i)
+ (nop ?) tc_true) tc_true).
+
+definition R_exec_move_i ≝ λsig,n,i,t1,t2.
+∀a,ls,rs.
+ t1 = midtape ? ls a rs →
+ (∀i.regular_trace sig n a ls rs i) →
+ nth n ? (vec … a) (blank ?) = head ? →
+ no_head_in … ls →
+ no_head_in … rs →
+ ∃ls1,a1,rs1.
+ t2 = midtape (multi_sig sig n) ls1 a1 rs1 ∧
+ (∀i.regular_trace sig n a1 ls1 rs1 i) ∧
+ rb_trace_i sig n ls1 (vec … a1) rs1 i =
+ tape_move ? (rb_trace_i sig n ls (vec … a) rs i) (get_move sig n a i) ∧
+ ∀j. j ≤n → j ≠ i →
+ rb_trace_i sig n ls1 (vec … a1) rs1 j =
+ rb_trace_i sig n ls (vec … a) rs j.
+
+lemma tape_move_N: ∀A,t. tape_move A t N = t.
+// qed.
+
+lemma sem_exec_move_i: ∀sig,n,i. i < n →
+ exec_move_i sig n i ⊨ R_exec_move_i sig n i.
+#sig #n #i #ltin
+@(sem_if_app … (sem_test_char …)
+ (sem_move_R_i … ltin)
+ (sem_if … (sem_test_char …)
+ (sem_move_L_i … ltin) (sem_nop ?)))
+#tin #tout #t1 *
+ [* * * #c * #Hc #HR #Ht1 #HMR
+ #a #ls #rs #Htin >Htin in Hc; whd in ⊢ (??%?→?); #H destruct (H)
+ >(\P HR) @HMR >Ht1 @Htin
+ |* * #HR #Ht1 >Ht1 *
+ [* #t2 * * * #c * #Hc #HL #Ht2 #HML
+ #a #ls #rs #Htin >Htin in Hc; whd in ⊢ (??%?→?); #H destruct (H)
+ >(\P HL) @HML >Ht2 @Htin
+ |* #t2 * * #HL #Ht2 >Ht2 whd in ⊢ (%→?); #Htout >Htout
+ #a #ls #rs #Htin >Htin in HR; #HR >Htin in HL; #HL
+ cut (get_move sig n a i = N)
+ [lapply (HR a (refl … )) lapply (HL a (refl … ))
+ cases (get_move sig n a i) normalize
+ [#H destruct (H) |#_ #H destruct (H) | //]]
+ #HN >HN >tape_move_N #Hreg #_ #_ #_
+ %{ls} %{a} %{rs}
+ %[%[%[%|@Hreg] |%] | //]
+ ]
+ ]
+qed.
+
+axiom reg_inv : ∀sig,n,a,ls,rs,a1,ls1,rs1.
+ regular_trace sig n a1 ls1 rs1 n →
+ (rb_trace_i sig n ls1 (vec … a1) rs1 n =
+ rb_trace_i sig n ls (vec … n a) rs n) →
+ nth n ? (vec … a) (blank ?) = head ? →
+ no_head_in … ls →
+ no_head_in … rs →
+ nth n ? (vec … a1) (blank ?) = head ? ∧
+ no_head_in … ls1 ∧ no_head_in … rs1.
+
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projects / helm.git / blobdiff