--- /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/unistep_aux.ma".
+
+axiom sem_unistep: ∀M,cf. unistep ⊨ R_unistep_high M cf.
+
+definition loop_uni ≝
+ ifTM ?? (inject_TM ? (test_char ? (λc.c == bit false)) 2 cfg)
+ (single_finalTM ?? unistep)
+ (nop …) tc_true.
+
+definition lu_acc : states ?? loop_uni ≝ (inr … (inl … (inr … start_nop))).
+
+lemma sem_loop_uni: ∀M,cf.
+ loop_uni ⊨ [lu_acc: R_unistep_high M cf, λt1,t2.t1=t2].
+#M #cf
+@(acc_sem_if_app ????????????
+ (sem_test_char_multi ? (λc.c == bit false) 2 cfg (le_S ?? (le_n 1)))
+ (sem_unistep M cf)
+ (sem_nop …))
+[#t1 #t2 #t3 * #_ #Ht3 >Ht3 // |#t1 #t2 #t3 * #_ #Ht3 whd in ⊢ (%→?); //]
+qed.
+
+definition universalTM ≝ whileTM ?? loop_uni lu_acc.
+
+definition low_R ≝ λM,R.λt1,t2:Vector (tape FSUnialpha) 3.
+ ∀cin. t1 = low_tapes M cin →
+ ∃cout. R (ctape … cin) (ctape … cout) ∧
+ halt ? M (cstate … cout) = true ∧ t2 = low_tapes M cout.
+
+theorem sem_universal: ∀M:normalTM. ∀startc.
+ universalTM ⊫ (low_R M (R_TM FinBool M startc)).
+#M #cin #intape #i #outc #Hloop
+lapply (sem_while … (sem_loop_uni intape i outc Hloop)
+ [@daemon] -Hloop
+* #ta * #Hstar generalize in match q; -q
+@(star_ind_l ??????? Hstar)
+ [#tb #q0 whd in ⊢ (%→?); #Htb #tape1 #Htb1
+ cases (Htb … Htb1) -Htb #Hhalt #Htb
+ <Htb >Htb1 @ex_intro
+ [|%{tape1} %
+ [ %
+ [ whd @(ex_intro … 1) @(ex_intro … (mk_config … q0 tape1))
+ % [|%] whd in ⊢ (??%?); >Hhalt %
+ | @Hhalt ]
+ | % ]
+ ]
+ |#tb #tc #td whd in ⊢ (%→?); #Htc #Hstar1 #IH
+ #q #Htd #tape1 #Htb
+ lapply (IH (\fst (trans ? M 〈q,current ? tape1〉)) Htd) -IH
+ #IH cases (Htc … Htb); -Htc #Hhaltq
+ whd in match (step FinBool ??); >(eq_pair_fst_snd ?? (trans ???))
+ #Htc change with (mk_config ????) in Htc:(???(??%));
+ cases (IH ? Htc) #q1 * #tape2 * * #HRTM #Hhaltq1 #Houtc
+ @(ex_intro … q1) @(ex_intro … tape2) %
+ [%
+ [cases HRTM #k * #outc1 * #Hloop #Houtc1
+ @(ex_intro … (S k)) @(ex_intro … outc1) %
+ [>loopM_unfold >loop_S_false [2://] whd in match (step FinBool ??);
+ >(eq_pair_fst_snd ?? (trans ???)) @Hloop
+ |@Houtc1
+ ]
+ |@Hhaltq1]
+ |@Houtc
+ ]
+ ]
+*)
+qed.
+
+qed.
+
+theorem sem_universal2: ∀M:normalTM. ∀R.
+ M ⊫ R → universalTM ⊫ (low_R M (start ? M) R).
+#M #R #HMR lapply (sem_universal … M (start ? M)) @WRealize_to_WRealize
+#t1 #t2 whd in ⊢ (%→%); #H #tape1 #Htape1 cases (H ? Htape1)
+#q * #tape2 * * #HRTM #Hhalt #Ht2 @(ex_intro … q) @(ex_intro … tape2)
+% [% [@(R_TM_to_R … HRTM) @HMR | //] | //]
+qed.
+
+axiom terminate_UTM: ∀M:normalTM.∀t.
+ M ↓ t → universalTM ↓ (low_config M (mk_config ?? (start ? M) t)).
+
+
+
+
+
+
+
+
+lemma current_embedding: ∀c.
+ high_c (〈match c with [None ⇒ null | Some b ⇒ bit b],false〉) = c.
+ * normalize // qed.
+
+lemma tape_embedding: ∀ls,c,rs.
+ high_tape
+ (map ?? (λb.〈bit b,false〉) ls)
+ (〈match c with [None ⇒ null | Some b ⇒ bit b],false〉)
+ (map ?? (λb.〈bit b,false〉) rs) = mk_tape ? ls c rs.
+#ls #c #rs >high_tape_eq >bool_embedding >bool_embedding
+>current_embedding %
+qed.
+
+definition high_move ≝ λc,mv.
+ match c with
+ [ bit b ⇒ Some ? 〈b,move_of_unialpha mv〉
+ | _ ⇒ None ?
+ ].
+
+definition map_move ≝
+ λc,mv.match c with [ null ⇒ None ? | _ ⇒ Some ? 〈c,false,move_of_unialpha mv〉 ].
+
+definition low_step_R_true ≝ λt1,t2.
+ ∀M:normalTM.
+ ∀c: nconfig (no_states M).
+ t1 = low_config M c →
+ halt ? M (cstate … c) = false ∧
+ t2 = low_config M (step ? M c).
+
+definition low_tape_aux : ∀M:normalTM.tape FinBool → tape STape ≝
+λM:normalTM.λt.
+ let current_low ≝ match current … t with
+ [ None ⇒ None ? | Some b ⇒ Some ? 〈bit b,false〉] in
+ let low_left ≝ map … (λb.〈bit b,false〉) (left … t) in
+ let low_right ≝ map … (λb.〈bit b,false〉) (right … t) in
+ mk_tape STape low_left current_low low_right.
+
+lemma left_of_low_tape: ∀M,t.
+ left ? (low_tape_aux M t) = map … (λb.〈bit b,false〉) (left … t).
+#M * //
+qed.
+
+lemma right_of_low_tape: ∀M,t.
+ right ? (low_tape_aux M t) = map … (λb.〈bit b,false〉) (right … t).
+#M * //
+qed.
+
+definition low_move ≝ λaction:option (bool × move).
+ match action with
+ [None ⇒ None ?
+ |Some act ⇒ Some ? (〈〈bit (\fst act),false〉,\snd act〉)].
+
+(* simulation lemma *)
+lemma low_tape_move : ∀M,action,t.
+ tape_move STape (low_tape_aux M t) (low_move action) =
+ low_tape_aux M (tape_move FinBool t action).
+#M * // (* None *)
+* #b #mv #t cases mv cases t //
+ [#ls #c #rs cases ls //|#ls #c #rs cases rs //]
+qed.
+
+lemma left_of_lift: ∀ls,c,rs. left ? (lift_tape ls c rs) = ls.
+#ls * #c #b #rs cases c // cases ls // cases rs //
+qed.
+
+lemma right_of_lift: ∀ls,c,rs. legal_tape ls c rs →
+ right ? (lift_tape ls c rs) = rs.
+#ls * #c #b #rs * #_ cases c // cases ls cases rs // #a #tll #b #tlr
+#H @False_ind cases H [* [#H1 /2/ |#H1 destruct] |#H1 destruct]
+qed.
+
+
+lemma current_of_lift: ∀ls,c,b,rs. legal_tape ls 〈c,b〉 rs →
+ current STape (lift_tape ls 〈c,b〉 rs) =
+ match c with [null ⇒ None ? | _ ⇒ Some ? 〈c,b〉].
+#ls #c #b #rs cases c // whd in ⊢ (%→?); * #_
+* [* [#Hnull @False_ind /2/ | #Hls >Hls whd in ⊢ (??%%); cases rs //]
+ |#Hrs >Hrs whd in ⊢ (??%%); cases ls //]
+qed.
+
+lemma current_of_lift_None: ∀ls,c,b,rs. legal_tape ls 〈c,b〉 rs →
+ current STape (lift_tape ls 〈c,b〉 rs) = None ? →
+ c = null.
+#ls #c #b #rs #Hlegal >(current_of_lift … Hlegal) cases c normalize
+ [#b #H destruct |// |3,4,5:#H destruct ]
+qed.
+
+lemma current_of_lift_Some: ∀ls,c,c1,rs. legal_tape ls c rs →
+ current STape (lift_tape ls c rs) = Some ? c1 →
+ c = c1.
+#ls * #c #cb #b #rs #Hlegal >(current_of_lift … Hlegal) cases c normalize
+ [#b1 #H destruct // |#H destruct |3,4,5:#H destruct //]
+qed.
+
+lemma current_of_low_None: ∀M,t. current FinBool t = None ? →
+ current STape (low_tape_aux M t) = None ?.
+#M #t cases t // #l #b #r whd in ⊢ ((??%?)→?); #H destruct
+qed.
+
+lemma current_of_low_Some: ∀M,t,b. current FinBool t = Some ? b →
+ current STape (low_tape_aux M t) = Some ? 〈bit b,false〉.
+#M #t cases t
+ [#b whd in ⊢ ((??%?)→?); #H destruct
+ |#b #l #b1 whd in ⊢ ((??%?)→?); #H destruct
+ |#b #l #b1 whd in ⊢ ((??%?)→?); #H destruct
+ |#c #c1 #l #r whd in ⊢ ((??%?)→?); #H destruct %
+ ]
+qed.
+
+lemma current_of_low:∀M,tape,ls,c,rs. legal_tape ls c rs →
+ lift_tape ls c rs = low_tape_aux M tape →
+ c = 〈match current … tape with
+ [ None ⇒ null | Some b ⇒ bit b], false〉.
+#M #tape #ls * #c #cb #rs #Hlegal #Hlift
+cut (current ? (lift_tape ls 〈c,cb〉 rs) = current ? (low_tape_aux M tape))
+ [@eq_f @Hlift] -Hlift #Hlift
+cut (current … tape = None ? ∨ ∃b.current … tape = Some ? b)
+ [cases (current … tape) [%1 // | #b1 %2 /2/ ]] *
+ [#Hcurrent >Hcurrent normalize
+ >(current_of_low_None …Hcurrent) in Hlift; #Hlift
+ >(current_of_lift_None … Hlegal Hlift)
+ @eq_f cases Hlegal * * #Hmarks #_ #_ #_ @(Hmarks 〈c,cb〉) @memb_hd
+ |* #b #Hcurrent >Hcurrent normalize
+ >(current_of_low_Some …Hcurrent) in Hlift; #Hlift
+ @(current_of_lift_Some … Hlegal Hlift)
+ ]
+qed.
+
+(*
+lemma current_of_low:∀M,tape,ls,c,rs. legal_tape ls c rs →
+ lift_tape ls c rs = low_tape_aux M tape →
+ c = 〈match current … tape with
+ [ None ⇒ null | Some b ⇒ bit b], false〉.
+#M #tape #ls * #c #cb #rs * * #_ #H cases (orb_true_l … H)
+ [cases c [2,3,4,5: whd in ⊢ ((??%?)→?); #Hfalse destruct]
+ #b #_ #_ cases tape
+ [whd in ⊢ ((??%%)→?); #H destruct
+ |#a #l whd in ⊢ ((??%%)→?); #H destruct
+ |#a #l whd in ⊢ ((??%%)→?); #H destruct
+ |#a #l #r whd in ⊢ ((??%%)→?); #H destruct //
+ ]
+ |cases c
+ [#b whd in ⊢ ((??%?)→?); #Hfalse destruct
+ |3,4,5:whd in ⊢ ((??%?)→?); #Hfalse destruct]
+ #_ * [* [#Habs @False_ind /2/
+ |#Hls >Hls whd in ⊢ ((??%%)→?); *)
+
+
+(* sufficent conditions to have a low_level_config *)
+lemma is_low_config: ∀ls,c,rs,M,s,tape,qhd,q_tl,table.
+legal_tape ls c rs →
+table = flatten ? (tuples_list (no_states M) (nhalt M) (graph_enum ?? (ntrans M))) →
+lift_tape ls c rs = low_tape_aux M tape →
+〈qhd,false〉::q_tl = m_bits_of_state (no_states M) (nhalt M) s →
+midtape STape (〈grid,false〉::ls)
+ 〈qhd,false〉
+ (q_tl@c::〈grid,false〉::table@〈grid,false〉::rs) =
+ low_config M (mk_config ?? s tape).
+#ls #c #rs #M #s #tape #qhd #q_tl #table #Hlegal #Htable
+#Hlift #Hstate whd in match (low_config ??); <Hstate
+@eq_f3
+ [@eq_f <(left_of_lift ls c rs) >Hlift //
+ | cut (∀A.∀a,b:A.∀l1,l2. a::l1 = b::l2 → a=b)
+ [#A #a #b #l1 #l2 #H destruct (H) %] #Hcut
+ @(Hcut …Hstate)
+ |@eq_f <(current_of_low … Hlegal Hlift) @eq_f @eq_f <Htable @eq_f @eq_f
+ <(right_of_lift ls c rs Hlegal) >Hlift @right_of_low_tape
+ ]
+qed.
+
+lemma unistep_true_to_low_step: ∀t1,t2.
+ R_uni_step_true t1 t2 → low_step_R_true t1 t2.
+#t1 #t2 (* whd in ⊢ (%→%); *) #Huni_step * #n #posn #t #h * #qin #tape #eqt1
+cases (low_config_eq … eqt1)
+#low_left * #low_right * #table * #q_low_hd * #q_low_tl * #current_low
+***** #Hlow_left #Hlow_right #Htable #Hq_low #Hcurrent_low #Ht1
+letin trg ≝ (t 〈qin,current ? tape〉)
+letin qout_low ≝ (m_bits_of_state n h (\fst trg))
+letin qout_low_hd ≝ (hd ? qout_low 〈bit true,false〉)
+letin qout_low_tl ≝ (tail ? qout_low)
+letin low_act ≝ (low_action (\snd (t 〈qin,current ? tape〉)))
+letin low_cout ≝ (\fst low_act)
+letin low_m ≝ (\snd low_act)
+lapply (Huni_step n table q_low_hd (\fst qout_low_hd)
+ current_low low_cout low_left low_right q_low_tl qout_low_tl low_m … Ht1)
+ [@daemon
+ |>Htable
+ @(trans_to_match n h t 〈qin,current ? tape〉 … (refl …))
+ >Hq_low >Hcurrent_low whd in match (mk_tuple ?????);
+ >(eq_pair_fst_snd … (t …)) whd in ⊢ (??%?);
+ >(eq_pair_fst_snd … (low_action …)) %
+ |//
+ |@daemon
+ ]
+-Ht1 #Huni_step lapply (Huni_step ? (refl …)) -Huni_step *
+#q_low_head_false * #ls1 * #rs1 * #c2 * *
+#Ht2 #Hlift #Hlegal %
+ [whd in ⊢ (??%?); >q_low_head_false in Hq_low;
+ whd in ⊢ ((???%)→?); generalize in match (h qin);
+ #x #H destruct (H) %
+ |>Ht2 whd in match (step FinBool ??);
+ whd in match (trans ???);
+ >(eq_pair_fst_snd … (t ?))
+ @is_low_config // >Hlift
+ <low_tape_move @eq_f2
+ [>Hlow_left >Hlow_right >Hcurrent_low whd in ⊢ (??%%);
+ cases (current …tape) [%] #b whd in ⊢ (??%%); %
+ |whd in match low_cout; whd in match low_m; whd in match low_act;
+ generalize in match (\snd (t ?)); * [%] * #b #mv
+ whd in ⊢ (??(?(???%)?)%); cases mv %
+ ]
+ ]
+qed.
+
+definition low_step_R_false ≝ λt1,t2.
+ ∀M:normalTM.
+ ∀c: nconfig (no_states M).
+ t1 = low_config M c → halt ? M (cstate … c) = true ∧ t1 = t2.
+
+lemma unistep_false_to_low_step: ∀t1,t2.
+ R_uni_step_false t1 t2 → low_step_R_false t1 t2.
+#t1 #t2 (* whd in ⊢ (%→%); *) #Huni_step * #n #posn #t #h * #qin #tape #eqt1
+cases (low_config_eq … eqt1) #low_left * #low_right * #table * #q_low_hd * #q_low_tl * #current_low
+***** #_ #_ #_ #Hqin #_ #Ht1 whd in match (halt ???);
+cases (Huni_step (h qin) ?) [/2/] >Ht1 whd in ⊢ (??%?); @eq_f
+normalize in Hqin; destruct (Hqin) %
+qed.
+
+definition low_R ≝ λM,qstart,R,t1,t2.
+ ∀tape1. t1 = low_config M (mk_config ?? qstart tape1) →
+ ∃q,tape2.R tape1 tape2 ∧
+ halt ? M q = true ∧ t2 = low_config M (mk_config ?? q tape2).
+
+lemma sem_uni_step1:
+ uni_step ⊨ [us_acc: low_step_R_true, low_step_R_false].
+qed.
+
+definition universalTM ≝ whileTM ? uni_step us_acc.
+
+theorem sem_universal: ∀M:normalTM. ∀qstart.
+ universalTM ⊫ (low_R M qstart (R_TM FinBool M qstart)).
+qed.
+
+theorem sem_universal2: ∀M:normalTM. ∀R.
+ M ⊫ R → universalTM ⊫ (low_R M (start ? M) R).
+#M #R #HMR lapply (sem_universal … M (start ? M)) @WRealize_to_WRealize
+#t1 #t2 whd in ⊢ (%→%); #H #tape1 #Htape1 cases (H ? Htape1)
+#q * #tape2 * * #HRTM #Hhalt #Ht2 @(ex_intro … q) @(ex_intro … tape2)
+% [% [@(R_TM_to_R … HRTM) @HMR | //] | //]
+qed.
+
+axiom terminate_UTM: ∀M:normalTM.∀t.
+ M ↓ t → universalTM ↓ (low_config M (mk_config ?? (start ? M) t)).
+
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