--- /dev/null
+
+include "turing/multi_to_mono/exec_moves.ma".
+
+definition Mono_realize ≝ Realize.
+
+definition to_sig ≝ λQ,sig.λc:sig_ext (TA Q sig).
+ match c with
+ [None ⇒ None ?
+ |Some a ⇒ match a with
+ [inl x ⇒ None ? (* this is not possible *)
+ |inr x ⇒ Some ? x]].
+
+definition to_sig_inv ≝ λQ,sig.λc1:(option sig) × move.λc2.
+ match (fst ?? c1) with
+ [None ⇒ c2
+ |Some a ⇒ Some ? (inr Q ? a)].
+
+definition transf ≝ λQ,sig:FinSet.λn.
+ λt: Q × (Vector (option sig) n) → Q × (Vector ((option sig) × move) n).
+ λa:MTA (HC Q n) sig (S n).
+ let qM ≝ nth n ? (vec … a) (blank ?) in
+ let a1 ≝ resize_vec ? (S n) a n (blank ?) in
+ let a2 ≝ vec_map ?? (to_sig ? sig) n a1 in
+ match qM with
+ [None ⇒ all_blanks (TA (HC Q n) sig) (S n)
+ |Some p ⇒
+ match p with
+ [inl p1 ⇒
+ let 〈q1,actions〉 ≝ t 〈fst ?? p1, a2〉 in
+ let moves ≝ vec_map ?? (snd ??) n actions in
+ let new_chars ≝ pmap_vec ??? (to_sig_inv (HC Q n) ?) n actions a1 in
+ vec_cons_right ? (Some ? (inl ?? 〈q1,moves〉)) n new_chars
+ |inr p2 ⇒ all_blanks (TA (HC Q n) sig) (S n)]
+ ].
+
+lemma transf_eq: ∀Q,sig,n,t.∀a:MTA (HC Q n) sig (S n).
+ ∀a1,a2,q,m,q1,actions,moves,new_chars.
+ nth n ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉) →
+ a1 = resize_vec ? (S n) a n (blank ?) →
+ a2 = vec_map ?? (to_sig ? sig) n a1 →
+ t 〈q, a2〉 = 〈q1,actions〉 →
+ moves = vec_map ?? (snd ??) n actions →
+ new_chars = pmap_vec ??? (to_sig_inv (HC Q n) ?) n actions a1 →
+ transf Q sig n t a =
+ vec_cons_right ? (Some ? (inl ?? 〈q1,moves〉)) n new_chars.
+#Q #sig #n #t #a #a1 #a2 #q #m #q1 #actions #moves #new_chars
+#Hnth #Ha1 #Ha2 #Ht #Hmoves #Hnew_chars
+whd in ⊢ (??%?); >Hnth whd in ⊢ (??%?); <Ha1 <Ha2 >Ht
+@eq_cons_right [<Hmoves // |<Hnew_chars // ]
+qed.
+
+lemma transf_eq_ex: ∀Q,sig,n,t.∀a:MTA (HC Q n) sig (S n).
+ ∀q,m.nth n ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉)→
+ ∃a1.a1 = resize_vec ? (S n) a n (blank ?) ∧
+ ∃a2. a2 = vec_map ?? (to_sig ? sig) n a1 ∧
+ ∃q1,actions. t 〈q, a2〉 = 〈q1,actions〉 ∧
+ ∃moves.moves = vec_map ?? (snd ??) n actions ∧
+ ∃new_chars.new_chars = pmap_vec ??? (to_sig_inv (HC Q n) ?) n actions a1 ∧
+ transf Q sig n t a =
+ vec_cons_right ? (Some ? (inl ?? 〈q1,moves〉)) n new_chars.
+#Q #sig #n #t #a #q #m #H
+%{(resize_vec ? (S n) a n (blank ?))} % [%]
+% [2:% [%] | skip] % [2: % [ 2: % [@eq_pair_fst_snd ] |skip] | skip]
+% [2:% [%] | skip] % [2:% [%] | skip] whd in ⊢ (??%?); >H whd in ⊢ (??%?);
+>(eq_pair_fst_snd … (t …)) %
+qed.
+
+(*
+lemma nth_transf: ∀Q,sig,n,t.∀a:MTA (HC Q n) sig (S n).
+ ∀a1,a2,q,m,q1,actions,moves.
+ nth n ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉) →
+ a1 = resize_vec ? (S n) a n (blank ?) →
+ a2 = vec_map ?? (to_sig ? sig) n a1 →
+ t 〈q, a2〉 = 〈q1,actions〉 →
+ moves = vec_map ?? (snd ??) n actions →
+ nth n ? (vec … (transf Q sig n t a)) (blank ?) = Some ? (inl ?? 〈q1,moves〉).
+#Q #sig #n #t #a #a1 #a2 #q #m #q1 #actions #moves
+#Hnth #Ha1 #Ha2 #Ht #Hmoves
+*)
+
+(*********************************** stepM ************************************)
+definition optf ≝ λA,B:Type[0].λf:A →B.λd,oa.
+ match oa with
+ [ None ⇒ d
+ | Some a ⇒ f a ].
+
+lemma optf_Some : ∀A,B,f,a,d. optf A B f d (Some A a) = f a.
+// qed.
+
+definition stepM ≝ λQ,sig,n,trans.
+ writef ? (optf ?? (transf Q sig (S n) trans) (all_blanks …)) ·
+ exec_moves Q sig (S n) (S n).
+
+let rec to_sig_map Q sig l on l ≝
+ match l with
+ [ nil ⇒ nil ?
+ | cons a tl ⇒ match to_sig Q sig a with
+ [ None ⇒ nil ?
+ | Some c ⇒ c::(to_sig_map Q sig tl)]].
+
+
+definition to_sig_tape ≝ λQ,sig,t.
+ match t with
+ [ niltape ⇒ niltape ?
+ | leftof a l ⇒ match to_sig Q sig a with
+ [ None ⇒ niltape ?
+ | Some x ⇒ leftof ? x (to_sig_map Q sig l) ]
+ | rightof a l ⇒ match to_sig Q sig a with
+ [ None ⇒ niltape ?
+ | Some x ⇒ rightof ? x (to_sig_map Q sig l) ]
+ | midtape ls a rs ⇒ match to_sig Q sig a with
+ [ None ⇒ niltape ?
+ | Some x ⇒ midtape ? (to_sig_map Q sig ls) x (to_sig_map Q sig rs) ]
+ ].
+
+definition rb_tapes ≝ λQ,sig,n,ls.λa:MTA Q sig (S n).λrs.
+ vec_map ?? (to_sig_tape ??) n (readback ? (S n) ls (vec … a) rs n).
+
+(* q0 is a default value *)
+definition get_state ≝ λQ,sig,n.λa:MTA (HC Q n) sig (S n).λq0.
+ match nth n ? (vec … a) (blank ?) with
+ [ None ⇒ q0 (* impossible *)
+ | Some qM ⇒ match qM with
+ [inl qM1 ⇒ fst ?? qM1
+ |inr _ ⇒ q0 (* impossible *) ]].
+
+definition get_chars ≝ λQ,sig,n.λa:MTA (HC Q n) sig (S n).
+ let a1 ≝ resize_vec ? (S n) a n (blank ?) in
+ vec_map ?? (to_sig ? sig) n a1.
+
+lemma get_chars_def : ∀Q,sig,n.∀a:MTA (HC Q n) sig (S n).
+ get_chars … a =
+ vec_map ?? (to_sig ? sig) n (resize_vec ? (S n) a n (blank ?)).
+// qed.
+
+include "turing/turing.ma".
+
+definition readback_config ≝
+ λQ,sig,n,q0,ls.λa:MTA (HC Q (S n)) sig (S (S n)).λrs.
+ mk_mconfig sig Q n
+ (get_state Q sig (S n) a q0)
+ (rb_tapes (HC Q (S n)) sig (S n) ls a rs).
+
+lemma state_readback : ∀Q,sig,n,q0,ls,a,rs.
+ cstate … (readback_config Q sig n q0 ls a rs) =
+ get_state Q sig (S n) a q0.
+// qed.
+
+lemma tapes_readback : ∀Q,sig,n,q0,ls,a,rs.
+ ctapes … (readback_config Q sig n q0 ls a rs) =
+ rb_tapes (HC Q (S n)) sig (S n) ls a rs.
+// qed.
+
+definition R_stepM ≝ λsig.λn.λM:mTM sig n.λt1,t2.
+∀a,ls,rs.
+ t1 = midtape ? ls a rs →
+ (∀i.regular_trace (TA (HC (states … M) (S n)) sig) (S(S n)) a ls rs i) →
+ is_head ?? (nth (S n) ? (vec … a) (blank ?)) = true →
+ no_head_in … ls →
+ no_head_in … rs →
+ ∃ls1,a1,rs1.
+ t2 = midtape (MTA (HC (states … M) (S n)) sig (S(S n))) ls1 a1 rs1 ∧
+ (∀i.regular_trace (TA (HC (states … M) (S n)) sig) (S(S n)) a1 ls1 rs1 i) ∧
+ readback_config ? sig n (start … M) ls1 a1 rs1 =
+ step sig n M (readback_config ? sig n (start … M) ls a rs).
+
+lemma nth_vec_map_lt :
+ ∀A,B,f,i,n.∀v:Vector A n.∀d1,d2.i < n →
+ nth i ? (vec_map A B f n v) d1 = f (nth i ? v d2).
+#A #B #f #i #n #v #d1 #d2 #ltin >(nth_default B i n ? d1 (f d2) ltin) @sym_eq @nth_vec_map
+qed.
+
+lemma ctapes_mk_config : ∀sig,Q,n,s,t.
+ ctapes sig Q n (mk_mconfig sig Q n s t) = t.
+// qed.
+
+lemma cstate_rb: ∀sig,n.∀M:mTM sig n.∀ls,a,rs.∀q,m.
+ nth (S n) ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉) →
+ cstate sig (states sig n M) n
+ (readback_config (states sig n M) sig n (start sig n M) ls a rs) = q.
+#sig #n #M #ls #a #rs #q #m #H
+>state_readback whd in ⊢ (??%?); >H %
+qed.
+
+axiom eq_cstate_get_state: ∀sig,n.∀M:mTM sig n.∀ls,a,rs.
+ is_head ?? (nth (S n) ? (vec … a) (blank ?)) = true →
+ cstate sig (states sig n M) n
+ (readback_config (states sig n M) sig n (start sig n M) ls a rs) =
+ get_state (states sig n M) sig (S n) a (start sig n M).
+
+axiom eq_current_chars_resize: ∀sig,n.∀M:mTM sig n.∀ls,a,rs.
+ current_chars sig n
+ (ctapes sig (states sig n M) n
+ (readback_config (states sig n M) sig n (start sig n M) ls a rs)) =
+ get_chars … a.
+
+(*
+lemma rb_trans : ∀sig,Q,n.∀M:mTM sig n.∀ls,a,rs,q,m.
+ nth (S n) ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉) →
+ transf (states sig n M) sig (S n) (trans sig n M) a =
+ trans sig n M 〈q,get_chars … a〉.
+*)
+
+lemma nth_pmap_lt :
+ ∀A,B,C,f,i,n.∀v1:Vector A n.∀v2:Vector B n.∀d1,d2,d3.i < n→
+ f (nth i ? v1 d1) (nth i ? v2 d2) = nth i ? (pmap_vec A B C f n v1 v2) d3.
+#A #B #C #f #i elim i
+[ *
+ [ #v1 #v2 #d1 #d2 #d3 #Hlt @False_ind /2/
+ | #n0 #v1 #v2 #d1 #d2 #d3 #_ >(vec_expand … v1)
+ >(vec_expand … v2) >(nth_default … d3 (f d1 d2)) [% | // ]]
+| #i0 #IH *
+ [ #v1 #v2 #d1 #d2 #d3 #Hlt @False_ind /2/
+ | #n #v1 #v2 #d1 #d2 #d3 #Hlt>(vec_expand … v1)
+ >(vec_expand … v2)
+ whd in match (nth …d1); whd in match (tail ??);
+ whd in match (nth … d2); whd in match (tail B ?);
+ whd in match (nth … d3); whd in match (tail C ?);
+ <(IH n ?? d1 d2 d3) [2:@le_S_S_to_le @Hlt] %
+ ] ]
+qed.
+
+lemma tape_move_mono_def : ∀sig,t,a,m.
+ tape_move_mono sig t 〈a,m〉 = tape_move sig (tape_write sig t a) m.
+// qed.
+
+axiom to_sig_move : ∀Q,sig,n,t,m.
+ to_sig_tape (HC Q (S n)) sig
+ (tape_move (sig_ext (TA (HC Q (S n)) sig)) t m)
+ = tape_move sig (to_sig_tape ?? t) m.
+
+definition to_sig_conv :∀Q,sig:FinSet.∀n.
+ option sig → option (sig_ext (TA (HC Q (S n)) sig))
+≝ λQ,sig,n,c.
+ match c with
+ [None ⇒ None ?
+ |Some a ⇒ Some ? (Some ? (inr (HC Q (S n)) ? a))].
+
+axiom to_sig_write : ∀Q,sig,n,t,c.
+ to_sig_tape (HC Q (S n)) sig
+ (tape_write ? t (to_sig_conv ??? c))
+ = tape_write sig (to_sig_tape ?? t) c.
+
+axiom rb_write: ∀sig,n,ls,a,rs,i,c1,c2.
+ rb_trace_i ? n ls c1 rs i =
+ tape_write ? (rb_trace_i sig n ls a rs i) c2.
+
+lemma sem_stepM : ∀sig,n.∀M:mTM sig n.
+ stepM (states sig n M) sig n (trans sig n M) ⊨
+ R_stepM sig n M.
+#sig #n #M
+@(sem_seq_app … (sem_writef … ) (sem_exec_moves … (le_n ?)))
+#tin #tout * #t1 * whd in ⊢ (%→?); #Hwrite #Hmoves
+#a #ls #rs #Htin #H1 #H2 #H3 #H4 >Htin in Hwrite; #Hwrite
+lapply (Hwrite … (refl …)) -Hwrite whd in match (right ??); whd in match (left ??);
+>optf_Some #Ht1
+cut (∃q,m. nth (S n) ? (vec … a) (blank ?) = Some ? (inl ?? 〈q,m〉))
+ [lapply H2 cases (nth (S n) ? (vec … a) (blank ?))
+ [whd in ⊢ (??%?→?); #H destruct (H)
+ |* #x whd in ⊢ (??%?→?); #H destruct (H) cases x #q #m %{q} %{m} %
+ ]
+ ] * #q * #m #HaSn
+(*
+lapply (transf_eq … HaSn (refl ??) (refl ??) (eq_pair_fst_snd …) (refl ??) (refl ??))
+*)
+lapply (Hmoves … Ht1 ?? H3 H4)
+ [>(transf_eq … HaSn (refl ??) (refl ??) (eq_pair_fst_snd …) (refl ??) (refl ??))
+ >nth_cons_right %
+ | (* regularity is preserved *) @daemon
+ |* #ls1 * #a1 * #rs1 * * * #Htout #Hreg #Hrb #HtrSn
+ lapply (HtrSn (S n) (le_n ?) (le_n ?)) -HtrSn #HtrSn
+ cut (nth (S n) ? (vec … a1) (blank ?) =
+ nth (S n ) ? (vec … (transf (states sig n M) sig (S n) (trans sig n M) a)) (blank ?))
+ [@daemon (* from HtrSn *)] #Ha1
+ %{ls1} %{a1} %{rs1}
+ %[%[@Htout |@Hreg]
+ |(* commutation *)
+ lapply(transf_eq_ex … (trans sig n M) … HaSn)
+ * #c1 * #Hc1 * #c2 * #Hc2 * #q1 * #actions *
+ #Htrans * #moves * #Hmoves * #new_chars * #Hnew_chars #Htransf
+ @mconfig_eq
+ [(* state *) >state_readback whd in match (step ????);
+ >(cstate_rb … HaSn) >eq_current_chars_resize >get_chars_def
+ <Hc1 <Hc2 >Htrans whd in ⊢ (???%);
+ whd in ⊢ (??%?); >Ha1 >HaSn >Htransf >nth_cons_right %
+ |>tapes_readback whd in match (step ????);
+ >(cstate_rb … HaSn) >eq_current_chars_resize >get_chars_def
+ <Hc1 <Hc2 >Htrans >ctapes_mk_config
+ @(eq_vec … (niltape ?)) #i #lti
+ >nth_vec_map_lt [2:@lti |3:@niltape]
+ >Hrb <nth_pmap_lt [2:@lti|3:@N|4:@niltape]
+ >tapes_readback
+ >Htransf whd in match (vec_moves ?????);
+ >get_moves_cons_right >resize_id [2:@(len ?? moves)]
+ (* lhs *)
+ <nth_pmap_lt [2:@lti|3:%[@None|@N]|4:@niltape]
+ >nth_vec_map_lt [2:@lti |3:@niltape]
+ >(eq_pair_fst_snd … (nth i ? actions ?))
+ >tape_move_mono_def
+ cut (snd ?? (nth i ? actions 〈None sig,N〉) = nth i ? moves N)
+ [>Hmoves @nth_vec_map] #Hmoves1 >Hmoves1
+ >(nth_readback … lti) >(nth_readback … lti)
+ >to_sig_move @eq_f2 [2://]
+ <to_sig_write @eq_f @rb_write (* finto *)
+ ]
+ ]
+ ]
+qed.
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projects / helm.git / commitdiff