include "turing/multi_to_mono/exec_trace_move.ma".
(* include "turing/turing.ma". *)
-
-(**************************** Vector Operations *******************************)
-
-let rec resize A (l:list A) i d on i ≝
- match i with
- [ O ⇒ [ ]
- | S j ⇒ (hd ? l d)::resize A (tail ? l) j d
- ].
-
-lemma length_resize : ∀A,l,i,d. |resize A l i d| = i.
-#A #l #i lapply l -l elim i
- [#l #d %
- |#m #Hind #l cases l
- [#d normalize @eq_f @Hind
- |#a #tl #d normalize @eq_f @Hind
- ]
- ]
-qed.
-
-lemma resize_id : ∀A,n,l,d. |l| = n →
- resize A l n d = l.
-#A #n elim n
- [#l #d #H >(lenght_to_nil … H) //
- |#m #Hind * #d normalize
- [#H destruct |#a #tl #H @eq_f @Hind @injective_S // ]
- ]
-qed.
-
-definition resize_vec :∀A,n.Vector A n → ∀i.A→Vector A i.
-#A #n #a #i #d @(mk_Vector A i (resize A a i d) ) @length_resize
-qed.
-
-axiom nth_resize_vec :∀A,n.∀v:Vector A n.∀d,i,j. i < j →i < n →
- nth i ? (resize_vec A n v j d) d = nth i ? v d.
-
-lemma resize_vec_id : ∀A,n.∀v:Vector A n.∀d.
- resize_vec A n v n d = v.
-#A #n #v #d @(eq_vec … d) #i #ltin @nth_resize_vec //
-qed.
-
-definition vec_single: ∀A,a. Vector A 1 ≝ λA,a.
- mk_Vector A 1 [a] (refl ??).
-
-definition vec_cons_right : ∀A.∀a:A.∀n. Vector A n → Vector A (S n).
-#A #a #n #v >(plus_n_O n) >plus_n_Sm @(vec_append … v (vec_single A a))
->length_append >(len A n v) //
-qed.
-
-lemma eq_cons_right : ∀A,a1,a2.∀n.∀v1,v2:Vector A n.
- a1 = a2 → v1 = v2 → vec_cons_right A a1 n v1 = vec_cons_right A a2 n v2.
-// qed.
-
-axiom nth_cons_right: ∀A.∀a:A.∀n.∀v:Vector A n.∀d.
- nth n ? (vec_cons_right A a n v) d = a.
-(*
-#A #a #n elim n
- [#v #d >(vector_nil … v) //
- |#m #Hind #v #d >(vec_expand … v) whd in ⊢ (??%?);
- whd in match (vec_cons_right ????);
-*)
-
-lemma get_moves_cons_right: ∀Q,sig,n,q,moves,a.
- get_moves Q sig (S n)
- (vec_cons_right ? (Some ? (inl ?? 〈q,moves〉)) (S n) a) = moves.
-#Q #sig #n #q #moves #a whd in ⊢(??%?); >nth_cons_right //
-qed.
-
-axiom resize_cons_right: ∀A.∀a:A.∀n.∀v:Vector A n.∀d.
- resize_vec ? (S n) (vec_cons_right A a n v) n d = v.
let rec exec_moves Q sig n i on i : TM (MTA (HC Q n) sig (S n)) ≝
match i with
∃ls1,a1,rs1.
t2 = midtape (MTA (HC Q n) sig (S n)) ls1 a1 rs1 ∧
(∀i.regular_trace (TA (HC Q n) sig) (S n) a1 ls1 rs1 i) ∧
- readback ? (S n) ls1 (vec … a1) rs1 i =
- tape_moves ?? (readback ? (S n) ls (vec … a) rs i) (vec_moves Q sig n a i) ∧
- (∀j. i ≤ j → j ≤ n →
+ (∀j. j < i → j ≤ n →
+ rb_trace_i ? (S n) ls1 (vec … a1) rs1 j =
+ tape_move ? (rb_trace_i ? (S n) ls (vec … a) rs j) (get_move Q sig n a j)) ∧
+ (* tape_moves ?? (readback ? (S n) ls (vec … a) rs i) (vec_moves Q sig n a i) ∧ *)
+ (∀j. i ≤ j → j ≤ n →
rb_trace_i ? (S n) ls1 (vec … a1) rs1 j =
rb_trace_i ? (S n) ls (vec … a) rs j).
(* alias id "Realize_to_Realize" =
"cic:/matita/turing/mono/Realize_to_Realize.def(13)". *)
+(*
lemma nth_readback: ∀sig,n,ls,a,rs,j,i. i < j →
nth i ? (readback sig n ls a rs j) (niltape ?) =
rb_trace_i sig n ls a rs (j - S i).
|#m #Hmj >minus_S_S @Hind @le_S_S_to_le //
]
]
-qed.
+qed. *)
lemma sem_exec_moves: ∀Q,sig,n,i. i ≤ n →
exec_moves Q sig n i ⊨ R_exec_moves Q sig n i.
#Q #sig #n #i elim i
[#_ @(Realize_to_Realize … (sem_nop …))
#t1 #t2 #H #a #ls #rs #Ht1 #Hreg #H1 #H2 #H3
- %{ls} %{a} %{rs} %[% >H [%[@Ht1|@Hreg]| %]|//]
+ %{ls} %{a} %{rs} %[% >H [%[@Ht1|@Hreg]| #j #ltjO @False_ind /2/]|//]
|#j #Hind #lttj lapply (lt_to_le … lttj) #ltj
@(sem_seq_app … (Hind ltj) (sem_exec_move_i … lttj)) -Hind
#int #outt * #t1 * #H1 #H2
cut (∀i. get_move Q sig n a i = get_move Q sig n a1 i)
[@daemon] #aa1
%[%[%[@Houtt|@Hregout]
- |whd in ⊢ (??%?); @Vector_eq >(vec_moves_cons … lttj)
- >tape_moves_def >pmap_vec_cons @eq_f2
- [<H10 [>aa1 @Hrboutt |@ltj |@le_n]
- |<tape_moves_def <H9 (* mitico *) @eq_f
- @(eq_vec … (niltape ?)) #i #ltij
- >(nth_readback … ltij) >(nth_readback … ltij) @Hrbid
- [@(transitive_le … ltj) // |@lt_to_not_eq @lt_plus_to_minus //]
+ |#k #ltk cases (le_to_or_lt_eq … ltk) #Hk #lekn
+ [>(Hrbid … lekn) [2:@lt_to_not_eq @le_S_S_to_le @Hk]
+ @(H9 k ? lekn) @le_S_S_to_le @Hk
+ |destruct (Hk) <H10 // @Hrboutt
]
]
|#a #Hja #Han >(Hrbid … Han)
definition Rmove_R_i ≝ λA,sig,n,i.
R_guarded_M ? (S n) i (Rmtil A sig n i).
+(**************************** Vector Operations *******************************)
+let rec resize A (l:list A) i d on i ≝
+ match i with
+ [ O ⇒ [ ]
+ | S j ⇒ (hd ? l d)::resize A (tail ? l) j d
+ ].
+
+lemma length_resize : ∀A,l,i,d. |resize A l i d| = i.
+#A #l #i lapply l -l elim i
+ [#l #d %
+ |#m #Hind #l cases l
+ [#d normalize @eq_f @Hind
+ |#a #tl #d normalize @eq_f @Hind
+ ]
+ ]
+qed.
+
+lemma resize_id : ∀A,n,l,d. |l| = n →
+ resize A l n d = l.
+#A #n elim n
+ [#l #d #H >(lenght_to_nil … H) //
+ |#m #Hind * #d normalize
+ [#H destruct |#a #tl #H @eq_f @Hind @injective_S // ]
+ ]
+qed.
+
+definition resize_vec :∀A,n.Vector A n → ∀i.A→Vector A i.
+#A #n #a #i #d @(mk_Vector A i (resize A a i d) ) @length_resize
+qed.
+
+axiom nth_resize_vec :∀A,n.∀v:Vector A n.∀d,i,j. i < j →i < n →
+ nth i ? (resize_vec A n v j d) d = nth i ? v d.
+
+lemma resize_vec_id : ∀A,n.∀v:Vector A n.∀d.
+ resize_vec A n v n d = v.
+#A #n #v #d @(eq_vec … d) #i #ltin @nth_resize_vec //
+qed.
+
+definition vec_single: ∀A,a. Vector A 1 ≝ λA,a.
+ mk_Vector A 1 [a] (refl ??).
+
+definition vec_cons_right : ∀A.∀a:A.∀n. Vector A n → Vector A (S n).
+#A #a #n #v >(plus_n_O n) >plus_n_Sm @(vec_append … v (vec_single A a))
+>length_append >(len A n v) //
+qed.
+
+lemma eq_cons_right : ∀A,a1,a2.∀n.∀v1,v2:Vector A n.
+ a1 = a2 → v1 = v2 → vec_cons_right A a1 n v1 = vec_cons_right A a2 n v2.
+// qed.
+
+axiom nth_cons_right_n: ∀A.∀a:A.∀n.∀v:Vector A n.∀d.
+ nth n ? (vec_cons_right A a n v) d = a.
+
+axiom nth_cons_right_lt: ∀A.∀a:A.∀n.∀v:Vector A n.∀d.∀i. i < n →
+ nth i ? (vec_cons_right A a n v) d = nth i ? v d.
+(*
+#A #a #n elim n
+ [#v #d >(vector_nil … v) //
+ |#m #Hind #v #d >(vec_expand … v) whd in ⊢ (??%?);
+ whd in match (vec_cons_right ????);
+*)
+
+axiom resize_cons_right: ∀A.∀a:A.∀n.∀v:Vector A n.∀d.
+ resize_vec ? (S n) (vec_cons_right A a n v) n d = v.
(*************************** readback of the tape *****************************)
definition opt_cur ≝ λsig,a.
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)
- ].
-
+(*
+definition readback :∀sig,n,ls,a,rs,i.Vector (tape (sig_ext sig)) i ≝
+vec_map (rb_trace_i *)
+
lemma orb_false_l: ∀b1,b2:bool.
(b1 ∨ b2) = false → (b1 = false) ∧ (b2 = false).
* * normalize /2/ qed.
definition get_move ≝ λQ,sig,n.λa:MTA (HC Q n) sig (S n).λi.
nth i ? (vec … (get_moves Q sig n a)) N.
+
+lemma get_moves_cons_right: ∀Q,sig,n,q,moves,a.
+ get_moves Q sig (S n)
+ (vec_cons_right ? (Some ? (inl ?? 〈q,moves〉)) (S n) a) = moves.
+#Q #sig #n #q #moves #a whd in ⊢(??%?); >nth_cons_right_n //
+qed.
definition exec_move_i ≝ λQ,sig,n,i.
(ifTM ? (test_char ? (λa. (get_move Q sig n a i == R)))
| 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).
-
+let rec rb_tapes Q sig n ls (a:MTA Q sig (S n)) rs i on i ≝
+ match i with
+ [ O ⇒ mk_Vector ? 0 (nil ?) (refl ??)
+ | S j ⇒ vec_cons_right ? (to_sig_tape ?? (rb_trace_i ? (S n) ls (vec … a) rs j)) j
+ (rb_tapes Q sig n ls a rs j)].
+
+lemma nth_rb_tapes : ∀Q,sig,n,ls.∀a:MTA Q sig (S n).∀rs,j,i. i < j →
+ nth i ? (rb_tapes Q sig n ls (a:MTA Q sig (S n)) rs j) (niltape ?) =
+ to_sig_tape ?? (rb_trace_i ? (S n) ls (vec … a) rs i).
+#Q #sig #n #ls #a #rs #j elim j
+ [#i #H @False_ind /2/
+ |#k #Hind #i #lti cases (le_to_or_lt_eq … (le_S_S_to_le … lti))
+ [#Hlt >nth_cons_right_lt [@Hind //|//]
+ |#Heq >Heq @nth_cons_right_n
+ ]
+ ]
+qed.
+
+
(* 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
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).
+ (rb_tapes (HC Q (S n)) sig (S n) ls a rs (S n)).
lemma state_readback : ∀Q,sig,n,q0,ls,a,rs.
cstate … (readback_config Q sig n q0 ls a rs) =
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.
+ rb_tapes (HC Q (S n)) sig (S n) ls a rs (S n).
// qed.
definition R_stepM ≝ λsig.λn.λM:mTM sig n.λt1,t2.
(tape_write ? t (to_sig_conv ??? c))
= tape_write sig (to_sig_tape ?? t) c.
+definition opt ≝ λA.λoc1: option A.λc2.
+ match oc1 with [None ⇒ c2 | Some c1 ⇒ c1].
+
axiom rb_write: ∀sig,n,ls,a,rs,i,c1,c2.
+ nth i ? c1 (None ?) = opt ? c2 (nth i ? c1 (None ?)) →
rb_trace_i ? n ls c1 rs i =
tape_write ? (rb_trace_i sig n ls a rs i) c2.
*)
lapply (Hmoves … Ht1 ?? H3 H4)
[>(transf_eq … HaSn (refl ??) (refl ??) (eq_pair_fst_snd …) (refl ??) (refl ??))
- >nth_cons_right %
+ >nth_cons_right_n %
| (* regularity is preserved *) @daemon
|* #ls1 * #a1 * #rs1 * * * #Htout #Hreg #Hrb #HtrSn
lapply (HtrSn (S n) (le_n ?) (le_n ?)) -HtrSn #HtrSn
[(* 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 %
+ whd in ⊢ (??%?); >Ha1 >HaSn >Htransf >nth_cons_right_n %
|>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)]
+ >nth_rb_tapes [2:@lti]
+ >Hrb [2:@lt_to_le @lti|3:@lti]
+ >Htransf whd in match (get_move ?????); (* whd in match (vec_moves ?????); *)
+ >get_moves_cons_right
(* 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 ?))
+ >ctapes_mk_config >nth_rb_tapes [2:@lti]
+ (* >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 *)
+ <to_sig_write @eq_f @rb_write
+ >nth_cons_right_lt [2:@lti]
+ >Hnew_chars <nth_pmap_lt [2:@lti|3:@None |4:%[@None|@N]]
+ whd in ⊢ (??%%);
+ inversion (\fst (nth i ? actions 〈None sig,N〉))
+ [#Hcase whd in ⊢ (??%%); >Hcase % |#c #Hcase % ]
]
]
]
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