include "turing/mono.ma".
+lemma le_to_eq : ∀m,n.m ≤ n → ∃k. n = m + k. /3 by plus_minus, ex_intro/
+qed.
+
+lemma minus_tech : ∀a,b.a + b - a = b. // qed.
+
+lemma loop_incr2 : ∀sig,M,m,n,cfg,cfg'.m ≤ n →
+ loopM sig M m cfg = Some ? cfg' → loopM sig M n cfg = Some ? cfg'.
+#sig #M #m #n #cfg #cfg' #H cases (le_to_eq … H) #k #Hk >Hk
+>commutative_plus @loop_incr
+qed.
+
(* given a FinSet F:
- get its cardinality
- return its nth element
- return the index of a given element
*)
-axiom FS_crd : FinSet → nat.
-axiom FS_nth : ∀F:FinSet.nat → option F.
-axiom index_of_FS : ∀F:FinSet.F → nat.
+definition FS_crd ≝ λF:FinSet.|enum F|.
+definition FS_nth ≝ λF:FinSet.λn.nth_opt ? n (enum F).
+definition index_of_FS_aux ≝ λF:FinSet.λf.position_of ? (λx.x==f) (enum F).
+
+lemma index_of_FS_aux_None :
+ ∀F,f.index_of_FS_aux F f = None ? → False.
+#F #f #e cut (memb ? f (enum F) = false)
+[ generalize in match e; -e normalize in ⊢ (%→?); generalize in match O;
+ elim (enum F) //
+ #hd #tl #IH #n whd in ⊢ (??%?→?); cases (true_or_false (hd==f))
+ #Hbool >Hbool normalize
+ [ #H destruct (H)
+ | #H >(\bf ?) [| @sym_not_eq @(\Pf Hbool) ] @IH // ]
+| >enum_complete #H destruct (H) ]
+qed.
+
+definition index_of_FS : ∀F:FinSet.F → nat ≝ λF,f.
+match index_of_FS_aux F f
+return (λx:option nat.index_of_FS_aux F f = x → nat) with
+[ None ⇒ λe.?
+| Some n ⇒ λe.n ] (refl ??).cases (index_of_FS_aux_None … e)
+qed.
(* unary bit representation (with a given length) of a certain number *)
-axiom unary_of_nat : nat → nat → (list bool).
+let rec unary_of_nat n k on n ≝
+ match n with [ O ⇒ [ ] | S q ⇒ (eqb q k)::unary_of_nat q k].
+
+lemma lt_FS_index_crd_aux : ∀sig,c,n.index_of_FS_aux sig c = Some ? n → n < FS_crd sig.
+#sig #c #n whd in ⊢ (??%?→?); >(?:FS_crd sig = O + FS_crd sig) //
+generalize in match O; normalize in match (FS_crd sig); elim (enum sig)
+normalize [ #n0 #H destruct (H) ]
+#hd #tl #IH #n0 cases (hd==c) normalize
+[ #H destruct (H) //
+| #H lapply (IH ? H) // ]
+qed.
+
+lemma index_of_FS_def : ∀sig,c,n.index_of_FS sig c = n → index_of_FS_aux sig c = Some ? n.
+#sig #c #n whd in ⊢ (??%?→?); lapply (refl ? (index_of_FS_aux sig c))
+cases (index_of_FS_aux sig c) in ⊢ (???%→??(match % return ? with [ _ ⇒ ? | _ ⇒ ? ] ?)?→%);
+[ #e cases (index_of_FS_aux_None ?? e)
+| normalize // ]
+qed.
+
+lemma index_of_FS_def2 : ∀sig,c.index_of_FS_aux sig c = Some ? (index_of_FS sig c)./2/
+qed.
+
+lemma lt_FS_index_crd: ∀sig,c.index_of_FS sig c < FS_crd sig.
+#sig #c @(lt_FS_index_crd_aux sig c ? (index_of_FS_def2 …))
+qed.
+
+lemma le_position_of_aux : ∀T,f,l,k,n.position_of_aux T f l k = Some ? n → k ≤ n.
+#T #f #l elim l normalize
+[ #k #n #H destruct (H)
+| #hd #tl #IH #k #n cases (f hd) normalize
+ [ #H destruct (H) %
+ | #H lapply (IH … H) /2 by lt_to_le/ ]
+]
+qed.
+
+lemma nth_index_of_FS_aux :
+∀sig,a,n.index_of_FS_aux sig a = Some ? n → FS_nth sig n = Some ? a.
+#sig #a #n normalize >(?:n = O + n) in ⊢ (%→?); //
+lapply O lapply n -n elim (enum sig) normalize
+[ #n #k #H destruct (H)
+| #hd #tl #IH #n #k cases (true_or_false (hd==a)) #Ha >Ha normalize
+ [ #H destruct (H) >(?:n = O) // >(\P Ha) //
+ | cases n
+ [ <plus_n_O #H @False_ind lapply (le_position_of_aux … H) #H1
+ cases (not_le_Sn_n k) /2/
+ | #n0 #Hrec @(IH ? (S k)) >Hrec /2 by eq_f/ ]
+ ]
+]
+qed.
+
+lemma nth_index_of_FS : ∀sig,a.FS_nth sig (index_of_FS ? a) = Some ? a.
+#sig #a @nth_index_of_FS_aux >index_of_FS_def2 %
+qed.
+
+definition bin_char ≝ λsig,ch.unary_of_nat (FS_crd sig) (index_of_FS sig ch).
+
+definition opt_bin_char ≝ λsig,c.match c with
+[ None ⇒ [ ] | Some c0 ⇒ bin_char sig c0 ].
+
+lemma eq_length_bin_char_FS_crd : ∀sig,c.|bin_char sig c| = FS_crd sig.
+#sig #c whd in ⊢ (??(??%)?); elim (FS_crd sig) //
+#n #IH <IH in ⊢ (???%); %
+qed.
+
+lemma bin_char_FS_nth_tech :
+ ∀sig,c,l1,b,l2.bin_char sig c = l1@b::l2 → b = (((|l2|):DeqNat) == index_of_FS sig c).
+#sig #c #l1 #b #l2 #Hbin lapply (eq_length_bin_char_FS_crd sig c)
+>Hbin #Hlen lapply Hbin lapply Hlen -Hlen -Hbin
+whd in match (bin_char ??); lapply l2 lapply c lapply l1 -l2 -c -l1
+elim (FS_crd sig)
+[ #l1 #b #l2 normalize in ⊢ (??%?→?); cases l1
+ [ normalize #H destruct (H) | #hd #tl normalize #H destruct (H) ]
+| #n #IH #l1 #b #l2 whd in ⊢ (?→??%?→?); cases l1
+ [ whd in ⊢ (??%?→???%→?); #Hlen destruct (Hlen)
+ #H <(cons_injective_l ????? H) @eq_f2 //
+ | #b0 #l10 #Hlen #H lapply (cons_injective_r ????? H) -H #H @(IH … H)
+ normalize in Hlen; destruct (Hlen) % ]
+]
+qed.
-axiom lt_FS_index_crd : ∀sig,c.index_of_FS sig c < FS_crd sig.
+lemma nth_opt_memb : ∀T:DeqSet.∀l,n,t.nth_opt T n l = Some ? t → memb T t l = true.
+#T #l elim l normalize [ #n #t #H destruct (H) ]
+#hd #tl #IH #n #t cases n normalize
+[ #Ht destruct (Ht) >(\b (refl ? t)) %
+| #n0 #Ht cases (t==hd) // @(IH … Ht) ]
+qed.
+
+lemma FS_nth_neq :
+∀sig,m,n. m ≠ n →
+∀s1,s2.FS_nth sig m = Some ? s1 → FS_nth sig n = Some ? s2 → s1 ≠ s2.
+#sig #m #n #Hneq #s1 #s2 lapply (enum_unique sig) lapply Hneq
+lapply n lapply m -n -m normalize elim (enum sig)
+[ #m #n #_ #_ normalize #H destruct (H)
+| #hd #tl #IH #m #n #Hneq whd in ⊢ (??%?→?);
+ cases (true_or_false (hd ∈ tl)) #Hbool >Hbool normalize in ⊢ (%→?);
+ [ #H destruct (H)
+ | #H cases m in Hneq;
+ [ #Hneq whd in ⊢ (??%?→?); #H1 destruct (H1) cases n in Hneq;
+ [ * #H cases (H (refl ??))
+ | #n0 #_ whd in ⊢ (??%?→?); #Htl % #Heq destruct (Heq)
+ >(nth_opt_memb … Htl) in Hbool; #Hfalse destruct (Hfalse)
+ ]
+ | #m0 #Hneq whd in ⊢ (??%?→?); #H1
+ whd in ⊢ (??%?→?); cases n in Hneq;
+ [ #_ whd in ⊢ (??%?→?); #H2 destruct (H2) % #Heq destruct (Heq)
+ >(nth_opt_memb … H1) in Hbool; #Hfalse destruct (Hfalse)
+ | #n0 #Hneq whd in ⊢ (??%?→?); @(IH m0 n0 ? H … H1)
+ % #Heq cases Hneq /2/
+ ]
+ ]
+ ]
+]
+qed.
+
+lemma nth_opt_Some : ∀T,l,n.n < |l| → ∃t.nth_opt T n l = Some ? t.
+#T #l elim l
+[ normalize #n #H @False_ind cases (not_le_Sn_O n) /2/
+| #hd #tl #IH #n normalize cases n
+ [ #_ %{hd} //
+ | #n0 #Hlt cases (IH n0 ?) [| @le_S_S_to_le // ]
+ #t #Ht normalize %{t} // ]
+]
+qed.
+
+corollary FS_nth_Some : ∀sig,n.n < FS_crd sig → ∃s.FS_nth sig n = Some ? s.
+#sig #n @nth_opt_Some
+qed.
+
+lemma bin_char_FS_nth :
+ ∀sig,c,l1,b,l2.bin_char sig c = l1@b::l2 → b = (FS_nth sig (|l2|) == Some ? c).
+#sig #c #l1 #b #l2 #H >(bin_char_FS_nth_tech … H)
+cases (true_or_false (((|l2|):DeqNat)==index_of_FS sig c)) #Hbool >Hbool
+[ >(?:(|l2|)=index_of_FS sig c) [|change with ((|l2|):DeqNat) in ⊢ (??%?); @(\P Hbool) ]
+ @sym_eq @(\b ?) @nth_index_of_FS
+| <nth_index_of_FS @sym_eq @(\bf ?) % #Hfalse
+ cases (FS_nth_Some sig (|l2|) ?) [| <(eq_length_bin_char_FS_crd sig c) >H >length_append normalize // ]
+ #s1 #H1
+ cases (FS_nth_Some sig (index_of_FS sig c) ?) [|//]
+ #s2 #H2
+ cases (FS_nth_neq … H1 H2) [| @(\Pf Hbool) ]
+ #Hfalse2 @Hfalse2 <Hfalse in H2; >H1 #HSome destruct (HSome) %
+]
+qed.
+
+corollary binary_to_bin_char :∀sig,csl,csr,a.
+ csl@true::csr=bin_char sig a → FS_nth ? (length ? csr) = Some ? a.
+#sig #csl #csr #a #H @(\P ?) @sym_eq @bin_char_FS_nth //
+qed.
(* axiom FinVector : Type[0] → nat → FinSet.*)
(〈start sig M,bin0,None ?,FS_crd sig〉) (halt_binaryTM sig M).
/2 by lt_S_to_lt/ qed.
-definition bin_char ≝ λsig,ch.unary_of_nat (FS_crd sig) (index_of_FS sig ch).
-
-axiom eq_length_bin_char_FS_crd : ∀sig,c.|bin_char sig c| = FS_crd sig.
-axiom bin_char_FS_nth :
- ∀sig,c,l1,b,l2.bin_char sig c = l1@b::l2 → b = (FS_nth sig (|l2|) == Some ? c).
-
-definition opt_bin_char ≝ λsig,c.match c with
-[ None ⇒ [ ] | Some c0 ⇒ bin_char sig c0 ].
-
definition bin_list ≝ λsig,l.flatten ? (map ?? (bin_char sig) l).
definition rev_bin_list ≝ λsig,l.flatten ? (map ?? (λc.reverse ? (bin_char sig c)) l).
>Hcur %
qed.
-(* to be checked *)
-axiom binary_to_bin_char :∀sig,csl,csr,a.
- csl@true::csr=bin_char sig a → FS_nth ? (length ? csr) = Some ? a.
-
-axiom daemon : ∀P:Prop.P.
-
lemma binaryTM_phase0_midtape_aux :
∀sig,M,q,ls,a,rs,k.
halt sig M q=false →
t = mk_tape ? (reverse ? csl@ls) (option_hd ? (csr@rs)) (tail ? (csr@rs)) →
csl@csr = bin_char sig a →
|csl@csr| = FS_crd sig →
- (index_of_FS ? a < |csl| → ch = Some ? a) →
+ (|csr| ≤ index_of_FS ? a → ch = Some ? a) →
loopM ? (mk_binaryTM sig M) (S (length ? csr) + k)
(mk_config ?? (〈q,bin0,ch,length ? csr〉) t)
= loopM ? (mk_binaryTM sig M) k
| #c1 #csr1 normalize >rev_append_def >rev_append_def >reverse_append % ]
| /2 by lt_S_to_lt/
|]
- #H whd in match (plus ??); >H @eq_f @eq_f2 %
+ #H whd in match (plus ??); >Ha >H @eq_f @eq_f2 %
| #csr0 #IH #csl #t #ch #Hlen #Ht #Heq #Hcrd #Hch >loopM_unfold >loop_S_false [|normalize //]
<loopM_unfold >binaryTM_bin0_false [| >Ht % ]
lapply (IH (csl@[false]) (tape_move FinBool t R) ??????)
[6: @ch
- | (* by cases: if index < |csl|, then Hch, else False *)
- @daemon
+ | #Hle cases (le_to_or_lt_eq … Hle) [ @Hch ]
+ #Hindex lapply (bin_char_FS_nth … (sym_eq … Heq)) >Hindex
+ >(nth_index_of_FS sig a) >(\b (refl ? (Some sig a))) #H destruct (H)
| >associative_append @Hcrd
| >associative_append @Heq
| >Ht whd in match (option_hd ??) in ⊢ (??%?); whd in match (tail ??) in ⊢ (??%?);
]
qed.
-lemma le_to_eq : ∀m,n.m ≤ n → ∃k. n = m + k. /3 by plus_minus, ex_intro/
-qed.
-
-lemma minus_tech : ∀a,b.a + b - a = b. // qed.
-
lemma binaryTM_phase0_midtape :
∀sig,M,t,q,ls,a,rs,ch.
O < FS_crd sig →
cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >(minus_tech (S (FS_crd sig)))
cut (∃c,cl.bin_char sig a = c::cl)
[ lapply (refl ? (|bin_char ? a|)) >eq_length_bin_char_FS_crd in ⊢ (???%→?);
- cases (bin_char ? a) normalize /3 by ex_intro/ #H
+ cases (bin_char ? a) [|/3 by ex_intro/] normalize in ⊢ (??%?→?); #H
<H in Hcrd; -H #H cases (not_le_Sn_O O) #Hfalse cases (Hfalse H) ]
* #c * #cl #Ha >Ha
cut (FS_crd sig = |bin_char sig a|) [/2 by plus_minus_m_m/] #Hlen
(mk_config ?? 〈q,bin0,〈ch,|c::cl|〉〉 t)))
[ @le_S_S <Ha <Hlen // | @eq_f2 // @eq_f2 // @eq_f <Ha >Hlen % ]
>(binaryTM_phase0_midtape_aux ? M q ls a rs ? ? (c::cl) [ ] t ch) //
-[| normalize #Hfalse @False_ind cases (not_le_Sn_O ?) /2/
+[| <Ha <Hlen lapply (lt_FS_index_crd sig a) #Hlt #Hle
+ lapply (transitive_le ??? Hlt Hle) #H cases (not_le_Sn_n (index_of_FS ? a))
+ #H1 @False_ind /2/
| <Ha >Hlen %
| >Ha %
| >Ht >Ha %
(mk_tape ? (csl@ls) (option_hd ? (csr@rs)) (tail ? (csr@rs))))
= loopM ? (mk_binaryTM sig M) k
(mk_config ?? (〈qn,bin3,ch,displ_of_move sig mv〉)
- (mk_tape ? (reverse ? (bin_char sig chn)@ls) (option_hd ? rs) (tail ? rs)))) [1,2: @le_S_S /2 by le_S/]
+ (mk_tape ? (reverse ? (bin_char sig chn)@ls) (option_hd ? rs) (tail ? rs)))) [1,2: @le_S_S [/2 by lt_to_le/|/2 by le_S/] ]
[ #sig #M #q #ch #qn #chn #mv #ls #rs #k #csr #Htrans elim csr
[ #csl #Hcount #Hcrd * #fs #Hfs >loopM_unfold >loop_S_false // normalize in match (length ? [ ]);
>(binaryTM_bin2_O … Htrans) <loopM_unfold @eq_f @eq_f @eq_f3 //
cases fs in Hfs; // #f0 #fs0 #H lapply (eq_f ?? (length ?) … H)
>length_append >(?:|bin_char sig chn| = FS_crd sig) [|//]
- <Hcrd >length_reverse #H1 cut (O = |f0::fs0|) [ /2/ ]
+ <Hcrd >length_reverse >length_append whd in match (|[]|); #H1 cut (O = |f0::fs0|) [ /2 by plus_to_minus/ ]
normalize #H1 destruct (H1)
| #b0 #bs0 #IH #csl #Hcount #Hcrd * #fs #Hfs
>loopM_unfold >loop_S_false // >(binaryTM_bin2_S_Some … Htrans)
(option_hd ? (bs0@rs)) (tail ? (bs0@rs)))
in match (tape_move ? (tape_write ???) ?);
[| cases bs0 // cases rs // ] @IH
- [ whd in Hcount:(?%?); /2 by lt_S_to_lt/
- | <Hcrd >length_append >length_append normalize //
+ [ <Hcrd >length_append >length_append normalize //
| cases fs in Hfs;
[ #Hfalse cut (|bin_char ? chn| = |csl|) [ >Hfalse >length_append >length_reverse // ] -Hfalse >(?:|bin_char sig chn| = FS_crd sig) [|//]
<Hcrd >length_append normalize >(?:|csl| = |csl|+ O) in ⊢ (???%→?); //
]
]
| #Hcut #sig #M #q #ch #qn #chn #mv #ls #cs #rs #Htrans #Hcrd #k #Hk
- cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >minus_tech @trans_eq
+ cases (le_to_eq … Hk) #k0 #Hk0 >Hk0 >(?:S (FS_crd sig) +k0-S (FS_crd sig) = k0) [|@minus_tech]
+ @trans_eq
[3: @(trans_eq ???? (Hcut ??????? ls ?? cs Htrans [ ] …)) //
[ normalize % // | normalize @Hcrd | >Hcrd // ]
|| @eq_f2 [ >Hcrd % | @eq_f2 // @eq_f cases Hcrd // ] ] ]
mk_tape ? [ ] (option_hd ? rs) (tail ? rs). #sig * //
qed.
-axiom loop_increase : ∀sig,M,m,n,cfg,cfg'.m < n →
- loopM sig M m cfg = Some ? cfg' → loopM sig M n cfg = Some ? cfg'.
-
lemma binaryTM_loop :
∀sig,M,i,tf,qf. O < FS_crd sig →
∀t,q.∃k.i ≤ k ∧
∀u1.t1 = tape_bin_lift sig u1 →
∃u2.t2 = tape_bin_lift sig u2 ∧ R u1 u2.
-(*
-∀sig,M,i,tf,qf. O < FS_crd sig →
- ∀t,q.∃k.i ≤ k ∧
- ((loopM sig M i (mk_config ?? q t) = Some ? (mk_config ?? qf tf) →
- loopM ? (mk_binaryTM sig M) k
- (mk_config ?? (state_bin_lift ? M q) (tape_bin_lift ? t)) =
- Some ? (mk_config ?? (state_bin_lift ? M qf) (tape_bin_lift ? tf))) ∧
- (loopM sig M i (mk_config ?? q t) = None ? →
- loopM ? (mk_binaryTM sig M) k
- (mk_config ?? (state_bin_lift ? M q) (tape_bin_lift ? t)) = None ?)).
- *)
-axiom loop_incr : ∀sig,M,m,n,cfg,cfg'.m ≤ n →
- loopM sig M m cfg = Some ? cfg' → loopM sig M n cfg = Some ? cfg'.
-
theorem sem_binaryTM :
∀sig,M,R.O < FS_crd sig → M ⊫ R → mk_binaryTM sig M ⊫ R_bin_lift ? R.
#sig #M #R #Hcrd #HM #t #k #outc #Hloopbin #u #Ht
[ #H cases (binaryTM_loop ? M k u (start ? M) Hcrd u (start ? M))
#k0 * #Hlt * #_ #H1 lapply (H1 H) -H -H1 <Ht
whd in match (initc ???) in Hloopbin; whd in match (start ??) in Hloopbin;
- >state_bin_lift_unfold >(loop_incr … Hlt Hloopbin) #H destruct (H)
+ >state_bin_lift_unfold >(loop_incr2 … Hlt Hloopbin) #H destruct (H)
| * #qf #tf #H cases (binaryTM_loop ? M k tf qf Hcrd u (start ? M))
#k0 * #Hlt * #H1 #_ lapply (H1 H) -H1 <Ht
whd in match (initc ???) in Hloopbin; whd in match (start ??) in Hloopbin;
- >state_bin_lift_unfold >(loop_incr … Hlt Hloopbin) #Heq destruct (Heq)
+ >state_bin_lift_unfold >(loop_incr2 … Hlt Hloopbin) #Heq destruct (Heq)
% [| % [%]] @(HM … H)
qed.
\ No newline at end of file
projects / helm.git / commitdiff