*)
definition trans_compare_step ≝
- λi,j.λsig:FinSet.λn.
+ λi,j.λsig:FinSet.λn.λis_endc.
λp:compare_states × (Vector (option sig) (S n)).
let 〈q,a〉 ≝ p in
match pi1 … q with
[ None ⇒ 〈comp2,null_action ? n〉
| Some ai ⇒ match nth j ? a (None ?) with
[ None ⇒ 〈comp2,null_action ? n〉
- | Some aj ⇒ if ai == aj
+ | Some aj ⇒ if notb (is_endc ai) ∧ ai == aj
then 〈comp1,change_vec ? (S n)
(change_vec ? (S n) (null_action ? n) (Some ? 〈ai,R〉) i)
(Some ? 〈aj,R〉) j〉
| S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ].
definition compare_step ≝
- λi,j,sig,n.
- mk_mTM sig n compare_states (trans_compare_step i j sig n)
+ λi,j,sig,n,is_endc.
+ mk_mTM sig n compare_states (trans_compare_step i j sig n is_endc)
comp0 (λq.q == comp1 ∨ q == comp2).
definition R_comp_step_true ≝
- λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
+ λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
∃x.
+ is_endc x = false ∧
current ? (nth i ? int (niltape ?)) = Some ? x ∧
current ? (nth j ? int (niltape ?)) = Some ? x ∧
outt = change_vec ??
(tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) j.
definition R_comp_step_false ≝
- λi,j:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
- (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
+ λi,j:nat.λsig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+ ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+ current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
current ? (nth i ? int (niltape ?)) = None ? ∨
current ? (nth j ? int (niltape ?)) = None ?) ∧ outt = int.
lemma comp_q0_q2_null :
- ∀i,j,sig,n,v.i < S n → j < S n →
+ ∀i,j,sig,n,is_endc,v.i < S n → j < S n →
(nth i ? (current_chars ?? v) (None ?) = None ? ∨
nth j ? (current_chars ?? v) (None ?) = None ?) →
- step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v)
+ step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v)
= mk_mconfig ??? comp2 v.
-#i #j #sig #n #v #Hi #Hj
+#i #j #sig #n #is_endc #v #Hi #Hj
whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?);
* #Hcurrent
[ @eq_f2
qed.
lemma comp_q0_q2_neq :
- ∀i,j,sig,n,v.i < S n → j < S n →
- nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?) →
- step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v)
+ ∀i,j,sig,n,is_endc,v.i < S n → j < S n →
+ ((∃x.nth i ? (current_chars ?? v) (None ?) = Some ? x ∧ is_endc x = true) ∨
+ nth i ? (current_chars ?? v) (None ?) ≠ nth j ? (current_chars ?? v) (None ?)) →
+ step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v)
= mk_mconfig ??? comp2 v.
-#i #j #sig #n #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
+#i #j #sig #n #is_endc #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
cases (nth i ?? (None ?)) in ⊢ (???%→?);
[ #Hnth #_ @comp_q0_q2_null // % //
| #ai #Hai lapply (refl ? (nth j ?(current_chars ?? v)(None ?)))
cases (nth j ?? (None ?)) in ⊢ (???%→?);
[ #Hnth #_ @comp_q0_q2_null // %2 //
- | #aj #Haj #Hneq
- whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
- [ whd in match (trans ????); >Hai >Haj
- whd in ⊢ (??(???%)?); >(\bf ?) // @(not_to_not … Hneq) //
- | whd in match (trans ????); >Hai >Haj
- whd in ⊢ (??(???????(???%))?); >(\bf ?) /2 by not_to_not/
- @tape_move_null_action
-] ]
+ | #aj #Haj *
+ [ * #c * >Hai #Heq #Hendc whd in ⊢ (??%?);
+ >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+ [ whd in match (trans ????); >Hai >Haj destruct (Heq)
+ whd in ⊢ (??(???%)?); >Hendc //
+ | whd in match (trans ????); >Hai >Haj destruct (Heq)
+ whd in ⊢ (??(???????(???%))?); >Hendc @tape_move_null_action
+ ]
+ | #Hneq
+ whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+ [ whd in match (trans ????); >Hai >Haj
+ whd in ⊢ (??(???%)?); cut ((¬is_endc ai∧ai==aj)=false)
+ [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) // |#Hcut >Hcut //]
+ | whd in match (trans ????); >Hai >Haj
+ whd in ⊢ (??(???????(???%))?); cut ((¬is_endc ai∧ai==aj)=false)
+ [>(\bf ?) /2 by not_to_not/ cases (is_endc ai) //
+ |#Hcut >Hcut @tape_move_null_action
+ ]
+ ]
+ ]
+ ]
+]
qed.
lemma comp_q0_q1 :
- ∀i,j,sig,n,v,a.i ≠ j → i < S n → j < S n →
- nth i ? (current_chars ?? v) (None ?) = Some ? a →
+ ∀i,j,sig,n,is_endc,v,a.i ≠ j → i < S n → j < S n →
+ nth i ? (current_chars ?? v) (None ?) = Some ? a → is_endc a = false →
nth j ? (current_chars ?? v) (None ?) = Some ? a →
- step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v) =
+ step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) =
mk_mconfig ??? comp1
(change_vec ? (S n)
(change_vec ?? v
(tape_move ? (nth i ? v (niltape ?)) (Some ? 〈a,R〉)) i)
(tape_move ? (nth j ? v (niltape ?)) (Some ? 〈a,R〉)) j).
-#i #j #sig #n #v #a #Heq #Hi #Hj #Ha1 #Ha2
+#i #j #sig #n #is_endc #v #a #Heq #Hi #Hj #Ha1 #Hnotendc #Ha2
whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
[ whd in match (trans ????);
- >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) //
+ >Ha1 >Ha2 whd in ⊢ (??(???%)?); >Hnotendc >(\b ?) //
| whd in match (trans ????);
- >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >(\b ?) //
+ >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >Hnotendc >(\b ?) //
change with (change_vec ?????) in ⊢ (??(???????%)?);
<(change_vec_same … v j (niltape ?)) in ⊢ (??%?);
<(change_vec_same … v i (niltape ?)) in ⊢ (??%?);
qed.
lemma sem_comp_step :
- ∀i,j,sig,n.i ≠ j → i < S n → j < S n →
- compare_step i j sig n ⊨
- [ comp1: R_comp_step_true i j sig n,
- R_comp_step_false i j sig n ].
-#i #j #sig #n #Hneq #Hi #Hj #int
+ ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n →
+ compare_step i j sig n is_endc ⊨
+ [ comp1: R_comp_step_true i j sig n is_endc,
+ R_comp_step_false i j sig n is_endc ].
+#i #j #sig #n #is_endc #Hneq #Hi #Hj #int
lapply (refl ? (current ? (nth i ? int (niltape ?))))
cases (current ? (nth i ? int (niltape ?))) in ⊢ (???%→?);
[ #Hcuri %{2} %
[ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2 <Hcurj in ⊢ (???%);
@sym_eq @nth_vec_map
| normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % >Ha >Hcurj % % % #H destruct (H) ] ]
- | #b #Hb %{2} cases (true_or_false (a == b)) #Hab
+ | #_ % // >Ha >Hcurj % % %2 % #H destruct (H) ] ]
+ | #b #Hb %{2}
+ cases (true_or_false (is_endc a)) #Haendc
[ %
- [| % [ %
- [whd in ⊢ (??%?); >(comp_q0_q1 … a Hneq Hi Hj) //
- [>(\P Hab) <Hb @sym_eq @nth_vec_map
- |<Ha @sym_eq @nth_vec_map ]
- | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) // ]
- | * #H @False_ind @H %
- ] ]
- | %
[| % [ %
[whd in ⊢ (??%?); >comp_q0_q2_neq //
- <(nth_vec_map ?? (current …) i ? int (niltape ?))
- <(nth_vec_map ?? (current …) j ? int (niltape ?)) >Ha >Hb
- @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
+ % %{a} % // <Ha @sym_eq @nth_vec_map
| normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % // % % >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
+ | #_ % // % % % >Ha %{a} % // ]
+ ]
+ |cases (true_or_false (a == b)) #Hab
+ [ %
+ [| % [ %
+ [whd in ⊢ (??%?); >(comp_q0_q1 … a Hneq Hi Hj) //
+ [>(\P Hab) <Hb @sym_eq @nth_vec_map
+ |<Ha @sym_eq @nth_vec_map ]
+ | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) % // ]
+ | * #H @False_ind @H %
+ ] ]
+ | %
+ [| % [ %
+ [whd in ⊢ (??%?); >comp_q0_q2_neq //
+ <(nth_vec_map ?? (current …) i ? int (niltape ?))
+ <(nth_vec_map ?? (current …) j ? int (niltape ?)) %2 >Ha >Hb
+ @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
+ | normalize in ⊢ (%→?); #H destruct (H) ]
+ | #_ % // % % %2 >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
+ ]
]
]
]
qed.
-definition compare ≝ λi,j,sig,n.
- whileTM … (compare_step i j sig n) comp1.
+definition compare ≝ λi,j,sig,n,is_endc.
+ whileTM … (compare_step i j sig n is_endc) comp1.
definition R_compare ≝
- λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
- ((current ? (nth i ? int (niltape ?))
- ≠ current ? (nth j ? int (niltape ?)) ∨
+ λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+ ((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+ (current ? (nth i ? int (niltape ?)) ≠ current ? (nth j ? int (niltape ?)) ∨
current ? (nth i ? int (niltape ?)) = None ? ∨
current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
- (∀ls,x,xs,ci,rs,ls0,cj,rs0.
+ (∀ls,x,xs,ci,rs,ls0,rs0.
nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → ci ≠ cj →
+ nth j ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
+ (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
+ (rs0 = [ ] →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
+ (mk_tape sig (reverse ? xs@x::ls0) (None ?) []) j) ∨
+ ∀cj,rs1.rs0 = cj::rs1 →
+ (is_endc ci = true ∨ ci ≠ cj) →
outt = change_vec ??
(change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
(midtape sig (reverse ? xs@x::ls0) cj rs0) j).
-lemma wsem_compare : ∀i,j,sig,n.i ≠ j → i < S n → j < S n →
- compare i j sig n ⊫ R_compare i j sig n.
-#i #j #sig #n #Hneq #Hi #Hj #ta #k #outc #Hloop
-lapply (sem_while … (sem_comp_step i j sig n Hneq Hi Hj) … Hloop) //
+lemma wsem_compare : ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n →
+ compare i j sig n is_endc ⊫ R_compare i j sig n is_endc.
+#i #j #sig #n #is_endc #Hneq #Hi #Hj #ta #k #outc #Hloop
+lapply (sem_while … (sem_comp_step i j sig n is_endc Hneq Hi Hj) … Hloop) //
-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc whd in ⊢ (%→?); * * [ *
- [ #Hcicj #Houtc %
+[ #tc whd in ⊢ (%→?); * * [ * [ *
+ [* #curi * #Hcuri #Hendi #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj #Hnotendc
+ @False_ind
+ >Hnthi in Hcuri; normalize in ⊢ (%→?); #H destruct (H)
+ >(Hnotendc ? (memb_hd … )) in Hendi; #H destruct (H)
+ ]
+ |#Hcicj #Houtc %
[ #_ @Houtc
| #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi #Hnthj
>Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
- ]
+ ]]
| #Hci #Houtc %
[ #_ @Houtc
| #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci;
[ #_ @Houtc
| #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj;
normalize in ⊢ (%→?); #H destruct (H) ] ]
- | #tc #td #te * #x * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
+ | #tc #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
#IH1 #IH2 %
- [ >Hci >Hcj * [* [* #H @False_ind @H % | #H destruct (H)] | #H destruct (H)]
+ [ >Hci >Hcj * [* #x0 * #H destruct (H) >Hendcx #H destruct (H)
+ |* [* #H @False_ind [cases H -H #H @H % | destruct (H)] | #H destruct (H)]]
| #ls #c0 #xs #ci #rs #ls0 #cj #rs0 cases xs
- [ #Hnthi #Hnthj #Hcicj >IH1
+ [ #Hnthi #Hnthj #Hnotendc #Hcicj >IH1
[ >Hd @eq_f3 //
[ @eq_f3 // >(?:c0=x) [ >Hnthi % ]
>Hnthi in Hci;normalize #H destruct (H) %
| >(?:c0=x) [ >Hnthj % ]
>Hnthi in Hci;normalize #H destruct (H) % ]
| >Hd >nth_change_vec // >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // >Hnthi >Hnthj normalize %1 %1 @(not_to_not ??? Hcicj)
- #H destruct (H) % ]
- | #x0 #xs0 #Hnthi #Hnthj #Hcicj
+ >nth_change_vec // >Hnthi >Hnthj normalize
+ cases Hcicj #Hcase
+ [%1 %{ci} % // | %2 %1 %1 @(not_to_not ??? Hcase) #H destruct (H) % ]
+ ]
+ | #x0 #xs0 #Hnthi #Hnthj #Hnotendc #Hcicj
>(IH2 (c0::ls) x0 xs0 ci rs (c0::ls0) cj rs0 … Hcicj)
[ >Hd >change_vec_commute in ⊢ (??%?); //
>change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
@sym_not_eq //
+ | #c1 #Hc1 @Hnotendc @memb_cons @Hc1
| >Hd >nth_change_vec // >Hnthj normalize
>Hnthi in Hci;normalize #H destruct (H) %
| >Hd >nth_change_vec_neq [|@sym_not_eq //] >Hnthi
]]]
qed.
-lemma terminate_compare : ∀i,j,sig,n,t.
+lemma terminate_compare : ∀i,j,sig,n,is_endc,t.
i ≠ j → i < S n → j < S n →
- compare i j sig n ↓ t.
-#i #j #sig #n #t #Hneq #Hi #Hj
+ compare i j sig n is_endc ↓ t.
+#i #j #sig #n #is_endc #t #Hneq #Hi #Hj
@(terminate_while … (sem_comp_step …)) //
<(change_vec_same … t i (niltape ?))
cases (nth i (tape sig) t (niltape ?))
-[ % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
-|2,3: #a0 #al0 % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+[ % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+|2,3: #a0 #al0 % #t1 * #x * * * #_ >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
| #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
- [#t #ls #c % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?);
+ [#t #ls #c % #t1 * #x * * * #Hendcx >nth_change_vec // normalize in ⊢ (%→?);
#H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 %
- #t2 * #x0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
+ #t2 * #x0 * * * #Hendcx0 >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
>nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
|#r0 #rs0 #IH #t #ls #c % #t1 * #x * * >nth_change_vec //
normalize in ⊢ (%→?); #H destruct (H) #Hcur
]
qed.
-lemma sem_compare : ∀i,j,sig,n.
+lemma sem_compare : ∀i,j,sig,n,is_endc.
i ≠ j → i < S n → j < S n →
- compare i j sig n ⊨ R_compare i j sig n.
-#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize /2/
+ compare i j sig n is_endc ⊨ R_compare i j sig n is_endc.
+#i #j #sig #n #is_endc #Hneq #Hi #Hj @WRealize_to_Realize /2/
qed.
(*
*)
definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
- compare src dst sig n ·
+ compare src dst sig n is_endc ·
(ifTM ?? (inject_TM ? (test_char ? (λa.is_endc a == false)) n src)
(single_finalTM ??
(parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
definition Rtc_multi_false ≝
λalpha,test,n,i.λt1,t2:Vector ? (S n).
(∀c. current alpha (nth i ? t1 (niltape ?)) = Some ? c → test c = false) ∧ t2 = t1.
-
+
+definition R_match_step_false ≝
+ λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+ ∀ls,x,xs,end,rs.
+ nth src ? int (niltape ?) = midtape sig ls x (xs@end::rs) →
+ (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) → is_endc end = true →
+ ((current sig (nth dst (tape sig) int (niltape sig)) = None ? ) ∧ outt = int) ∨
+ (∃ls0,rs0.
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+ ∀rsj,end,c.
+ rs0 = c::rsj →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rs) src)
+ (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
+(*
definition R_match_step_false ≝
λsrc,dst,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
- ((current ? (nth src ? int (niltape ?)) ≠ current ? (nth dst ? int (niltape ?)) ∨
+ (((∃x.current ? (nth src ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
current sig (nth src (tape sig) int (niltape sig)) = None ? ∨
current sig (nth dst (tape sig) int (niltape sig)) = None ? ) ∧ outt = int) ∨
- ∃ls,ls0,rs,rs0,x,xs. ∀rsi,rsj,end,c.
- rs = end::rsi → rs0 = c::rsj →
- is_endc end = true ∧
- nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧
+ (∃ls,ls0,rs,rs0,x,xs.
+ nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧ is_endc x = false ∧
nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+ ∀rsi,rsj,end,c.
+ rs = end::rsi → rs0 = c::rsj →
+ (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) ∧ is_endc end = true ∧
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@c::rsj) ∧
outt = change_vec ??
(change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rsi) src)
- (midtape sig (reverse ? xs@x::ls0) c rsj) dst.
+ (midtape sig (reverse ? xs@x::ls0) c rsj) dst).
+*)
definition R_match_step_true ≝
λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
- is_startc s = true →
- (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 →
- s ≠ s1 →
+ is_startc s = true →
+ (∀c.c ∈ right ? (nth src (tape sig) int (niltape sig)) = true → is_startc c = false) →
+ (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →
outt = change_vec ?? int
(tape_move … (nth dst ? int (niltape ?)) (Some ? 〈s1,R〉)) dst ∧ is_endc s = false) ∧
(∀ls,x,xs,ci,rs,ls0,cj,rs0.
nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → ci ≠ cj →
+ (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
outt = change_vec ?? int
(tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false).
| @sym_eq @Hnth_j @sym_not_eq // ] ] ]
qed.
-axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S. ∃l,tl1,tl2.
- l1 = l@tl1 ∧ l2 = l@tl2 ∧ ∀a,b,tla,tlb. tl1 = a::tla → tl2 = b::tlb → a≠b.
+axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2.
+ l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧
+ ∀a,tla. tl1 = a::tla → is_endc a = true ∨ (∀b,tlb.tl2 = b::tlb → a≠b).
axiom daemon : ∀X:Prop.X.
-
+
lemma sem_match_step :
∀src,dst,sig,n,is_startc,is_endc.src ≠ dst → src < S n → dst < S n →
match_step src dst sig n is_startc is_endc ⊨
R_match_step_true src dst sig n is_startc is_endc,
R_match_step_false src dst sig n is_endc ].
#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst
-@(acc_sem_seq_app sig n … (sem_compare src dst sig n Hneq Hsrc Hdst)
+@(acc_sem_seq_app sig n … (sem_compare src dst sig n is_endc Hneq Hsrc Hdst)
(acc_sem_if ? n … (sem_test_char_multi sig (λa.is_endc a == false) n src (le_S_S_to_le … Hsrc))
(sem_seq …
(sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
(sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
(sem_nop …)))
[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * * * #c * #Hcurtc #Hcend #Htd >Htd -Htd
- #Htb #s #Hcurta_src #Hstart %
+ #Htb #s #Hcurta_src #Hstart #Hnotstart %
[ #s1 #Hcurta_dst #Hneqss1
lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta)
- [|@Hcomp1 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
+ [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
#Hcurtc * #te * * #_ #Hte >Hte // whd in ⊢ (%→?); * * #_ #Htbdst #Htbelse %
[ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
[ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
| >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Htbelse @sym_not_eq // ]
| >Hcurtc in Hcurta_src; #H destruct (H) cases (is_endc s) in Hcend;
normalize #H destruct (H) // ]
- |#ls #x #xs #ci #rs #ls0 #cj #rs0 #Htasrc_mid #Htadst_mid #Hcicj
- lapply (Hcomp2 … Htasrc_mid Htadst_mid Hcicj) -Hcomp2 #Hcomp2
+ |#ls #x #xs #ci #rs #ls0 #cj #rs0 #Htasrc_mid #Htadst_mid #Hcicj #Hnotendc
+ lapply (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc (or_intror ?? Hcicj))
+ -Hcomp2 #Hcomp2
cases Htb #td * * #Htd #_ >Htasrc_mid in Hcurta_src; normalize in ⊢ (%→?);
#H destruct (H)
>(Htd ls ci (reverse ? xs) rs s ??? ls0 cj (reverse ? xs) s rs0 (refl ??)) //
[| >Hcomp2 >nth_change_vec //
- | @daemon
+ | #c0 #Hc0 @(Hnotstart c0) >Htasrc_mid
+ cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [@memb_append_l2 >(\P Hc0) @memb_hd
+ |@memb_append_l1 <(reverse_reverse …xs) @memb_reverse //
+ ]
| >Hcomp2 >nth_change_vec_neq [|@sym_not_eq // ] @nth_change_vec // ]
* * #_ #Htbdst #Htbelse %
[ @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
| >nth_change_vec_neq [|@sym_not_eq //]
<Htbelse [|@sym_not_eq // ]
>nth_change_vec_neq [|@sym_not_eq //]
- STOP.
-
- >nth_change_vec in Htbdst; // #Htbdst >(Htbdst
- … Htadst_mid) >Hta_mid cases rs //
- | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Htbelse @sym_not_eq // ]
- | >Hcurtc in Hcurta_src; #H destruct (H) cases (is_endc s) in Hcend;
+ (* STOP. *)
+ cases (decidable_eq_nat i src) #Hisrc
+ [ >Hisrc >nth_change_vec // >Htasrc_mid //
+ | >nth_change_vec_neq [|@sym_not_eq //]
+ <(Htbelse i) [|@sym_not_eq // ]
+ >Hcomp2 >nth_change_vec_neq [|@sym_not_eq // ]
+ >nth_change_vec_neq [|@sym_not_eq // ] //
+ ]
+ ]
+ | >Hcomp2 in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // whd in ⊢ (??%?→?);
+ #H destruct (H) cases (is_endc c) in Hcend;
normalize #H destruct (H) // ]
+ ]
|#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
+ whd in ⊢ (%→?); #Hout >Hout >Htb whd
+ #ls #c_src #xs #end #rs #Hmid_src #Hnotend #Hend
+ lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
+ cases (current … (nth dst ? intape (niltape ?))) in Hcomp1;
+ [#Hcomp1 #_ %1 % [% | @Hcomp1 %2 %2 % ]
+ |#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
+ [#_ #Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
+ #ls_dst * #rs_dst #Hmid_dst %2
+ cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * *
+ #Hrs_src #Hrs_dst #Hnotendc #Hneq
+ %{ls_dst} %{rsj} %
+ [<Hrs_dst >(\P Hceq) // ]]
+ #rsi0 #rsj0 #end #c #Hend #Hc_dst
+ >Hrs_src in Hmid_src; >Hend #Hmid_src
+ >Hrs_dst in Hmid_dst; >Hc_dst <(\P Hceq) #Hmid_dst
+ cut (is_endc end = true ∨ end ≠ c)
+ [cases (Hneq … Hend) /2/ -Hneq #Hneq %2 @(Hneq … Hc_dst) ] #Hneq
+ lapply (Hcomp2 … Hmid_src Hmid_dst ? Hneq)
+ [#c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) //
+ | @Hnotendc // ]
+ ]
+ -Hcomp2 #Hcomp2 <Hcomp2
+ % // % [
+ >Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // #H lapply (H ? (refl …))
+ cases (is_endc end) [|normalize #H destruct (H) ]
+ #_ % // #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) // | @Hnotendc // ]
+ |@Hmid_dst]
+ ]
+ |#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc
+ %1 %
+ [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1
+ [| %2 % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ]
+ cases (is_endc c_src) //
+ >Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H)
+ |@Hcomp1 %2 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
+ ]
+ ]
+ ]
+ ]
+qed.
+
+#intape #outtape #ta * #Hcomp1 #Hcomp2 * #tb * * #Hc #Htb
whd in ⊢ (%→?); #Hout >Hout >Htb whd
lapply (current_to_midtape sig (nth src ? intape (niltape ?)))
cases (current … (nth src ? intape (niltape ?))) in Hcomp1;
- [#Hcomp1 #_ %1 % [%1 %2 // | @Hcomp1 %1 %2 %]
+ [#Hcomp1 #_ %1 % [%1 %2 // | @Hcomp1 %2 %1 %2 %]
|#c_src lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
cases (current … (nth dst ? intape (niltape ?)))
- [#_ #Hcomp1 #_ %1 % [%2 % | @Hcomp1 %2 %]
+ [#_ #Hcomp1 #_ %1 % [%2 % | @Hcomp1 %2 % % % #H destruct (H)]
|#c_dst cases (true_or_false (c_src == c_dst)) #Hceq
[#Hmid_dst cases (Hmid_dst c_dst (refl …)) -Hmid_dst
- #ls_dst * #rs_dst #Hmid_dst #_
+ #ls_dst * #rs_dst #Hmid_dst #Hcomp1
#Hmid_src cases (Hmid_src c_src (refl …)) -Hmid_src
- #ls_src * #rs_src #Hmid_src %2
- cases (comp_list … rs_src rs_dst) #xs * #rsi * #rsj * *
- #Hrs_src #Hrs_dst #Hneq
- %{ls_src} %{ls_dst} %{rsi} %{rsj} %{c_src} %{xs}
- #rsi0 #rsj0 #end #c #Hend #Hc_dst
- >Hrs_src in Hmid_src; >Hend #Hmid_src
- >Hrs_dst in Hmid_dst; >Hc_dst <(\P Hceq) #Hmid_dst
- lapply(Hcomp2 … Hmid_src Hmid_dst ?)
- [@(Hneq … Hend Hc_dst)]
- -Hcomp2 #Hcomp2 <Hcomp2
- % // % [ %
- [>Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // #H lapply (H ? (refl …))
- cases (is_endc end) normalize //
- |@Hmid_src]
- |@Hmid_dst]
- |#_ #Hcomp1 #_ %1 %
- [% % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
- |@Hcomp1 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
+ #ls_src * #rs_src #Hmid_src
+ cases (true_or_false (is_endc c_src)) #Hc_src
+ [ % % [ % % %{c_src} % // | @Hcomp1 % %{c_src} % // ]
+ | %2 cases (comp_list … rs_src rs_dst is_endc) #xs * #rsi * #rsj * * *
+ #Hrs_src #Hrs_dst #Hnotendc #Hneq
+ %{ls_src} %{ls_dst} %{rsi} %{rsj} %{c_src} %{xs} %
+ [% [% // <Hrs_src //|<Hrs_dst >(\P Hceq) // ]]
+ #rsi0 #rsj0 #end #c #Hend #Hc_dst
+ >Hrs_src in Hmid_src; >Hend #Hmid_src
+ >Hrs_dst in Hmid_dst; >Hc_dst <(\P Hceq) #Hmid_dst
+ cut (is_endc end = true ∨ end ≠ c)
+ [cases (Hneq … Hend) /2/ -Hneq #Hneq %2 @(Hneq … Hc_dst) ] #Hneq
+ lapply (Hcomp2 … Hmid_src Hmid_dst ? Hneq)
+ [#c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) //
+ | @Hnotendc // ]
+ ]
+ -Hcomp2 #Hcomp2 <Hcomp2
+ % // % [
+ >Hcomp2 in Hc; >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec // #H lapply (H ? (refl …))
+ cases (is_endc end) [|normalize #H destruct (H) ]
+ #_ % // #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) // | @Hnotendc // ]
+ |@Hmid_dst]
+ ]
+ |#_ #Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc
+ %1 %
+ [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1
+ [| %2 % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ]
+ cases (is_endc c_src) //
+ >Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H)
+ |@Hcomp1 %2 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
]
]
]
]
+qed.
+definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
+ whileTM … (match_step src dst sig n is_startc is_endc)
+ (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
-2:#t1 #t2 #t3 whd in ⊢ (%→?); * #Hc #H #H1 whd #ls #c #rs #Ht1 %
- [lapply(Hc c ?) [>Ht1 %] #Hgrid @injective_notb @Hgrid |>H1 @H]
-
+definition R_match_m ≝
+ λi,j,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+ (((∃x.current ? (nth i ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
+ current ? (nth i ? int (niltape ?)) = None ? ∨
+ current ? (nth j ? int (niltape ?)) = None ?) → outt = int) ∧
+ (∀ls,x,xs,ci,rs,ls0,x0,rs0.
+ (∀x. is_startc x ≠ is_endc x) →
+ is_startc x = true → is_endc ci = true →
+ (∀z. memb ? z (x::xs) = true → is_endc x = false) →
+ nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+ nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+ (∃l,l1.x0::rs0 = l@x::xs@l1 ∧
+ ∀cj,l2.l1=cj::l2 →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
+ (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨
+ ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
+(*
+axiom sub_list_dec: ∀A.∀l,ls:list A.
+ ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
+*)
+lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
+src ≠ dst → src < S n → dst < S n →
+ match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
+#src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
+lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ #tc whd in ⊢ (%→%); *
+ [ * * [ *
+ [ * #cur_src * #H1 #H2 #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #Hnotend #Hnthi
+ @False_ind
+ >Hnthi in H1; whd in ⊢ (??%?→?); #H destruct (H) cases (Hdiff cur_src)
+ #Habs @Habs //
+ ]
+ | #Hci #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #Hnotend
+ #Hnthi >Hnthi in Hci; normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | #Hcj #Houtc %
+ [ #_ @Houtc
+ | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hdiff #Hstartc #Hendc #_ #_ #Hnthj >Hnthj in Hcj;
+ normalize in ⊢ (%→?); #H destruct (H) ]
+ ]
+ |* #ls * #ls0 * #rs * #rs0 * #x0 * #xs * * * #Hsrc #Hx0 #Hdst #H %
+ [>Hsrc *
+ [* [* #x * whd in ⊢ (??%?→?); #Habs destruct (Habs) >Hx0 #Habs destruct (Habs)
+ |whd in ⊢ (??%?→?); #Habs destruct (Habs) ]
+ |>Hdst whd in ⊢ (??%?→?); #Habs destruct (Habs) ]
+ |#ls1 #x1 #xs1 #ci #rsi #ls2 #x2 #rs2
+ #Hdiff #Hstart #Hend #Hnotend
+ >Hsrc #Hsrc1 destruct (Hsrc1) >Hdst #Hdst1 destruct (Hdst1)
+ %1 %{[ ]} %{rs0} normalize in ⊢ (%→?); #Heq #cj #l2 #Hl1
+ cut (xs=xs1)
+ [@(append_l1_injective_r … rs0 rs0 (refl …)) @(cons_injective_r …Heq)]
+ #eqxs <eqxs
+ whd in match (append ? [ ] (x2::xs)); >reverse_cons >associative_append
+ normalize in match (append ? [x2] ls2);
+ cases (H rsi l2 ci cj ? Hl1)
+ [* #_ #_ #H3 @H3
+ |>eqxs in e0; #e0 @(append_l2_injective … e0) //
+ ]
+ ]
+ ]
+|#tc #td #te #Hd #Hstar #IH #He lapply (IH He) -IH *
+ #IH1 #IH2 % [@IH1]
+
+
+ cases (comp_list ? (x1::xs1@ci::rsi) (x2::rs2) is_endc)
+ #l * #tl1 * #tl2 * * * #H1 #H2 #H3 #H4
projects / helm.git / blobdiff