(* *)
(**************************************************************************)
-include "turing/multi_universal/moves.ma".
-include "turing/if_multi.ma".
-include "turing/inject.ma".
-include "turing/basic_machines.ma".
+include "turing/multi_universal/compare.ma".
+include "turing/multi_universal/par_test.ma".
+include "turing/multi_universal/moves_2.ma".
-definition compare_states ≝ initN 3.
-
-definition comp0 : compare_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
-definition comp1 : compare_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
-definition comp2 : compare_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
-
-(*
-
-0) (x,x) → (x,x)(R,R) → 1
- (x,y≠x) → None 2
-1) (_,_) → None 1
-2) (_,_) → None 2
-
-*)
-
-definition trans_compare_step ≝
- λi,j.λsig:FinSet.λn.λis_endc.
- λp:compare_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in
- match pi1 … q with
- [ O ⇒ match nth i ? a (None ?) with
- [ None ⇒ 〈comp2,null_action ? n〉
- | Some ai ⇒ match nth j ? a (None ?) with
- [ None ⇒ 〈comp2,null_action ? n〉
- | 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〉
- else 〈comp2,null_action ? n〉 ]
- ]
- | S q ⇒ match q with
- [ O ⇒ (* 1 *) 〈comp1,null_action ? n〉
- | S _ ⇒ (* 2 *) 〈comp2,null_action ? n〉 ] ].
-
-definition compare_step ≝
- 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,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 ??
- (change_vec ?? int
- (tape_move ? (nth i ? int (niltape ?)) (Some ? 〈x,R〉)) i)
- (tape_move ? (nth j ? int (niltape ?)) (Some ? 〈x,R〉)) j.
-
-definition R_comp_step_false ≝
- 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,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 is_endc) (mk_mconfig ??? comp0 v)
- = mk_mconfig ??? comp2 v.
-#i #j #sig #n #is_endc #v #Hi #Hj
-whd in ⊢ (? → ??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (?→??%?);
-* #Hcurrent
-[ @eq_f2
- [ whd in ⊢ (??(???%)?); >Hcurrent %
- | whd in ⊢ (??(???????(???%))?); >Hcurrent @tape_move_null_action ]
-| @eq_f2
- [ whd in ⊢ (??(???%)?); >Hcurrent cases (nth i ?? (None sig)) //
- | whd in ⊢ (??(???????(???%))?); >Hcurrent
- cases (nth i ?? (None sig)) [|#x] @tape_move_null_action ] ]
-qed.
-
-lemma comp_q0_q2_neq :
- ∀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 #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 *
- [ * #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,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 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 #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 ⊢ (??(???%)?); >Hnotendc >(\b ?) //
-| whd in match (trans ????);
- >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 ⊢ (??%?);
- >pmap_change >pmap_change >tape_move_null_action
- @eq_f2 // @eq_f2 // >nth_change_vec_neq //
-]
-qed.
-
-lemma sem_comp_step :
- ∀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/ % <Hcuri in ⊢ (???%);
- @sym_eq @nth_vec_map
- | normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % // % %2 // ] ]
-| #a #Ha lapply (refl ? (current ? (nth j ? int (niltape ?))))
- cases (current ? (nth j ? int (niltape ?))) in ⊢ (???%→?);
- [ #Hcurj %{2} %
- [| % [ %
- [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2 <Hcurj in ⊢ (???%);
- @sym_eq @nth_vec_map
- | normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % // >Ha >Hcurj % % %2 % #H destruct (H) ] ]
- | #b #Hb %{2}
- cases (true_or_false (is_endc a)) #Haendc
- [ %
- [| % [ %
- [whd in ⊢ (??%?); >comp_q0_q2_neq //
- % %{a} % // <Ha @sym_eq @nth_vec_map
- | normalize in ⊢ (%→?); #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,is_endc.
- whileTM … (compare_step i j sig n is_endc) comp1.
-
-definition R_compare ≝
- 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.
- nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) →
- (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
- (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,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 ⊢ (%→?); * * [ * [ *
- [* #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;
- normalize in ⊢ (%→?); #H destruct (H) ] ]
- | #Hcj #Houtc %
- [ #_ @Houtc
- | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj;
- normalize in ⊢ (%→?); #H destruct (H) ] ]
- | #tc #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
- #IH1 #IH2 %
- [ >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 #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
- 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
- >nth_change_vec // normalize
- >Hnthi in Hci;normalize #H destruct (H) %
- ]
-]]]
-qed.
-
-lemma terminate_compare : ∀i,j,sig,n,is_endc,t.
- i ≠ j → i < S n → j < S n →
- 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
-| #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
- [#t #ls #c % #t1 * #x * * * #Hendcx >nth_change_vec // normalize in ⊢ (%→?);
- #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 %
- #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
- >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
- ]
-]
-qed.
-
-lemma sem_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 @WRealize_to_Realize /2/
-qed.
-
-(*
- |conf1 $
- |confin 0/1 confout move
-
- match machine step ≝
- compare;
- if (cur(src) != $)
- then
- parmoveL;
- moveR(dst);
- else nop
- *)
-
-definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
- compare src dst sig n ·
- (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)))
- (nop …)
- tc_true).
-
definition Rtc_multi_true ≝
λalpha,test,n,i.λt1,t2:Vector ? (S n).
(∃c. current alpha (nth i ? t1 (niltape ?)) = Some ? c ∧ test c = true) ∧ t2 = t1.
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).
- (((∃x.current ? (nth src ? int (niltape ?)) = Some ? x ∧ is_endc x = true) ∨
- (* current ? (nth src ? int (niltape ?)) ≠ current ? (nth dst ? int (niltape ?)) ∨ *)
- 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 x = false ∧ is_endc end = true ∧
- nth src ? int (niltape ?) = midtape sig ls x (xs@rs) ∧
- nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
- outt = change_vec ??
- (change_vec ?? int (midtape sig (reverse ? xs@x::ls) end rsi) src)
- (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 →
- (∀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 →
- outt = change_vec ?? int
- (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false).
-
lemma sem_test_char_multi :
∀alpha,test,n,i.i ≤ n →
inject_TM ? (test_char ? test) n i ⊨
| @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.
+definition Rm_test_null_true ≝
+ λalpha,n,i.λt1,t2:Vector ? (S n).
+ current alpha (nth i ? t1 (niltape ?)) ≠ None ? ∧ t2 = t1.
+
+definition Rm_test_null_false ≝
+ λalpha,n,i.λt1,t2:Vector ? (S n).
+ current alpha (nth i ? t1 (niltape ?)) = None ? ∧ t2 = t1.
+
+lemma sem_test_null_multi : ∀alpha,n,i.i ≤ n →
+ inject_TM ? (test_null ?) n i ⊨
+ [ tc_true : Rm_test_null_true alpha n i, Rm_test_null_false alpha n i ].
+#alpha #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_null alpha) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * #Hcur #Hnth_i #Hnth_j % //
+ @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @sym_eq @Hnth_i | @sym_eq @Hnth_j @sym_not_eq // ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Hcur #Hnth_i #Hnth_j %
+ [ @Hcur
+ | @(eq_vec … (niltape ?)) #i0 #Hi0 cases (decidable_eq_nat i0 i) //
+ #Hi0i @sym_eq @Hnth_j @sym_not_eq // ] ]
+qed.
+
+definition match_test ≝ λsrc,dst.λsig:DeqSet.λn.λv:Vector ? n.
+ match (nth src (option sig) v (None ?)) with
+ [ None ⇒ false
+ | Some x ⇒ notb (nth dst (DeqOption sig) v (None ?) == None ?) ].
+
+definition mmove_states ≝ initN 2.
+
+definition mmove0 : mmove_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
+definition mmove1 : mmove_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
+
+definition trans_mmove ≝
+ λi,sig,n,D.
+ λp:mmove_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in match (pi1 … q) with
+ [ O ⇒ 〈mmove1,change_vec ? (S n) (null_action ? n) (〈None ?,D〉) i〉
+ | S _ ⇒ 〈mmove1,null_action sig n〉 ].
+
+definition mmove ≝
+ λi,sig,n,D.
+ mk_mTM sig n mmove_states (trans_mmove i sig n D)
+ mmove0 (λq.q == mmove1).
+
+definition Rm_multi ≝
+ λalpha,n,i,D.λt1,t2:Vector ? (S n).
+ t2 = change_vec ? (S n) t1 (tape_move alpha (nth i ? t1 (niltape ?)) D) i.
+
+lemma sem_move_multi :
+ ∀alpha,n,i,D.i ≤ n →
+ mmove i alpha n D ⊨ Rm_multi alpha n i D.
+#alpha #n #i #D #Hin #int %{2}
+%{(mk_mconfig ? mmove_states n mmove1 ?)}
+[| %
+ [ whd in ⊢ (??%?); @eq_f whd in ⊢ (??%?); @eq_f %
+ | whd >tape_move_multi_def
+ <(change_vec_same … (ctapes …) i (niltape ?))
+ >pmap_change <tape_move_multi_def >tape_move_null_action % ] ]
+ qed.
+
+definition rewind ≝ λsrc,dst,sig,n.
+ parmove src dst sig n L · mmove src sig n R · mmove dst sig n R.
+
+definition R_rewind ≝ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ (∀x,x0,xs,rs.
+ nth src ? int (niltape ?) = midtape sig (xs@[x0]) x rs →
+ ∀ls0,y,y0,target,rs0.|xs| = |target| →
+ nth dst ? int (niltape ?) = midtape sig (target@y0::ls0) y rs0 →
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig [] x0 (reverse ? xs@x::rs)) src)
+ (midtape sig ls0 y0 (reverse ? target@y::rs0)) dst).
+
+theorem accRealize_to_Realize :
+ ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
+ M ⊨ [ acc: Rtrue, Rfalse ] → M ⊨ Rtrue ∪ Rfalse.
+#sig #n #M #Rtrue #Rfalse #acc #HR #t
+cases (HR t) #k * #outc * * #Hloop
+#Htrue #Hfalse %{k} %{outc} % //
+cases (true_or_false (cstate sig (states sig n M) n outc == acc)) #Hcase
+[ % @Htrue @(\P Hcase) | %2 @Hfalse @(\Pf Hcase) ]
+qed.
+
+lemma sem_rewind : ∀src,dst,sig,n.
+ src ≠ dst → src < S n → dst < S n →
+ rewind src dst sig n ⊨ R_rewind src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst
+@(sem_seq_app sig n ????? (sem_parmoveL src dst sig n Hneq Hsrc Hdst) ?)
+[| @(sem_seq_app sig n ????? (sem_move_r_multi …) (sem_move_r_multi …)) //
+ @le_S_S_to_le // ]
+#ta #tb * #tc * * #Htc #_ * #td * whd in ⊢ (%→%→?); #Htd #Htb
+#x #x0 #xs #rs #Hmidta_src #ls0 #y #y0 #target #rs0 #Hlen #Hmidta_dst
+>(Htc ??? Hmidta_src ls0 y (target@[y0]) rs0 ??) in Htd;
+[|>Hmidta_dst //
+|>length_append >length_append >Hlen % ] * #_
+[ whd in ⊢ (%→?); * #x1 * #x2 * *
+ >change_vec_commute in ⊢ (%→?); // >nth_change_vec //
+ cases (reverse sig (xs@[x0])@x::rs)
+ [|#z #zs] normalize in ⊢ (%→?); #H destruct (H)
+| whd in ⊢ (%→?); * #_ #Htb >Htb -Htb FAIL
-axiom daemon : ∀X:Prop.X.
+ normalize in ⊢ (%→?);
+ (sem_parmove_step src dst sig n R Hneq Hsrc Hdst))
+ (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
+ (sem_seq …
+ (sem_parmoveL ???? Hneq Hsrc Hdst)
+ (sem_inject … dst (le_S_S_to_le … Hdst) (sem_move_r ? )))
+ (sem_nop …)))
+
+
+definition match_step ≝ λsrc,dst,sig,n.
+ compare src dst sig n ·
+ (ifTM ?? (partest sig n (match_test src dst sig ?))
+ (single_finalTM ??
+ (rewind src dst sig n · (inject_TM ? (move_r ?) n dst)))
+ (nop …)
+ partest1).
+
+definition R_match_step_false ≝
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ ∀ls,x,xs.
+ nth src ? int (niltape ?) = midtape sig ls x xs →
+ ((current sig (nth dst (tape sig) int (niltape sig)) = None ?) ∧ outt = int) ∨
+ (∃ls0,rs0,xs0. nth dst ? int (niltape ?) = midtape sig ls0 x rs0 ∧
+ xs = rs0@xs0 ∧
+ current sig (nth dst (tape sig) outt (niltape sig)) = None ?) ∨
+ (∃ls0,rs0.
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) ∧
+ (* ∀rsj,c.
+ rs0 = c::rsj → *)
+ outt = change_vec ??
+ (change_vec ?? int (mk_tape sig (reverse ? xs@x::ls) (None ?) [ ]) src)
+ (mk_tape sig (reverse ? xs@x::ls0) (option_hd ? rs0) (tail ? rs0)) dst).
+(*definition R_match_step_true ≝
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ ∀s,rs.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
+ current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
+ (∀s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 → s ≠ s1 →
+ outt = change_vec ?? int
+ (tape_move_mono … (nth dst ? int (niltape ?)) (〈Some ? s1,R〉)) dst) ∧
+ (∀ls,x,xs,ci,rs,ls0,rs0.
+ nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
+ rs0 ≠ [] ∧
+ ∀cj,rs1.rs0 = cj::rs1 →
+ ci ≠ cj →
+ (outt = change_vec ?? int
+ (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst)).
+*)
+definition R_match_step_true ≝
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ ∀s.current sig (nth src (tape sig) int (niltape sig)) = Some ? s →
+ ∃s1.current sig (nth dst (tape sig) int (niltape sig)) = Some ? s1 ∧
+ (left ? (nth src ? int (niltape ?)) = [ ] →
+ (s ≠ s1 →
+ outt = change_vec ?? int
+ (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst) ∧
+ (∀xs,ci,rs,ls0,rs0.
+ nth src ? int (niltape ?) = midtape sig [] s (xs@ci::rs) →
+ nth dst ? int (niltape ?) = midtape sig ls0 s (xs@rs0) →
+ rs0 ≠ [] ∧
+ ∀cj,rs1.rs0 = cj::rs1 →
+ ci ≠ cj →
+ (outt = change_vec ?? int
+ (tape_move_mono … (nth dst ? int (niltape ?)) (〈None ?,R〉)) dst))).
+
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 ⊨
+ ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
+ match_step src dst sig n ⊨
[ inr ?? (inr ?? (inl … (inr ?? start_nop))) :
- 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
+ R_match_step_true src dst sig n,
+ R_match_step_false src dst sig n ].
+#src #dst #sig #n #Hneq #Hsrc #Hdst
@(acc_sem_seq_app sig n … (sem_compare src dst sig n 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))
+ (acc_sem_if ? n … (sem_partest sig n (match_test src dst sig ?))
(sem_seq …
- (sem_parmoveL ???? is_startc Hneq Hsrc Hdst)
+ (sem_parmoveL ???? 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 #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) % ]
- #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)
- #ls * #rs #Hta_mid >(Htbdst … Hta_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;
- 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
- 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 //
- | #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 //
+[#ta #tb #tc * #Hcomp1 #Hcomp2 * #td * #Htest
+ * #te * #Hte #Htb #s #Hcurta_src whd
+ cut (∃s1.current sig (nth dst (tape sig) ta (niltape sig))=Some sig s1)
+ [ lapply Hcomp1 -Hcomp1
+ lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
+ [ #Hcurta_dst #Hcomp1 >Hcomp1 in Htest; // *
+ change with (vec_map ?????) in match (current_chars ???); whd in ⊢ (??%?→?);
+ <(nth_vec_map ?? (current ?) src ? ta (niltape ?))
+ <(nth_vec_map ?? (current ?) dst ? ta (niltape ?))
+ >Hcurta_src >Hcurta_dst whd in ⊢ (??%?→?); #H destruct (H)
+ | #s1 #_ #_ %{s1} % ] ]
+ * #s1 #Hcurta_dst %{s1} % // #Hleftta %
+ [ #Hneqss1 -Hcomp2 cut (tc = ta)
+ [@Hcomp1 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //]
+ #H destruct (H) -Hcomp1 cut (td = ta)
+ [ cases Htest -Htest // ] #Htdta destruct (Htdta)
+ cases Hte -Hte #Hte #_
+ cases (current_to_midtape … Hcurta_src) #ls * #rs #Hmidta_src
+ cases (current_to_midtape … Hcurta_dst) #ls0 * #rs0 #Hmidta_dst
+ >Hmidta_src in Hleftta; normalize in ⊢ (%→?); #Hls destruct (Hls)
+ >(Hte s [ ] rs Hmidta_src ls0 s1 [ ] rs0 (refl ??) Hmidta_dst) in Htb;
+ * whd in ⊢ (%→?);
+ mid
+
+ in Htb;
+ cut (te = ta)
+ [ cases Htest -Htest #Htest #Htdta <Htdta @Hte %1 >Htdta @Hcurta_src %{s} % //]
+ -Hte #H destruct (H) %
+ [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
+ #i #Hi cases (decidable_eq_nat i dst) #Hidst
+ [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
+ #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
+ | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
+ | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
+ >Hcurta_src in Htest; whd in ⊢ (??%?→?);
+ cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq //
]
- | >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
- [ >Hidst >nth_change_vec // >Htadst_mid >(Htbdst ls0 s (xs@cj::rs0))
- [ cases xs //
- | >nth_change_vec // ]
- | >nth_change_vec_neq [|@sym_not_eq //]
- <Htbelse [|@sym_not_eq // ]
- >nth_change_vec_neq [|@sym_not_eq //]
- (* 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 // ] //
+ <(nth_vec_map ?? (current ?) dst ? tc (niltape ?))
+ >Hcurta_src normalize
+ lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
+ [| #s1 #Hcurta_dst %
+ [ % #Hfalse destruct (Hfalse)
+ | #s1' #Hs1 destruct (Hs1) #Hneqss1 -Hcomp2
+ cut (tc = ta)
+ [@Hcomp1 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //]
+ #H destruct (H) -Hcomp1 cases Hte -Hte #_ #Hte
+ cut (te = ta) [ cases Htest -Htest #Htest #Htdta <Htdta @Hte %1 %{s} % //] -Hte #H destruct (H) %
+ [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
+ #i #Hi cases (decidable_eq_nat i dst) #Hidst
+ [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
+ #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
+ | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
+ | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
+ >Hcurta_src in Htest; whd in ⊢ (??%?→?);
+ cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_eq //
]
+
+ ]
+ #Hcurta_dst >Hcomp1 in Htest; [| %2 %2 //]
+ whd in ⊢ (??%?→?); change with (current ? (niltape ?)) in match (None ?);
+ <nth_vec_map >Hcurta_src whd in ⊢ (??%?→?); <nth_vec_map
+ >Hcurta_dst cases (is_endc s) normalize in ⊢ (%→?); #H destruct (H)
+ | #Hstart #Hnotstart %
+ [ #s1 #Hcurta_dst #Hneqss1 -Hcomp2
+ cut (tc = ta)
+ [@Hcomp1 %2 %1 %1 >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) //]
+ #H destruct (H) -Hcomp1 cases Hte #_ -Hte #Hte
+ cut (te = ta) [@Hte %1 %1 %{s} % //] -Hte #H destruct (H) %
+ [cases Htb * #_ #Hmove #Hmove1 @(eq_vec … (niltape … ))
+ #i #Hi cases (decidable_eq_nat i dst) #Hidst
+ [ >Hidst >nth_change_vec // cases (current_to_midtape … Hcurta_dst)
+ #ls * #rs #Hta_mid >(Hmove … Hta_mid) >Hta_mid cases rs //
+ | >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @Hmove1 @sym_not_eq // ]
+ | whd in Htest:(??%?); >(nth_vec_map ?? (current sig)) in Hcurta_src; #Hcurta_src
+ >Hcurta_src in Htest; whd in ⊢ (??%?→?);
+ cases (is_endc s) // whd in ⊢ (??%?→?); #H @sym_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) // ]
+ |#ls #x #xs #ci #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc
+ cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc)
+ [ * #Hrs00 #Htc >Htc in Htest; whd in ⊢ (??%?→?);
+ <(nth_vec_map ?? (current sig) ??? (niltape ?))
+ >change_vec_commute // >nth_change_vec // whd in ⊢ (??%?→?);
+ cases (is_endc ci)
+ [ whd in ⊢ (??%?→?); #H destruct (H)
+ | <(nth_vec_map ?? (current sig) ??? (niltape ?))
+ >change_vec_commute [| @sym_not_eq // ] >nth_change_vec //
+ >(?:current ? (mk_tape ?? (None ?) ?) = None ?)
+ [ whd in ⊢ (??%?→?); #H destruct (H)
+ | cases (reverse sig xs@x::ls0) normalize // ] ] ]
+ * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2 % [ %
+ [ cases (true_or_false (is_endc ci)) //
+ #Hendci >(Hcomp2 (or_introl … Hendci)) in Htest;
+ whd in ⊢ (??%?→?); <(nth_vec_map ?? (current sig) ??? (niltape ?))
+ >change_vec_commute // >nth_change_vec // whd in ⊢ (??%?→?);
+ >Hendci normalize //
+ | % #H destruct (H) ] ] #cj #rs1 #H destruct (H) #Hcicj
+ lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc %
+ [ cases Hte -Hte #Hte #_ whd in Hte;
+ >Htasrc_mid in Hcurta_src; whd in ⊢ (??%?→?); #H destruct (H)
+ lapply (Hte ls ci (reverse ? xs) rs s ??? ls0 cj (reverse ? xs) s rs1 (refl ??) ?) //
+ [ >Htc >nth_change_vec //
+ | #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 //
+ ]
+ | >Htc >change_vec_commute // >nth_change_vec // ] -Hte
+ >Htc >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec #Hte
+ >Hte in Htb; * * #_ >reverse_reverse #Htbdst1 #Htbdst2 -Hte @(eq_vec … (niltape ?))
+ #i #Hi cases (decidable_eq_nat i dst) #Hidst
+ [ >Hidst >nth_change_vec // >(Htbdst1 ls0 s (xs@cj::rs1))
+ [| >nth_change_vec // ]
+ >Htadst_mid cases xs //
+ | >nth_change_vec_neq [|@sym_not_eq // ]
+ <Htbdst2 [| @sym_not_eq // ] >nth_change_vec_neq [| @sym_not_eq // ]
+ <Htasrc_mid >change_vec_same % ]
+ | >Hcurta_src in Htest; whd in ⊢(??%?→?);
+ >Htc >change_vec_commute //
+ change with (current ? (niltape ?)) in match (None ?);
+ <nth_vec_map >nth_change_vec // whd in ⊢ (??%?→?);
+ cases (is_endc ci) whd in ⊢ (??%?→?); #H destruct (H) %
+ ]
+ ]
]
|#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 %]
- |#c_src lapply (current_to_midtape sig (nth dst ? intape (niltape ?)))
- cases (current … (nth dst ? intape (niltape ?)))
- [#_ #Hcomp1 #_ %1 % [%2 % | @Hcomp1 %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 #_
- #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 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls * #rs #Hsrc
- %1 %
- [% % %{c_src} % // lapply (Hc c_src) -Hc >Hcomp1
- [| % % @(not_to_not ??? (\Pf Hceq)) #H destruct (H) // ]
- cases (is_endc c_src) //
- >Hsrc #Hc lapply (Hc (refl ??)) normalize #H destruct (H)
- |@Hcomp1 %1 %1 @(not_to_not ??? (\Pf Hceq)) #H destruct (H) //
- ]
- ]
- ]
- ]
+ #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
+ cases (comp_list … (xs@end::rs) rs_dst is_endc) #xs1 * #rsi * #rsj * * *
+ #Hrs_src #Hrs_dst #Hnotendxs1 #Hneq >Hrs_dst in Hmid_dst; #Hmid_dst
+ cut (∃r1,rs1.rsi = r1::rs1)
+ [cases rsi in Hrs_src;
+ [ >append_nil #H <H in Hnotendxs1; #Hnotendxs1
+ >(Hnotendxs1 end) in Hend; [ #H1 destruct (H1) ]
+ @memb_append_l2 @memb_hd
+ | #r1 #rs1 #_ %{r1} %{rs1} % ] ]
+ * #r1 * #rs1 #Hrs1 >Hrs1 in Hrs_src;
+ #Hrs_src >Hrs_src in Hmid_src; #Hmid_src <(\P Hceq) in Hmid_dst; #Hmid_dst
+ lapply (Hcomp2 ??????? Hmid_src Hmid_dst ?)
+ [ #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) @Hnotend @memb_hd | @Hnotendxs1 //] ]
+ *
+ [ * #Hrsj >Hrsj #Hta % %2 >Hta >nth_change_vec //
+ %{ls_dst} %{xs1} cut (∃xs0.xs = xs1@xs0)
+ [lapply Hnotendxs1 -Hnotendxs1 lapply Hrs_src lapply xs elim xs1
+ [ #l #_ #_ %{l} %
+ | #x2 #xs2 #IH *
+ [ whd in ⊢ (??%%→?); #H destruct (H) #Hnotendxs2
+ >Hnotendxs2 in Hend; [ #H destruct (H) |@memb_hd ]
+ | #x2' #xs2' whd in ⊢ (??%%→?); #H destruct (H)
+ #Hnotendxs2 cases (IH xs2' e0 ?)
+ [ #xs0 #Hxs2 %{xs0} @eq_f //
+ |#c #Hc @Hnotendxs2 @memb_cons // ]
+ ]
+ ]
+ ] * #xs0 #Hxs0 %{xs0} % [ %
+ [ >Hmid_dst >Hrsj >append_nil %
+ | @Hxs0 ]
+ | cases (reverse ? xs1) // ]
+ | * #cj * #rs2 * #Hrsj #Hta lapply (Hta ?)
+ [ cases (Hneq ?? Hrs1) /2/ * #_ #Hr1 %2 @(Hr1 ?? Hrsj) ] -Hta #Hta
+ %2 >Hta in Hc; whd in ⊢ (??%?→?);
+ change with (current ? (niltape ?)) in match (None ?);
+ <nth_vec_map >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+ whd in ⊢ (??%?→?); #Hc cut (is_endc r1 = true)
+ [ cases (is_endc r1) in Hc; whd in ⊢ (??%?→?); //
+ change with (current ? (niltape ?)) in match (None ?);
+ <nth_vec_map >nth_change_vec // normalize #H destruct (H) ]
+ #Hendr1 cut (xs = xs1)
+ [ lapply Hnotendxs1 lapply Hnotend lapply Hrs_src lapply xs1
+ -Hnotendxs1 -Hnotend -Hrs_src -xs1 elim xs
+ [ * normalize in ⊢ (%→?); //
+ #x2 #xs2 normalize in ⊢ (%→?); #Heq destruct (Heq) #_ #Hnotendxs1
+ lapply (Hnotendxs1 ? (memb_hd …)) >Hend #H destruct (H)
+ | #x2 #xs2 #IH *
+ [ normalize in ⊢ (%→?); #Heq destruct (Heq) #Hnotendc
+ >Hnotendc in Hendr1; [| @memb_cons @memb_hd ]
+ normalize in ⊢ (%→?); #H destruct (H)
+ | #x3 #xs3 normalize in ⊢ (%→?); #Heq destruct (Heq)
+ #Hnotendc #Hnotendcxs1 @eq_f @IH
+ [ @(cons_injective_r … Heq)
+ | #c0 #Hc0 @Hnotendc cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) @memb_hd
+ | @memb_cons @memb_cons // ]
+ | #c #Hc @Hnotendcxs1 @memb_cons // ]
+ ]
+ ]
+ | #Hxsxs1 destruct (Hxsxs1) >Hmid_dst %{ls_dst} %{rsj} % //
+ #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0)
+ lapply (append_l2_injective … Hrs_src) // #Hrs' destruct (Hrs') %
+ ]
+ ]
+ |#Hcomp1 #Hsrc cases (Hsrc ? (refl ??)) -Hsrc #ls0 * #rs0 #Hdst
+ @False_ind lapply (Hcomp1 ?) [%2 %1 %1 >Hmid_src normalize
+ @(not_to_not ??? (\Pf Hceq)) #H destruct //] #Hintape >Hintape in Hc;
+ whd in ⊢(??%?→?); >Hmid_src
+ change with (current ? (niltape ?)) in match (None ?);
+ <nth_vec_map >Hmid_src whd in ⊢ (??%?→?);
+ >(Hnotend c_src) [|@memb_hd]
+ change with (current ? (niltape ?)) in match (None ?);
+ <nth_vec_map >Hmid_src whd in ⊢ (??%?→?); >Hdst normalize #H destruct (H)
+ ]
+ ]
+]
qed.
definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
(inr ?? (inr ?? (inl … (inr ?? start_nop)))).
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.
- is_startc x = true → is_endc ci = true →
- nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- nth j ? int (niltape ?) = midtape sig ls0 x0 rs0 →
- ∃l,cj,l1.x0::rs0 = l@x::xs@cj::l1 ∧
- outt = change_vec ??
- (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) i)
- (midtape sig ((reverse ? (l@x::xs))@ls0) cj l1) j).
+ λsrc,dst,sig,n,is_startc,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 →
+ (∀c0. memb ? c0 (xs@end::rs) = true → is_startc c0 = false) →
+ (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
+ (is_startc x = true →
+ (∀ls0,x0,rs0.
+ nth dst ? 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) end rs) src)
+ (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) dst) ∨
+ ∀l,l1.x0::rs0 ≠ l@x::xs@l1)).
+
+lemma not_sub_list_merge :
+ ∀T.∀a,b:list T. (∀l1.a ≠ b@l1) → (∀t,l,l1.a ≠ t::l@b@l1) → ∀l,l1.a ≠ l@b@l1.
+#T #a #b #H1 #H2 #l elim l normalize //
+qed.
+
+lemma not_sub_list_merge_2 :
+ ∀T:DeqSet.∀a,b:list T.∀t. (∀l1.t::a ≠ b@l1) → (∀l,l1.a ≠ l@b@l1) → ∀l,l1.t::a ≠ l@b@l1.
+#T #a #b #t #H1 #H2 #l elim l //
+#t0 #l1 #IH #l2 cases (true_or_false (t == t0)) #Htt0
+[ >(\P Htt0) % normalize #H destruct (H) cases (H2 l1 l2) /2/
+| normalize % #H destruct (H) cases (\Pf Htt0) /2/ ]
+qed.
+
lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
src ≠ dst → src < S n → dst < S n →
#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 #_ #_ #Hnthi #Hnthj
- >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
+[ #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart
+ cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse
+ [(* current dest = None *) *
+ [ * #Hcur_dst #Houtc %
+ [#_ >Houtc //
+ |#Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcur_dst;
+ normalize in ⊢ (%→?); #H destruct (H)
]
- | #Hci #Houtc %
- [ #_ @Houtc
- | #ls #x #xs #ci #rs #ls0 #cj #rs0 #Hnthi >Hnthi in Hci;
- normalize in ⊢ (%→?); #H destruct (H) ] ]
- | #Hcj #Houtc %
- [ #_ @Houtc
- | #ls #x #xs #ci #rs #ls0 #cj #rs0 #_ #Hnthj >Hnthj in Hcj;
- normalize in ⊢ (%→?); #H destruct (H) ] ]
-
-
-
-[ #tc whd in ⊢ (%→?); * * [ *
+ | * #ls0 * #rs0 * #xs0 * * #Htc_dst #Hrs0 #HNone %
+ [ >Htc_dst normalize in ⊢ (%→?); #H destruct (H)
+ | #Hstart #ls1 #x1 #rs1 >Htc_dst #H destruct (H)
+ >Hrs0 cases xs0
+ [ % %{[ ]} %{[ ]} % [ >append_nil >append_nil %]
+ #cj #ls2 #H destruct (H)
+ | #x2 #xs2 %2 #l #l1 % #Habs lapply (eq_f ?? (length ?) ?? Habs)
+ >length_append whd in ⊢ (??%(??%)→?); >length_append
+ >length_append normalize >commutative_plus whd in ⊢ (???%→?);
+ #H destruct (H) lapply e0 >(plus_n_O (|rs1|)) in ⊢ (??%?→?);
+ >associative_plus >associative_plus
+ #e1 lapply (injective_plus_r ??? e1) whd in ⊢ (???%→?);
+ #e2 destruct (e2)
+ ]
+ ]
+ ]
+ |* #ls0 * #rs0 * #Hmid_dst #HFalse %
+ [ >Hmid_dst normalize in ⊢ (%→?); #H destruct (H)
+ | #Hstart #ls1 #x1 #rs1 >Hmid_dst #H destruct (H)
+ %1 %{[ ]} %{rs0} % [%] #cj #l2 #Hnotnil
+ >reverse_cons >associative_append @(HFalse ?? Hnotnil)
+ ]
+ ]
+|-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
+ #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart
+ lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?);
+ [#Hmid_dst %
+ [#_ whd in Htrue; >Hmid_src in Htrue; #Htrue
+ cases (Htrue x (refl … )) -Htrue * #Htaneq #_
+ @False_ind >Hmid_dst in Htaneq; /2/
+ |#Hstart #ls0 #x0 #rs0 #Hmid_dst2 >Hmid_dst2 in Hmid_dst; normalize in ⊢ (%→?);
+ #H destruct (H)
+ ]
+ | #c #Hcurta_dst % [ >Hcurta_dst #H destruct (H) ]
+ #Hstart #ls0 #x0 #rs0 #Hmid_dst >Hmid_dst in Hcurta_dst; normalize in ⊢ (%→?);
+ #H destruct (H) whd in Htrue; >Hmid_src in Htrue; #Htrue
+ cases (Htrue x (refl …)) -Htrue #_ #Htrue cases (Htrue Hstart Hnotstart) -Htrue
+ cases (true_or_false (x==c)) #eqx
+ [ lapply (\P eqx) -eqx #eqx destruct (eqx)
+ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
+ #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
+ cases tl1 in Hxs;
+ [>append_nil #Hx1 <Hx1 in Hnotendx1; #Hnotendx1
+ lapply (Hnotendx1 end ?) [ @memb_append_l2 @memb_hd ]
+ >Hend #H destruct (H) ]
+ #ci -tl1 #tl1 #Hxs #H cases (H … (refl … )) -H
+ [ #Hendci % >Hrs0 in Hmid_dst; cut (ci = end ∧ x1 = xs)
+ [ lapply Hxs lapply Hnotendx1 lapply x1 elim xs in Hnotend;
+ [ #_ *
+ [ #_ normalize #H destruct (H) /2/
+ | #x2 #xs2 #Hnotendx2 normalize #H destruct (H)
+ >(Hnotendx2 ? (memb_hd …)) in Hend; #H destruct (H) ]
+ | #x2 #xs2 #IH #Hnotendx2 *
+ [ #_ normalize #H destruct (H) >(Hnotendx2 ci ?) in Hendci;
+ [ #H destruct (H)
+ | @memb_cons @memb_hd ]
+ | #x3 #xs3 #Hnotendx3 normalize #H destruct (H)
+ cases (IH … e0)
+ [ #H1 #H2 /2/
+ | #c0 #Hc0 @Hnotendx2 cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) @memb_hd
+ | @memb_cons @memb_cons @Hc0 ]
+ | #c0 #Hc0 @Hnotendx3 @memb_cons @Hc0 ]
+ ]
+ ]
+ | * #Hcieq #Hx1eq >Hx1eq #Hmid_dst
+ cases (Htrue ??????? (refl ??) Hmid_dst Hnotend)
+ <Hcieq >Hendci * #H destruct (H) ]
+ |cases tl2 in Hrs0;
+ [ >append_nil #Hrs0 destruct (Hrs0) * #Hcifalse#_ %2
+ cut (∃l.xs = x1@ci::l)
+ [lapply Hxs lapply Hnotendx1 lapply Hnotend lapply xs
+ -Hxs -xs -Hnotendx1 elim x1
+ [ *
+ [ #_ #_ normalize #H1 destruct (H1) >Hend in Hcifalse;
+ #H1 destruct (H1)
+ | #x2 #xs2 #_ #_ normalize #H >(cons_injective_l ????? H) %{xs2} % ]
+ | #x2 #xs2 #IHin *
+ [ #_ #Hnotendxs2 normalize #H destruct (H)
+ >(Hnotendxs2 ? (memb_hd …)) in Hend; #H destruct (H)
+ | #x3 #xs3 #Hnotendxs3 #Hnotendxs2 normalize #H destruct (H)
+ cases (IHin ??? e0)
+ [ #xs4 #Hxs4 >Hxs4 %{xs4} %
+ | #c0 #Hc0 cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) @Hnotendxs3 @memb_hd
+ | @Hnotendxs3 @memb_cons @memb_cons @Hc0 ]
+ | #c0 #Hc0 @Hnotendxs2 @memb_cons @Hc0 ]
+ ]
+ ]
+ ] * #l #Hxs' >Hxs'
+ #l0 #l1 % #H lapply (eq_f ?? (length ?) ?? H) -H
+ >length_append normalize >length_append >length_append
+ normalize >commutative_plus normalize #H destruct (H) -H
+ >associative_plus in e0; >associative_plus
+ >(plus_n_O (|x1|)) in ⊢(??%?→?); #H lapply (injective_plus_r … H)
+ -H normalize #H destruct (H)
+ | #cj #tl2' #Hrs0 * #Hcifalse #Hcomp
+ lapply (Htrue ls c x1 ci tl1 ls0 (cj::tl2') ???)
+ [ #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
+ [ @Hnotend >(\P Hc0) @memb_hd
+ | @Hnotendx1 // ]
+ | >Hmid_dst >Hrs0 %
+ | >Hxs %
+ | * * #_ #_ -Htrue #Htrue lapply (Htrue ?? (refl ??) ?) [ @(Hcomp ?? (refl ??)) ]
+ * #Htb >Htb #Hendci >Hrs0 >Hxs
+ cases (IH ls c xs end rs ? Hnotend Hend Hnotstart) -IH
+ [| >Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src ]
+ #_ #IH lapply Hxs lapply Hnotendx1 -Hxs -Hnotendx1 cases x1 in Hrs0;
+ [ #Hrs0 #_ whd in ⊢ (???%→?); #Hxs
+ cases (IH Hstart (c::ls0) cj tl2' ?)
+ [ -IH * #l * #l1 * #Hll1 #IH % %{(c::l)} %{l1}
+ % [ @eq_f @Hll1 ]
+ #cj0 #l2 #Hcj0 >(IH … Hcj0) >Htb
+ >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq // ] @eq_f3 //
+ >reverse_cons >associative_append %
+ | #IH %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%]
+ @not_sub_list_merge
+ [ #l2 cut (∃xs'.xs = ci::xs')
+ [ cases xs in Hxs;
+ [ normalize #H destruct (H) >Hend in Hendci; #H destruct (H)
+ | #ci' #xs' normalize #H lapply (cons_injective_l ????? H)
+ #H1 >H1 %{xs'} % ]
+ ]
+ * #xs' #Hxs' >Hxs' normalize % #H destruct (H)
+ lapply (Hcomp … (refl ??)) * /2/
+ |#t #l2 #l3 % normalize #H lapply (cons_injective_r ????? H)
+ -H #H >H in IH; #IH cases (IH l2 l3) -IH #IH @IH % ]
+ | >Htb >nth_change_vec // >Hmid_dst >Hrs0 % ]
+ | #x2 #xs2 normalize in ⊢ (%→?); #Hrs0 #Hnotendxs2 normalize in ⊢ (%→?);
+ #Hxs cases (IH Hstart (c::ls0) x2 (xs2@cj::tl2') ?)
+ [ -IH * #l * #l1 * #Hll1 #IH % %{(c::l)} %{l1}
+ % [ @eq_f @Hll1 ]
+ #cj0 #l2 #Hcj0 >(IH … Hcj0) >Htb
+ >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq // ] @eq_f3 //
+ >reverse_cons >associative_append %
+ | -IH #IH %2 #l #l1 >(?:l@c::xs@l1 = l@(c::xs)@l1) [|%]
+ @not_sub_list_merge_2 [| @IH]
+ cut (∃l2.xs = (x2::xs2)@ci::l2)
+ [lapply Hnotendxs2
+ lapply Hnotend -Hnotend lapply Hxs
+ >(?:x2::xs2@ci::tl1 = (x2::xs2)@ci::tl1) [|%]
+ lapply (x2::xs2) elim xs
+ [ *
+ [ normalize in ⊢ (%→?); #H1 destruct (H1)
+ >Hendci in Hend; #Hend destruct (Hend)
+ | #x3 #xs3 normalize in ⊢ (%→?); #H1 destruct (H1)
+ #_ #Hnotendx3 >(Hnotendx3 ? (memb_hd …)) in Hend;
+ #Hend destruct (Hend)
+ ]
+ | #x3 #xs3 #IHin *
+ [ normalize in ⊢ (%→?); #Hxs3 destruct (Hxs3) #_ #_
+ %{xs3} %
+ | #x4 #xs4 normalize in ⊢ (%→?); #Hxs3xs4 #Hnotend
+ #Hnotendxs4 destruct (Hxs3xs4) cases (IHin ? e0 ??)
+ [ #l0 #Hxs3 >Hxs3 %{l0} %
+ | #c0 #Hc0 @Hnotend cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ >(\P Hc0) @memb_hd
+ | @memb_cons @memb_cons @Hc0 ]
+ | #c0 #Hc0 @Hnotendxs4 @memb_cons //
+ ]
+ ]
+ ]
+ ] * #l2 #Hxs'
+ >Hxs' #l3 normalize >associative_append normalize % #H
+ destruct (H) lapply (append_l2_injective ?????? e1) //
+ #H1 destruct (H1) cases (Hcomp ?? (refl ??)) /2/
+ | >Htb >nth_change_vec // >Hmid_dst >Hrs0 % ]
+ ]
+ ]
+ ]
+ ]
+ |lapply (\Pf eqx) -eqx #eqx >Hmid_dst #Htrue
+ cases (Htrue ? (refl ??) eqx) -Htrue #Htb #Hendcx #_
+ cases rs0 in Htb;
+ [ #_ %2 #l #l1 cases l
+ [ normalize cases xs
+ [ cases l1
+ [ normalize % #H destruct (H) cases eqx /2/
+ | #tmp1 #l2 normalize % #H destruct (H) ]
+ | #tmp1 #l2 normalize % #H destruct (H) ]
+ | #tmp1 #l2 normalize % #H destruct (H)cases l2 in e0;
+ [ normalize #H1 destruct (H1)
+ | #tmp2 #l3 normalize #H1 destruct (H1) ]
+ ]
+ | #r1 #rs1 normalize in ⊢ (???(????%?)→?); #Htb >Htb in IH; #IH
+ cases (IH ls x xs end rs ? Hnotend Hend Hnotstart)
+ [| >Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src ] -IH
+ #_ #IH cases (IH Hstart (c::ls0) r1 rs1 ?)
+ [|| >nth_change_vec // ] -IH
+ [ * #l * #l1 * #Hll1 #Hout % %{(c::l)} %{l1} % >Hll1 //
+ >reverse_cons >associative_append #cj0 #ls #Hl1 >(Hout ?? Hl1)
+ >change_vec_commute in ⊢ (??(???%??)?); // @sym_not_eq //
+ | #IH %2 @(not_sub_list_merge_2 ?? (x::xs)) normalize [|@IH]
+ #l1 % #H destruct (H) cases eqx /2/
+ ]
+ ]
+ ]
+]
+qed.
+definition Pre_match_m ≝
+ λsrc,sig,n,is_startc,is_endc.λt: Vector (tape sig) (S n).
+ ∃start,xs,end.
+ nth src (tape sig) t (niltape sig) = midtape ? [] start (xs@[end]) ∧
+ is_startc start = true ∧
+ (∀c.c ∈ (xs@[end]) = true → is_startc c = false) ∧
+ (∀c.c ∈ (start::xs) = true → is_endc c = false) ∧
+ is_endc end = true.
+
+lemma terminate_match_m :
+ ∀src,dst,sig,n,is_startc,is_endc,t.
+ src ≠ dst → src < S n → dst < S n →
+ Pre_match_m src sig n is_startc is_endc t →
+ match_m src dst sig n is_startc is_endc ↓ t.
+#src #dst #sig #n #is_startc #is_endc #t #Hneq #Hsrc #Hdst * #start * #xs * #end
+* * * * #Hmid_src #Hstart #Hnotstart #Hnotend #Hend
+@(terminate_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst)) //
+<(change_vec_same … t dst (niltape ?))
+lapply (refl ? (nth dst (tape sig) t (niltape ?)))
+cases (nth dst (tape sig) t (niltape ?)) in ⊢ (???%→?);
+[ #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
+ >Hmid_src #HR cases (HR ? (refl ??)) -HR
+ >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
+ * #H @False_ind @H %
+| #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
+ >Hmid_src #HR cases (HR ? (refl ??)) -HR
+ >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
+ * #H @False_ind @H %
+| #x0 #xs0 #Htape_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
+ >Hmid_src #HR cases (HR ? (refl ??)) -HR
+ >nth_change_vec // >Htape_dst normalize in ⊢ (%→?);
+ * #H @False_ind @H %
+| #ls #s #rs lapply s -s lapply ls -ls lapply Hmid_src lapply t -t elim rs
+ [#t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?); >nth_change_vec_neq [|@sym_not_eq //]
+ >Hmid_src >nth_change_vec // >Hmid_dst #HR cases (HR ? (refl ??)) -HR #_
+ #HR cases (HR Hstart Hnotstart)
+ cases (true_or_false (start == s)) #Hs
+ [ lapply (\P Hs) -Hs #Hs <Hs #_ #Htrue
+ cut (∃ci,xs1.xs@[end] = ci::xs1)
+ [ cases xs
+ [ %{end} %{[]} %
+ | #x1 #xs1 %{x1} %{(xs1@[end])} % ] ] * #ci * #xs1 #Hxs
+ >Hxs in Htrue; #Htrue
+ cases (Htrue [ ] start [ ] ? xs1 ? [ ] (refl ??) (refl ??) ?)
+ [ * #_ * #H @False_ind @H % ]
+ #c0 #Hc0 @Hnotend >(memb_single … Hc0) @memb_hd
+ | lapply (\Pf Hs) -Hs #Hs #Htrue #_
+ cases (Htrue ? (refl ??) Hs) -Htrue #Ht1 #_ %
+ #t2 whd in ⊢ (%→?); #HR cases (HR start ?)
+ [ >Ht1 >nth_change_vec // normalize in ⊢ (%→?); * #H @False_ind @H %
+ | >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src % ]
+ ]
+ |#r0 #rs0 #IH #t #Hmid_src #ls #s #Hmid_dst % #t1 whd in ⊢ (%→?);
+ >nth_change_vec_neq [|@sym_not_eq //] >Hmid_src
+ #Htrue cases (Htrue ? (refl ??)) -Htrue #_ #Htrue
+ <(change_vec_same … t1 dst (niltape ?))
+ cases (Htrue Hstart Hnotstart) -Htrue
+ cases (true_or_false (start == s)) #Hs
+ [ lapply (\P Hs) -Hs #Hs <Hs #_ #Htrue
+ cut (∃ls0,xs0,ci,rs,rs0.
+ nth src ? t (niltape ?) = midtape sig [ ] start (xs0@ci::rs) ∧
+ nth dst ? t (niltape ?) = midtape sig ls0 s (xs0@rs0) ∧
+ (is_endc ci = true ∨ (is_endc ci = false ∧ (∀b,tlb.rs0 = b::tlb → ci ≠ b))))
+ [cases (comp_list ? (xs@[end]) (r0::rs0) is_endc) #xs0 * #xs1 * #xs2
+ * * * #Hxs #Hrs #Hxs0notend #Hcomp >Hrs
+ cut (∃y,ys. xs1 = y::ys)
+ [ lapply Hxs0notend lapply Hxs lapply xs0 elim xs
+ [ *
+ [ normalize #Hxs1 <Hxs1 #_ %{end} %{[]} %
+ | #z #zs normalize in ⊢ (%→?); #H destruct (H) #H
+ lapply (H ? (memb_hd …)) -H >Hend #H1 destruct (H1)
+ ]
+ | #y #ys #IH0 *
+ [ normalize in ⊢ (%→?); #Hxs1 <Hxs1 #_ %{y} %{(ys@[end])} %
+ | #z #zs normalize in ⊢ (%→?); #H destruct (H) #Hmemb
+ @(IH0 ? e0 ?) #c #Hc @Hmemb @memb_cons // ] ] ] * #y * #ys #Hxs1
+ >Hxs1 in Hxs; #Hxs >Hmid_src >Hmid_dst >Hxs >Hrs
+ %{ls} %{xs0} %{y} %{ys} %{xs2}
+ % [ % // | @Hcomp // ] ]
+ * #ls0 * #xs0 * #ci * #rs * #rs0 * * #Hmid_src' #Hmid_dst' #Hcomp
+ <Hmid_src in Htrue; >nth_change_vec // >Hs #Htrue destruct (Hs)
+ lapply (Htrue ??????? Hmid_src' Hmid_dst' ?) -Htrue
+ [ #c0 #Hc0 @Hnotend cases (orb_true_l … Hc0) -Hc0 #Hc0
+ [ whd in ⊢ (??%?); >Hc0 %
+ | @memb_cons >Hmid_src in Hmid_src'; #Hmid_src' destruct (Hmid_src')
+ lapply e0 -e0 @(list_elim_left … rs)
+ [ #e0 destruct (e0) lapply (append_l1_injective_r ?????? e0) //
+ | #x1 #xs1 #_ >append_cons in ⊢ (???%→?);
+ <associative_append #e0 lapply (append_l1_injective_r ?????? e0) //
+ #e1 >e1 @memb_append_l1 @memb_append_l1 // ] ]
+ | * * #Hciendc cases rs0 in Hcomp;
+ [ #_ * #H @False_ind @H %
+ | #r1 #rs1 * [ >Hciendc #H destruct (H) ]
+ * #_ #Hcomp lapply (Hcomp ?? (refl ??)) -Hcomp #Hcomp #_ #Htrue
+ cases (Htrue ?? (refl ??) Hcomp) #Ht1 #_ >Ht1 @(IH ?? (s::ls) r0)
+ [ >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
+ | >nth_change_vec // >Hmid_dst % ] ] ]
+ | >Hmid_dst >nth_change_vec // lapply (\Pf Hs) -Hs #Hs #Htrue #_
+ cases (Htrue ? (refl ??) Hs) #Ht1 #_ >Ht1 @(IH ?? (s::ls) r0)
+ [ >nth_change_vec_neq [|@sym_not_eq //]
+ >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
+ | >nth_change_vec // ] ] ] ]
+qed.
\ No newline at end of file
projects / helm.git / blobdiff