(* *)
(**************************************************************************)
-include "turing/multi_universal/moves.ma".
-include "turing/if_multi.ma".
-include "turing/inject.ma".
-include "turing/basic_machines.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,rs0.
- nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- 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 rs1) 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 #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 #rs0 #Hnthi #Hnthj
- >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
- ]]
- | #Hci #Houtc %
- [ #_ @Houtc
- | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi >Hnthi in Hci;
- normalize in ⊢ (%→?); #H destruct (H) ] ]
- | #Hcj #Houtc %
- [ #_ @Houtc
- | #ls #x #xs #ci #rs #ls0 #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 #rs0 cases xs
- [ #Hnthi #Hnthj #Hnotendc cases rs0 in Hnthj;
- [ #Hnthj % % // >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 %2 %2 >nth_change_vec // >Hnthj % ]
- | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // *
- [ #Hendci >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 % %{ci} % //
- ]
- |#Hcir1 >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 %2 % % >nth_change_vec //
- >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // >Hnthi >Hnthj normalize @(not_to_not … Hcir1)
- #H destruct (H) % ]
- ]
- ]
- |#x0 #xs0 #Hnthi #Hnthj #Hnotendc
- cut (c0 = x) [ >Hnthi in Hci; normalize #H destruct (H) // ]
- #Hcut destruct (Hcut) cases rs0 in Hnthj;
- [ #Hnthj % % //
- cases (IH2 (x::ls) x0 xs0 ci rs (x::ls0) [ ] ???) -IH2
- [ * #_ #IH2 >IH2 >Hd >change_vec_commute in ⊢ (??%?); //
- >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
- @sym_not_eq //
- | * #cj * #rs1 * #H destruct (H)
- | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
- >Hnthi %
- | >Hd >nth_change_vec // >Hnthj %
- | #c0 #Hc0 @Hnotendc @memb_cons @Hc0 ]
- | #r1 #rs1 #Hnthj %2 %{r1} %{rs1} % // #Hcir1
- cases(IH2 (x::ls) x0 xs0 ci rs (x::ls0) (r1::rs1) ???)
- [ * #H destruct (H)
- | * #r1' * #rs1' * #H destruct (H) #Hc1r1 >Hc1r1 //
- >Hd >change_vec_commute in ⊢ (??%?); //
- >change_vec_change_vec >change_vec_commute in ⊢ (??%?); //
- @sym_not_eq //
- | >Hd >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
- >Hnthi //
- | >Hd >nth_change_vec // >Hnthi >Hnthj %
- | #c0 #Hc0 @Hnotendc @memb_cons @Hc0
-]]]]]
-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
- *)
+include "turing/multi_universal/compare.ma".
definition Rtc_multi_true ≝
λalpha,test,n,i.λt1,t2:Vector ? (S n).
axiom daemon : ∀X:Prop.X.
-(*
-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,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_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) →
- (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧
- (∀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,rs0.
- nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- nth dst ? int (niltape ?) = midtape sig ls0 x (xs@rs0) →
- (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
- (∀cj,rs1.rs0 = cj::rs1 → ci ≠ cj →
- (outt = change_vec ?? int
- (tape_move … (nth dst ? int (niltape ?)) (Some ? 〈x,R〉)) dst ∧ is_endc ci = false)) ∧
- (rs0 = [ ] →
- outt = change_vec ??
- (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs) src)
- (mk_tape sig (reverse ? xs@x::ls0) (None ?) [ ]) 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 ⊨
- [ 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
-@(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 #Hnotstart % [ %
- [#Hdst_none @daemon
- | #s1 #Hcurta_dst #Hneqss1
- lapply Htb lapply Hcurtc -Htb -Hcurtc >(?:tc=ta)
- [|@Hcomp1 %2 % % >Hcurta_src >Hcurta_dst @(not_to_not … Hneqss1) #H destruct (H) % ]
- #Hcurtc * #te * * #_ #Hte >Hte [2: %1 %1 %{s} % //]
- 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 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc
- cases rs00 in Htadst_mid;
- [(* case rs empty *) #Htadst_mid % [ #cj #rs1 #H destruct (H) ]
- #_ cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) -Hcomp2
- [2: * #x0 * #rs1 * #H destruct (H) ]
- * #_ #Htc cases Htb #td * * #_ #Htd >Htasrc_mid in Hcurta_src;
- normalize in ⊢ (%→?); #H destruct (H)
- >Htd [2: %2 >Htc >nth_change_vec // cases (reverse sig ?) //]
- >Htc * * >nth_change_vec // #Htbdst #_ #Htbelse
- @(eq_vec … (niltape ?)) #i #Hi cases (decidable_eq_nat i dst) #Hidst
- [ >Hidst >nth_change_vec // <Htbdst // cases (reverse sig ?) //
- |@sym_eq @Htbelse @sym_not_eq //
- ]
- |#cj0 #rs0 #Htadst_mid % [| #H destruct (H) ]
- #cj #rs1 #H destruct (H) #Hcicj
- cases (Hcomp2 … Htasrc_mid Htadst_mid Hnotendc) [ * #H destruct (H) ]
- * #cj' * #rs0' * #Hcjrs0 destruct (Hcjrs0) -Hcomp2 #Hcomp2
- lapply (Hcomp2 (or_intror ?? Hcicj)) -Hcomp2 #Htc
- 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 ??)) //
- [| >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 >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 //]
- 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 // ]
- >Htc >nth_change_vec_neq [|@sym_not_eq // ]
- >nth_change_vec_neq [|@sym_not_eq // ] //
- ]
- ]
- | >Htc 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 #Hnotendxs1 #Hneq %{ls_dst} %{rsj} >Hrs_dst in Hmid_dst; #Hmid_dst
- cut (∃r1,rs1.rsi = r1::rs1) [@daemon] * #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 #Hta %
- [ >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
- #Hc lapply (Hc ? (refl ??)) #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 >Hmid_dst >Hxsxs1 % ]
- | #rsj0 #c >Hrsj #Hrsj0 destruct (Hrsj0) ]
- | * #cj * #rs2 * #Hrs2 #Hta lapply (Hta ?)
- [ cases (Hneq … Hrs1) /2/ #H %2 @(H ?? Hrs2) ]
- -Hta #Hta >Hta in Hc; >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // #Hc lapply (Hc ? (refl ??)) #Hendr1
- (* lemmatize this proof *) 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 >Hmid_dst >Hxsxs1 % //
- #rsj0 #c #Hcrsj destruct (Hxsxs1 Hrs2 Hcrsj) @eq_f3 //
- @eq_f3 // lapply (append_l2_injective ?????? Hrs_src) //
- #Hendr1 destruct (Hendr1) % ]
- ]
- ]
- (* STOP *)
- |#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; >Hmid_src #Hc lapply (Hc ? (refl …)) -Hc
- >(Hnotend c_src) // normalize #H destruct (H)
- ]
- ]
-]
-qed.
-*)
definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
compare src dst sig n is_endc ·
projects / helm.git / commitdiff