-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
- *)
-
-definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
- 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)))
- (nop …)
- tc_true).
-