*)
definition trans_compare_step ≝
- λi,j.λsig:FinSet.λn.λis_endc.
+ λi,j.λsig:FinSet.λn.
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〉
+ [ None ⇒ 〈comp2,null_action sig n〉
| Some ai ⇒ match nth j ? a (None ?) with
[ None ⇒ 〈comp2,null_action ? n〉
- | Some aj ⇒ if notb (is_endc ai) ∧ ai == aj
+ | Some aj ⇒ if ai == aj
then 〈comp1,change_vec ? (S n)
- (change_vec ? (S n) (null_action ? n) (Some ? 〈ai,R〉) i)
- (Some ? 〈aj,R〉) j〉
+ (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) i)
+ (〈None ?,R〉) j〉
else 〈comp2,null_action ? n〉 ]
]
| S q ⇒ match q with
| 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)
+ λi,j,sig,n.
+ mk_mTM sig n compare_states (trans_compare_step i j sig n)
comp0 (λq.q == comp1 ∨ q == comp2).
definition R_comp_step_true ≝
- λi,j,sig,n,is_endc.λint,outt: Vector (tape sig) (S n).
+ λi,j,sig,n.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.
+ (tape_move_right ? (nth i ? int (niltape ?))) i)
+ (tape_move_right ? (nth j ? int (niltape ?))) 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.
+ λi,j:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
+ (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 →
+ ∀i,j,sig,n,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)
+ step sig n (compare_step i j sig n) (mk_mconfig ??? comp0 v)
= mk_mconfig ??? comp2 v.
-#i #j #sig #n #is_endc #v #Hi #Hj
+#i #j #sig #n #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 ]
+ | whd in ⊢ (??(????(???%))?); >Hcurrent @tape_move_null_action ]
| @eq_f2
[ whd in ⊢ (??(???%)?); >Hcurrent cases (nth i ?? (None sig)) //
- | whd in ⊢ (??(???????(???%))?); >Hcurrent
+ | 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)
+ ∀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)
= mk_mconfig ??? comp2 v.
-#i #j #sig #n #is_endc #v #Hi #Hj lapply (refl ? (nth i ?(current_chars ?? v)(None ?)))
+#i #j #sig #n #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
- ]
- ]
+ | #aj #Haj * #Hneq
+ whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
+ [ whd in match (trans ????); >Hai >Haj
+ whd in ⊢ (??(???%)?); cut ((ai==aj)=false)
+ [>(\bf ?) /2 by not_to_not/ % #Haiaj @Hneq
+ >Hai >Haj //
+ | #Haiaj >Haiaj % ]
+ | whd in match (trans ????); >Hai >Haj
+ whd in ⊢ (??(????(???%))?); cut ((ai==aj)=false)
+ [>(\bf ?) /2 by not_to_not/ % #Haiaj @Hneq
+ >Hai >Haj //
+ |#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 →
+ ∀i,j,sig,n,v,a.i ≠ j → i < S n → j < S n →
+ nth i ? (current_chars ?? v) (None ?) = Some ? a →
nth j ? (current_chars ?? v) (None ?) = Some ? a →
- step sig n (compare_step i j sig n is_endc) (mk_mconfig ??? comp0 v) =
+ step sig n (compare_step i j sig n) (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
+ (tape_move_right ? (nth i ? v (niltape ?))) i)
+ (tape_move_right ? (nth j ? v (niltape ?))) j).
+#i #j #sig #n #v #a #Heq #Hi #Hj #Ha1 #Ha2
whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
[ whd in match (trans ????);
- >Ha1 >Ha2 whd in ⊢ (??(???%)?); >Hnotendc >(\b ?) //
+ >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) //
| whd in match (trans ????);
- >Ha1 >Ha2 whd in ⊢ (??(???????(???%))?); >Hnotendc >(\b ?) //
- change with (change_vec ?????) in ⊢ (??(???????%)?);
+ >Ha1 >Ha2 whd in ⊢ (??(????(???%))?); >(\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 //
+ >tape_move_multi_def
+ >pmap_change >pmap_change <tape_move_multi_def
+ >tape_move_null_action
+ @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
+ ∀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
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
+ [ whd in ⊢ (??%?); >comp_q0_q2_null /2/
| 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
+ [ whd in ⊢ (??%?); >comp_q0_q2_null /2/ %2
| normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % // >Ha >Hcurj % % %2 % #H destruct (H) ] ]
- | #b #Hb %{2}
- cases (true_or_false (is_endc a)) #Haendc
+ | #_ % // >Ha >Hcurj % % % #H destruct (H) ] ]
+ | #b #Hb %{2} cases (true_or_false (a == b)) #Hab
[ %
+ [| % [ %
+ [whd in ⊢ (??%?); >(comp_q0_q1 … a Hneq Hi Hj) //
+ >(\P Hab) <Hb @sym_eq @nth_vec_map
+ | #_ whd >(\P Hab) %{b} % // % // <(\P Hab) // ]
+ | * #H @False_ind @H %
+ ] ]
+ | %
[| % [ %
[whd in ⊢ (??%?); >comp_q0_q2_neq //
- % %{a} % // <Ha @sym_eq @nth_vec_map
+ <(nth_vec_map ?? (current …) i ? int (niltape ?))
+ <(nth_vec_map ?? (current …) j ? int (niltape ?)) >Ha >Hb
+ @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
| 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) % ] ]
- ]
+ | #_ % // % % >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 compare ≝ λi,j,sig,n.
+ whileTM … (compare_step i j sig n) comp1.
+(* (∃rs'.rs = rs0@rs' ∧ current ? (nth j ? outt (niltape ?)) = None ?) ∨
+ (∃rs0'.rs0 = rs@rs0' ∧
+ outt = change_vec ??
+ (change_vec ?? int
+ (mk_tape sig (reverse sig rs@x::ls) (None sig) []) i)
+ (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs0')
+ (tail sig rs0')) j) ∨
+ (∃xs,ci,cj,rs',rs0'.ci ≠ cj ∧ rs = xs@ci::rs' ∧ rs0 = xs@cj::rs0' ∧
+ outt = change_vec ??
+ (change_vec ?? int (midtape sig (reverse ? xs@x::ls) ci rs') i)
+ (midtape sig (reverse ? xs@x::ls0) cj rs0') j)).*)
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 ?)) ∨
+ λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
+ ((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 = [ ] ∧
+ (∀ls,x,rs,ls0,rs0.
+(* nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) → *)
+ nth i ? int (niltape ?) = midtape sig ls x rs →
+ nth j ? int (niltape ?) = midtape sig ls0 x rs0 →
+ (∃rs'.rs = rs0@rs' ∧
+ outt = change_vec ??
+ (change_vec ?? int
+ (mk_tape sig (reverse sig rs0@x::ls) (option_hd sig rs') (tail ? rs')) i)
+ (mk_tape sig (reverse sig rs0@x::ls0) (None ?) [ ]) j) ∨
+ (∃rs0'.rs0 = rs@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)).
+ (change_vec ?? int
+ (mk_tape sig (reverse sig rs@x::ls) (None sig) []) i)
+ (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs0')
+ (tail sig rs0')) j) ∨
+ (∃xs,ci,cj,rs',rs0'.ci ≠ cj ∧ rs = xs@ci::rs' ∧ rs0 = xs@cj::rs0' ∧
+ 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) //
+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) //
-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ 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) ] ]
- | #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.
+[ whd in ⊢ (%→?); * * [ *
+ [ #Hcicj #Houtc %
+ [ #_ @Houtc
+ | #ls #x #rs #ls0 #rs0 #Hnthi #Hnthj
+ >Hnthi in Hcicj; >Hnthj normalize in ⊢ (%→?); * #H @False_ind @H %
+ ]
+ | #Hci #Houtc %
+ [ #_ @Houtc
+ | #ls #x #rs #ls0 #rs0 #Hnthi >Hnthi in Hci;
+ normalize in ⊢ (%→?); #H destruct (H) ] ]
+ | #Hcj #Houtc %
+ [ #_ @Houtc
+ | #ls #x #rs #ls0 #rs0 #_ #Hnthj >Hnthj in Hcj;
+ normalize in ⊢ (%→?); #H destruct (H) ] ]
+| #td #te * #x * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
+ #IH1 #IH2 %
+ [ >Hci >Hcj * [ *
+ [ * #H @False_ind @H % | #H destruct (H)] | #H destruct (H)]
+ | #ls #c0 #rs #ls0 #rs0 cases rs
+ [ -IH2 #Hnthi #Hnthj % %2 %{rs0} % [%]
+ >Hnthi in Hd; #Hd >Hd in IH1; #IH1 >IH1
+ [| % %2 >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec // % ]
+ >Hnthj cases rs0 [| #r1 #rs1 ] %
+ | #r1 #rs1 #Hnthi cases rs0
+ [ -IH2 #Hnthj % % %{(r1::rs1)} % [%]
+ >Hnthj in Hd; #Hd >Hd in IH1; #IH1 >IH1
+ [| %2 >nth_change_vec // ]
+ >Hnthi >Hnthj %
+ | #r2 #rs2 #Hnthj lapply IH2; >Hd in IH1; >Hnthi >Hnthj
+ >nth_change_vec //
+ >nth_change_vec_neq [| @sym_not_eq // ] >nth_change_vec //
+ cases (true_or_false (r1 == r2)) #Hr1r2
+ [ >(\P Hr1r2) #_ #IH2 cases (IH2 … (refl ??) (refl ??)) [ *
+ [ * #rs' * #Hrs1 #Hcurout_j % % %{rs'}
+ >Hrs1 %
+ [ %
+ | >Hcurout_j >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute // @sym_not_eq // ]
+ | * #rs0' * #Hrs2 #Hcurout_i % %2 %{rs0'}
+ >Hrs2 >Hcurout_i % //
+ >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute [|@sym_not_eq//] >change_vec_change_vec
+ >reverse_cons >associative_append >associative_append % ]
+ | * #xs * #ci * #cj * #rs' * #rs0' * * * #Hcicj #Hrs1 #Hrs2
+ >change_vec_commute // >change_vec_change_vec
+ >change_vec_commute [| @sym_not_eq ] // >change_vec_change_vec
+ #Houtc %2 %{(r2::xs)} %{ci} %{cj} %{rs'} %{rs0'}
+ % [ % [ % [ // | >Hrs1 // ] | >Hrs2 // ]
+ | >reverse_cons >associative_append >associative_append >Houtc % ] ]
+ | lapply (\Pf Hr1r2) -Hr1r2 #Hr1r2 #IH1 #_ %2
+ >IH1 [| % % normalize @(not_to_not … Hr1r2) #H destruct (H) % ]
+ %{[]} %{r1} %{r2} %{rs1} %{rs2} % [ % [ % /2/ | % ] | % ] ]]]]]
+qed.
-lemma terminate_compare : ∀i,j,sig,n,is_endc,t.
+lemma terminate_compare : ∀i,j,sig,n,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
+ compare i j sig n ↓ t.
+#i #j #sig #n #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 * * * #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 //]
+ [#t #ls #c % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?);
+ #H1 destruct (H1) #_ >change_vec_change_vec #Ht1 %
+ #t2 * #x0 * * >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,is_endc.
+lemma sem_compare : ∀i,j,sig,n.
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/
+ compare i j sig n ⊨ R_compare i j sig n.
+#i #j #sig #n #Hneq #Hi #Hj @WRealize_to_Realize
+ [/2/| @wsem_compare // ]
qed.
projects / helm.git / blobdiff