-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A||
+ \ / This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ V_____________________________________________________________*)
-include "turing/turing.ma".
+include "turing/multi_universal/moves.ma".
+include "turing/if_multi.ma".
include "turing/inject.ma".
-include "turing/while_multi.ma".
+include "turing/basic_machines.ma".
definition copy_states ≝ initN 3.
definition copy1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
definition copy2 : copy_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
-(*
-
-src: a b c ... z # ---→ a b c ... z #
- ^ ^
-
-dst: _ _ _ ... _ d ---→ a b c ... z d
- ^ ^
-
-0) (x ≠ sep,_) → (x,x)(R,R) → 1
- (sep,d) → None 2
-1) (_,_) → None 1
-2) (_,_) → None 2
-
-*)
definition trans_copy_step ≝
- λsrc,dst,sig,n,is_sep.
+ λsrc,dst.λsig:FinSet.λn.
λp:copy_states × (Vector (option sig) (S n)).
let 〈q,a〉 ≝ p in
match pi1 … q with
[ O ⇒ match nth src ? a (None ?) with
- [ None ⇒ 〈copy2,null_action ? n〉
- | Some a0 ⇒ if is_sep a0 then 〈copy2,null_action ? n〉
- else 〈copy1,change_vec ? (S n)
- (change_vec ?(S n)
- (null_action ? n) (〈Some ? a0,R〉) src)
- (〈Some ? a0,R〉) dst〉 ]
+ [ None ⇒ 〈copy2,null_action sig n〉
+ | Some ai ⇒ match nth dst ? a (None ?) with
+ [ None ⇒ 〈copy2,null_action ? n〉
+ | Some aj ⇒
+ 〈copy1,change_vec ? (S n)
+ (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
+ (〈Some ? ai,R〉) dst〉
+ ]
+ ]
| S q ⇒ match q with
[ O ⇒ (* 1 *) 〈copy1,null_action ? n〉
| S _ ⇒ (* 2 *) 〈copy2,null_action ? n〉 ] ].
definition copy_step ≝
- λsrc,dst,sig,n,is_sep.
- mk_mTM sig n copy_states (trans_copy_step src dst sig n is_sep)
+ λsrc,dst,sig,n.
+ mk_mTM sig n copy_states (trans_copy_step src dst sig n)
copy0 (λq.q == copy1 ∨ q == copy2).
-definition R_copy_step_true ≝
- λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n).
- ∃x1.
- current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
- is_sep x1 = false ∧
+definition R_comp_step_true ≝
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ ∃x,y.
+ current ? (nth src ? int (niltape ?)) = Some ? x ∧
+ current ? (nth dst ? int (niltape ?)) = Some ? y ∧
outt = change_vec ??
(change_vec ?? int
- (tape_move_mono ? (nth src ? int (niltape ?)) (〈Some ? x1,R〉)) src)
- (tape_move_mono ? (nth dst ? int (niltape ?)) (〈Some ? x1,R〉)) dst.
+ (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
+ (tape_move_mono ? (nth dst ? int (niltape ?)) 〈Some ? x, R〉) dst.
-definition R_copy_step_false ≝
- λsrc,dst:nat.λsig,n,is_sep.λint,outt: Vector (tape sig) (S n).
- (∃x1.
- current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
- is_sep x1 = true ∧ outt = int) ∨
- current ? (nth src ? int (niltape ?)) = None ? ∧
- outt = int.
+definition R_comp_step_false ≝
+ λsrc,dst:nat.λsig,n.λint,outt: Vector (tape sig) (S n).
+ (current ? (nth src ? int (niltape ?)) = None ? ∨
+ current ? (nth dst ? int (niltape ?)) = None ?) ∧ outt = int.
lemma copy_q0_q2_null :
- ∀src,dst,sig,n,is_sep,v,t.src < S n → dst < S n →
- current ? t = None ? →
- step sig n (copy_step src dst sig n is_sep)
- (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) =
- mk_mconfig ??? copy2 (change_vec ? (S n) v t src).
-#src #dst #sig #n #is_sep #v #t #Hsrc #Hdst #Hcurrent
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent %
-| >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent @tape_move_null_action
-]
-qed.
-
-lemma copy_q0_q2_sep :
- ∀src,dst,sig,n,is_sep,v,t.src < S n → dst < S n →
- ∀s.current ? t = Some ? s → is_sep s = true →
- step sig n (copy_step src dst sig n is_sep)
- (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) =
- mk_mconfig ??? copy2 (change_vec ? (S n) v t src).
-#src #dst #sig #n #is_sep #v #t #Hsrc #Hdst #s #Hcurrent #Hsep
-whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent whd in ⊢ (??(???%)?); >Hsep %
-| >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent whd in ⊢ (??(????(???%))?);
- >Hsep @tape_move_null_action
-]
+ ∀src,dst,sig,n,v.src < S n → dst < S n →
+ (nth src ? (current_chars ?? v) (None ?) = None ? ∨
+ nth dst ? (current_chars ?? v) (None ?) = None ?) →
+ step sig n (copy_step src dst sig n) (mk_mconfig ??? copy0 v)
+ = mk_mconfig ??? copy2 v.
+#src #dst #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 ]
+| @eq_f2
+ [ whd in ⊢ (??(???%)?); >Hcurrent cases (nth src ?? (None sig)) //
+ | whd in ⊢ (??(????(???%))?); >Hcurrent
+ cases (nth src ?? (None sig)) [|#x] @tape_move_null_action ] ]
qed.
-axiom copy_q0_q1 :
- ∀src,dst,sig,n,is_sep,v,t.src ≠ dst → src < S n → dst < S n →
- ∀s.current ? t = Some ? s → is_sep s = false →
- step sig n (copy_step src dst sig n is_sep)
- (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) =
+lemma copy_q0_q1 :
+ ∀src,dst,sig,n,v,a,b.src ≠ dst → src < S n → dst < S n →
+ nth src ? (current_chars ?? v) (None ?) = Some ? a →
+ nth dst ? (current_chars ?? v) (None ?) = Some ? b →
+ step sig n (copy_step src dst sig n) (mk_mconfig ??? copy0 v) =
mk_mconfig ??? copy1
(change_vec ? (S n)
(change_vec ?? v
- (tape_move_mono ? t (〈Some ? s,R〉)) src)
- (tape_move_mono ? (nth dst ? v (niltape ?)) (〈Some ? s,R〉)) dst).
-(*
-#src #dst #sig #n #is_sep #v #t #Hneq #Hsrc #Hdst #s #Hcurrent #Hsep
+ (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
+ (tape_move_mono ? (nth dst ? v (niltape ?)) 〈Some ? a, R〉) dst).
+#src #dst #sig #n #v #a #b #Heq #Hsrc #Hdst #Ha1 #Ha2
whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
-[ >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent whd in ⊢ (??(???%)?); >Hsep %
-| >current_chars_change_vec // whd in match (trans ????);
- >nth_change_vec // >Hcurrent whd in ⊢ (??(????(???%))?);
- >Hsep whd in ⊢ (??(????(???%))?); >change_vec_commute // >pmap_change
- >change_vec_commute // @eq_f3 //
- <(change_vec_same ?? v dst (niltape ?)) in ⊢(??%?);
- >pmap_change @eq_f3 //
+[ whd in match (trans ????);
+ >Ha1 >Ha2 whd in ⊢ (??(???%)?); >(\b ?) //
+| whd in match (trans ????);
+ >Ha1 >Ha2 whd in ⊢ (??(????(???%))?); >(\b ?) //
+ change with (change_vec ?????) in ⊢ (??(????%)?);
+ <(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
+ <(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
+ >tape_move_multi_def
+ >pmap_change >pmap_change <tape_move_multi_def
+ >tape_move_null_action
+ @eq_f2 // >nth_change_vec_neq //
]
-qed.*)
+qed.
lemma sem_copy_step :
- ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst < S n →
- copy_step src dst sig n is_sep ⊨
- [ copy1: R_copy_step_true src dst sig n is_sep,
- R_copy_step_false src dst sig n is_sep ].
-#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #int
+ ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
+ copy_step src dst sig n ⊨
+ [ copy1: R_comp_step_true src dst sig n,
+ R_comp_step_false src dst sig n ].
+#src #dst #sig #n #Hneq #Hsrc #Hdst #int
lapply (refl ? (current ? (nth src ? int (niltape ?))))
cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?);
-[ #Hcur <(change_vec_same … int src (niltape ?)) %{2} %
- [| % [ %
+[ #Hcur_src %{2} %
+ [| % [ %
[ whd in ⊢ (??%?); >copy_q0_q2_null /2/
| normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ %2 >nth_change_vec >Hcur // % // ] ]
-| #c #Hcur cases (true_or_false (is_sep c)) #Hsep
- [ <(change_vec_same … int src (niltape ?)) %{2} %
+ | #_ % // % // ] ]
+| #a #Ha lapply (refl ? (current ? (nth dst ? int (niltape ?))))
+ cases (current ? (nth dst ? int (niltape ?))) in ⊢ (???%→?);
+ [ #Hcur_dst %{2} %
[| % [ %
- [ whd in ⊢ (??%?); >copy_q0_q2_sep /2/
- | normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % >nth_change_vec // %{c} % [ % /2/ | // ] ] ]
- | %{2} % [| % [ %
- [ whd in ⊢ (??%?);
- <(change_vec_same … int src (niltape ?)) in ⊢ (??%?);
- >Hcur in ⊢ (??%?); whd in ⊢ (??%?); >(copy_q0_q1 … Hsep) /2/
- | #_ whd %{c} % % /2/ ]
- | * #Hfalse @False_ind /2/ ] ] ] ]
+ [ whd in ⊢ (??%?); >copy_q0_q2_null /2/
+ | normalize in ⊢ (%→?); #H destruct (H) ]
+ | #_ % // %2 >Hcur_dst % ] ]
+ | #b #Hb %{2} %
+ [| % [ %
+ [whd in ⊢ (??%?); >(copy_q0_q1 … a b Hneq Hsrc Hdst) //
+ | #_ %{a} %{b} % // % //]
+ | * #H @False_ind @H %
+ ]
+ ]
+ ]
+]
qed.
-definition copy ≝ λsrc,dst,sig,n,is_sep.
- whileTM … (copy_step src dst sig n is_sep) copy1.
+definition copy ≝ λsrc,dst,sig,n.
+ whileTM … (copy_step src dst sig n) copy1.
definition R_copy ≝
- λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n).
- (∀ls,x,xs,rs,sep.
- nth src ? int (niltape ?) = midtape sig ls x (xs@sep::rs) →
- (∀c.memb ? c (x::xs) = true → is_sep c = false) → is_sep sep = true →
- ∀ls0,x0,target,c,rs0.|xs| = |target| →
- nth dst ? int (niltape ?) = midtape sig ls0 x0 (target@c::rs0) →
- outt = change_vec ??
- (change_vec ?? int (midtape sig (reverse ? xs@x::ls) sep rs) src)
- (midtape sig (reverse ? xs@x::ls0) c rs0) dst) ∧
- (∀c.current ? (nth src ? int (niltape ?)) = Some ? c → is_sep c = true →
- outt = int) ∧
- (current ? (nth src ? int (niltape ?)) = None ? → outt = int).
-
-lemma wsem_copy : ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst < S n →
- copy src dst sig n is_sep ⊫ R_copy src dst sig n is_sep.
-#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
-lapply (sem_while … (sem_copy_step src dst sig n is_sep Hneq Hsrc Hdst) … Hloop) //
--Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -ta
-[ whd in ⊢ (%→?); *
- [ * #x * * #Hx #Hsep #Houtc % [ %
- [ #ls #x0 #xs #rs #sep #Hsrctc #Hnosep >Hsrctc in Hx; normalize in ⊢ (%→?);
- #Hx0 destruct (Hx0) lapply (Hnosep ? (memb_hd …)) >Hsep
- #Hfalse destruct (Hfalse)
- | #c #Hc #Hsepc @Houtc ]
- | #_ @Houtc ]
- | * #Hcur #Houtc % [ %
- [ #ls #x0 #xs #rs #sep #Hsrctc >Hsrctc in Hcur; normalize in ⊢ (%→?);
- #Hcur destruct (Hcur)
- | #c #Hc #Hsepc @Houtc ]
- | #_ @Houtc ]
+ λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+ ((current ? (nth src ? int (niltape ?)) = None ? ∨
+ current ? (nth dst ? int (niltape ?)) = None ?) → outt = int) ∧
+ (∀ls,x,x0,rs,ls0,rs0.
+ nth src ? int (niltape ?) = midtape sig ls x rs →
+ nth dst ? int (niltape ?) = midtape sig ls0 x0 rs0 →
+ (∃rs01,rs02.rs0 = rs01@rs02 ∧ |rs01| = |rs| ∧
+ outt = change_vec ??
+ (change_vec ?? int
+ (mk_tape sig (reverse sig rs@x::ls) (None sig) []) src)
+ (mk_tape sig (reverse sig rs@x::ls0) (option_hd sig rs02)
+ (tail sig rs02)) dst) ∨
+ (∃rs1,rs2.rs = rs1@rs2 ∧ |rs1| = |rs0| ∧
+ outt = change_vec ??
+ (change_vec ?? int
+ (mk_tape sig (reverse sig rs1@x::ls) (option_hd sig rs2)
+ (tail sig rs2)) src)
+ (mk_tape sig (reverse sig rs1@x::ls0) (None sig) []) dst)).
+
+axiom daemon : ∀P:Prop.P.
+
+lemma wsem_copy : ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n →
+ copy src dst sig n ⊫ R_copy src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
+lapply (sem_while … (sem_copy_step src dst sig n Hneq Hsrc Hdst) … Hloop) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ whd in ⊢ (%→?); * #Hnone #Hout %
+ [#_ @Hout
+ |#ls #x #x0 #rs #ls0 #rs0 #Hsrc1 #Hdst1 @False_ind cases Hnone
+ [>Hsrc1 normalize #H destruct (H) | >Hdst1 normalize #H destruct (H)]
]
-| #td #te * #c0 * * #Hc0 #Hc0nosep #Hd #Hstar #IH #He
- lapply (IH He) -IH * * #IH1 #IH2 #IH3 % [ %
- [ #ls #x #xs #rs #sep #Hsrc_tc #Hnosep #Hsep #ls0 #x0 #target
- #c #rs0 #Hlen #Hdst_tc
- >Hsrc_tc in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
- <(change_vec_same … td src (niltape ?)) in Hd:(???(???(???%??)??));
- <(change_vec_same … td dst (niltape ?)) in ⊢(???(???(???%??)??)→?);
- >Hdst_tc >Hsrc_tc >(change_vec_change_vec ?) >change_vec_change_vec
- >(change_vec_commute ?? td ?? dst src) [|@(sym_not_eq … Hneq)]
- >change_vec_change_vec @(list_cases2 … Hlen)
- [ #Hxsnil #Htargetnil #Hd>(IH2 … Hsep)
- [ >Hd -Hd >Hxsnil >Htargetnil @(eq_vec … (niltape ?))
- #i #Hi cases (decidable_eq_nat i src) #Hisrc
- [ >Hisrc >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
- >nth_change_vec // >nth_change_vec //
- >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
- >nth_change_vec // whd in ⊢ (??%?); %
- | cases (decidable_eq_nat i dst) #Hidst
- [ >Hidst >nth_change_vec // >nth_change_vec //
- >nth_change_vec_neq // >Hdst_tc >Htargetnil %
- | >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
- >nth_change_vec_neq [|@(sym_not_eq … Hisrc)]
- >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
- >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] % ]
- ]
- | >Hd >nth_change_vec_neq [|@(sym_not_eq … Hneq)]
- >nth_change_vec // >nth_change_vec // >Hxsnil % ]
- |#hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd
- >(IH1 (c0::ls) hd1 tl1 rs sep ?? Hsep (c0::ls0) hd2 tl2 c rs0)
- [ >Hd >(change_vec_commute … ?? td ?? src dst) //
- >change_vec_change_vec
- >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //]
- >change_vec_change_vec
- >reverse_cons >associative_append >associative_append %
- | >Hd >nth_change_vec // >nth_change_vec_neq // >Hdst_tc >Htarget //
- | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) //
- | <Hxs #c1 #Hc1 @Hnosep @memb_cons //
- | >Hd >nth_change_vec_neq [|@sym_not_eq //]
- >nth_change_vec // >nth_change_vec // ]
- ]
- | #c #Hc #Hsepc >Hc in Hc0; #Hcc0 destruct (Hcc0) >Hc0nosep in Hsepc;
- #H destruct (H)
- ]
-| #HNone >HNone in Hc0; #Hc0 destruct (Hc0) ] ]
+|#tc #td * #x * #y * * #Hcx #Hcy #Htd #Hstar #IH #He lapply (IH He) -IH *
+ #IH1 #IH2 %
+ [* [>Hcx #H destruct (H) | >Hcy #H destruct (H)]
+ |#ls #x' #y' #rs #ls0 #rs0 #Hnth_src #Hnth_dst
+ >Hnth_src in Hcx; whd in ⊢ (??%?→?); #H destruct (H)
+ >Hnth_dst in Hcy; whd in ⊢ (??%?→?); #H destruct (H)
+ >Hnth_src in Htd; >Hnth_dst -Hnth_src -Hnth_dst
+ cases rs
+ [(* the source tape is empty after the move *)
+ lapply (IH1 ?) [@daemon]
+ #Hout (* whd in match (tape_move ???); *) #Htemp %1 %{([])} %{rs0} %
+ [% [// | // ]
+ |whd in match (reverse ??); whd in match (reverse ??);
+ >Hout >Htemp @eq_f2 // cases rs0 //
+ ]
+ |#c1 #tl1 cases rs0
+ [(* the dst tape is empty after the move *)
+ lapply (IH1 ?) [@daemon]
+ #Hout (* whd in match (tape_move ???); *) #Htemp %2 %{[ ]} %{(c1::tl1)} %
+ [% [// | // ]
+ |whd in match (reverse ??); whd in match (reverse ??);
+ >Hout >Htemp @eq_f2 //
+ ]
+ |#c2 #tl2 whd in match (tape_move_mono ???); whd in match (tape_move_mono ???);
+ #Htd
+
+
+
+ [ >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 // ]
+ >nth_change_vec //
+ | #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 normalize % //
+ | * #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_copy : ∀src,dst,sig,n,is_sep,t.
- src ≠ dst → src < S n → dst < S n →
- copy src dst sig n is_sep ↓ t.
-#src #dst #sig #n #is_sep #t #Hneq #Hsrc #Hdst
-@(terminate_while … (sem_copy_step …)) //
-<(change_vec_same … t src (niltape ?))
-cases (nth src (tape sig) t (niltape ?))
-[ % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+lemma terminate_compare : ∀i,j,sig,n,t.
+ i ≠ j → i < S n → j < S n →
+ 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
| #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
[#t #ls #c % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?);
- #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 %
+ #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) #Hxsep
+ normalize in ⊢ (%→?); #H destruct (H) #Hcur
>change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
]
]
qed.
-lemma sem_copy : ∀src,dst,sig,n,is_sep.
- src ≠ dst → src < S n → dst < S n →
- copy src dst sig n is_sep ⊨ R_copy src dst sig n is_sep.
-#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst @WRealize_to_Realize /2/
-qed.
\ No newline at end of file
+lemma sem_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 @WRealize_to_Realize
+ [/2/| @wsem_compare // ]
+qed.
projects / helm.git / commitdiff