(* *)
(**************************************************************************)
-include "turing/turing.ma".
-include "turing/inject.ma".
-include "turing/while_multi.ma".
+include "turing/multi_universal/compare.ma".
+include "turing/multi_universal/par_test.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 …)).
+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.
-(*
-
-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.
- 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 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.
- 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.λint,outt: Vector (tape sig) (S n).
- ∃x.
- 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.
+lemma sem_test_char_multi :
+ ∀alpha,test,n,i.i ≤ n →
+ inject_TM ? (test_char ? test) n i ⊨
+ [ tc_true : Rtc_multi_true alpha test n i, Rtc_multi_false alpha test n i ].
+#alpha #test #n #i #Hin #int
+cases (acc_sem_inject … Hin (sem_test_char alpha test) int)
+#k * #outc * * #Hloop #Htrue #Hfalse %{k} %{outc} % [ %
+[ @Hloop
+| #Hqtrue lapply (Htrue Hqtrue) * * * #c *
+ #Hcur #Htestc #Hnth_i #Hnth_j %
+ [ %{c} % //
+ | @(eq_vec … (niltape ?)) #i0 #Hi0
+ cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @Hnth_i
+ | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+| #Hqfalse lapply (Hfalse Hqfalse) * * #Htestc #Hnth_i #Hnth_j %
+ [ @Htestc
+ | @(eq_vec … (niltape ?)) #i0 #Hi0
+ cases (decidable_eq_nat i0 i) #Hi0i
+ [ >Hi0i @Hnth_i
+ | @sym_eq @Hnth_j @sym_not_eq // ] ] ]
+qed.
-definition R_comp_step_false ≝
- 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.
+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 comp_q0_q2_null :
- ∀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) (mk_mconfig ??? comp0 v)
- = mk_mconfig ??? comp2 v.
-#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 ]
-| @eq_f2
- [ whd in ⊢ (??(???%)?); >Hcurrent cases (nth i ?? (None sig)) //
- | whd in ⊢ (??(???????(???%))?); >Hcurrent
- cases (nth i ?? (None sig)) [|#x] @tape_move_null_action ] ]
+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.
-lemma comp_q0_q2_neq :
- ∀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 #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 #Hneq
- whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
- [ whd in match (trans ????); >Hai >Haj
- whd in ⊢ (??(???%)?); >(\bf ?) // @(not_to_not … Hneq) //
- | whd in match (trans ????); >Hai >Haj
- whd in ⊢ (??(???????(???%))?); >(\bf ?) /2 by not_to_not/
- @tape_move_null_action
-] ]
-qed.
+axiom comp_list: ∀S:DeqSet. ∀l1,l2:list S.∀is_endc. ∃l,tl1,tl2.
+ l1 = l@tl1 ∧ l2 = l@tl2 ∧ (∀c.c ∈ l = true → is_endc c = false) ∧
+ ∀a,tla. tl1 = a::tla → is_endc a = true ∨ (∀b,tlb.tl2 = b::tlb → a≠b).
+
+axiom daemon : ∀X:Prop.X.
-lemma comp_q0_q1 :
- ∀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) (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 #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 ⊢ (??(???%)?); >(\b ?) //
-| whd in match (trans ????);
- >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 //
-]
-qed.
+definition match_test ≝ λsrc,dst.λsig:DeqSet.λn,is_endc.λv:Vector ? n.
+ match (nth src (option sig) v (None ?)) with
+ [ None ⇒ false
+ | Some x ⇒ notb ((is_endc x) ∨ (nth dst (DeqOption sig) v (None ?) == None ?))].
-lemma sem_comp_step :
- ∀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
- | 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 % % % #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
- |<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 ?)) >Ha >Hb
- @(not_to_not ??? (\Pf Hab)) #H destruct (H) %
- | normalize in ⊢ (%→?); #H destruct (H) ]
- | #_ % // % % >Ha >Hb @(not_to_not ??? (\Pf Hab)) #H destruct (H) % ] ]
+definition match_step ≝ λsrc,dst,sig,n,is_startc,is_endc.
+ compare src dst sig n is_endc ·
+ (ifTM ?? (partest sig n (match_test src dst sig ? is_endc))
+ (single_finalTM ??
+ (parmove src dst sig n L is_startc · (inject_TM ? (move_r ?) n dst)))
+ (nop …)
+ partest1).
+
+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,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 (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 →
+ current sig (nth dst (tape sig) int (niltape sig)) ≠ None ? ∧
+ (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,cj,rs,ls0,rs0.
+ nth src ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
+ nth dst ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) →
+ (∀c0. memb ? c0 (x::xs) = true → is_endc c0 = false) →
+ 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_partest sig n (match_test src dst sig ? is_endc))
+ (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 * * #Htest #Htd >Htd -Htd
+ * #te * #Hte #Htb whd
+ #s #Hcurta_src %
+ [ lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
+ cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→%);
+ [| #c #_ % #Hfalse destruct (Hfalse) ]
+ #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 //
+ ]
+ |#ls #x #xs #ci #cj #rs #ls0 #rs00 #Htasrc_mid #Htadst_mid #Hnotendc #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 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 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 >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'::rs0'))
+ [| >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
+ #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) [@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 >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 compare ≝ λi,j,sig,n.
- whileTM … (compare_step i j sig n) comp1.
+definition match_m ≝ λsrc,dst,sig,n,is_startc,is_endc.
+ whileTM … (match_step src dst sig n is_startc is_endc)
+ (inr ?? (inr ?? (inl … (inr ?? start_nop)))).
+
+definition R_match_m ≝
+ λsrc,dst,sig,n,is_startc,is_endc.λint,outt: Vector (tape sig) (S n).
+(* (current sig (nth dst (tape sig) int (niltape sig)) = None ? → outt = int) ∧ *)
+ ∀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) ∧
+ (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)).
-definition R_compare ≝
- λi,j,sig,n.λint,outt: Vector (tape sig) (S n).
- (current sig (nth i (tape sig) int (niltape sig))
- ≠current sig (nth j (tape sig) int (niltape sig)) →
- outt = int) ∧
- (∀ls,x,xs,ci,rs,ls0,cj,rs0.
+(*
+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.
+ (∀x. is_startc x ≠ is_endc x) →
+ is_startc x = true → is_endc ci = true →
+ (∀z. memb ? z (x::xs) = true → is_endc x = false) →
nth i ? int (niltape ?) = midtape sig ls x (xs@ci::rs) →
- nth j ? int (niltape ?) = midtape sig ls0 x (xs@cj::rs0) → 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).
+ nth j ? 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) ci rs) i)
+ (midtape sig ((reverse ? (l@x::xs))@ls0) cj l2) j) ∨
+ ∀l,l1.x0::rs0 ≠ l@x::xs@l1).
+*)
-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) //
+(*
+axiom sub_list_dec: ∀A.∀l,ls:list A.
+ ∃l1,l2. l = l1@ls@l2 ∨ ∀l1,l2. l ≠ l1@ls@l2.
+*)
+
+lemma wsem_match_m : ∀src,dst,sig,n,is_startc,is_endc.
+src ≠ dst → src < S n → dst < S n →
+ match_m src dst sig n is_startc is_endc ⊫ R_match_m src dst sig n is_startc is_endc.
+#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 ⊢ (%→?); * * [ *
- [ #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 * * #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
- #IH1 #IH2 %
- [ >Hci >Hcj * #H @False_ind @H %
- | #ls #c0 #xs #ci #rs #ls0 #cj #rs0 cases xs
- [ #Hnthi #Hnthj #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 @(not_to_not ??? Hcicj)
- #H destruct (H) % ]
- | #x0 #xs0 #Hnthi #Hnthj #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 //
- | >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) %
+[ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
+ 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)
+ ]
+ | * #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)
]
-]]]
-qed.
-
-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 %
- #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
- >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
+ ]
+ ]
+ |* #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)
+ ]
]
-]
-qed.
+|#ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
+ #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend
+ 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 ?) -Htrue
+ [2: #z #membz @daemon (*aggiungere l'ipotesi*)]
+ cases (true_or_false (x==c)) #eqx
+ [ #_ #Htrue cases (comp_list ? (xs@end::rs) rs0 is_endc)
+ #x1 * #tl1 * #tl2 * * * #Hxs #Hrs0 #Hnotendx1
+ cases tl1 in Hxs;
+ [>append_nil #Hx1 @daemon (* absurd by Hx1 e notendx1 *)]
+ #ci -tl1 #tl1 #Hxs #H cases (H … (refl … ))
+ [(* this is absurd, since Htrue conlcudes is_endc ci =false *)
+ #Hend_ci @daemon (* lapply(Htrue … (refl …)) -Htrue *)
+ |cases tl2 in Hrs0;
+ [
+ | #cj #tl2' #Hrs0 #Hcomp lapply (Htrue ls x x1 ci cj tl1 ls0 tl2' ????)
+ [ @(Hcomp ?? (refl ??))
+ | #c0 #Hc0 cases (orb_true_l … Hc0) #Hc0
+ [ @Hnotend >(\P Hc0) @memb_hd
+ | @Hnotendx1 // ]
+ | >Hmid_dst >Hrs0 >(\P eqx) %
+ | >Hxs %
+ | * #Htb >Htb #Hendci %2 >Hrs0 >Hxs
+ cases (IH ls x xs end rs ? Hnotend Hend) [|
+ STOP
+
-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/
+
+ >nth_change_vec_neq [|@sym_not_eq //] >nth_change_vec //
+
+ >Hrs0 in Hmid_dst; #Hmid_dst
+ cases(Htrue ???????? Hmid_dst) -Htrue #Htb #Hendx
+ whd in IH;
+ cases(IH ls x xs end rs ? Hstart Hnotend Hend)
+ [* #H1 #H2 >Htb in H1; >nth_change_vec //
+ >Hmid_dst cases rs0 [2: #a #tl normalize in ⊢ (%→?); #H destruct (H)]
+ #_ %2 @daemon (* si dimostra *)
+ |@daemon
+ |>Htb >nth_change_vec_neq [|@sym_not_eq //] @Hmid_src
+ ]
+ ]
+ ]
+]
qed.
projects / helm.git / blobdiff