V_____________________________________________________________*)
-(* COMPARE BIT
-
-*)
-
include "turing/universal/tuples.ma".
definition write_states ≝ initN 2.
+definition wr0 : write_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
+definition wr1 : write_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
+
definition write ≝ λalpha,c.
mk_TM alpha write_states
(λp.let 〈q,a〉 ≝ p in
- match q with
- [ O ⇒ 〈1,Some ? 〈c,N〉〉
- | S _ ⇒ 〈1,None ?〉 ])
- O (λx.x == 1).
+ match pi1 … q with
+ [ O ⇒ 〈wr1,Some ? 〈c,N〉〉
+ | S _ ⇒ 〈wr1,None ?〉 ])
+ wr0 (λx.x == wr1).
definition R_write ≝ λalpha,c,t1,t2.
∀ls,x,rs.t1 = midtape alpha ls x rs → t2 = midtape alpha ls c rs.
definition R_copy_step_subcase ≝
λalpha,c,RelseM,t1,t2.
- ∀ls,x,rs.t1 = midtape (FinProd … alpha FinBool) ls 〈x,true〉 rs →
- (x = c ∧
- ∀a,l1,x0,a0,l2,l3. (∀c.memb ? c l1 = true → is_marked ? c = false) →
- ls = l1@〈a0,false〉::〈x0,true〉::l2 →
- rs = 〈a,false〉::l3 →
- t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨
+ ∀a,l1,x0,a0,l2,x,l3.
+ t1 = midtape (FinProd … alpha FinBool) (l1@〈a0,false〉::〈x0,true〉::l2)
+ 〈x,true〉 (〈a,false〉::l3) →
+ (∀c.memb ? c l1 = true → is_marked ? c = false) →
+ (x = c ∧ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x,false〉::l2) 〈a,true〉 l3) ∨
(x ≠ c ∧ RelseM t1 t2).
axiom sem_copy_step_subcase :
mark;
move_r;
advance_to_mark_r;
-else nop
+else if current = null
+ then advance_mark_r;
+ move_l;
+ advance_to_mark_l
+ adv_mark_r;
+ move_r;
+ advance_to_mark_r
*)
+definition nocopy_subcase ≝
+ ifTM STape (test_char ? (λx:STape.x == 〈null,true〉))
+ (seq ? (adv_mark_r …)
+ (seq ? (move_l …)
+ (seq ? (adv_to_mark_l … (is_marked ?))
+ (seq ? (adv_mark_r …)
+ (seq ? (move_r …) (adv_to_mark_r … (is_marked ?)))))))
+ (nop ?) tc_true.
+
+definition R_nocopy_subcase ≝
+ λt1,t2.
+ ∀a,l1,x0,a0,l2,x,l3.
+ t1 = midtape STape (l1@〈a0,false〉::〈x0,true〉::l2)
+ 〈x,true〉 (〈a,false〉::l3) →
+ (∀c.memb ? c l1 = true → is_marked ? c = false) →
+ (x = null ∧
+ t2 = midtape ? (〈x,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3) ∨
+ (x ≠ null ∧ t2 = t1).
+
+axiom sem_nocopy_subcase : Realize ? nocopy_subcase R_nocopy_subcase.
+(* #intape
+cases (sem_if ? (test_char ? (λx:STape.x == 〈null,true〉)) ?????? tc_true
+ (sem_test_char ? (λx:STape.x == 〈null,true〉))
+ (sem_seq … (sem_adv_mark_r …)
+ (sem_seq … (sem_move_l …)
+ (sem_seq … (sem_adv_to_mark_l … (is_marked ?))
+ (sem_seq … (sem_adv_mark_r …)
+ (sem_seq … (sem_move_r …) (sem_adv_to_mark_r … (is_marked ?))
+ ))))) (sem_nop ?) intape)
+#k * #outc * #Hloop #HR @(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop] -Hloop
+cases HR -HR
+[| * #ta * whd in ⊢ (%→%→?); #Hta #Houtc
+ #ls #x #rs #Hintape %2 >Hintape in Hta; #Hta cases (Hta ? (refl ??)) -Hta #Hx #Hta %
+ [ lapply (\Pf Hx) @not_to_not #Hx' >Hx' %
+ | <Hta @Houtc ] ]
+@daemon
+qed. *)
+
definition copy_step ≝
ifTM ? (test_char STape (λc.is_bit (\fst c)))
(single_finalTM ? (copy_step_subcase FSUnialpha (bit false)
- (copy_step_subcase FSUnialpha (bit true) (nop ?))))
+ (copy_step_subcase FSUnialpha (bit true) nocopy_subcase)))
(nop ?)
tc_true.
definition R_copy_step_true ≝
λt1,t2.
- ∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls 〈c,true〉 rs →
- ∃x. c = bit x ∧
- (∀a,l1,c0,a0,l2,l3. (∀y.memb ? y l1 = true → is_marked ? y = false) →
- ls = l1@〈a0,false〉::〈c0,true〉::l2 →
- rs = 〈a,false〉::l3 →
- t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3).
-
+ ∀ls,c,rs. t1 = midtape STape ls 〈c,true〉 rs →
+ bit_or_null c = true ∧
+ (∀a,l1,x0,a0,l2,l3.
+ ls = (l1@〈a0,false〉::〈x0,true〉::l2) →
+ rs = (〈a,false〉::l3) →
+ no_marks l1 →
+ ((∃x. c = bit x ∧
+ t2 = midtape STape (〈bit x,false〉::l1@〈a0,true〉::〈bit x,false〉::l2) 〈a,true〉 l3) ∨
+ (c = null ∧
+ t2 = midtape ? (〈null,false〉::l1@〈a0,true〉::〈x0,false〉::l2) 〈a,true〉 l3))).
+
definition R_copy_step_false ≝
λt1,t2.
∀ls,c,rs.t1 = midtape (FinProd … FSUnialpha FinBool) ls c rs →
- is_bit (\fst c) = false ∧ t2 = t1.
+ bit_or_null (\fst c) = false ∧ t2 = t1.
-axiom sem_comp_step :
- accRealize ? copy_step (inr … (inl … (inr … 0))) R_copy_step_true R_copy_step_false.
-
-definition copy ≝ whileTM ? copy_step (inr … (inl … (inr … 0))).
+axiom sem_copy_step :
+ accRealize ? copy_step (inr … (inl … (inr … start_nop))) R_copy_step_true R_copy_step_false.
-definition R_copy ≝ λt1,t2.
- ∀ls,c,rs.t1 = midtape ? ls 〈c,true〉 rs →
- (∀l1,d,l2,l3,l4.
- 〈c,false〉::rs = l1@〈d,false〉::l2 → only_bits l1 → is_bit d = false →
- ls = l3@l4@〈c0,true〉::l5 → |l4| = |l1@[〈d,false〉]|
-
-
-
-axiom no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
(*
-l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2
- ^ ^
-
-if current (* x *) = #
- then
- else if x = 0
- then move_right; ----
- adv_to_mark_r;
- if current (* x0 *) = 0
- then advance_mark ----
- adv_to_mark_l;
- advance_mark
- else STOP
- else x = 1 (* analogo *)
-
+1) il primo carattere è marcato
+2) l'ultimo carattere è l'unico che può essere null, gli altri sono bit
+3) il terminatore non è né bit, né null
*)
-
-
-(*
- MARK NEXT TUPLE machine
- (partially axiomatized)
- marks the first character after the first bar (rightwards)
- *)
-
-definition bar_or_grid ≝ λc:STape.is_bar (\fst c) ∨ is_grid (\fst c).
-
-definition mark_next_tuple ≝
- seq ? (adv_to_mark_r ? bar_or_grid)
- (ifTM ? (test_char ? (λc:STape.is_bar (\fst c)))
- (move_right_and_mark ?) (nop ?) 1).
+definition copy0 ≝ whileTM ? copy_step (inr … (inl … (inr … start_nop))).
+
+let rec merge_config (l1,l2:list STape) ≝
+ match l1 with
+ [ nil ⇒ nil ?
+ | cons p1 l1' ⇒ match l2 with
+ [ nil ⇒ nil ?
+ | cons p2 l2' ⇒
+ let 〈c1,b1〉 ≝ p1 in let 〈c2,b2〉 ≝ p2 in
+ match c2 with
+ [ null ⇒ p1
+ | _ ⇒ p2 ] :: merge_config l1' l2' ] ].
+
+lemma merge_config_append :
+ ∀l1,l2,l3,l4.|l1| = |l2| →
+ merge_config (l1@l3) (l2@l4) = merge_config l1 l2@merge_config l3 l4.
+#l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen)
+[normalize //
+| #t1 #t2 * #c1 #b1 * #c2 #b2 #IH whd in ⊢ (??%%); >IH % ]
+qed.
-definition R_mark_next_tuple ≝
- λt1,t2.
- ∀ls,c,rs1,rs2.
- (* c non può essere un separatore ... speriamo *)
- t1 = midtape STape ls c (rs1@〈grid,false〉::rs2) →
- no_marks rs1 → no_grids rs1 → bar_or_grid c = false →
- (∃rs3,rs4,d,b.rs1 = rs3 @ 〈bar,false〉 :: rs4 ∧
- no_bars rs3 ∧
- Some ? 〈d,b〉 = option_hd ? (rs4@〈grid,false〉::rs2) ∧
- t2 = midtape STape (〈bar,false〉::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@〈grid,false〉::rs2)))
- ∨
- (no_bars rs1 ∧ t2 = midtape ? (reverse ? rs1@c::ls) 〈grid,false〉 rs2).
+definition R_copy0 ≝ λt1,t2.
+ ∀ls,c,c0,rs,l1,l3,l4.
+ t1 = midtape STape (l3@l4@〈c0,true〉::ls) 〈c,true〉 (l1@rs) →
+ no_marks l1 → no_marks (l3@l4) → |l1| = |l4| →
+ ∀l1',bv.〈c,false〉::l1 = l1'@[〈comma,bv〉] → only_bits_or_nulls l1' →
+ ∀l4',bg.l4@[〈c0,false〉] = 〈grid,bg〉::l4' → only_bits_or_nulls l4' →
+ (c = comma ∧ t2 = t1) ∨
+ (c ≠ comma ∧
+ t2 = midtape ? (reverse ? l1'@l3@〈grid,true〉::
+ merge_config l4' (reverse ? l1')@ls)
+ 〈comma,true〉 rs).
-axiom tech_split :
- ∀A:DeqSet.∀f,l.
- (∀x.memb A x l = true → f x = false) ∨
- (∃l1,c,l2.f c = true ∧ l = l1@c::l2 ∧ ∀x.memb ? x l1 = true → f x = false).
-(*#A #f #l elim l
-[ % #x normalize #Hfalse *)
-
-theorem sem_mark_next_tuple :
- Realize ? mark_next_tuple R_mark_next_tuple.
-#intape
-lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid)
- (ifTM ? (test_char ? (λc:STape.is_bar (\fst c))) (move_right_and_mark ?) (nop ?) 1) ????)
-[@sem_if [5: // |6: @sem_move_right_and_mark |7: // |*:skip]
-| //
-|||#Hif cases (Hif intape) -Hif
- #j * #outc * #Hloop * #ta * #Hleft #Hright
- @(ex_intro ?? j) @ex_intro [|% [@Hloop] ]
- -Hloop
- #ls #c #rs1 #rs2 #Hrs #Hrs1 #Hrs1' #Hc
- cases (Hleft … Hrs)
- [ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
- | * #_ #Hta cases (tech_split STape (λc.is_bar (\fst c)) rs1)
- [ #H1 lapply (Hta rs1 〈grid,false〉 rs2 (refl ??) ? ?)
- [ * #x #b #Hx whd in ⊢ (??%?); >(Hrs1' … Hx) >(H1 … Hx) %
- | %
- | -Hta #Hta cases Hright
- [ * #tb * whd in ⊢ (%→?); #Hcurrent
- @False_ind cases (Hcurrent 〈grid,false〉 ?)
- [ normalize #Hfalse destruct (Hfalse)
- | >Hta % ]
- | * #tb * whd in ⊢ (%→?); #Hcurrent
- cases (Hcurrent 〈grid,false〉 ?)
- [ #_ #Htb whd in ⊢ (%→?); #Houtc
- %2 %
- [ @H1
- | >Houtc >Htb >Hta % ]
- | >Hta % ]
- ]
- ]
- | * #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
- % @(ex_intro ?? rs3) @(ex_intro ?? rs4)
- lapply (Hta rs3 c0 (rs4@〈grid,false〉::rs2) ???)
- [ #x #Hrs3' whd in ⊢ (??%?); >Hsplit in Hrs1;>Hsplit in Hrs3;
- #Hrs3 #Hrs1 >(Hrs1 …) [| @memb_append_l1 @Hrs3'|]
- >(Hrs3 … Hrs3') @Hrs1' >Hsplit @memb_append_l1 //
- | whd in ⊢ (??%?); >Hc0 %
- | >Hsplit >associative_append % ] -Hta #Hta
- cases Hright
- [ * #tb * whd in ⊢ (%→?); #Hta'
- whd in ⊢ (%→?); #Htb
- cases (Hta' c0 ?)
- [ #_ #Htb' >Htb' in Htb; #Htb
- generalize in match Hsplit; -Hsplit
- cases rs4 in Hta;
- [ #Hta #Hsplit >(Htb … Hta)
- >(?:c0 = 〈bar,false〉)
- [ @(ex_intro ?? grid) @(ex_intro ?? false)
- % [ % [ %
- [(* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ]
- | (* Hc0 *) @daemon ]
- | #r5 #rs5 >(eq_pair_fst_snd … r5)
- #Hta #Hsplit >(Htb … Hta)
- >(?:c0 = 〈bar,false〉)
- [ @(ex_intro ?? (\fst r5)) @(ex_intro ?? (\snd r5))
- % [ % [ % [ (* Hc0, Hsplit *) @daemon | (*Hrs3*) @daemon ] | % ]
- | % ] | (* Hc0 *) @daemon ] ] | >Hta % ]
- | * #tb * whd in ⊢ (%→?); #Hta'
- whd in ⊢ (%→?); #Htb
- cases (Hta' c0 ?)
- [ #Hfalse @False_ind >Hfalse in Hc0;
- #Hc0 destruct (Hc0)
- | >Hta % ]
-]]]]
+lemma inj_append_singleton_l1 :
+ ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → l1 = l2.
+#A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
+>reverse_append >reverse_append normalize #H1 destruct
+lapply (eq_f … (reverse ?) … e0) >reverse_reverse >reverse_reverse //
qed.
-definition init_current ≝
- seq ? (adv_to_mark_l ? (is_marked ?))
- (seq ? (clear_mark ?)
- (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (seq ? (move_r ?) (mark ?)))).
-
-definition R_init_current ≝ λt1,t2.
- ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 → is_grid c = false →
- Some ? 〈c0,b0〉 = option_hd ? (reverse ? (〈c,true〉::l2)) →
- t1 = midtape STape (l1@〈c,true〉::l2@〈grid,b〉::l3) 〈c1,false〉 rs →
- t2 = midtape STape (〈grid,b〉::l3) 〈c0,true〉
- ((tail ? (reverse ? (l1@〈c,false〉::l2))@〈c1,false〉::rs)).
-
-lemma sem_init_current : Realize ? init_current R_init_current.
-#intape
-cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
- (sem_seq ????? (sem_clear_mark ?)
- (sem_seq ????? (sem_adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
- (sem_seq ????? (sem_move_r ?) (sem_mark ?)))) intape)
-#k * #outc * #Hloop #HR
-@(ex_intro ?? k) @(ex_intro ?? outc) % [@Hloop]
-cases HR -HR #ta * whd in ⊢ (%→?); #Hta
-* #tb * whd in ⊢ (%→?); #Htb
-* #tc * whd in ⊢ (%→?); #Htc
-* #td * whd in ⊢ (%→%→?); #Htd #Houtc
-#l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hc0 #Hintape
-cases (Hta … Hintape) [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
--Hta * #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%]
--Hta #Hta lapply (Htb … Hta) -Htb #Htb cases (Htc … Htb) [ >Hc -Hc * #Hc destruct (Hc) ]
--Htc * #_ #Htc lapply (Htc … (refl ??) (refl ??) ?) [@Hl2]
--Htc #Htc lapply (Htd … Htc) -Htd
->reverse_append >reverse_cons
->reverse_cons in Hc0; cases (reverse … l2)
-[ normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
- #Htd >(Houtc … Htd) %
-| * #c2 #b2 #tl2 normalize in ⊢ (%→?);
- #Hc0 #Htd >(Houtc … Htd)
- whd in ⊢ (???%); destruct (Hc0)
- >associative_append >associative_append %
-]
+lemma inj_append_singleton_l2 :
+ ∀A.∀l1,l2:list A.∀a1,a2.l1@[a1] = l2@[a2] → a1 = a2.
+#A #l1 #l2 #a1 #a2 #H lapply (eq_f … (reverse ?) … H)
+>reverse_append >reverse_append normalize #H1 destruct %
qed.
-definition match_tuple_step ≝
- ifTM ? (test_char ? (λc:STape.¬ is_grid (\fst c)))
- (single_finalTM ?
- (seq ? compare
- (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
- (nop ?)
- (seq ? mark_next_tuple
- (ifTM ? (test_char ? (λc:STape.is_grid (\fst c)))
- (mark ?) (seq ? (move_l ?) init_current) tc_true)) tc_true)))
- (nop ?) tc_true.
-
-definition R_match_tuple_step_true ≝ λt1,t2.
- ∀ls,c,l1,l2,c1,l3,l4,rs,n.
- is_bit c = true → only_bits l1 → no_grids l2 → is_bit c1 = true →
- only_bits l3 → n = |l1| → |l1| = |l3| →
- table_TM (S n) (〈c1,true〉::l3@〈comma,false〉::l4) →
- t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
- (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈comma,false〉::l4@〈grid,false〉::rs) →
- (* facciamo match *)
- (〈c,false〉::l1 = 〈c1,false〉::l3 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
- (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,true〉::l4@〈grid,false〉::rs))
- ∨
- (* non facciamo match e marchiamo la prossima tupla *)
- ((〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧
- ∃c2,l5,l6,l7.l4 = l5@〈bar,false〉::〈c2,false〉::l6@〈comma,false〉::l7 ∧
- (* condizioni su l5 l6 l7 *)
- t2 = midtape STape (〈grid,false〉::ls) 〈c,true〉
- (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,true〉::l3@〈comma,false〉::
- l5@〈bar,false〉::〈c2,true〉::l6@〈comma,false〉::l7))
- ∨
- (* non facciamo match e non c'è una prossima tupla:
- non specifichiamo condizioni sul nastro di output, perché
- non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
- (〈c,false〉::l1 ≠ 〈c1,false〉::l3 ∧ no_bars l4 ∧ current ? t2 = Some ? 〈grid,true〉)).
-
-definition R_match_tuple_step_false ≝ λt1,t2.
- ∀ls,c,rs.t1 = midtape STape ls c rs → is_grid (\fst c) = true ∧ t2 = t1.
-
-include alias "basics/logic.ma".
-
-(*
-lemma eq_f4: ∀A1,A2,A3,A4,B.∀f:A1 → A2 →A3 →A4 →B.
- ∀x1,x2,x3,x4,y1,y2,y3,y4. x1 = y1 → x2 = y2 →x3=y3 →x4 = y4 →
- f x1 x2 x3 x4 = f y1 y2 y3 y4.
-//
-qed-. *)
-
-lemma some_option_hd: ∀A.∀l:list A.∀a.∃b.
- Some ? b = option_hd ? (l@[a]) .
-#A #l #a cases l normalize /2/
+axiom length_reverse : ∀A,l.|reverse A l| = |l|.
+
+lemma wsem_copy0 : WRealize ? copy0 R_copy0.
+#intape #k #outc #Hloop
+lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop
+* #ta * #Hstar @(star_ind_l ??????? Hstar)
+[ #tb whd in ⊢ (%→?); #Hleft
+ #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
+ #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits
+ cases (Hleft … Htb) -Hleft #Hc #Houtc % %
+ [ generalize in match Hl1bits; -Hl1bits cases l1' in Hl1;
+ [ normalize #Hl1 #c1 destruct (Hl1) %
+ | * #c' #b' #l0 #Heq normalize in Heq; destruct (Heq)
+ #Hl1bits lapply (Hl1bits 〈c',false〉 ?) [ @memb_hd ]
+ >Hc #Hfalse destruct ]
+ | @Houtc ]
+| #tb #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd
+ lapply (Hind Htd) -Hind #Hind
+ #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
+ #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits %2
+ cases (Htc … Htb) -Htc #Hcbitnull #Htc
+ % [ % #Hc' >Hc' in Hcbitnull; normalize #Hfalse destruct (Hfalse) ]
+ cut (|l1| = |reverse ? l4|) [@daemon] #Hlen1
+ @(list_cases_2 … Hlen1)
+ [ (* case l1 = [] is discriminated because l1 contains at least comma *)
+ #Hl1nil @False_ind >Hl1nil in Hl1; cases l1' normalize
+ [ #Hl1 destruct normalize in Hcbitnull; destruct (Hcbitnull)
+ | #p0 #l0 normalize #Hfalse destruct (Hfalse) cases l0 in e0;
+ [ normalize #Hfalse1 destruct (Hfalse1)
+ | #p0' #l0' normalize #Hfalse1 destruct (Hfalse1) ] ]
+ | (* case c::l1 = c::a::l1'' *)
+ * #a #ba * #a0 #ba0 #l1'' #l4'' #Hl1cons #Hl4cons
+ lapply (eq_f ?? (reverse ?) ?? Hl4cons) >reverse_reverse >reverse_cons -Hl4cons #Hl4cons
+ cut (ba = false)
+ [ >Hl1cons in Hl1nomarks; #Hl1nomarks lapply (Hl1nomarks 〈a,ba〉 ?)
+ [ @memb_hd | normalize // ] ] #Hba
+ cut (ba0 = false)
+ [ >Hl4cons in Hl3l4nomarks; #Hl3l4nomarks lapply (Hl3l4nomarks 〈a0,ba0〉 ?)
+ [ @memb_append_l2 @memb_append_l2 @memb_hd | normalize // ] ] #Hba0
+ >Hba0 in Hl4cons; >Hba in Hl1cons; -Hba0 -Hba #Hl1cons #Hl4cons
+ >Hl4cons in Htc; >Hl1cons #Htc
+ lapply (Htc a (l3@reverse ? l4'') c0 a0 ls (l1''@rs) ? (refl ??) ?)
+ [ #x #Hx @Hl3l4nomarks >Hl4cons <associative_append
+ @memb_append_l1 @Hx
+ | >associative_append >associative_append %
+ | -Htc
+ cut (∃la.l1' = 〈c,false〉::la)
+ [ >Hl1cons in Hl1; cases l1'
+ [normalize #Hfalse destruct (Hfalse)
+ | #p #la normalize #Hla destruct (Hla) @(ex_intro ?? la) % ] ]
+ * #la #Hla
+ cut (∃lb.l4' = lb@[〈c0,false〉])
+ [ >Hl4cons in Hl4;
+ @(list_elim_left … l4')
+ [ #Heq lapply (eq_f … (length ?) … Heq)
+ >length_append >length_append
+ >commutative_plus normalize >commutative_plus normalize
+ #Hfalse destruct
+ | #a1 #tl #_ #Heq
+ >(inj_append_singleton_l2 ? (reverse ? l4''@[〈a0,false〉]) (〈grid,bg〉::tl) 〈c0,false〉 a1 Heq)
+ @ex_intro //
+ ] ] * #lb #Hlb
+ cut (|lb| = |reverse ? la|)
+ [ >Hla in Hl1; >Hlb in Hl4; #Hl4 #Hl1
+ >(?:l1 = la@[〈comma,bv〉]) in Hlen;
+ [|normalize in Hl1; destruct (Hl1) %]
+ >(?:l4 = 〈grid,bg〉::lb)
+ [|@(inj_append_singleton_l1 ?? (〈grid,bg〉::lb) ?? Hl4) ]
+ >length_append >commutative_plus >length_reverse
+ normalize #Hlalb destruct (Hlalb) //
+ ] #Hlen2
+ *
+ (* by hyp on the first iteration step,
+ we consider whether c = bit x or c = null *)
+ (* c = bit x *)
+ [ * #x * #Hx #Htc
+ lapply (Hind (〈bit x,false〉::ls) a a0 rs l1''
+ (〈bit x,false〉::l3) (reverse ? l4'') ????)
+ [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus
+ normalize #Hlen destruct (Hlen) //
+ | #x0 #Hx0 cases (orb_true_l … Hx0)
+ [ #Hx0eq >(\P Hx0eq) %
+ | -Hx0 #Hx0 @Hl3l4nomarks >Hl4cons
+ <associative_append @memb_append_l1 // ]
+ | #x0 #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
+ | >Htc >associative_append %
+ | -Hind
+ <Hl1cons <Hl4cons #Hind lapply (Hind la bv ?? lb bg ??)
+ [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
+ | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
+ @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
+ | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
+ | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
+ destruct (Hl1) // ] -Hind
+ (* by IH, we proceed by cases, whether a = comma
+ (consequently several lists = []) or not *)
+ *
+ [ * #Ha #Houtc1
+(* cut (l1 = [〈a,false〉])
+ [ cases l1'' in Hl1cons; // #y #ly #Hly
+ >Hly in Hl1; cases l1' in Hl1bits;
+ [ #_ normalize #Hfalse destruct (Hfalse)
+ | #p #lp #Hl1bits normalize #Heq destruct (Heq)
+ @False_ind lapply (Hl1bits 〈a,false〉 ?)
+ [ cases lp in e0;
+ [ normalize #Hfalse destruct (Hfalse)
+ | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
+ @memb_cons @memb_hd ]
+ | >Ha normalize #Hfalse destruct (Hfalse) ]
+ ]
+ ] #Hl1a
+ cut (l4 = [〈a0,false〉])
+ [ generalize in match Hl4bits; cases l4' in Hl4;
+ [ >Hl4cons #Hfalse #_
+ lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
+ cases (reverse ? l4'') normalize
+ [ #Hfalse1 | #p0 #lp0 #Hfalse1 ] destruct (Hfalse1)
+ | #p #lp
+
+ cases l4'' in Hl4cons; // #y #ly #Hly
+ >Hly in Hl4; cases l4' in Hl4bits;
+ [ #_ >reverse_cons #Hfalse
+ lapply (inj_append_singleton_l1 ?? [] ?? Hfalse)
+ -Hfalse cases ly normalize
+ [ #Hfalse | #p #Hp #Hfalse ] destruct (Hfalse)
+
+ | #p #lp #Hl1bits normalize #Heq destruct (Heq)
+ @False_ind lapply (Hl1bits 〈a,false〉 ?)
+ [ cases lp in e0;
+ [ normalize #Hfalse destruct (Hfalse)
+ | #p0 #lp0 normalize in ⊢ (%→?); #Heq destruct (Heq)
+ @memb_cons @memb_hd ]
+ | >Ha normalize #Hfalse destruct (Hfalse) ]
+ ]
+ ] #Hl1a
+
+ >Hla normalize #Hl1 destruct (Hl1) lapply (inj_append_ @False_ind
+
+ cut (l1'' = [] ∧ l4'' = [])
+ [ % [ >Hla in Hl1; normalize #Hl1 destruct (Hl1)
+
+ cases l1'' in Hl1bits;
+
+ [ #_ normalize #H *)
+ cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
+ [ @daemon ] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
+ >Hl1cons in Hl1; >Hla
+ >Houtc1 >Htc #Hl1
+ >Hl4cons in Hl4; >Hlb #Hl4
+ >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hx
+ cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
+ normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
+ | * #Ha #Houtc1 >Houtc1 @eq_f3 //
+ >Hla >reverse_cons >associative_append @eq_f
+ >Hx whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
+ >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
+ ]
+ ]
+ | (* c = null *)
+ * #Hc #Htc
+ lapply (Hind (〈c0,false〉::ls) a a0 rs l1'' (〈null,false〉::l3) (reverse ? l4'') ????)
+ [ >Hl1cons in Hlen; >Hl4cons >length_append >commutative_plus normalize
+ #Hlen destruct (Hlen) @e0
+ | #x0 #Hx0 cases (memb_append STape ? [〈null,false〉] (l3@reverse ? l4'') … Hx0) -Hx0 #Hx0
+ [ >(memb_single … Hx0) %
+ | @Hl3l4nomarks cases (memb_append … Hx0) -Hx0 #Hx0
+ [ @memb_append_l1 //
+ | @memb_append_l2 >Hl4cons @memb_append_l1 // ]
+ ]
+ | >Hl1cons #x' #Hx0 @Hl1nomarks >Hl1cons @memb_cons //
+ | >Htc @eq_f3 // >associative_append % ] -Hind <Hl1cons <Hl4cons #Hind
+ lapply (Hind la bv ?? lb bg ??)
+ [ #x0 #Hx0 @Hl4bits >Hlb @memb_append_l1 //
+ | >Hlb in Hl4; normalize in ⊢ (%→?); #Hl4
+ @(inj_append_singleton_l1 ? l4 (〈grid,bg〉::lb) … Hl4)
+ | #x0 #Hx0 @Hl1bits >Hla @memb_cons //
+ | >Hla in Hl1; normalize in ⊢ (%→?); #Hl1
+ destruct (Hl1) // ] -Hind *
+ (* by IH, we proceed by cases, whether a = comma
+ (consequently several lists = []) or not *)
+ [ * #Ha #Houtc1 >Hl1cons in Hl1; >Hla
+ >Houtc1 >Htc #Hl1
+ >Hl4cons in Hl4; >Hlb #Hl4
+ cut (la = [] ∧ lb = [] ∧ l1'' = [] ∧ l4'' = [])
+ [@daemon] * * * #Hla1 #Hlb1 #Hl1nil #Hl4nil
+ >Hla1 >Hlb1 >Hl1nil >Hl4nil >Hc
+ cut (a0 = grid) [ @daemon ] #Ha0 <Ha <Ha0
+ normalize in ⊢ (??(??%?%)(??%?%)); >associative_append %
+ | * #Ha #Houtc1 >Houtc1 @eq_f3 //
+ >Hla >reverse_cons >associative_append @eq_f
+ >Hc whd in ⊢ (??%?); @eq_f whd in ⊢ (???%); @eq_f @eq_f
+ >Hlb >append_cons @eq_f2 // >(merge_config_append … Hlen2) %
+ ]
+ ]
+]]]
qed.
-lemma bit_not_grid: ∀d. is_bit d = true → is_grid d = false.
-* // normalize #H destruct
+definition merge_char ≝ λc1,c2.
+ match c2 with
+ [ null ⇒ c1
+ | _ ⇒ c2 ].
+
+lemma merge_cons :
+ ∀c1,c2,conf1,conf2.
+ merge_config (〈c1,false〉::conf1) (〈c2,false〉::conf2) =
+ 〈merge_char c1 c2,false〉::merge_config conf1 conf2.
+#c1 #c2 #conf1 #conf2 normalize @eq_f2 //
+cases c2 /2/
qed.
-lemma bit_not_bar: ∀d. is_bit d = true → is_bar d = false.
-* // normalize #H destruct
+lemma merge_config_c_nil :
+ ∀c.merge_config c [] = [].
+#c cases c normalize //
qed.
-axiom sem_match_tuple_step:
- accRealize ? match_tuple_step (inr … (inl … (inr … 0)))
- R_match_tuple_step_true R_match_tuple_step_false.
-(* @(acc_sem_if_app … (sem_test_char ? (λc:STape.¬ is_grid (\fst c))) …
- (sem_seq … sem_compare
- (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
- (sem_nop …)
- (sem_seq … sem_mark_next_tuple
- (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
- (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
- (sem_nop ?) …)
-[(* is_grid: termination case *)
- 2:#t1 #t2 #t3 whd in ⊢ (%→?); #H #H1 whd #ls #c #rs #Ht1
- cases (H c ?) [2: >Ht1 %] #Hgrid #Heq %
- [@injective_notb @Hgrid | <Heq @H1]
-|#tapea #tapeout #tapeb whd in ⊢ (%→?); #Htapea
- * #tapec * #Hcompare #Hor
- #ls #c #l1 #l2 #c1 #l3 #l4 #rs #n #Hc #Hl1 #Hl2 #Hc1 #Hl3 #eqn
- #eqlen #Htable #Htapea1 cases (Htapea 〈c,true〉 ?) >Htapea1 [2:%]
- #notgridc -Htapea -Htapea1 -tapea #Htapeb
- cases (Hcompare … Htapeb) -Hcompare -Htapeb * #_ #_ #Hcompare
- cases (Hcompare c c1 l1 l3 (l2@[〈bar,false〉]) (l4@〈grid,false〉::rs) eqlen … (refl …) Hc ?)
- -Hcompare
- [* #Htemp destruct (Htemp) #Htapec %1 % [%]
- >Htapec in Hor; -Htapec *
- [2: * #t3 * whd in ⊢ (%→?); #H @False_ind
- cases (H … (refl …)) whd in ⊢ ((??%?)→?); #H destruct (H)
- |* #taped * whd in ⊢ (%→?); #Htaped cases (Htaped ? (refl …)) -Htaped *
- #Htaped whd in ⊢ (%→?); #Htapeout >Htapeout >Htaped >associative_append
- %
- ]
- |* #la * #c' * #d' * #lb * #lc * * * #H1 #H2 #H3 #Htapec
- cut (〈c,false〉::l1 ≠ 〈c1,false〉::l3)
- [>H2 >H3 elim la
- [@(not_to_not …H1) normalize #H destruct %
- |#x #tl @not_to_not normalize #H destruct //
- ]
- ] #Hnoteq %2
- cut (is_bit d' = true)
- [cases la in H3;
- [normalize in ⊢ (%→?); #H destruct //
- |#x #tl #H @(Hl3 〈d',false〉)
- normalize in H; destruct @memb_append_l2 @memb_hd
- ]
- ] #Hd'
- >Htapec in Hor; -Htapec *
- [* #taped * whd in ⊢ (%→?); #H @False_ind
- cases (H … (refl …)) >Hd' #Htemp destruct (Htemp)
- |* #taped * whd in ⊢ (%→?); #H cases (H … (refl …)) -H #_
- #Htaped * #tapee * whd in ⊢ (%→?); #Htapee
- <(associative_append ? lc (〈comma,false〉::l4)) in Htaped; #Htaped
- lapply (Htapee … Htaped ???) -Htaped -Htapee
- [whd in ⊢ (??%?); >(bit_not_grid … Hd') >(bit_not_bar … Hd') %
- |#x #Hx cases (memb_append … Hx)
- [-Hx #Hx @bit_not_grid @Hl3 cases la in H3; normalize
- [#H3 destruct (H3) @Hx | #y #tl #H3 destruct (H3)
- @memb_append_l2 @memb_cons @Hx ]
- |-Hx #Hx @(no_grids_in_table … Htable)
- @memb_cons @memb_append_l2 @Hx
- ]
- |@daemon (* TODO *)
- |*
- [* #rs3 * * (* we proceed by cases on rs4 *)
- [* #d * #b * * * #Heq1 #Hnobars
- whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
- #Htapee *
- [* #tapef * whd in ⊢ (%→?); >Htapee -Htapee #Htapef
- cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
- whd in ⊢ (%→?); #H lapply (H … ???? (refl …)) #Htapeout
- %1 %
- [ //| @daemon]
- | >Htapeout %
- ]
- |* #tapef * whd in ⊢ (%→?); >Htapee -Htapee #Htapef
- cases (Htapef … (refl …)) whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
- ]
- |* #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
- cut (is_grid d2 = false) [@daemon (* ??? *)] #Hd2
- whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
- [* #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp)
- |* #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) #_ -Htapef #Htapef
- * #tapeg >Htapef -Htapef * whd in ⊢ (%→?);
- #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
- >Htapeg -Htapeg whd in ⊢ (%→?); #Htapeout
- %1 cases (some_option_hd ? (reverse ? (reverse ? la)) 〈c',true〉)
- * #c00 #b00 #Hoption
- lapply
- (Htapeout (reverse ? rs3 @〈d',false〉::reverse ? la@reverse ? (l2@[〈bar,false〉])@(〈grid,false〉::reverse ? lb))
- c' (reverse ? la) false ls bar (〈d2,true〉::rs3'@〈grid,false〉::rs) c00 b00 ?????) -Htapeout
- [whd in ⊢ (??(??%??)?); @eq_f3 [2:%|3: %]
- >associative_append
- generalize in match (〈c',true〉::reverse ? la@〈grid,false〉::ls); #l
- whd in ⊢ (???(???%)); >associative_append >associative_append
- %
- |@daemon
- |@daemon
- |@daemon
- |@daemon
- |@daemon
- ]
- ]
- ]
- |* #Hnobars #Htapee >Htapee -Htapee *
- [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
- whd in ⊢ (%→?); #Htapeout %2
- >(Htapeout … (refl …)) %
- [ %
- [ @daemon
- | @daemon
- ]
- | %
- ]
- |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
- cases (Htapef … (refl …)) -Htapef
- whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
- ]
- |
-
-
-
-
-
-
- ????? (refl …) Hc ?) -Hcompare
- #Hcompare
- is_bit c = true → only_bits l1 → no_grids l2 → is_bit c1 = true →
- only_bits l3 → n = |l2| → |l2| = |l3| →
- table_TM (S n) (〈c1,true〉::l3@〈comma,false〉::l4) →#x
-
- #intape
-cases
- (acc_sem_if … (sem_test_char ? (λc:STape.¬ is_grid (\fst c)))
- (sem_seq … sem_compare
- (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
- (sem_nop …)
- (sem_seq … sem_mark_next_tuple
- (sem_if … (sem_test_char ? (λc:STape.is_grid (\fst c)))
- (sem_mark ?) (sem_seq … (sem_move_l …) (sem_init_current …))))))
- (sem_nop ?) intape)
-#k * #outc * * #Hloop #H1 #H2
-@(ex_intro ?? k) @(ex_intro ?? outc) %
-[ % [@Hloop ] ] -Hloop
- *)
+axiom reverse_merge_config :
+ ∀c1,c2.|c1| = |c2| → reverse ? (merge_config c1 c2) =
+ merge_config (reverse ? c1) (reverse ? c2).
-(*
- MATCH TUPLE
+definition copy
+≝
+ seq STape (move_l …) (seq ? (adv_to_mark_l … (is_marked ?))
+ (seq ? (clear_mark …) (seq ? (adv_to_mark_r … (is_marked ?)) (clear_mark …)))).
- scrolls through the tuples in the transition table until one matching the
- current configuration is found
+(*
+ s0, s1 = caratteri di testa dello stato
+ c0 = carattere corrente del nastro oggetto
+ c1 = carattere in scrittura sul nastro oggetto
+
+ questa dimostrazione sfrutta il fatto che
+ merge_config (l0@[c0]) (l1@[c1]) = l1@[merge_char c0 c1]
+ se l0 e l1 non contengono null
*)
-definition match_tuple ≝ whileTM ? match_tuple_step (inr … (inl … (inr … 0))).
-
-definition R_match_tuple ≝ λt1,t2.
- ∀ls,c,l1,c1,l2,rs,n.
- is_bit c = true → only_bits l1 → is_bit c1 = true → n = |l1| →
- table_TM (S n) (〈c1,true〉::l2) →
- t1 = midtape STape (〈grid,false〉::ls) 〈c,true〉
- (l1@〈grid,false〉::〈c1,true〉::l2@〈grid,false〉::rs) →
- (* facciamo match *)
- (∃l3,newc,mv,l4.
- 〈c1,false〉::l2 = l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4 ∧
- t2 = midtape ? (reverse ? l1@〈c,false〉::〈grid,false〉::ls) 〈grid,false〉
- (l3@〈c,false〉::l1@〈comma,true〉::newc@〈comma,false〉::mv@l4@〈grid,false〉::rs))
- ∨
- (* non facciamo match su nessuna tupla;
- non specifichiamo condizioni sul nastro di output, perché
- non eseguiremo altre operazioni, quindi il suo formato non ci interessa *)
- (current ? t2 = Some ? 〈grid,true〉 ∧
- ∀l3,newc,mv,l4.
- 〈c1,false〉::l2 ≠ l3@〈c,false〉::l1@〈comma,false〉::newc@〈comma,false〉::mv@l4).
+definition R_copy ≝ λt1,t2.
+ ∀ls,s0,s1,c0,c1,rs,l1,l3,l4.
+ t1 = midtape STape (l3@〈grid,false〉::〈c0,false〉::l4@〈s0,true〉::ls) 〈s1,true〉 (l1@〈c1,false〉::〈comma,false〉::rs) →
+ no_marks l1 → no_marks l3 → no_marks l4 → |l1| = |l4| →
+ only_bits (l4@[〈s0,true〉]) → only_bits (〈s1,true〉::l1) →
+ bit_or_null c0 = true → bit_or_null c1 = true →
+ t2 = midtape STape (〈c1,false〉::reverse ? l1@〈s1,false〉::l3@〈grid,false〉::
+ 〈merge_char c0 c1,false〉::reverse ? l1@〈s1,false〉::ls)
+ 〈comma,false〉 rs.
+
+axiom sem_copy : Realize ? copy R_copy.
\ No newline at end of file
projects / helm.git / blobdiff