definition STape ≝ FinProd … FSUnialpha FinBool.
-definition only_bits ≝ λl.
- ∀c.memb STape c l = true → is_bit (\fst c) = true.
+definition only_bits_or_nulls ≝ λl.
+ ∀c.memb STape c l = true → bit_or_null (\fst c) = true.
+
+definition no_grids ≝ λl.
+ ∀c.memb STape c l = true → is_grid (\fst c) = false.
+definition no_bars ≝ λl.
+ ∀c.memb STape c l = true → is_bar (\fst c) = false.
+
+definition no_marks ≝ λl.
+ ∀c.memb STape c l = true → is_marked ? c = false.
+
+lemma bit_not_grid: ∀d. is_bit d = true → is_grid d = false.
+* // normalize #H destruct
+qed.
+
+lemma bit_or_null_not_grid: ∀d. bit_or_null d = true → is_grid d = false.
+* // normalize #H destruct
+qed.
+
+lemma bit_not_bar: ∀d. is_bit d = true → is_bar d = false.
+* // normalize #H destruct
+qed.
+
+lemma bit_or_null_not_bar: ∀d. bit_or_null d = true → is_bar d = false.
+* // normalize #H destruct
+qed.
+
+(* by definition, a tuple is not marked *)
+definition tuple_TM : nat → list STape → Prop ≝
+ λn,t.∃qin,qout,mv.
+ no_marks t ∧
+ only_bits_or_nulls qin ∧ only_bits_or_nulls qout ∧ bit_or_null mv = true ∧
+ |qin| = n ∧ |qout| = n (* ∧ |mv| = ? *) ∧
+ t = qin@〈comma,false〉::qout@〈comma,false〉::[〈mv,false〉].
+
+inductive table_TM : nat → list STape → Prop ≝
+| ttm_nil : ∀n.table_TM n []
+| ttm_cons : ∀n,t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@〈bar,false〉::T).
+
+lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
+#n #l #t elim t
+ [normalize #n #x #H destruct
+ |#m #t1 #t2 * #qin * #qout * #mv * * * * * *
+ #Hmarks #Hqin #Hqout #Hmv #_ #_ #Heq #Ht2 #Hind
+ whd >Heq #x #membx
+ cases (memb_append … membx) -membx #membx
+ [cases (memb_append … membx) -membx #membx
+ [@bit_or_null_not_grid @Hqin //
+ |cases (orb_true_l … membx) -membx #membx
+ [>(\P membx) //
+ |cases (memb_append … membx) -membx #membx
+ [@bit_or_null_not_grid @Hqout //
+ |cases (orb_true_l … membx) -membx #membx
+ [>(\P membx) //
+ |@bit_or_null_not_grid >(memb_single … membx) @Hmv
+ ]
+ ]
+ ]
+ ]
+ |cases (orb_true_l … membx) -membx #membx
+ [>(\P membx) //
+ |@Hind //
+ ]
+ ]
+ ]
+qed.
+
+lemma no_marks_in_table: ∀n.∀l.table_TM n l → no_marks l.
+#n #l #t elim t
+ [normalize #n #x #H destruct
+ |#m #t1 #t2 * #qin * #qout * #mv * * * * * *
+ #Hmarks #_ #_ #_ #_ #_ #_ #Ht2 #Hind
+ #x #Hx cases (memb_append … Hx) -Hx #Hx
+ [@Hmarks //
+ |cases (orb_true_l … Hx) -Hx #Hx
+ [>(\P Hx) //
+ |@Hind //
+ ]
+ ]
+ ]
+qed.
+
+axiom last_of_table: ∀n,l,b.¬ table_TM n (l@[〈bar,b〉]).
+
(*
l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2
^ ^
marks the first character after the first bar (rightwards)
*)
-check unialpha
-
-axiom is_bar : FinProd … myalpha FinBool → bool.
-axiom is_grid : FinProd … myalpha FinBool → bool.
-definition bar_or_grid ≝ λc.is_bar c ∨ is_grid c.
-axiom bar : FinProd … myalpha FinBool.
-axiom grid : FinProd … myalpha FinBool.
+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 ? is_bar)
- (move_r_and_mark ?) (nop ?) 1).
+ (ifTM ? (test_char ? (λc:STape.is_bar (\fst c)))
+ (move_right_and_mark ?) (nop ?) 1).
definition R_mark_next_tuple ≝
λt1,t2.
∀ls,c,rs1,rs2.
(* c non può essere un separatore ... speriamo *)
- t1 = midtape ? ls c (rs1@grid::rs2) →
- memb ? grid rs1 = false → bar_or_grid c = false →
- (∃rs3,rs4,d,b.rs1 = rs3 @ bar :: rs4 ∧
- memb ? bar rs3 = false ∧
- Some ? 〈d,b〉 = option_hd ? (rs4@grid::rs2) ∧
- t2 = midtape ? (bar::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (rs4@grid::rs2)))
+ 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)))
∨
- (memb ? bar rs1 = false ∧
- t2 = midtape ? (reverse ? rs1@c::ls) grid rs2).
+ (no_bars rs1 ∧ t2 = midtape ? (reverse ? rs1@c::ls) 〈grid,false〉 rs2).
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 c = 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 *)
Realize ? mark_next_tuple R_mark_next_tuple.
#intape
lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid)
- (ifTM ? (test_char ? is_bar) (mark ?) (nop ?) 1) ????)
-[@sem_if //
+ (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 #Hc
+ #ls #c #rs1 #rs2 #Hrs #Hrs1 #Hrs1' #Hc
cases (Hleft … Hrs)
[ * #Hfalse >Hfalse in Hc; #Htf destruct (Htf)
- | * #_ #Hta cases (tech_split ? is_bar rs1)
- [ #H1 lapply (Hta rs1 grid rs2 (refl ??) ? ?)
- [ (* Hrs1, H1 *) @daemon
- | (* bar_or_grid grid = true *) @daemon
+ | * #_ #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 ?)
- [ #Hfalse (* grid is not a bar *) @daemon
+ @False_ind cases (Hcurrent 〈grid,false〉 ?)
+ [ normalize #Hfalse destruct (Hfalse)
| >Hta % ]
| * #tb * whd in ⊢ (%→?); #Hcurrent
- cases (Hcurrent grid ?)
+ cases (Hcurrent 〈grid,false〉 ?)
[ #_ #Htb whd in ⊢ (%→?); #Houtc
%2 %
- [ (* H1 *) @daemon
+ [ @H1
| >Houtc >Htb >Hta % ]
| >Hta % ]
]
]
| * #rs3 * #c0 * #rs4 * * #Hc0 #Hsplit #Hrs3
% @(ex_intro ?? rs3) @(ex_intro ?? rs4)
- lapply (Hta rs3 c0 (rs4@grid::rs2) ???)
- [ #x #Hrs3' (* Hrs1, Hrs3, Hsplit *) @daemon
- | (* bar → bar_or_grid *) @daemon
+ 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'
[ #_ #Htb' >Htb' in Htb; #Htb
generalize in match Hsplit; -Hsplit
cases rs4 in Hta;
- [ >(eq_pair_fst_snd … grid)
- #Hta #Hsplit >(Htb … Hta)
- >(?:c0 = bar)
- [ @(ex_intro ?? (\fst grid)) @(ex_intro ?? (\snd grid))
- % [ % [ % [ (* Hsplit *) @daemon |(*Hrs3*) @daemon ] | % ] | % ]
- | (* Hc0 *) @daemon ]
+ [ #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)
+ >(?:c0 = 〈bar,false〉)
[ @(ex_intro ?? (\fst r5)) @(ex_intro ?? (\snd r5))
% [ % [ % [ (* Hc0, Hsplit *) @daemon | (*Hrs3*) @daemon ] | % ]
| % ] | (* Hc0 *) @daemon ] ] | >Hta % ]
#Hc0 destruct (Hc0)
| >Hta % ]
]]]]
-qed.
\ No newline at end of file
+qed.
+
+definition init_current_on_match ≝
+ (seq ? (move_l ?)
+ (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
+ (seq ? (move_r ?) (mark ?)))).
+
+definition R_init_current_on_match ≝ λt1,t2.
+ ∀l1,l2,c,rs. no_grids l1 → is_grid c = false →
+ t1 = midtape STape (l1@〈c,false〉::〈grid,false〉::l2) 〈grid,false〉 rs →
+ t2 = midtape STape (〈grid,false〉::l2) 〈c,true〉 ((reverse ? l1)@〈grid,false〉::rs).
+
+lemma sem_init_current_on_match :
+ Realize ? init_current_on_match R_init_current_on_match.
+#intape
+cases (sem_seq ????? (sem_move_l ?)
+ (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] -Hloop
+#l1 #l2 #c #rs #Hl1 #Hc #Hintape
+cases HR -HR #ta * whd in ⊢ (%→?); #Hta lapply (Hta … Hintape) -Hta -Hintape
+generalize in match Hl1; cases l1
+ [#Hl1 whd in ⊢ ((???(??%%%))→?); #Hta
+ * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Hta
+ [* >Hc #Htemp destruct (Htemp) ]
+ * #_ #Htc lapply (Htc [ ] 〈grid,false〉 ? (refl ??) (refl …) Hl1)
+ whd in ⊢ ((???(??%%%))→?); -Htc #Htc
+ * #td * whd in ⊢ (%→?); #Htd lapply (Htd … Htc) -Htc -Htd
+ whd in ⊢ ((???(??%%%))→?); #Htd
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Htd) -Houtc #Houtc
+ >Houtc %
+ |#d #tl #Htl whd in ⊢ ((???(??%%%))→?); #Hta
+ * #tb * whd in ⊢ (%→?); #Htb cases (Htb … Hta) -Htb
+ [* >(Htl … (memb_hd …)) #Htemp destruct (Htemp)]
+ * #Hd >append_cons #Htb lapply (Htb … (refl ??) (refl …) ?)
+ [#x #membx cases (memb_append … membx) -membx #membx
+ [@Htl @memb_cons @membx | >(memb_single … membx) @Hc]]-Htb #Htb
+ * #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htb -Htc
+ >reverse_append >associative_append whd in ⊢ ((???(??%%%))→?); #Htc
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Htc) -Houtc #Houtc
+ >Houtc >reverse_cons >associative_append %
+ ]
+qed.
+
+(*
+definition init_current_gen ≝
+ seq ? (adv_to_mark_l ? (is_marked ?))
+ (seq ? (clear_mark ?)
+ (seq ? (move_l ?)
+ (seq ? (adv_to_mark_l ? (λc:STape.is_grid (\fst c)))
+ (seq ? (move_r ?) (mark ?))))).
+
+definition R_init_current_gen ≝ λt1,t2.
+ ∀l1,c,l2,b,l3,c1,rs,c0,b0. no_marks l1 → no_grids l2 →
+ 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_gen : Realize ? init_current_gen R_init_current_gen.
+#intape
+cases (sem_seq ????? (sem_adv_to_mark_l ? (is_marked ?))
+ (sem_seq ????? (sem_clear_mark ?)
+ (sem_seq ????? (sem_move_l ?)
+ (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] -Hloop
+#l1 #c #l2 #b #l3 #c1 #rs #c0 #b0 #Hl1 #Hl2 #Hc #Hintape
+cases HR -HR #ta * whd in ⊢ (%→?); #Hta cases (Hta … Hintape) -Hta -Hintape
+ [ * #Hfalse normalize in Hfalse; destruct (Hfalse) ]
+* #_ #Hta lapply (Hta l1 〈c,true〉 ? (refl ??) ??) [@Hl1|%] -Hta #Hta
+* #tb * whd in ⊢ (%→?); #Htb lapply (Htb … Hta) -Htb -Hta #Htb
+* #tc * whd in ⊢ (%→?); #Htc lapply (Htc … Htb) -Htc -Htb
+generalize in match Hc; generalize in match Hl2; cases l2
+ [#_ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp)
+ whd in ⊢ ((???(??%%%))→?); #Htc
+ * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
+ [2: * whd in ⊢ (??%?→?); #Htemp destruct (Htemp) ]
+ * #_ #Htd >Htd in Htc; -Htd #Htd
+ * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
+ >reverse_append >reverse_cons
+ whd in ⊢ ((???(??%%%))→?); #Hte
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
+ >Houtc %
+ |#d #tl #Htl #Hc0 whd in ⊢ ((???(??%%%))→?); #Htc
+ * #td * whd in ⊢ (%→?); #Htd cases (Htd … Htc) -Htd
+ [* >(Htl … (memb_hd …)) whd in ⊢ (??%?→?); #Htemp destruct (Htemp)]
+ * #Hd #Htd lapply (Htd … (refl ??) (refl ??) ?)
+ [#x #membx @Htl @memb_cons @membx] -Htd #Htd
+ * #te * whd in ⊢ (%→?); #Hte lapply (Hte … Htd) -Htd
+ >reverse_append >reverse_cons >reverse_cons
+ >reverse_cons in Hc0; >reverse_cons cases (reverse ? tl)
+ [normalize in ⊢ (%→?); #Hc0 destruct (Hc0) #Hte
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
+ >Houtc %
+ |* #c2 #b2 #tl2 normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
+ whd in ⊢ ((???(??%%%))→?); #Hte
+ whd in ⊢ (%→?); #Houtc lapply (Houtc … Hte) -Houtc -Hte #Houtc
+ >Houtc >associative_append >associative_append >associative_append %
+ ]
+ ]
+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 %
+]
+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.
+ bit_or_null c = true → only_bits_or_nulls l1 → no_marks l1 (* → no_grids l2 *) → bit_or_null c1 = true →
+ only_bits_or_nulls l3 → n = |l1| → |l1| = |l3| →
+ table_TM (S n) (l2@〈bar,false〉::〈c1,false〉::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.l4 = l5@〈bar,false〉::〈c2,false〉::l6 ∧
+ (* condizioni su l5 l6 l7 *)
+ t2 = midtape STape (〈grid,false〉::ls) 〈c,true〉
+ (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::
+ l5@〈bar,false〉::〈c2,true〉::l6@〈grid,false〉::rs))
+ ∨
+ (* 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/
+qed.
+
+axiom tech_split2 : ∀A,l1,l2,l3,l4,x.
+ memb A x l1 = false → memb ? x l3 = false →
+ l1@x::l2 = l3@x::l4 → l1 = l3 ∧ l2 = l4.
+
+axiom injective_append : ∀A,l.injective … (λx.append A x l).
+
+lemma 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 #Hl1bars #Hl1marks #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 Hl1bars Hl3 Hl1marks … (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 (bit_or_null 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 …)) >(bit_or_null_not_grid ? 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
+ cases (Htapee … Htaped ???) -Htaped -Htapee
+ [* #rs3 * * (* we proceed by cases on rs4 *)
+ [(* rs4 is empty : the case is absurd since the tape
+ cannot end with a bar *)
+ * #d * #b * * * #Heq1 @False_ind
+ cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
+ >Hcut in Htable; >H3 >associative_append
+ normalize >Heq1 >Hcut <associative_append >Hcut
+ <associative_append #Htable @(absurd … Htable)
+ @last_of_table
+ |(* rs4 not empty *)
+ * #d2 #b2 #rs3' * #d * #b * * * #Heq1 #Hnobars
+ cut (memb STape 〈d2,b2〉 (l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::l4) = true)
+ [@memb_append_l2 @memb_cons
+ cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut
+ >Hcut >H3 >associative_append @memb_append_l2
+ @memb_cons >Heq1 @memb_append_l2 @memb_cons @memb_hd] #d2intable
+ cut (is_grid d2 = false)
+ [@(no_grids_in_table … Htable … 〈d2,b2〉 d2intable)] #Hd2
+ cut (b2 = false)
+ [@(no_marks_in_table … Htable … 〈d2,b2〉 d2intable)] #Hb2
+ >Hb2 in Heq1; #Heq1 -Hb2 -b2
+ whd in ⊢ ((???%)→?); #Htemp destruct (Htemp) #Htapee >Htapee -Htapee *
+ [(* we know current is not grid *)
+ * #tapef * whd in ⊢ (%→?); #Htapef
+ cases (Htapef … (refl …)) >Hd2 #Htemp destruct (Htemp)
+ |* #tapef * whd in ⊢ (%→?); #Htapef
+ cases (Htapef … (refl …)) #_ -Htapef #Htapef
+ * #tapeg >Htapef -Htapef *
+ (* move_l *)
+ whd in ⊢ (%→?);
+ #H lapply (H … (refl …)) whd in ⊢ (???%→?); -H #Htapeg
+ >Htapeg -Htapeg
+ (* init_current *)
+ 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 %
+ |>reverse_cons @Hoption
+ |cases la in H2;
+ [normalize in ⊢ (%→?); #Htemp destruct (Htemp)
+ @injective_notb @notgridc
+ |#x #tl normalize in ⊢ (%→?); #Htemp destruct (Htemp)
+ @bit_or_null_not_grid @(Hl1bars 〈c',false〉) @memb_append_l2 @memb_hd
+ ]
+ |cut (only_bits_or_nulls (la@(〈c',false〉::lb)))
+ [<H2 whd #c0 #Hmemb cases (orb_true_l … Hmemb)
+ [#eqc0 >(\P eqc0) @Hc |@Hl1bars]
+ |#Hl1' #x #Hx @bit_or_null_not_grid @Hl1'
+ @memb_append_l1 @daemon
+ ]
+ |@daemon
+ |>reverse_append >reverse_cons >reverse_reverse
+ >reverse_append >reverse_reverse
+ >reverse_cons >reverse_append >reverse_reverse
+ >reverse_append >reverse_cons >reverse_reverse
+ >reverse_reverse
+ #Htapeout % [@Hnoteq]
+ @(ex_intro … d2)
+ cut (∃rs32.rs3 = lc@〈comma,false〉::rs32)
+ [ (*cases (tech_split STape (λc.c == 〈bar,false〉) l4)
+ [
+ | * #l41 * * #cbar #bfalse * #l42 * * #Hbar #Hl4 #Hl41
+ @(ex_intro ?? l41) >Hl4 in Heq1; #Heq1
+
+ cut (sublist … lc l3)
+ [ #x #Hx cases la in H3;
+ [ normalize #H3 destruct (H3) @Hx
+ | #p #la' normalize #Hla' destruct (Hla')
+ @memb_append_l2 @memb_cons @Hx ] ] #Hsublist*)
+ @daemon]
+ * #rs32 #Hrs3
+ (* cut
+ (〈c1,false〉::l3@〈comma,false〉::l4= la@〈d',false〉::rs3@〈bar,false〉::〈d2,b2〉::rs3')
+ [@daemon] #Hcut *)
+ cut (l4=rs32@〈bar,false〉::〈d2,false〉::rs3')
+ [ >Hrs3 in Heq1; @daemon ] #Hl4
+ @(ex_intro … rs32) @(ex_intro … rs3') %
+ [@Hl4
+ |>Htapeout @eq_f2
+ [@daemon
+ |(*>Hrs3 *)>append_cons
+ > (?:l1@〈grid,false〉::l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::rs32@〈bar,false〉::〈d2,true〉::rs3'@〈grid,false〉::rs
+ = (l1@〈grid,false〉::l2@〈bar,false〉::〈c1,false〉::l3@〈comma,false〉::rs32@[〈bar,false〉])@〈d2,true〉::rs3'@〈grid,false〉::rs)
+ [|>associative_append normalize
+ >associative_append normalize
+ >associative_append normalize
+ >associative_append normalize
+ % ]
+ @eq_f2 [|%]
+ @(injective_append … (〈d2,false〉::rs3'))
+ >(?:(la@[〈c',false〉])@((((lb@[〈grid,false〉])@l2@[〈bar,false〉])@la)@[〈d',false〉])@rs3
+ =((la@〈c',false〉::lb)@([〈grid,false〉]@l2@[〈bar,false〉]@la@[〈d',false〉]@rs3)))
+ [|>associative_append >associative_append
+ >associative_append >associative_append >associative_append
+ >associative_append >associative_append % ]
+ <H2 normalize (* <Hrs3 *)
+ >associative_append >associative_append >associative_append
+ @eq_f normalize @eq_f >associative_append
+ >associative_append @eq_f normalize @eq_f
+ >(append_cons ? 〈d',false〉) >associative_append
+ <Heq1 >Hl4 <associative_append <append_cons
+ <H3
+ >associative_append normalize
+ >associative_append normalize %
+ ]
+ ]
+ ]
+ ]
+ ]
+ |* #Hnobars #Htapee >Htapee -Htapee *
+ [whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
+ cases (Htapef … (refl …)) -Htapef #_ #Htapef >Htapef -Htapef
+ whd in ⊢ (%→?); #Htapeout %2
+ >(Htapeout … (refl …)) %
+ [ %
+ [ @Hnoteq
+ | whd #x #Hx @Hnobars @memb_append_l2 @memb_cons //
+ ]
+ | %
+ ]
+ |whd in ⊢ (%→?); * #tapef * whd in ⊢ (%→?); #Htapef
+ cases (Htapef … (refl …)) -Htapef
+ whd in ⊢ ((??%?)→?); #Htemp destruct (Htemp)
+ ]
+ |(* no marks in table *)
+ #x #membx @(no_marks_in_table … Htable)
+ @memb_append_l2 @memb_cons
+ cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
+ >H3 >associative_append @memb_append_l2 @memb_cons @membx
+ |(* no grids in table *)
+ #x #membx @(no_grids_in_table … Htable)
+ @memb_append_l2 @memb_cons
+ cut (∀A,l1,l2.∀a:A. a::l1@l2=(a::l1)@l2) [//] #Hcut >Hcut
+ >H3 >associative_append @memb_append_l2 @memb_cons @membx
+ |whd in ⊢ (??%?); >(bit_or_null_not_grid … Hd') >(bit_or_null_not_bar … Hd') %
+ ]
+ ]
+ |#x #membx @(no_marks_in_table … Htable)
+ @memb_append_l2 @memb_cons @memb_cons @memb_append_l1 @membx
+ |#x #membx @(no_marks_in_table … Htable)
+ cases (memb_append … membx) -membx #membx
+ [@memb_append_l1 @membx | @memb_append_l2 >(memb_single … membx) @memb_hd]
+ |>associative_append %
+ ]
+ ]
+qed.
+
+
+(*
+ MATCH TUPLE
+
+ scrolls through the tuples in the transition table until one matching the
+ current configuration is found
+*)
+
+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_or_nulls l1 → is_bit c1 = true → n = |l1| →
+ table_TM (S n) (〈c1,false〉::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).
projects / helm.git / blobdiff