+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.
+ 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) →
+ 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,true〉::l2 = 〈c1,true〉::l3 ∧
+ t2 = midtape ? (reverse ? l2@〈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,true〉::l2 ≠ 〈c1,true〉::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,true〉::l2 ≠ 〈c1,true〉::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.
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