From 1c878b2e2fcdc1d61c9604741f2c8f22e8cfee8a Mon Sep 17 00:00:00 2001
From: Wilmer Ricciotti <ricciott@cs.unibo.it>
Date: Mon, 14 May 2012 16:41:07 +0000
Subject: [PATCH] Added copy machine (mockup)

---
 matita/matita/lib/turing/universal/copy.ma   | 513 +++++++++++++++++++
 matita/matita/lib/turing/universal/tuples.ma |  34 +-
 2 files changed, 544 insertions(+), 3 deletions(-)
 create mode 100644 matita/matita/lib/turing/universal/copy.ma

diff --git a/matita/matita/lib/turing/universal/copy.ma b/matita/matita/lib/turing/universal/copy.ma
new file mode 100644
index 000000000..5775eab05
--- /dev/null
+++ b/matita/matita/lib/turing/universal/copy.ma
@@ -0,0 +1,513 @@
+(*
+    ||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_____________________________________________________________*)
+
+
+(* COMPARE BIT
+
+*)
+
+include "turing/universal/tuples.ma".
+
+definition write_states ≝ initN 2.
+
+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).
+  
+definition R_write ≝ λalpha,c,t1,t2.
+  ∀ls,x,rs.t1 = midtape alpha ls x rs → t2 = midtape alpha ls c rs.
+  
+axiom sem_write : ∀alpha,c.Realize ? (write alpha c) (R_write alpha c).
+
+definition copy_step_subcase ≝
+  λalpha,c,elseM.ifTM ? (test_char ? (λx.x == 〈c,true〉))
+    (seq (FinProd alpha FinBool) (adv_mark_r …)
+      (seq ? (move_l …)
+        (seq ? (adv_to_mark_l … (is_marked alpha))
+          (seq ? (write ? 〈c,false〉)
+            (seq ? (move_r …)
+              (seq ? (mark …)
+                (seq ? (move_r …) (adv_to_mark_r … (is_marked alpha)))))))))
+    elseM tc_true.
+
+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) ∨
+    (x ≠ c ∧ RelseM t1 t2).
+    
+axiom sem_copy_step_subcase : 
+  ∀alpha,c,elseM,RelseM.
+  Realize ? (copy_step_subcase alpha c elseM) (R_copy_step_subcase alpha c RelseM).
+    
+(*
+if current = 0,tt
+   then advance_mark_r;
+        move_l;
+        advance_to_mark_l;
+        write(0,ff)
+        move_r;
+        mark;
+        move_r;
+        advance_to_mark_r;
+else if current = 1,tt
+   then advance_mark_r;
+        move_l;
+        advance_to_mark_l;
+        write(1,ff)
+        move_r;
+        mark;
+        move_r;
+        advance_to_mark_r;
+else nop
+*)
+
+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 ?))))
+  (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).
+
+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.
+
+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))).
+
+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 *)
+
+*)
+
+
+(*
+   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 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).
+     
+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 % ]
+]]]]
+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 = |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/
+qed.
+
+lemma bit_not_grid: ∀d. is_bit 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.
+
+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
+  *)
+
+(* 
+  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 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).  
diff --git a/matita/matita/lib/turing/universal/tuples.ma b/matita/matita/lib/turing/universal/tuples.ma
index 10f5f3721..4224a155c 100644
--- a/matita/matita/lib/turing/universal/tuples.ma
+++ b/matita/matita/lib/turing/universal/tuples.ma
@@ -310,10 +310,10 @@ lemma some_option_hd: ∀A.∀l:list A.∀a.∃b.
 #A #l #a cases l normalize /2/
 qed.
 
-lemma sem_match_tuple_step: 
+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))) …
+(* @(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 …)
@@ -450,4 +450,32 @@ lemma sem_match_tuple_step:
  ]
 qed.
 
-  
\ No newline at end of file
+  *)
+
+(* 
+  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 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).  
-- 
2.39.2