From 0a8212f3e87b75e8ab47dc853e612a9a3e1d2544 Mon Sep 17 00:00:00 2001
From: Wilmer Ricciotti <ricciott@cs.unibo.it>
Date: Fri, 11 May 2012 17:03:27 +0000
Subject: [PATCH] Definition of the structure of the transition table of a
 simulated Turing Machine.

---
 matita/matita/lib/turing/universal/tuples.ma | 158 +++++++++++++++----
 1 file changed, 130 insertions(+), 28 deletions(-)

diff --git a/matita/matita/lib/turing/universal/tuples.ma b/matita/matita/lib/turing/universal/tuples.ma
index 3492f2e2c..2baf597e7 100644
--- a/matita/matita/lib/turing/universal/tuples.ma
+++ b/matita/matita/lib/turing/universal/tuples.ma
@@ -20,6 +20,25 @@ definition STape ≝ FinProd … FSUnialpha FinBool.
 
 definition only_bits ≝ λl.
   ∀c.memb STape c l = true → is_bit (\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.
+  
+definition tuple_TM : nat → list STape → Prop ≝ 
+ λn,t.∃qin,qout,mv,b1,b2.
+ only_bits qin ∧ only_bits qout ∧ only_bits mv ∧
+ |qin| = n ∧ |qout| = n (* ∧ |mv| = ? *) ∧ 
+ t = qin@〈comma,b1〉::qout@〈comma,b2〉::mv.
+ 
+inductive table_TM : nat → list STape → Prop ≝ 
+| ttm_nil  : ∀n.table_TM n [] 
+| ttm_cons : ∀n,t1,T,b.tuple_TM n t1 → table_TM n T → table_TM n (t1@〈bar,b〉::T).
 
 (*
 l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2
@@ -58,19 +77,19 @@ definition R_mark_next_tuple ≝
   λt1,t2.
     ∀ls,c,rs1,rs2.
     (* c non può essere un separatore ... speriamo *)
-    t1 = midtape ? ls c (rs1@〈grid,false〉::rs2) → 
-    only_bits rs1 → bar_or_grid c = false → 
+    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 ? (bar::reverse ? rs3@c::ls) 〈d,true〉 (tail ? (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,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 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 *)
      
@@ -78,39 +97,41 @@ 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 ? 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'
@@ -119,15 +140,15 @@ lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid)
          [ #_ #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 % ]
@@ -138,4 +159,85 @@ lapply (sem_seq ? (adv_to_mark_r ? bar_or_grid)
                  #Hc0 destruct (Hc0)
                | >Hta % ]
 ]]]]
-qed.
\ No newline at end of file
+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.
\ No newline at end of file
-- 
2.39.2