X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Funiversal%2Ftuples.ma;h=9207b8582fafb0a47f0bf195a91ab512ad74a264;hb=bed400cf37906a25129907986b10f24cb499dbb4;hp=59c993adaaecccac7477ac8dfd6f3be6620d0c20;hpb=b14ce4adebec4078cf662290a7d611c1d54bf388;p=helm.git

diff --git a/matita/matita/lib/turing/universal/tuples.ma b/matita/matita/lib/turing/universal/tuples.ma
index 59c993ada..9207b8582 100644
--- a/matita/matita/lib/turing/universal/tuples.ma
+++ b/matita/matita/lib/turing/universal/tuples.ma
@@ -10,139 +10,284 @@
       V_____________________________________________________________*)
 
 
-(* COMPARE BIT
+(****************************** table of tuples *******************************)
+include "turing/universal/normalTM.ma".
 
-*)
+(* a well formed table is a list of tuples *) 
+ 
+inductive table_TM (n:nat) : list STape → Prop ≝ 
+| ttm_nil  : table_TM n [] 
+| ttm_cons : ∀t1,T.tuple_TM n t1 → table_TM n T → table_TM n (t1@T).
 
-include "turing/universal/marks.ma".
+lemma wftable: ∀n,h,l.table_TM (S n) (flatten ? (tuples_list n h l)).
+#n #h #l elim l // -l #a #tl #Hind 
+whd in match (flatten … (tuples_list …));
+@ttm_cons //
+qed.
 
-definition STape ≝ FinProd … FSUnialpha FinBool.
+(*********************** general properties of tables *************************)
+lemma no_grids_in_table: ∀n.∀l.table_TM n l → no_grids l.
+#n #l #t elim t   
+  [normalize #c #H destruct
+  |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_grids_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
+   cases (memb_append … Hx) -Hx #Hx
+   [ @(Ht1 … Hx)
+   | @(IH … Hx) ] ]
+qed.
 
-definition only_bits ≝ λl.
-  ∀c.memb STape c l = true → is_bit (\fst c) = true.
+lemma no_marks_in_table: ∀n.∀l.table_TM n l → no_marks l.
+#n #l #t elim t   
+  [normalize #c #H destruct
+  |#t1 #t2 #Ht1 #Ht2 #IH lapply (no_marks_in_tuple … Ht1) -Ht1 #Ht1 #x #Hx
+   cases (memb_append … Hx) -Hx #Hx
+   [ @(Ht1 … Hx)
+   | @(IH … Hx) ] ] 
+qed.      
 
-(*
-l0 x* a l1 x0* a0 l2 ------> l0 x a* l1 x0 a0* l2
-   ^                               ^
+axiom last_of_table: ∀n,l,b.¬ table_TM n (l@[〈bar,b〉]).
+
+(************************** matching in a table *******************************)
+inductive match_in_table (n:nat) (qin:list STape) (cin: STape) 
+                         (qout:list STape) (cout:STape) (mv:STape) 
+: list STape → Prop ≝ 
+| mit_hd : 
+   ∀tb.
+   tuple_TM n (mk_tuple qin cin qout cout mv) → 
+   match_in_table n qin cin qout cout mv 
+     (mk_tuple qin cin qout cout mv @tb)
+| mit_tl :
+   ∀qin0,cin0,qout0,cout0,mv0,tb.
+   tuple_TM n (mk_tuple qin0 cin0 qout0 cout0 mv0) → 
+   match_in_table n qin cin qout cout mv tb → 
+   match_in_table n qin cin qout cout mv  
+     (mk_tuple qin0 cin0 qout0 cout0 mv0@tb).
+
+lemma tuple_to_match:  ∀n,h,l,qin,cin,qout,cout,mv,p.
+  p = mk_tuple qin cin qout cout mv 
+  → mem ? p (tuples_list n h l) →
+  match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_list n h l)).
+#n #h #l #qin #cin #qout #cout #mv #p
+#Hp elim l 
+  [whd in ⊢ (%→?); @False_ind
+  |#p1 #tl #Hind *
+    [#H whd in match (tuples_list ???);
+     <H >Hp @mit_hd //
+    |#H whd in match (tuples_list ???); 
+     cases (is_tuple n h p1) #qin1 * #cin1 * #qout1 * #cout1 * #mv1
+     * #_ #Htuplep1 >Htuplep1 @mit_tl // @Hind //
+    ]
+  ]
+qed.
+
+axiom match_decomp: ∀n,l,qin,cin,qout,cout,mv.
+  match_in_table (S n) qin cin qout cout mv l →
+  ∃l1,l2. l = l1@(mk_tuple qin cin qout cout mv)@l2 ∧
+    (∃q.|l1| = (tuple_length (S n))*q) ∧ 
+      tuple_TM (S n) (mk_tuple qin cin qout cout mv).
+
+lemma match_to_tuples_list: ∀n,h,l,qin,cin,qout,cout,mv.
+  match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_list n h l)) → 
+    ∃p. p = mk_tuple qin cin qout cout mv ∧ mem ? p (tuples_list n h l).
+#n #h #l #qin #cin #qout #cout #mv #Hmatch 
+@(ex_intro … (mk_tuple qin cin qout cout mv)) % //
+cases (match_decomp … Hmatch) #l1 * #l2 * * #Hflat #Hlen #Htuple
+@(flatten_to_mem … Hflat … Hlen)  
+  [// 
+  |#x #memx @length_of_tuple 
+   cases (mem_map ????? memx) #t * #memt #Ht <Ht // 
+  |@(length_of_tuple … Htuple) 
+  ]
+qed.
 
-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 *)
+lemma match_to_tuple: ∀n,h,l,qin,cin,qout,cout,mv.
+  match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_list n h l)) → 
+    ∃p. tuple_encoding n h p = mk_tuple qin cin qout cout mv ∧ mem ? p l.
+#n #h #l #qin #cin #qout #cout #mv #Hmatch 
+cases (match_to_tuples_list … Hmatch)
+#p * #eqp #memb 
+cases(mem_map … (λp.tuple_encoding n h p) … memb)
+#p1 * #Hmem #H @(ex_intro … p1) % /2/
+qed.
 
-*)
+lemma match_to_trans: 
+  ∀n.∀trans: trans_source n → trans_target n.
+  ∀h,qin,cin,qout,cout,mv.
+  match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_list n h (graph_enum ?? trans))) → 
+  ∃s,t. tuple_encoding n h 〈s,t〉 = mk_tuple qin cin qout cout mv 
+    ∧ trans s = t.
+#n #trans #h #qin #cin #qout #cout #mv #Hmatch
+cases (match_to_tuple … Hmatch) -Hmatch * #s #t * #Heq #Hmem
+@(ex_intro … s) @(ex_intro … t) % // @graph_enum_correct 
+@mem_to_memb @Hmem 
+qed.
 
+lemma trans_to_match:
+  ∀n.∀h.∀trans: trans_source n → trans_target n.
+  ∀s,t,qin,cin,qout,cout,mv. trans s = t →
+  tuple_encoding n h 〈s,t〉 = mk_tuple qin cin qout cout mv →
+  match_in_table (S n) qin cin qout cout mv (flatten ? (tuples_list n h (graph_enum ?? trans))).
+#n #h #trans #inp #outp #qin #cin #qout #cout #mv #Htrans #Htuple 
+@(tuple_to_match … (refl…)) <Htuple @mem_map_forward 
+@(memb_to_mem (FinProd (trans_source n) (trans_target n)))
+@graph_enum_complete //
+qed.
 
 (*
-   MARK NEXT TUPLE machine
-   (partially axiomatized)
-   
-   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 mark_next_tuple ≝ 
-  seq ? (adv_to_mark_r ? bar_or_grid)
-     (ifTM ? (test_char ? is_bar)
-       (move_r_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)))
-    ∨
-    (memb ? bar rs1 = false ∧ 
-     t2 = midtape ? (reverse ? rs1@c::ls) grid 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).
-(*#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 ? is_bar) (mark ?) (nop ?) 1) ????)
-[@sem_if //
-| //
-|||#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
-   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 #Hta cases Hright
-         [ * #tb * whd in ⊢ (%→?); #Hcurrent
-           @False_ind cases(Hcurrent grid ?)
-           [ #Hfalse (* grid is not a bar *) @daemon
-           | >Hta % ]
-         | * #tb * whd in ⊢ (%→?); #Hcurrent
-           cases (Hcurrent grid ?)
-           [  #_ #Htb whd in ⊢ (%→?); #Houtc
-             %2 %
-             [ (* H1 *) @daemon
-             | >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
-     | >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;
-           [ >(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 ]
-           | #r5 #rs5 >(eq_pair_fst_snd … r5)
-             #Hta #Hsplit >(Htb … Hta)
-             >(?:c0 = bar)
-             [ @(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.
\ No newline at end of file
+axiom append_eq_tech1 :
+  ∀A,l1,l2,l3,l4.l1@l2 = l3@l4 → |l1| < |l3| → ∃la:list A.l1@la = l3.
+axiom append_eq_tech2 :
+  ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false → ∃la:list A.l3 = l1@a::la.
+axiom list_decompose_cases : 
+  ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → ∃la,lb:list A.l3 = la@a::lb ∨ l4 = la@a::lb.
+axiom list_decompose_l :
+  ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l4 = false → 
+  ∃la,lb.l2 = la@lb ∧ l3 = l1@a::la.*)
+  
+lemma list_decompose_r :
+  ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → memb A a l3 = false → 
+  ∃la,lb.l1 = la@lb ∧ l4 = lb@a::l2.
+#A #l1 #l2 #l3 generalize in match l1; generalize in match l2; elim l3
+  [normalize #l1 #l2 #l4 #a #H #_ @(ex_intro … []) @(ex_intro … l2) /2/
+  |#b #tl #Hind #l1 #l2 #l4 #a cases l2 
+    [normalize #Heq destruct >(\b (refl … b)) normalize #Hfalse destruct
+    |#c #tl2 whd in ⊢ ((??%%)→?); #Heq destruct #Hmema 
+     cases (Hind l1 tl2 l4 a ??)
+      [#l5 * #l6 * #eql #eql4 
+       @(ex_intro … (b::l5)) @(ex_intro … l6) % /2/
+      |@e0
+      |cases (true_or_false (memb ? a tl)) [2://]
+       #H @False_ind @(absurd ?? not_eq_true_false)
+       <Hmema @sym_eq @memb_cons //
+      ]
+    ]
+  ]
+qed. 
+         
+(*axiom list_decompose_memb :
+  ∀A,l1,l2,l3,l4,a.l1@a::l2 = l3@l4 → |l1| < |l3| → memb A a l3 = true.*)
+
+lemma table_invert_r : ∀n,t,T.
+  tuple_TM n t → table_TM n (t@T) → table_TM n T.
+#n #t #T #Htuple #Htable inversion Htable
+[ cases Htuple #qin * #cin * #qout * #cout * #mv * #_ #Ht >Ht
+  normalize #Hfalse destruct (Hfalse)
+| #t0 #T0 #Htuple0 #Htable0 #_ #Heq 
+  lapply (append_l2_injective ?????? Heq)
+  [ >(length_of_tuple … Htuple) >(length_of_tuple … Htuple0) % ]
+  -Heq #Heq destruct (Heq) // ]
+qed.
+
+lemma match_in_table_to_tuple :
+  ∀n,T,qin,cin,qout,cout,mv.
+  match_in_table n qin cin qout cout mv T → table_TM n T → 
+  tuple_TM n (mk_tuple qin cin qout cout mv).
+#n #T #qin #cin #qout #cout #mv #Hmatch elim Hmatch
+[ //
+| #qin0 #cin0 #qout0 #cout0 #mv0 #tb #Htuple #Hmatch #IH #Htable
+  @IH @(table_invert_r ???? Htable) @Htuple
+]
+qed.
+
+lemma match_in_table_append :
+  ∀n,T,qin,cin,qout,cout,mv,t.
+  tuple_TM n t → 
+  match_in_table n qin cin qout cout mv (t@T) → 
+  t = mk_tuple qin cin qout cout mv ∨ match_in_table n qin cin qout cout mv T.
+#n #T #qin #cin #qout #cout #mv #t #Ht #Hmatch inversion Hmatch
+[ #T0 #H #H1 % >(append_l1_injective … H1) //
+  >(length_of_tuple … Ht) >(length_of_tuple … H) %
+| #qin0 #cin0 #qout0 #cout0 #mv0 #T0 #H #H1 #_ #H2 %2
+  >(append_l2_injective … H2) // >(length_of_tuple … Ht) >(length_of_tuple … H) %
+]
+qed.
+
+lemma generic_match_to_match_in_table_tech : 
+  ∀n,t,T0,T1,T2.tuple_TM n t → table_TM n (T1@〈bar,false〉::T2) → 
+   t@T0 = T1@〈bar,false〉::T2 → T1 = [] ∨ ∃T3.T1 = t@T3.
+#n #t #T0 #T1 #T2 #Ht cases T1
+[ #_ #_ % %
+| normalize #c #T1c #Htable #Heq %2
+  cases Ht in Heq; #qin * #cin * #qout * #cout * #mv **********
+  #Hqin1 #Hqout1 #Hqin2 #Hqout2 #Hcin #Hcout #Hmv #Hcoutmv #Hqinlen #Hqoutlen
+  #Heqt >Heqt whd in ⊢ (??%%→?); #Ht lapply (cons_injective_r ????? Ht)
+  #Ht' cases (list_decompose_r STape … (sym_eq … Ht') ?)
+  [ #la * #lb * #HT1c #HT0 %{lb} >HT1c @(eq_f2 ??? (append ?) (c::la)) //
+    >HT0 in Ht'; >HT1c >associative_append in ⊢ (???%→?); #Ht'
+    <(append_l1_injective_r … Ht') // <(cons_injective_l ????? Ht) %
+  |@(noteq_to_eqnot ? true) @(not_to_not … not_eq_true_false) #Hbar @sym_eq 
+   cases (memb_append … Hbar) -Hbar #Hbar
+    [@(Hqin2 … Hbar) 
+    |cases (orb_true_l … Hbar) -Hbar
+      [#Hbar lapply (\P Hbar) -Hbar #Hbar destruct (Hbar) @Hcin
+      |whd in ⊢ ((??%?)→?); #Hbar cases (memb_append … Hbar) -Hbar #Hbar
+        [@(Hqout2 … Hbar)
+        |cases (orb_true_l … Hbar) -Hbar
+          [#Hbar lapply (\P Hbar) -Hbar #Hbar destruct (Hbar) @Hcout
+          |#Hbar cases (orb_true_l … Hbar) -Hbar 
+            [whd in ⊢ ((??%?)→?); #Hbar @Hbar
+            |#Hbar lapply (memb_single … Hbar) -Hbar #Hbar destruct (Hbar) @Hmv
+            ]
+          ]
+        ]
+      ]
+    ]
+  ]
+qed.
+    
+lemma generic_match_to_match_in_table :
+  ∀n,T.table_TM n T → 
+  ∀qin,cin,qout,cout,mv.|qin| = n → |qout| = n → 
+  only_bits qin → only_bits qout → 
+  bit_or_null (\fst cin) = true → bit_or_null (\fst cout) = true → 
+  bit_or_null (\fst mv) = true →  
+  ∀t1,t2.
+  T = (t1@〈bar,false〉::qin@cin::〈comma,false〉::qout@cout::〈comma,false〉::[mv])@t2 → 
+  match_in_table n qin cin qout cout mv T.
+#n #T #Htable #qin #cin #qout #cout #mv #Hlenqin #Hlenqout
+#Hqinbits #Hqoutbits #Hcin #Hcout #Hmv
+elim Htable
+[ * [ #t2 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
+    | #c0 #t1 #t2 normalize in ⊢ (%→?); #Hfalse destruct (Hfalse) ]
+| #tuple #T0 #H1 #Htable0#IH #t1 #t2 #HT cases H1 #qin0 * #cin0 * #qout0 * #cout0 * #mv0
+  * * * * * * * * * *
+  #Hqin0marks #Hqout0marks #Hqin0bits #Hqout0bits #Hcin0 #Hcout0 #Hmv0 #Hcout0mv0
+  #Hlenqin0 #Hlenqout0 #Htuple 
+  lapply (generic_match_to_match_in_table_tech n ? T0 t1 
+           (qin@cin::〈comma,false〉::qout@[cout;〈comma,false〉;mv]@t2) H1) #Htmp
+  >Htuple in H1; #H1 
+  lapply (ttm_cons … T0 H1 Htable0) <Htuple in ⊢ (%→?); >HT
+  >associative_append normalize >associative_append normalize
+  >associative_append #Htable cases (Htmp Htable ?)
+  [ #Ht1 >Htuple in HT; >Ht1 normalize in ⊢ (??%%→?);
+    >associative_append >associative_append #HT
+    cut (qin0 = qin ∧ (〈cin0,false〉 = cin ∧ (qout0 = qout ∧ 
+         (〈cout0,false〉 = cout ∧ (〈mv0,false〉 = mv ∧ T0 = t2)))))
+    [ lapply (cons_injective_r ????? HT) -HT #HT 
+      lapply (append_l1_injective … HT) [ >Hlenqin @Hlenqin0 ]
+      #Hqin % [ @Hqin ] -Hqin
+      lapply (append_l2_injective … HT) [ >Hlenqin @Hlenqin0 ] -HT #HT
+      lapply (cons_injective_l ????? HT) #Hcin % [ @Hcin ] -Hcin
+      lapply (cons_injective_r ????? HT) -HT #HT 
+      lapply (cons_injective_r ????? HT) -HT
+      >associative_append >associative_append #HT
+      lapply (append_l1_injective … HT) [ >Hlenqout @Hlenqout0 ]
+      #Hqout % [ @Hqout ] -Hqout
+      lapply (append_l2_injective … HT) [ >Hlenqout @Hlenqout0 ] -HT normalize #HT
+      lapply (cons_injective_l ????? HT) #Hcout % [ @Hcout ] -Hcout
+      lapply (cons_injective_r ????? HT) -HT #HT 
+      lapply (cons_injective_r ????? HT) -HT #HT
+      lapply (cons_injective_l ????? HT) #Hmv % [ @Hmv ] -Hmv
+      @(cons_injective_r ????? HT) ]
+    -HT * #Hqin * #Hcin * #Hqout * #Hcout * #Hmv #HT0
+    >(?:〈bar,false〉::qin0@(〈cin0,false〉::〈comma,false〉::qout0@
+        [〈cout0,false〉;〈comma,false〉;〈mv0,false〉])@T0 = tuple@T0)
+    [ >Htuple >Hqin >Hqout >Hcin >Hcout >Hmv % //
+    | >Htuple normalize >associative_append normalize >associative_append
+      normalize >associative_append % ]
+  | * #T3 #HT3 >HT3 in HT; >associative_append; >associative_append #HT
+    lapply (append_l2_injective … HT) // -HT #HT %2 //
+    @(IH T3 t2) >HT >associative_append %
+  |>HT >associative_append normalize >associative_append normalize
+   >associative_append % ]
+]
+qed.