X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fwhile_machine.ma;h=d5782464400c710ef4dd7ac30ba04e068834aa7a;hb=5fc2b08d86038360e588b8fff333a623964efabe;hp=7cea9ed35976ebba536bc17919ea9da44d35abc1;hpb=f464217340074c5a81e4bda91814abb0611a02e0;p=helm.git

diff --git a/matita/matita/lib/turing/while_machine.ma b/matita/matita/lib/turing/while_machine.ma
index 7cea9ed35..d57824644 100644
--- a/matita/matita/lib/turing/while_machine.ma
+++ b/matita/matita/lib/turing/while_machine.ma
@@ -61,6 +61,12 @@ lemma halt_while_acc :
 cases (halt sig M acc) %
 qed.
 
+lemma halt_while_not_acc : 
+  ∀sig,M,acc,s.s == acc = false → halt sig (whileTM sig M acc) s = halt sig M s.
+#sig #M #acc #s #neqs normalize >neqs 
+cases (halt sig M s) %
+qed.
+
 lemma step_while_acc :
  ∀sig,M,acc,c.cstate ?? c = acc → 
    step sig (whileTM sig M acc) c = initc … (ctape ?? c).
@@ -123,204 +129,50 @@ cases (loop_split ?? (λc. halt sig M (cstate ?? c)) ????? Hloop)
  ]
 qed.
 
-(* inductive move_states : Type[0] ≝ 
-| start : move_states
-| q1 : move_states
-| q2 : move_states
-| q3 : move_states
-| qacc : move_states
-| qfail : move_states.
-
-definition 
-*)
-
-definition mystates : FinSet → FinSet ≝ λalpha:FinSet.FinProd (initN 5) alpha.
-
-definition move_char ≝ 
- λalpha:FinSet.λsep:alpha.
- mk_TM alpha (mystates alpha)
- (λp.let 〈q,a〉 ≝ p in
-  let 〈q',b〉 ≝ q in
-  match a with 
-  [ None ⇒ 〈〈4,sep〉,None ?〉 
-  | Some a' ⇒ 
-  match q' with
-  [ O ⇒ (* qinit *)
-    match a' == sep with
-    [ true ⇒ 〈〈4,sep〉,None ?〉
-    | false ⇒ 〈〈1,a'〉,Some ? 〈a',L〉〉 ]
-  | S q' ⇒ match q' with
-    [ O ⇒ (* q1 *)
-      〈〈2,a'〉,Some ? 〈b,R〉〉
-    | S q' ⇒ match q' with
-      [ O ⇒ (* q2 *)
-        〈〈3,sep〉,Some ? 〈b,R〉〉
-      | S q' ⇒ match q' with
-        [ O ⇒ (* qacc *)
-          〈〈3,sep〉,None ?〉
-        | S q' ⇒ (* qfail *)
-          〈〈4,sep〉,None ?〉 ] ] ] ] ])
-  〈0,sep〉
-  (λq.let 〈q',a〉 ≝ q in q' == 3 ∨ q' == 4).
-
-definition mk_tape : 
-  ∀sig:FinSet.list sig → option sig → list sig → tape sig ≝ 
-  λsig,lt,c,rt.match c with
-  [ Some c' ⇒ midtape sig lt c' rt
-  | None ⇒ match lt with 
-    [ nil ⇒ match rt with
-      [ nil ⇒ niltape ?
-      | cons r0 rs0 ⇒ leftof ? r0 rs0 ]
-    | cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
-    
-lemma cmove_q0_q1 : 
-  ∀alpha:FinSet.∀sep,a,ls,a0,rs.
-  a0 == sep = false → 
-  step alpha (move_char alpha sep)
-    (mk_config ?? 〈0,a〉 (mk_tape … ls (Some ? a0) rs)) =
-  mk_config alpha (states ? (move_char alpha sep)) 〈1,a0〉
-    (tape_move_left alpha ls a0 rs).
-#alpha #sep #a *
-[ #a0 #rs #Ha0 whd in ⊢ (??%?); 
-  normalize in match (trans ???); >Ha0 %
-| #a1 #ls #a0 #rs #Ha0 whd in ⊢ (??%?);
-  normalize in match (trans ???); >Ha0 %
-]
-qed.
-    
-lemma cmove_q1_q2 :
-  ∀alpha:FinSet.∀sep,a,ls,a0,rs.
-  step alpha (move_char alpha sep) 
-    (mk_config ?? 〈1,a〉 (mk_tape … ls (Some ? a0) rs)) = 
-  mk_config alpha (states ? (move_char alpha sep)) 〈2,a0〉 
-    (tape_move_right alpha ls a rs).
-#alpha #sep #a #ls #a0 * //
-qed.
-
-lemma cmove_q2_q3 :
-  ∀alpha:FinSet.∀sep,a,ls,a0,rs.
-  step alpha (move_char alpha sep) 
-    (mk_config ?? 〈2,a〉 (mk_tape … ls (Some ? a0) rs)) = 
-  mk_config alpha (states ? (move_char alpha sep)) 〈3,sep〉 
-    (tape_move_right alpha ls a rs).
-#alpha #sep #a #ls #a0 * //
-qed.
-
-definition option_hd ≝ 
-  λA.λl:list A. match l with
-  [ nil ⇒ None ?
-  | cons a _ ⇒ Some ? a ].
-
-definition Rmove_char_true ≝ 
-  λalpha,sep,t1,t2.
-   ∀a,b,ls,rs. b ≠ sep → 
-    t1 = midtape alpha (a::ls) b rs → 
-    t2 = mk_tape alpha (a::b::ls) (option_hd ? rs) (tail ? rs).
-
-definition Rmove_char_false ≝ 
-  λalpha,sep,t1,t2.
-    (current alpha t1 = None alpha → t2 = t1) ∧
-    (current alpha t1 = Some alpha sep → t2 = t1).
-    
-lemma loop_S_true : 
-  ∀A,n,f,p,a.  p a = true → 
-  loop A (S n) f p a = Some ? a. /2/
-qed.
-
-lemma loop_S_false : 
-  ∀A,n,f,p,a.  p a = false → 
-  loop A (S n) f p a = loop A n f p (f a).
-normalize #A #n #f #p #a #Hpa >Hpa %
-qed.
-
-notation < "𝐅" non associative with precedence 90 
- for @{'bigF}.
-notation < "𝐃" non associative with precedence 90 
- for @{'bigD}.
- 
-interpretation "FinSet" 'bigF = (mk_FinSet ???).
-interpretation "DeqSet" 'bigD = (mk_DeqSet ???).
-
-lemma trans_init_sep: 
-  ∀alpha,sep,x.
-  trans ? (move_char alpha sep) 〈〈0,x〉,Some ? sep〉 = 〈〈4,sep〉,None ?〉.
-#alpha #sep #x normalize >(\b ?) //
-qed.
- 
-lemma trans_init_not_sep: 
-  ∀alpha,sep,x,y.y == sep = false → 
-  trans ? (move_char alpha sep) 〈〈0,x〉,Some ? y〉 = 〈〈1,y〉,Some ? 〈y,L〉〉.
-#alpha #sep #x #y #H1 normalize >H1 //
-qed.
-
-lemma sem_move_char :
-  ∀alpha,sep.
-  accRealize alpha (move_char alpha sep) 
-    〈3,sep〉 (Rmove_char_true alpha sep) (Rmove_char_false alpha sep).
-#alpha #sep *
-[@(ex_intro ?? 2)  
-  @(ex_intro … (mk_config ?? 〈4,sep〉 (niltape ?)))
-  % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 whd % #_ % ]
-|#l0 #lt0 @(ex_intro ?? 2)  
-  @(ex_intro … (mk_config ?? 〈4,sep〉 (leftof ? l0 lt0)))
-  % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 whd % #_ % ]
-|#r0 #rt0 @(ex_intro ?? 2)  
-  @(ex_intro … (mk_config ?? 〈4,sep〉 (rightof ? r0 rt0)))
-  % [% [whd in ⊢ (??%?);% |#Hfalse destruct ] |#H1 whd % #_ % ]
-| #lt #c #rt cases (true_or_false (c == sep)) #Hc
-  [ @(ex_intro ?? 2) 
-    @(ex_intro ?? (mk_config ?? 〈4,sep〉 (midtape ? lt c rt)))
-    % 
-    [% 
-      [ >(\P Hc) >loop_S_false //
-       >loop_S_true 
-       [ @eq_f whd in ⊢ (??%?); >trans_init_sep %
-       |>(\P Hc) whd in ⊢(??(???(???%))?);
-         >trans_init_sep % ]
-     | #Hfalse destruct
+theorem terminate_while: ∀sig,M,acc,Rtrue,Rfalse,t.
+  halt sig M acc = true →
+  accRealize sig M acc Rtrue Rfalse → 
+  WF ? (inv … Rtrue) t → Terminate sig (whileTM sig M acc) t.
+#sig #M #acc #Rtrue #Rfalse #t #Hacctrue #HM #HWF elim HWF
+#t1 #H #Hind cases (HM … t1) #i * #outc * * #Hloop
+#Htrue #Hfalse cases (true_or_false (cstate … outc == acc)) #Hcase
+  [cases (Hind ? (Htrue … (\P Hcase))) #iwhile * #outcfinal
+   #Hloopwhile @(ex_intro … (i+iwhile)) 
+   @(ex_intro … outcfinal) @(loop_merge … outc … Hloopwhile)
+    [@(λc.halt sig M (cstate … c))
+    |* #s0 #t0 normalize cases (s0 == acc) normalize
+     [ cases (halt sig M s0) // 
+     | cases (halt sig M s0) normalize //
      ]
-    |#_ % #_ % ]
-  | @(ex_intro ?? 4)
-    cases lt
-    [ @ex_intro
-      [|%
-        [ %
-          [ >loop_S_false //
-            >cmove_q0_q1 //
-          | normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
-          ]
-        | normalize in ⊢ (%→?); #_ %
-          [ normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
-          | normalize in ⊢ (%→?); #Hfalse destruct (Hfalse)
-            @False_ind @(absurd ?? (\Pf Hc)) %
-          ]
-        ]
-      ]
-    | #l0 #lt @ex_intro
-      [| %
-       [ %
-         [ >loop_S_false //
-           >cmove_q0_q1 //
-         | #_ #a #b #ls #rs #Hb #Htape
-           destruct (Htape)
-           >cmove_q1_q2
-           >cmove_q2_q3
-           cases rs normalize //
-         ]
-       | normalize in ⊢ (% → ?); * #Hfalse
-         @False_ind /2/
-       ]
+    |@(loop_lift ?? i (λc.c) ? 
+                (step ? (whileTM ? M acc)) ? 
+                (λc.halt sig M (cstate ?? c)) ?? 
+                ?? Hloop)
+     [ #x %
+     | * #s #t #Hx whd in ⊢ (??%%); >while_trans_false
+       [%
+       |% #Hfalse <Hfalse in Hacctrue; >Hx #H0 destruct ]
      ]
+   |@step_while_acc @(\P Hcase)
+   |>(\P Hcase) @halt_while_acc
+   ]
+ |@(ex_intro … i) @(ex_intro … outc)
+  @(loop_lift_acc ?? i (λc.c) ?????? (λc.cstate ?? c == acc) ???? Hloop)
+   [#x #Hx >(\P Hx) //
+   |#x @halt_while_not_acc
+   |#x #H whd in ⊢ (??%%); >while_trans_false [%]
+    % #eqx >eqx in H; >Hacctrue #H destruct
+   |@Hcase
    ]
  ]
-]
 qed.
 
-definition R_while_cmove :
-  λalpha,sep,t1,t2.
-   ∀a,b,ls,rs. b ≠ sep → memb ? sep rs = false → 
-    t1 = midtape alpha (a::ls) b (rs@sep::rs') → 
-    t2 = midtape alpha (a::rev ? rs@b::ls) sep rs'.
+(*
+axiom terminate_while: ∀sig,M,acc,Rtrue,Rfalse,t.
+  halt sig M acc = true →
+  accRealize sig M acc Rtrue Rfalse → 
+  ∃t1. Rfalse t t1 → Terminate sig (whileTM sig M acc) t.
+*)
 
 (* (*