porting of move_char_c

[helm.git] / matita / matita / lib / turing / universal / marks.ma

diff --git a/matita/matita/lib/turing/universal/marks.ma b/matita/matita/lib/turing/universal/marks.ma
index 17992f95fca7cdef66e89e4f82c85abc174f338b..70000e3996690e2837ead12ca3c932cfa581ae12 100644 (file)
--- a/matita/matita/lib/turing/universal/marks.ma
+++ b/matita/matita/lib/turing/universal/marks.ma
@@ -30,17 +30,21 @@ include "turing/universal/tests.ma".
 
 definition atm_states ≝ initN 3.
 
+definition atm0 : atm_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition atm1 : atm_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition atm2 : atm_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
 definition atmr_step ≝ 
   λalpha:FinSet.λtest:alpha→bool.
   mk_TM alpha atm_states
   (λp.let 〈q,a〉 ≝ p in
    match a with
-   [ None ⇒ 〈1, None ?〉
+   [ None ⇒ 〈atm1, None ?〉
    | Some a' ⇒ 
      match test a' with
-     [ true ⇒ 〈1,None ?〉
-     | false ⇒ 〈2,Some ? 〈a',R〉〉 ]])
-  O (λx.notb (x == 0)).
+     [ true ⇒ 〈atm1,None ?〉
+     | false ⇒ 〈atm2,Some ? 〈a',R〉〉 ]])
+  atm0 (λx.notb (x == atm0)).
 
 definition Ratmr_step_true ≝ 
   λalpha,test,t1,t2.
@@ -57,43 +61,45 @@ definition Ratmr_step_false ≝
 lemma atmr_q0_q1 :
   ∀alpha,test,ls,a0,rs. test a0 = true → 
   step alpha (atmr_step alpha test)
-    (mk_config ?? 0 (midtape … ls a0 rs)) =
-  mk_config alpha (states ? (atmr_step alpha test)) 1
+    (mk_config ?? atm0 (midtape … ls a0 rs)) =
+  mk_config alpha (states ? (atmr_step alpha test)) atm1
     (midtape … ls a0 rs).
-#alpha #test #ls #a0 #ts #Htest normalize >Htest %
+#alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
+whd in match (trans … 〈?,?〉); >Htest %
 qed.
      
 lemma atmr_q0_q2 :
   ∀alpha,test,ls,a0,rs. test a0 = false → 
   step alpha (atmr_step alpha test)
-    (mk_config ?? 0 (midtape … ls a0 rs)) =
-  mk_config alpha (states ? (atmr_step alpha test)) 2
+    (mk_config ?? atm0 (midtape … ls a0 rs)) =
+  mk_config alpha (states ? (atmr_step alpha test)) atm2
     (mk_tape … (a0::ls) (option_hd ? rs) (tail ? rs)).
-#alpha #test #ls #a0 #ts #Htest normalize >Htest cases ts //
+#alpha #test #ls #a0 #ts #Htest whd in ⊢ (??%?);
+whd in match (trans … 〈?,?〉); >Htest cases ts //
 qed.
 
 lemma sem_atmr_step :
   ∀alpha,test.
   accRealize alpha (atmr_step alpha test) 
-    2 (Ratmr_step_true alpha test) (Ratmr_step_false alpha test).
+    atm2 (Ratmr_step_true alpha test) (Ratmr_step_false alpha test).
 #alpha #test *
 [ @(ex_intro ?? 2)
-  @(ex_intro ?? (mk_config ?? 1 (niltape ?))) %
-  [ % // #Hfalse destruct | #_ % // % % ]
-| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? 1 (leftof ? a al)))
-  % [ % // #Hfalse destruct | #_ % // % % ]
-| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? 1 (rightof ? a al)))
-  % [ % // #Hfalse destruct | #_ % // % % ]
+  @(ex_intro ?? (mk_config ?? atm1 (niltape ?))) %
+  [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
+| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (leftof ? a al)))
+  % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
+| #a #al @(ex_intro ?? 2) @(ex_intro ?? (mk_config ?? atm1 (rightof ? a al)))
+  % [ % // whd in ⊢ ((??%%)→?); #Hfalse destruct | #_ % // % % ]
 | #ls #c #rs @(ex_intro ?? 2)
   cases (true_or_false (test c)) #Htest
-  [ @(ex_intro ?? (mk_config ?? 1 ?))
+  [ @(ex_intro ?? (mk_config ?? atm1 ?))
     [| % 
       [ % 
         [ whd in ⊢ (??%?); >atmr_q0_q1 //
-        | #Hfalse destruct ]
+        | whd in ⊢ ((??%%)→?); #Hfalse destruct ]
       | #_ % // %2 @(ex_intro ?? c) % // ]
     ]
-  | @(ex_intro ?? (mk_config ?? 2 (mk_tape ? (c::ls) (option_hd ? rs) (tail ? rs))))
+  | @(ex_intro ?? (mk_config ?? atm2 (mk_tape ? (c::ls) (option_hd ? rs) (tail ? rs))))
     % 
     [ %
       [ whd in ⊢ (??%?); >atmr_q0_q2 //
@@ -116,7 +122,7 @@ definition R_adv_to_mark_r ≝ λalpha,test,t1,t2.
      t2 = midtape ? (reverse ? rs1@c::ls) b rs2))).
      
 definition adv_to_mark_r ≝ 
-  λalpha,test.whileTM alpha (atmr_step alpha test) 2.
+  λalpha,test.whileTM alpha (atmr_step alpha test) atm2.
 
 lemma wsem_adv_to_mark_r :
   ∀alpha,test.
@@ -188,15 +194,18 @@ qed.
  
 definition mark_states ≝ initN 2.
 
+definition ms0 : mark_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
+definition ms1 : mark_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
+
 definition mark ≝ 
   λalpha:FinSet.mk_TM (FinProd … alpha FinBool) mark_states
   (λp.let 〈q,a〉 ≝ p in
     match a with
-    [ None ⇒ 〈1,None ?〉
-    | Some a' ⇒ match q with
-      [ O ⇒ let 〈a'',b〉 ≝ a' in 〈1,Some ? 〈〈a'',true〉,N〉〉
-      | S q ⇒ 〈1,None ?〉 ] ])
-  O (λq.q == 1).
+    [ None ⇒ 〈ms1,None ?〉
+    | Some a' ⇒ match (pi1 … q) with
+      [ O ⇒ let 〈a'',b〉 ≝ a' in 〈ms1,Some ? 〈〈a'',true〉,N〉〉
+      | S q ⇒ 〈ms1,None ?〉 ] ])
+  ms0 (λq.q == ms1).
   
 definition R_mark ≝ λalpha,t1,t2.
   ∀ls,c,b,rs.
@@ -223,16 +232,18 @@ qed.
 *)
 
 definition move_states ≝ initN 2.
+definition move0 : move_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 2 (refl …)).
+definition move1 : move_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 2 (refl …)).
 
 definition move_r ≝ 
   λalpha:FinSet.mk_TM alpha move_states
   (λp.let 〈q,a〉 ≝ p in
     match a with
-    [ None ⇒ 〈1,None ?〉
-    | Some a' ⇒ match q with
-      [ O ⇒ 〈1,Some ? 〈a',R〉〉
-      | S q ⇒ 〈1,None ?〉 ] ])
-  O (λq.q == 1).
+    [ None ⇒ 〈move1,None ?〉
+    | Some a' ⇒ match (pi1 … q) with
+      [ O ⇒ 〈move1,Some ? 〈a',R〉〉
+      | S q ⇒ 〈move1,None ?〉 ] ])
+  move0 (λq.q == move1).
   
 definition R_move_r ≝ λalpha,t1,t2.
   ∀ls,c,rs.
@@ -263,11 +274,11 @@ definition move_l ≝
   λalpha:FinSet.mk_TM alpha move_states
   (λp.let 〈q,a〉 ≝ p in
     match a with
-    [ None ⇒ 〈1,None ?〉
-    | Some a' ⇒ match q with
-      [ O ⇒ 〈1,Some ? 〈a',L〉〉
-      | S q ⇒ 〈1,None ?〉 ] ])
-  O (λq.q == 1).
+    [ None ⇒ 〈move1,None ?〉
+    | Some a' ⇒ match pi1 … q with
+      [ O ⇒ 〈move1,Some ? 〈a',L〉〉
+      | S q ⇒ 〈move1,None ?〉 ] ])
+  move0 (λq.q == move1).
   
 definition R_move_l ≝ λalpha,t1,t2.
   ∀ls,c,rs.
@@ -297,18 +308,22 @@ qed.
  
 definition mrm_states ≝ initN 3.
 
+definition mrm0 : mrm_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition mrm1 : mrm_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition mrm2 : mrm_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
 definition move_right_and_mark ≝ 
   λalpha:FinSet.mk_TM (FinProd … alpha FinBool) mrm_states
   (λp.let 〈q,a〉 ≝ p in
     match a with
-    [ None ⇒ 〈2,None ?〉
-    | Some a' ⇒ match q with
-      [ O ⇒ 〈1,Some ? 〈a',R〉〉
+    [ None ⇒ 〈mrm2,None ?〉
+    | Some a' ⇒ match pi1 … q with
+      [ O ⇒ 〈mrm1,Some ? 〈a',R〉〉
       | S q ⇒ match q with
         [ O ⇒ let 〈a'',b〉 ≝ a' in
-              〈2,Some ? 〈〈a'',true〉,N〉〉
-        | S _ ⇒ 〈2,None ?〉 ] ] ])
-  O (λq.q == 2).
+              〈mrm2,Some ? 〈〈a'',true〉,N〉〉
+        | S _ ⇒ 〈mrm2,None ?〉 ] ] ])
+  mrm0 (λq.q == mrm2).
   
 definition R_move_right_and_mark ≝ λalpha,t1,t2.
   ∀ls,c,d,b,rs.
@@ -337,15 +352,19 @@ qed.
  
 definition clear_mark_states ≝ initN 3.
 
+definition clear0 : clear_mark_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition clear1 : clear_mark_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+definition claer2 : clear_mark_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 3 (refl …)).
+
 definition clear_mark ≝ 
   λalpha:FinSet.mk_TM (FinProd … alpha FinBool) clear_mark_states
   (λp.let 〈q,a〉 ≝ p in
     match a with
-    [ None ⇒ 〈1,None ?〉
-    | Some a' ⇒ match q with
-      [ O ⇒ let 〈a'',b〉 ≝ a' in 〈1,Some ? 〈〈a'',false〉,N〉〉
-      | S q ⇒ 〈1,None ?〉 ] ])
-  O (λq.q == 1).
+    [ None ⇒ 〈clear1,None ?〉
+    | Some a' ⇒ match pi1 … q with
+      [ O ⇒ let 〈a'',b〉 ≝ a' in 〈clear1,Some ? 〈〈a'',false〉,N〉〉
+      | S q ⇒ 〈clear1,None ?〉 ] ])
+  clear0 (λq.q == clear1).
   
 definition R_clear_mark ≝ λalpha,t1,t2.
   ∀ls,c,b,rs.
@@ -704,7 +723,7 @@ definition R_comp_step_false ≝
    is_marked ? c = false ∧ t2 = t1.
    
 lemma sem_comp_step : 
-  accRealize ? comp_step (inr … (inl … (inr … 0))) 
+  accRealize ? comp_step (inr … (inl … (inr … start_nop))) 
     R_comp_step_true R_comp_step_false.
 #intape
 cases (acc_sem_if … (sem_test_char ? (is_marked ?))
@@ -769,7 +788,7 @@ cases (acc_sem_if … (sem_test_char ? (is_marked ?))
 qed.
 
 definition compare ≝ 
-  whileTM ? comp_step (inr … (inl … (inr … 0))).
+  whileTM ? comp_step (inr … (inl … (inr … start_nop))).
 
 (*
 definition R_compare :=