Removed all axioms in unistep_aux

diff --git a/matita/matita/lib/turing/multi_universal/moves_2.ma b/matita/matita/lib/turing/multi_universal/moves_2.ma
index d86b89a6a00a769084280d483435b755b3b672d8..807823f747d0e42e68280e1f27f1645d5202e23e 100644 (file)
--- a/matita/matita/lib/turing/multi_universal/moves_2.ma
+++ b/matita/matita/lib/turing/multi_universal/moves_2.ma
@@ -321,6 +321,23 @@ lapply (sem_while … (sem_mte_step sig R) … Hloop) //
     | #ls1 #c1 #rs1 #H destruct (H) >reverse_cons >associative_append @IH % ] ] ]
 qed.
 
+lemma terminate_move_to_end_r :  ∀sig,t.move_to_end sig R ↓ t.
+#sig #t @(terminate_while … (sem_mte_step sig R …)) //
+cases t
+[ % #t1 * #ls * #c * #rs * #H destruct
+|2,3: #a0 #al0 % #t1 * #ls * #c * #rs * #H destruct
+| #ls #c #rs lapply c -c lapply ls -ls elim rs
+  [ #ls #c % #t1 * #ls0 * #c0 * #rs0 * #Hmid #Ht1 destruct %
+    #t2 * #ls1 * #c1 * #rs1 * normalize in ⊢ (%→?); #H destruct
+  | #r0 #rs0 #IH #ls #c % #t1 * #ls1 * #c1 * #rs1 * #Hmid #Ht1 destruct @IH
+  ]
+]
+qed.
+
+lemma sem_move_to_end_r : ∀sig. move_to_end sig R ⊨ R_move_to_end_r sig.
+#sig @WRealize_to_Realize //
+qed.
+
 definition R_move_to_end_l ≝ 
   λsig,int,outt.
   (current ? int = None ? → outt = int) ∧
@@ -340,4 +357,21 @@ lapply (sem_while … (sem_mte_step sig L) … Hloop) //
     lapply (IH Hfalse) -IH * #_ #IH %
     [ normalize in ⊢ (%→?); #H destruct (H)
     | #ls1 #c1 #rs1 #H destruct (H) >reverse_cons >associative_append @IH % ] ] ]
-qed.
\ No newline at end of file
+qed.
+
+lemma terminate_move_to_end_l :  ∀sig,t.move_to_end sig L ↓ t.
+#sig #t @(terminate_while … (sem_mte_step sig L …)) //
+cases t
+[ % #t1 * #ls * #c * #rs * #H destruct
+|2,3: #a0 #al0 % #t1 * #ls * #c * #rs * #H destruct
+| #ls elim ls
+  [ #c #rs % #t1 * #ls0 * #c0 * #rs0 * #Hmid #Ht1 destruct %
+    #t2 * #ls1 * #c1 * #rs1 * normalize in ⊢ (%→?); #H destruct
+  | #l0 #ls0 #IH #c #rs % #t1 * #ls1 * #c1 * #rs1 * #Hmid #Ht1 destruct @IH
+  ]
+]
+qed.
+
+lemma sem_move_to_end_l : ∀sig. move_to_end sig L ⊨ R_move_to_end_l sig.
+#sig @WRealize_to_Realize //
+qed.

diff --git a/matita/matita/lib/turing/multi_universal/unistep_aux.ma b/matita/matita/lib/turing/multi_universal/unistep_aux.ma
index b1f1114092836576b4ace3b161b7c07d2b8b405c..4f37564c0ea459d8aa8365813dec2084917e13e5 100644 (file)
--- a/matita/matita/lib/turing/multi_universal/unistep_aux.ma
+++ b/matita/matita/lib/turing/multi_universal/unistep_aux.ma
@@ -52,6 +52,60 @@ include "turing/multi_universal/tuples.ma".
   cfg_to_obj
 *)
 
+definition copy_char_states ≝ initN 3.
+
+definition cc0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
+definition cc1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
+
+definition trans_copy_char ≝ 
+ λsrc,dst.λsig:FinSet.λn.
+ λp:copy_char_states × (Vector (option sig) (S n)).
+ let 〈q,a〉 ≝ p in
+ match pi1 … q with
+ [ O ⇒ 〈cc1,change_vec ? (S n) 
+           (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
+           (〈nth src ? a (None ?),R〉) dst〉
+ | S _ ⇒ 〈cc1,null_action ? n〉 ].
+
+definition copy_char ≝ 
+  λsrc,dst,sig,n.
+  mk_mTM sig n copy_char_states (trans_copy_char src dst sig n) 
+    cc0 (λq.q == cc1).
+
+definition R_copy_char ≝ 
+  λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
+  outt = change_vec ?? 
+         (change_vec ?? int
+          (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
+          (tape_move_mono ? (nth dst ? int (niltape ?)) 
+           〈current ? (nth src ? int (niltape ?)), R〉) dst.
+
+lemma copy_char_q0_q1 :
+  ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n → 
+  step sig n (copy_char src dst sig n) (mk_mconfig ??? cc0 v) =
+    mk_mconfig ??? cc1 
+     (change_vec ? (S n) 
+       (change_vec ?? v
+         (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
+            (tape_move_mono ? (nth dst ? v (niltape ?)) 〈current ? (nth src ? v (niltape ?)), R〉) dst).
+#src #dst #sig #n #v #Heq #Hsrc #Hdst
+whd in ⊢ (??%?);
+<(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
+<(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
+>tape_move_multi_def @eq_f2 //
+>pmap_change >pmap_change <tape_move_multi_def
+>tape_move_null_action @eq_f2 // @eq_f2
+[ >change_vec_same %
+| >change_vec_same >change_vec_same // ]
+qed.
+
+lemma sem_copy_char:
+  ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → 
+  copy_char src dst sig n ⊨ R_copy_char src dst sig n.
+#src #dst #sig #n #Hneq #Hsrc #Hdst #int
+%{2} % [| % [ % | whd >copy_char_q0_q1 // ]]
+qed.
+
 definition obj ≝ (0:DeqNat).
 definition cfg ≝ (1:DeqNat).
 definition prg ≝ (2:DeqNat).
@@ -59,7 +113,7 @@ definition prg ≝ (2:DeqNat).
 definition obj_to_cfg ≝
   mmove cfg FSUnialpha 2 L ·
   (ifTM ?? (inject_TM ? (test_null ?) 2 obj)
-    (copy_step obj cfg FSUnialpha 2 ·
+    (copy_char obj cfg FSUnialpha 2 ·
      mmove cfg FSUnialpha 2 L ·
      mmove obj FSUnialpha 2 L) 
     (inject_TM ? (write FSUnialpha null) 2 cfg)
@@ -78,7 +132,6 @@ definition R_obj_to_cfg ≝ λt1,t2:Vector (tape FSUnialpha) 3.
          (mk_tape ? [ ] (option_hd FSUnialpha (reverse ? (null::ls))) 
            (tail ? (reverse ? (null::ls)))) cfg).
            
-axiom sem_move_to_end_l : ∀sig. move_to_end sig L ⊨ R_move_to_end_l sig.
 axiom accRealize_to_Realize :
   ∀sig,n.∀M:mTM sig n.∀Rtrue,Rfalse,acc.
   M ⊨ [ acc: Rtrue, Rfalse ] →  M ⊨ Rtrue ∪ Rfalse.
@@ -116,7 +169,7 @@ lemma sem_obj_to_cfg : obj_to_cfg ⊨  R_obj_to_cfg.
    (sem_seq ??????
     (sem_if ??????????
      (sem_test_null_multi ?? obj ?)
-      (sem_seq ?????? (accRealize_to_Realize … (sem_copy_step …))
+      (sem_seq ?????? (sem_copy_char …)
        (sem_seq ?????? (sem_move_multi ? 2 cfg L ?)
         (sem_move_multi ? 2 obj L ?)))
       (sem_inject ???? cfg ? (sem_write FSUnialpha null)))
@@ -126,8 +179,11 @@ lemma sem_obj_to_cfg : obj_to_cfg ⊨  R_obj_to_cfg.
 #tc * whd in ⊢ (%→?); #Htc *
 #td * *
 [ * #te * * #Hcurtc #Hte
-  * destruct (Hte) #te * *
-  [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte
+  * destruct (Hte) #te * whd in ⊢ (%→?); #Hte
+  cut (∃x.current ? (nth obj ? tc (niltape ?)) = Some ? x)
+  [ cases (current ? (nth obj ? tc (niltape ?))) in Hcurtc;
+    [ * #H @False_ind /2/ | #x #_ %{x} % ] ] * #x #Hcurtc'
+(*  [ whd in ⊢ (%→%→?); * #x * #y * * -Hcurtc #Hcurtc1 #Hcurtc2 #Hte *)
     * #tf * whd in ⊢ (%→%→?); #Htf #Htd
     * #tg * * * whd in ⊢ (%→%→%→%→?); #Htg1 #Htg2 #Htg3 #Htb
     #c #ls #Hta1 %
@@ -136,8 +192,8 @@ lemma sem_obj_to_cfg : obj_to_cfg ⊨  R_obj_to_cfg.
       cut (tg = change_vec ?? td (mk_tape ? [ ] (None ?) (reverse ? ls@[x])) cfg)
       [ lapply (eq_vec_change_vec ??????? (Htg2 ls x [ ] ?) Htg3) //
         >Htd >nth_change_vec_neq // >Htf >nth_change_vec //
-        >Hte >nth_change_vec // >Htc >nth_change_vec // ] -Htg1 -Htg2 -Htg3 #Htg
-      destruct 
+        >Hte >Hcurtc' >nth_change_vec // >Htc >nth_change_vec // ] 
+      -Htg1 -Htg2 -Htg3 #Htg destruct 
       >change_vec_change_vec >change_vec_change_vec
       >change_vec_commute // >change_vec_change_vec
       >change_vec_commute [|@sym_not_eq //] >change_vec_change_vec
@@ -149,18 +205,12 @@ lemma sem_obj_to_cfg : obj_to_cfg ⊨  R_obj_to_cfg.
         [ #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same
         | #r1 #rs1 #Hta2 whd in match (tape_move ???); <Hta2 @change_vec_same ]
       | >tape_move_mk_tape_R [| #_ % %] >reverse_cons
-        >nth_change_vec_neq in Hcurtc1; [|@sym_not_eq //] >Hta2
+        >nth_change_vec_neq in Hcurtc'; [|@sym_not_eq //] >Hta2
         normalize in ⊢ (%→?); #H destruct (H) %
       ]
-    | #Hta2 >Htc in Hcurtc1; >nth_change_vec_neq [| @sym_not_eq //]
+    | #Hta2 >Htc in Hcurtc'; >nth_change_vec_neq [| @sym_not_eq //]
       >Hta2 #H destruct (H)
     ]
-  | * #Hcurtc0 #Hte #_ #_ #c #ls #Hta1 >Hta1 in Htc; >eq_mk_tape_rightof
-    whd in match (tape_move ???); #Htc >Htc in Hcurtc0; *
-    [ >Htc in Hcurtc; >nth_change_vec_neq [|@sym_not_eq //]
-      #Hcurtc #Hcurtc0 >Hcurtc0 in Hcurtc; * #H @False_ind @H %
-    | >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H) ]
-  ]
 | * #te * * #Hcurtc #Hte
   * whd in ⊢ (%→%→?); #Htd1 #Htd2
   * #tf * * * #Htf1 #Htf2 #Htf3 whd in ⊢ (%→?); #Htb
@@ -203,60 +253,6 @@ lemma sem_test_null_char :
   | <Houtc % ] ]
 qed.
 
-definition copy_char_states ≝ initN 3.
-
-definition cc0 : copy_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 3 (refl …)).
-definition cc1 : copy_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 3 (refl …)).
-
-definition trans_copy_char ≝ 
- λsrc,dst.λsig:FinSet.λn.
- λp:copy_char_states × (Vector (option sig) (S n)).
- let 〈q,a〉 ≝ p in
- match pi1 … q with
- [ O ⇒ 〈cc1,change_vec ? (S n) 
-           (change_vec ? (S n) (null_action ? n) (〈None ?,R〉) src)
-           (〈nth src ? a (None ?),R〉) dst〉
- | S _ ⇒ 〈cc1,null_action ? n〉 ].
-
-definition copy_char ≝ 
-  λsrc,dst,sig,n.
-  mk_mTM sig n copy_char_states (trans_copy_char src dst sig n) 
-    cc0 (λq.q == cc1).
-
-definition R_copy_char ≝ 
-  λsrc,dst,sig,n.λint,outt: Vector (tape sig) (S n).
-  outt = change_vec ?? 
-         (change_vec ?? int
-          (tape_move_mono ? (nth src ? int (niltape ?)) 〈None ?, R〉) src)
-          (tape_move_mono ? (nth dst ? int (niltape ?)) 
-           〈current ? (nth src ? int (niltape ?)), R〉) dst.
-
-lemma copy_char_q0_q1 :
-  ∀src,dst,sig,n,v.src ≠ dst → src < S n → dst < S n → 
-  step sig n (copy_char src dst sig n) (mk_mconfig ??? cc0 v) =
-    mk_mconfig ??? cc1 
-     (change_vec ? (S n) 
-       (change_vec ?? v
-         (tape_move_mono ? (nth src ? v (niltape ?)) 〈None ?, R〉) src)
-            (tape_move_mono ? (nth dst ? v (niltape ?)) 〈current ? (nth src ? v (niltape ?)), R〉) dst).
-#src #dst #sig #n #v #Heq #Hsrc #Hdst
-whd in ⊢ (??%?);
-<(change_vec_same … v dst (niltape ?)) in ⊢ (??%?);
-<(change_vec_same … v src (niltape ?)) in ⊢ (??%?);
->tape_move_multi_def @eq_f2 //
->pmap_change >pmap_change <tape_move_multi_def
->tape_move_null_action @eq_f2 // @eq_f2
-[ >change_vec_same %
-| >change_vec_same >change_vec_same // ]
-qed.
-
-lemma sem_copy_char:
-  ∀src,dst,sig,n.src ≠ dst → src < S n → dst < S n → 
-  copy_char src dst sig n ⊨ R_copy_char src dst sig n.
-#src #dst #sig #n #Hneq #Hsrc #Hdst #int
-%{2} % [| % [ % | whd >copy_char_q0_q1 // ]]
-qed.
-
 definition cfg_to_obj ≝
   mmove cfg FSUnialpha 2 L ·
   (ifTM ?? (inject_TM ? test_null_char 2 cfg)