copy machine (multi-tape) completed

diff --git a/matita/matita/lib/turing/multi_universal/copy.ma b/matita/matita/lib/turing/multi_universal/copy.ma
index a68c2262401179f8ec64d5b8f9d2c86828ee24af..e4931202652279a6abe6745086cf2b99444d1508 100644 (file)
--- a/matita/matita/lib/turing/multi_universal/copy.ma
+++ b/matita/matita/lib/turing/multi_universal/copy.ma
@@ -14,6 +14,7 @@
 
 include "turing/turing.ma".
 include "turing/inject.ma".
+include "turing/while_multi.ma".
 
 definition copy_states ≝ initN 3.
 
@@ -57,17 +58,23 @@ definition copy_step ≝
   mk_mTM sig n copy_states (trans_copy_step src dst sig n is_sep) 
     copy0 (λq.q == copy1 ∨ q == copy2).
 
-definition R_copy_step ≝ 
+definition R_copy_step_true ≝ 
   λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n).
-  (∀x1,x2,xls,xrs.
-   nth src ? int (niltape ?) = midtape sig xls x1 (x2::xrs) →
-   (is_sep x1 = true ∧ outt = int) ∨
-   (is_sep x1 = false ∧
-    outt = change_vec ?? 
-            (change_vec ?? int (midtape sig (x1::xls) x2 xrs) src)
-            (tape_move ? (nth dst ? int (niltape ?)) (Some ? 〈x1,R〉)) dst)) ∧
-  (current ? (nth src ? int (niltape ?)) = None ? → 
-   outt = int).
+  ∃x1.
+   current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
+   is_sep x1 = false ∧
+   outt = change_vec ?? 
+            (change_vec ?? int
+              (tape_move ? (nth src ? int (niltape ?)) (Some ? 〈x1,R〉)) src)
+            (tape_move ? (nth dst ? int (niltape ?)) (Some ? 〈x1,R〉)) dst.
+
+definition R_copy_step_false ≝ 
+  λsrc,dst:nat.λsig,n,is_sep.λint,outt: Vector (tape sig) (S n).
+  (∃x1.
+   current ? (nth src ? int (niltape ?)) = Some ? x1 ∧
+   is_sep x1 = true ∧ outt = int) ∨
+   current ? (nth src ? int (niltape ?)) = None ? ∧
+   outt = int.
 
 lemma copy_q0_q2_null :
   ∀src,dst,sig,n,is_sep,v,t.src < S n → dst < S n → 
@@ -100,39 +107,180 @@ whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
 ]
 qed.
 
+lemma change_vec_commute : ∀A,n,v,a,b,i,j. i ≠ j → 
+  change_vec A n (change_vec A n v a i) b j
+  = change_vec A n (change_vec A n v b j) a i.
+#A #n #v #a #b #i #j #Hij @(eq_vec … a)
+#k #Hk cases (decidable_eq_nat k i) #Hki
+[ >Hki >nth_change_vec // >(nth_change_vec_neq ??????? (sym_not_eq … Hij))
+  >nth_change_vec //
+| cases (decidable_eq_nat k j) #Hkj
+  [ >Hkj >nth_change_vec // >(nth_change_vec_neq ??????? Hij) >nth_change_vec //
+  | >(nth_change_vec_neq ??????? (sym_not_eq … Hki))
+    >(nth_change_vec_neq ??????? (sym_not_eq … Hkj))
+    >(nth_change_vec_neq ??????? (sym_not_eq … Hki))
+    >(nth_change_vec_neq ??????? (sym_not_eq … Hkj)) //
+  ]
+]
+qed.
+
 lemma copy_q0_q1 :
-  ∀src,dst,sig,n,is_sep,v,t.src < S n → dst < S n → 
+  ∀src,dst,sig,n,is_sep,v,t.src ≠ dst → src < S n → dst < S n → 
   ∀s.current ? t = Some ? s → is_sep s = false → 
   step sig n (copy_step src dst sig n is_sep)
     (mk_mconfig ??? copy0 (change_vec ? (S n) v t src)) =
     mk_mconfig ??? copy1 
      (change_vec ? (S n) 
        (change_vec ?? v
-         (tape_move ? (nth src ? v (niltape ?)) (Some ? 〈s,R〉)) src)
+         (tape_move ? t (Some ? 〈s,R〉)) src)
        (tape_move ? (nth dst ? v (niltape ?)) (Some ? 〈s,R〉)) dst).
-#src #dst #sig #n #is_sep #v #t #Hsrc #Hdst #s #Hcurrent #Hsep
+#src #dst #sig #n #is_sep #v #t #Hneq #Hsrc #Hdst #s #Hcurrent #Hsep
 whd in ⊢ (??%?); >(eq_pair_fst_snd … (trans ????)) whd in ⊢ (??%?); @eq_f2
 [ >current_chars_change_vec // whd in match (trans ????);
   >nth_change_vec // >Hcurrent whd in ⊢ (??(???%)?); >Hsep %
 | >current_chars_change_vec // whd in match (trans ????);
   >nth_change_vec // >Hcurrent whd in ⊢ (??(???????(???%))?);
-  >Hsep whd in ⊢ (??(???????(???%))?); >pmap_change
-  (* le due change commutano *)
+  >Hsep whd in ⊢ (??(???????(???%))?); >change_vec_commute // >pmap_change
+  >change_vec_commute // @eq_f3 //
+  <(change_vec_same ?? v dst (niltape ?)) in ⊢(??%?);
+  >pmap_change @eq_f3 //
 ]
+qed.
 
 lemma sem_copy_step :
-  ∀src,dst,sig,n,is_sep.src < S n → dst < S n → 
-  copy_step src dst sig n is_sep ⊨ R_copy_step src dst sig n is_sep.
-#src #dst #sig #n #is_sep #Hsrc #Hdst #int
-<(change_vec_same ?? int src (niltape ?)) cases (nth src ? int (niltape ?))
-[ %{2} % [| % 
-  [ whd in ⊢ (??%?); >copy_q0_q2 // 
-  | % // #x1 #x2 #xls #xrs >nth_change_vec // #Hfalse destruct ] ]
-| #a #al %{2} % [| % 
-  [ whd in ⊢ (??%?); >copy_q0_q2 // 
-  | % // #x1 #x2 #xls #xrs >nth_change_vec // #Hfalse destruct ] ]
-| #a #al %{2} % [| % 
-  [ whd in ⊢ (??%?); >copy_q0_q2 // 
-  | % // #x1 #x2 #xls #xrs >nth_change_vec // #Hfalse destruct ] ]
-| #ls #c #rs %{2} % [| %
-  [  
\ No newline at end of file
+  ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst < S n → 
+  copy_step src dst sig n is_sep ⊨ 
+    [ copy1: R_copy_step_true src dst sig n is_sep, 
+             R_copy_step_false src dst sig n is_sep ].
+#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #int
+lapply (refl ? (current ? (nth src ? int (niltape ?))))
+cases (current ? (nth src ? int (niltape ?))) in ⊢ (???%→?);
+[ #Hcur <(change_vec_same … int src (niltape ?)) %{2} %
+  [| % [ % 
+    [ whd in ⊢ (??%?); >copy_q0_q2_null /2/ 
+    | normalize in ⊢ (%→?); #H destruct (H) ]
+    | #_ %2 >nth_change_vec >Hcur // % // ] ]
+| #c #Hcur cases (true_or_false (is_sep c)) #Hsep
+  [ <(change_vec_same … int src (niltape ?)) %{2} % 
+    [| % [ %
+      [ whd in ⊢ (??%?); >copy_q0_q2_sep /2/
+      | normalize in ⊢ (%→?); #H destruct (H) ]
+      | #_ % >nth_change_vec // %{c} % [ % /2/ | // ] ] ]
+  | %{2} % [| % [ %
+    [ whd in ⊢ (??%?);
+      <(change_vec_same … int src (niltape ?)) in ⊢ (??%?);
+      >Hcur in ⊢ (??%?); whd in ⊢ (??%?); >(copy_q0_q1 … Hsep) /2/
+    | #_ whd %{c} % % /2/ ]
+    | * #Hfalse @False_ind /2/ ] ] ] ]
+qed.
+
+definition copy ≝ λsrc,dst,sig,n,is_sep.
+  whileTM … (copy_step src dst sig n is_sep) copy1.
+
+definition R_copy ≝ 
+  λsrc,dst,sig,n,is_sep.λint,outt: Vector (tape sig) (S n).
+  (∀ls,x,xs,rs,sep.
+    nth src ? int (niltape ?) = midtape sig ls x (xs@sep::rs) →
+    (∀c.memb ? c (x::xs) = true → is_sep c = false) → is_sep sep = true → 
+    ∀ls0,x0,target,c,rs0.|xs| = |target| → 
+    nth dst ? int (niltape ?) = midtape sig ls0 x0 (target@c::rs0) → 
+    outt = change_vec ?? 
+           (change_vec ?? int (midtape sig (reverse ? xs@x::ls) sep rs) src)
+           (midtape sig (reverse ? xs@x::ls0) c rs0) dst) ∧
+  (∀c.current ? (nth src ? int (niltape ?)) = Some ? c → is_sep c = true → 
+   outt = int) ∧
+  (current ? (nth src ? int (niltape ?)) = None ? → outt = int).
+  
+lemma change_vec_change_vec : ∀A,n,v,a,b,i. 
+  change_vec A n (change_vec A n v a i) b i = change_vec A n v b i.
+#A #n #v #a #b #i @(eq_vec … a) #i0 #Hi0
+cases (decidable_eq_nat i i0) #Hii0
+[ >Hii0 >nth_change_vec // >nth_change_vec //
+| >nth_change_vec_neq // >nth_change_vec_neq //
+  >nth_change_vec_neq // ]
+qed.
+
+lemma wsem_copy : ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst < S n → 
+  copy src dst sig n is_sep ⊫ R_copy src dst sig n is_sep.
+#src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
+lapply (sem_while … (sem_copy_step src dst sig n is_sep Hneq Hsrc Hdst) … Hloop) //
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
+[ #tc whd in ⊢ (%→?); *
+  [ * #x * * #Hx #Hsep #Houtc % [ %
+    [ #ls #x0 #xs #rs #sep #Hsrctc #Hnosep >Hsrctc in Hx; normalize in ⊢ (%→?);
+      #Hx0 destruct (Hx0) lapply (Hnosep ? (memb_hd …)) >Hsep
+      #Hfalse destruct (Hfalse)
+    | #c #Hc #Hsepc @Houtc ]
+    | #_ @Houtc ]
+  | * #Hcur #Houtc % [ %
+    [ #ls #x0 #xs #rs #sep #Hsrctc >Hsrctc in Hcur; normalize in ⊢ (%→?); 
+      #Hcur destruct (Hcur)
+    | #c #Hc #Hsepc @Houtc ]
+    | #_ @Houtc ]
+  ]
+| #tc #td #te * #c0 * * #Hc0 #Hc0nosep #Hd #Hstar #IH #He 
+  lapply (IH He) -IH * * #IH1 #IH2 #IH3 % [ %
+  [ #ls #x #xs #rs #sep #Hsrc_tc #Hnosep #Hsep #ls0 #x0 #target
+    #c #rs0 #Hlen #Hdst_tc
+    >Hsrc_tc in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
+    <(change_vec_same … tc src (niltape ?)) in Hd:(???(???(???%??)??));
+    <(change_vec_same … tc dst (niltape ?)) in ⊢(???(???(???%??)??)→?);
+    >Hdst_tc >Hsrc_tc >change_vec_change_vec >change_vec_change_vec
+    >(change_vec_commute ?? tc ?? dst src) [|@(sym_not_eq … Hneq)]
+    >change_vec_change_vec @(list_cases2 … Hlen)
+    [ #Hxsnil #Htargetnil #Hd>(IH2 … Hsep)
+      [ >Hd -Hd >Hxsnil >Htargetnil @(eq_vec … (niltape ?))
+        #i #Hi cases (decidable_eq_nat i src) #Hisrc
+        [ >Hisrc >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
+          >nth_change_vec // >nth_change_vec //
+          >(nth_change_vec_neq … src dst) [|@(sym_not_eq … Hneq)]
+          >nth_change_vec // whd in ⊢ (??%?); %
+        | cases (decidable_eq_nat i dst) #Hidst
+          [ >Hidst >nth_change_vec // >nth_change_vec //
+            >nth_change_vec_neq // >Hdst_tc >Htargetnil %
+          | >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
+            >nth_change_vec_neq [|@(sym_not_eq … Hisrc)]
+            >nth_change_vec_neq [|@(sym_not_eq … Hidst)]
+            >nth_change_vec_neq [|@(sym_not_eq … Hisrc)] % ]
+         ]
+      | >Hd >nth_change_vec_neq [|@(sym_not_eq … Hneq)]
+        >nth_change_vec // >nth_change_vec // >Hxsnil % ]
+    |#hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd
+     >(IH1 (c0::ls) hd1 tl1 rs sep ?? Hsep (c0::ls0) hd2 tl2 c rs0)
+     [ >Hd >(change_vec_commute … ?? tc ?? src dst) //
+       >change_vec_change_vec
+       >(change_vec_commute … ?? tc ?? dst src) [|@sym_not_eq //]
+       >change_vec_change_vec
+       >reverse_cons >associative_append >associative_append % 
+     | >Hd >nth_change_vec // >nth_change_vec_neq // >Hdst_tc >Htarget //
+     | >Hxs in Hlen; >Htarget normalize #Hlen destruct (Hlen) //
+     | <Hxs #c1 #Hc1 @Hnosep @memb_cons //
+     | >Hd >nth_change_vec_neq [|@sym_not_eq //]
+       >nth_change_vec // >nth_change_vec // ]
+   ]
+ | #c #Hc #Hsepc >Hc in Hc0; #Hcc0 destruct (Hcc0) >Hc0nosep in Hsepc;
+   #H destruct (H)
+ ]
+| #HNone >HNone in Hc0; #Hc0 destruct (Hc0) ] ]
+qed.
+ 
+lemma terminate_copy :  ∀src,dst,sig,n,is_sep,t.
+  src ≠ dst → src < S n → dst < S n → 
+  copy src dst sig n is_sep ↓ t.
+#src #dst #sig #n #is_sep #t #Hneq #Hsrc #Hdst
+@(terminate_while … (sem_copy_step …)) //
+<(change_vec_same … t src (niltape ?))
+cases (nth src (tape sig) t (niltape ?))
+[ % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct 
+|2,3: #a0 #al0 % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?); #Hx destruct
+| #ls #c #rs lapply c -c lapply ls -ls lapply t -t elim rs
+  [#t #ls #c % #t1 * #x * * >nth_change_vec // normalize in ⊢ (%→?);
+   #H1 destruct (H1) #Hxsep >change_vec_change_vec #Ht1 % 
+   #t2 * #x0 * * >Ht1 >nth_change_vec_neq [|@sym_not_eq //]
+   >nth_change_vec // normalize in ⊢ (%→?); #H destruct (H)
+  |#r0 #rs0 #IH #t #ls #c % #t1 * #x * * >nth_change_vec //
+   normalize in ⊢ (%→?); #H destruct (H) #Hxsep
+   >change_vec_change_vec >change_vec_commute // #Ht1 >Ht1 @IH
+  ]
+]
+qed.
\ No newline at end of file