From 04b536f1693534e450fde5dc824022321d93d039 Mon Sep 17 00:00:00 2001
From: Wilmer Ricciotti <ricciott@cs.unibo.it>
Date: Mon, 7 Oct 2013 14:05:18 +0000
Subject: [PATCH] Starting proof about semantics of binary machines.

---
 .../lib/turing/multi_universal/binaryTM.ma    | 135 ++++++++++++------
 1 file changed, 91 insertions(+), 44 deletions(-)

diff --git a/matita/matita/lib/turing/multi_universal/binaryTM.ma b/matita/matita/lib/turing/multi_universal/binaryTM.ma
index 668ac00f6..34c9ae4b8 100644
--- a/matita/matita/lib/turing/multi_universal/binaryTM.ma
+++ b/matita/matita/lib/turing/multi_universal/binaryTM.ma
@@ -24,26 +24,29 @@ axiom FS_nth : ∀F:FinSet.nat → option F.
 axiom index_of_FS : ∀F:FinSet.F → nat.
 
 (* unary bit representation (with a given length) of a certain number *)
-axiom unary_of_nat : nat → nat → nat.
+axiom unary_of_nat : nat → nat → (list bool).
 
 axiom FinVector : Type[0] → nat → FinSet.
 
-definition binary_base_states ≝ initN 7.
+definition binary_base_states ≝ initN 6.
 
-definition bin0 : binary_base_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 7 (refl …)).
-definition bin1 : binary_base_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 7 (refl …)).
-definition bin2 : binary_base_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 7 (refl …)).
-definition bin3 : binary_base_states ≝ mk_Sig ?? 3 (leb_true_to_le 4 7 (refl …)).
-definition bin4 : binary_base_states ≝ mk_Sig ?? 4 (leb_true_to_le 5 7 (refl …)).
-definition bin5 : binary_base_states ≝ mk_Sig ?? 5 (leb_true_to_le 6 7 (refl …)).
-definition bin6 : binary_base_states ≝ mk_Sig ?? 6 (leb_true_to_le 7 7 (refl …)).
+definition bin0 : binary_base_states ≝ mk_Sig ?? 0 (leb_true_to_le 1 6 (refl …)).
+definition bin1 : binary_base_states ≝ mk_Sig ?? 1 (leb_true_to_le 2 6 (refl …)).
+definition bin2 : binary_base_states ≝ mk_Sig ?? 2 (leb_true_to_le 3 6 (refl …)).
+definition bin3 : binary_base_states ≝ mk_Sig ?? 3 (leb_true_to_le 4 6 (refl …)).
+definition bin4 : binary_base_states ≝ mk_Sig ?? 4 (leb_true_to_le 5 6 (refl …)).
+definition bin5 : binary_base_states ≝ mk_Sig ?? 5 (leb_true_to_le 6 6 (refl …)).
 
 definition states_binaryTM : FinSet → FinSet → FinSet ≝ λsig,states.
  FinProd (FinProd states binary_base_states) 
-         (FinProd (FinOption sig) (initN (2 * (FS_crd sig)))).
+         (FinProd (FinOption sig) (initN (S (2 * (FS_crd sig))))).
 
 axiom daemon : ∀T:Type[0].T.
-definition initN_pred ≝ λn.λm:initN n.(pred (pi1 … m) : initN n).
+
+definition to_initN : ∀n,m.n < m → initN m ≝ λn,m,Hn.mk_Sig … n ….// qed.
+
+definition initN_pred : ∀n.∀m:initN n.initN n ≝ λn,m.mk_Sig … (pred (pi1 … m)) …. 
+cases m #m0 /2 by le_to_lt_to_lt/ qed.
 
 (* controllare i contatori, molti andranno incrementati di uno *)
 definition trans_binaryTM : ∀sig,states:FinSet.
@@ -53,69 +56,113 @@ definition trans_binaryTM : ∀sig,states:FinSet.
 ≝ λsig,states,trans,p.
   let 〈s,a〉 ≝ p in
   let 〈s0,phase,ch,count〉 ≝ s in
+  let (H1 : O < S (2*FS_crd sig)) ≝ ? in
+  let (H2 : FS_crd sig < S (2*FS_crd sig)) ≝ ? in
   match pi1 … phase with
   [ O ⇒ (*** PHASE 0: read ***)
       match a with
       [ Some a0 ⇒ 
-        match count with
-        [ O ⇒ 〈〈s0,1,ch,FS_crd sig〉,None ?,N〉
+        match pi1 … count with
+        [ O ⇒ 〈〈s0,bin1,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
         | S k ⇒ if (a0 == true) 
-                then 〈〈s0,0,FS_nth sig k,k〉, None ?,R〉
-                else 〈〈s0,0,ch,k〉,None ?,R〉 ]
+                then 〈〈s0,bin0,FS_nth sig k,initN_pred … count〉, None ?,R〉
+                else 〈〈s0,bin0,ch,initN_pred … count〉,None ?,R〉 ]
       | None ⇒ (* Overflow position! *)
-          〈〈s0,4,None ?,0〉,None ?,R〉 ]
+          〈〈s0,bin4,None ?,to_initN 0 ? H1〉,None ?,R〉 ]
   | S phase ⇒ match phase with
   [ O ⇒ (*** PHASE 1: restart ***)
-      match count with
-      [ O ⇒ 〈〈s0,2,ch,FS_crd sig〉,None ?,N〉
-      | S k ⇒ 〈〈s0,1,ch,k〉,None ?,L〉 ]
+      match pi1 … count with
+      [ O ⇒ 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
+      | S k ⇒ 〈〈s0,bin1,ch,initN_pred … count〉,None ?,L〉 ]
   | S phase ⇒ match phase with
   [ O ⇒ (*** PHASE 2: write ***)
       let 〈s',a',mv〉 ≝ trans 〈s0,ch〉 in
-      match count with
+      match pi1 … count with
       [ O ⇒ let mv' ≝ match mv with [ R ⇒ N | _ ⇒ L ] in
             let count' ≝ match mv with [ R ⇒ 0 | N ⇒ FS_crd sig | L ⇒ 2*(FS_crd sig) ] in
-             〈〈s',3,ch,count'〉,None ?,mv'〉
+             〈〈s',bin3,ch,to_initN count' ??〉,None ?,mv'〉
       | S k ⇒ match a' with
-         [ None ⇒ 〈〈s0,2,ch,k〉,None ?,R〉
-         | Some a0' ⇒ let out ≝ (FS_nth k == a') in
-                      〈〈s0,2,ch,k〉,Some ? out,R〉 ]
+         [ None ⇒ 〈〈s0,bin2,ch,initN_pred … count〉,None ?,R〉
+         | Some a0' ⇒ let out ≝ (FS_nth ? k == a') in
+                      〈〈s0,bin2,ch,initN_pred … count〉,Some ? out,R〉 ]
       ]
   | S phase ⇒ match phase with
   [ O ⇒ (*** PHASE 3: move head left ***)
-      match count with
-      [ O ⇒ 〈〈s0,6,ch,O〉, None ?,N〉
-      | S k ⇒ 〈〈s0,3,ch,k〉, None ?,L〉 ]
+      match pi1 … count with
+      [ O ⇒ 〈〈s0,bin0,None ?,to_initN (FS_crd sig) ? H2〉, None ?,N〉 (* the end: restart *)
+      | S k ⇒ 〈〈s0,bin3,ch,initN_pred … count〉, None ?,L〉 ]
   | S phase ⇒ match phase with
   [ O ⇒ (*** PHASE 4: check position ***)
       match a with
-      [ None ⇒ (* niltape/rightof: we can write *) 〈〈s0,2,ch,FS_crd sig〉,None ?,N〉
+      [ None ⇒ (* niltape/rightof: we can write *) 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
       | Some _ ⇒ (* leftof *)
         let 〈s',a',mv〉 ≝ trans 〈s0,ch〉 in
         match a' with
-        [ None ⇒ (* we don't write anything: go to end of 2 *) 〈〈s0,2,ch,0〉,None ?,N〉
-        | Some _ ⇒ (* extend tape *) 〈〈s0,5,ch,FS_crd sig〉,None ?,L〉 ]
+        [ None ⇒ (* we don't write anything: go to end of 2 *) 〈〈s0,bin2,ch,to_initN 0 ? H1〉,None ?,N〉
+        | Some _ ⇒ (* extend tape *) 〈〈s0,bin5,ch,to_initN (FS_crd sig) ? H2〉,None ?,L〉 ]
       ]
-  | S phase ⇒ match phase with
-  [ O ⇒ (*** PHASE 5: left extension ***)
+  | S _ ⇒ (*** PHASE 5: left extension ***)
       match pi1 … count with
-      [ O ⇒ 〈〈s0,bin2,ch,FS_crd sig〉,None ?,N〉
-      | S k ⇒ 〈〈s0,bin5,ch,k〉,Some ? false,L〉 ]
-  | S _ ⇒ (*** PHASE 6: stop ***) 〈s,None ?,N〉 ]]]]]].  
+      [ O ⇒ 〈〈s0,bin2,ch,to_initN (FS_crd sig) ? H2〉,None ?,N〉
+      | S k ⇒ 〈〈s0,bin5,ch,initN_pred … count〉,Some ? false,L〉 ]]]]]].
+[2,3: //]
+whd in match count'; cases mv whd in ⊢ (?%?); //
+qed.
+
+definition halt_binaryTM : ∀sig,M.states_binaryTM sig (states sig M) → bool ≝ 
+  λsig,M,s.let 〈s0,phase,ch,count〉 ≝ s in
+  pi1 … phase == O ∧ halt sig M s0.
 
 (*
  * Una mk_binaryTM prende in input una macchina M e produce una macchina che:
  * - ha per alfabeto FinBool
- * - ha stati di tipo (states … M) × (initN 3) × (initN (dimensione dell'alfabeto di M))
+ * - ha stati di tipo ((states … M) × (initN 7)) × 
+       ((option sig) × (initN (2*dimensione dell'alfabeto di M + 1))
  *   dove il primo elemento corrisponde allo stato della macchina input,
  *   il secondo identifica la fase (lettura, scrittura, spostamento)
- *   il terzo è un contatore
- * - (la funzione di transizione è complessa al punto di rendere discutibile 
+ *   il terzo identifica il carattere oggetto letto
+ *   il quarto è un contatore
+ * - la funzione di transizione viene prodotta da trans_binaryTM
+ * - la funzione di arresto viene prodotta da halt_binaryTM
  *)
 definition mk_binaryTM ≝ 
-  λsig.λM:TM sig.mk_TM FinBool (FinProd (states … M) (FinProd (initN 3) (initN
-{ no_states : nat;
-  pos_no_states : (0 < no_states); 
-  ntrans : trans_source no_states → trans_target no_states;
-  nhalt : initN no_states → bool
-}.
\ No newline at end of file
+  λsig.λM:TM sig.
+  mk_TM FinBool (states_binaryTM sig (states sig M)) 
+    (trans_binaryTM sig (states sig M) (trans sig M)) 
+    (〈start sig M,bin0,None ?,FS_crd sig〉) (halt_binaryTM sig M).// qed.
+
+definition bin_current ≝ λsig,t.match current ? t with
+[ None ⇒ [ ] | Some c ⇒ unary_of_nat (FS_crd sig) (index_of_FS sig c) ].
+
+definition tape_bin_lift ≝ λsig,t.
+let ls' ≝ flatten ? (map ?? (unary_of_nat (FS_crd sig) ∘ (index_of_FS sig)) (left ? t)) in
+let c' ≝ option_hd ? (bin_current sig t) in
+let rs' ≝ tail ? (bin_current sig t)@flatten ? (map ?? (unary_of_nat (FS_crd sig) ∘ (index_of_FS sig)) (right ? t)) in
+ mk_tape ? ls' c' rs'.
+
+definition R_bin_lift ≝ λsig,R,t1,t2.
+  ∃u1.t1 = tape_bin_lift sig u1 → 
+  ∃u2.t2 = tape_bin_lift sig u2 ∧ R u1 u2.
+  
+definition state_bin_lift :
+  ∀sig.∀M:TM sig.states sig M → states ? (mk_binaryTM ? M)
+ ≝ λsig,M,q.〈q,bin0,None ?,FS_crd sig〉.// qed.
+
+lemma binaryTM_loop :
+ ∀sig,M,i,t,q,tf,qf.
+ loopM sig M i (mk_config ?? q t) = Some ? (mk_config ?? qf tf) →
+ ∃k.loopM ? (mk_binaryTM sig M) k 
+  (mk_config ?? (state_bin_lift ? M q) (tape_bin_lift ? t)) = 
+  Some ? (mk_config ?? (state_bin_lift ? M qf) (tape_bin_lift ? tf)).
+#sig #M #i elim i
+[ #t #q #qf #tf change with (None ?) in ⊢ (??%?→?); #H destruct (H)
+| -i #i #IH #t #q #tf #qf
+
+
+(*
+theorem sem_binaryTM : ∀sig,M.
+  mk_binaryTM sig M ⊫ R_bin_lift ? (R_TM ? M (start ? M)).
+#sig #M #t #i generalize in match t; -t
+@(nat_elim1 … i) #m #IH #intape #outc #Hloop
+
+*)
\ No newline at end of file
-- 
2.39.2