X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmono.ma;h=3477508c15528dad59bdc44fd0cc654db3e4ab20;hb=600fba840c748f67593838673a6eb40eab9b68e5;hp=37ce2f2dee192414fc174a9783f0fd0fe4a5dd89;hpb=5535cd4e08fd8d1e7e6e067eac1bb6c1bf8fcbbf;p=helm.git

diff --git a/matita/matita/lib/turing/mono.ma b/matita/matita/lib/turing/mono.ma
index 37ce2f2de..3477508c1 100644
--- a/matita/matita/lib/turing/mono.ma
+++ b/matita/matita/lib/turing/mono.ma
@@ -9,7 +9,10 @@
      \ /   GNU General Public License Version 2   
       V_____________________________________________________________*)
 
+include "basics/core_notation/fintersects_2.ma".
+include "basics/finset.ma".
 include "basics/vectors.ma".
+include "basics/finset.ma".
 (* include "basics/relations.ma". *)
 
 (******************************** tape ****************************************)
@@ -47,6 +50,21 @@ definition mk_tape :
       | cons r0 rs0 ⇒ leftof ? r0 rs0 ]
     | cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
 
+lemma right_mk_tape : 
+  ∀sig,ls,c,rs.(c = None ? → ls = [ ] ∨ rs = [ ]) → right ? (mk_tape sig ls c rs) = rs.
+#sig #ls #c #rs cases c // cases ls 
+[ cases rs // 
+| #l0 #ls0 #H normalize cases (H (refl ??)) #H1 [ destruct (H1) | >H1 % ] ]
+qed-.
+
+lemma left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls.
+#sig #ls #c #rs cases c // cases ls // cases rs //
+qed.
+
+lemma current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c.
+#sig #ls #c #rs cases c // cases ls // cases rs //
+qed.
+
 lemma current_to_midtape: ∀sig,t,c. current sig t = Some ? c →
   ∃ls,rs. t = midtape ? ls c rs.
 #sig *
@@ -62,36 +80,102 @@ qed.
 
 inductive move : Type[0] ≝
   | L : move | R : move | N : move.
+  
+(*************************** turning moves into a DeqSet **********************)
+
+definition move_eq ≝ λm1,m2:move.
+  match m1 with
+  [R ⇒ match m2 with [R ⇒ true | _ ⇒ false]
+  |L ⇒ match m2 with [L ⇒ true | _ ⇒ false]
+  |N ⇒ match m2 with [N ⇒ true | _ ⇒ false]].
+
+lemma move_eq_true:∀m1,m2.
+  move_eq m1 m2 = true ↔ m1 = m2.
+*
+  [* normalize [% #_ % |2,3: % #H destruct ]
+  |* normalize [1,3: % #H destruct |% #_ % ]
+  |* normalize [1,2: % #H destruct |% #_ % ]
+qed.
+
+definition DeqMove ≝ mk_DeqSet move move_eq move_eq_true.
+
+unification hint 0 ≔ ;
+    X ≟ DeqMove
+(* ---------------------------------------- *) ⊢ 
+    move ≡ carr X.
+
+unification hint  0 ≔ m1,m2; 
+    X ≟ DeqMove
+(* ---------------------------------------- *) ⊢ 
+    move_eq m1 m2 ≡ eqb X m1 m2.
+
+
+(************************ turning DeqMove into a FinSet ***********************)
+
+definition move_enum ≝ [L;R;N].
 
+lemma move_enum_unique: uniqueb ? [L;R;N] = true.
+// qed.
+
+lemma move_enum_complete: ∀x:move. memb ? x [L;R;N] = true.
+* // qed.
+
+definition FinMove ≝ 
+  mk_FinSet DeqMove [L;R;N] move_enum_unique move_enum_complete.
+
+unification hint  0 ≔ ; 
+    X ≟ FinMove
+(* ---------------------------------------- *) ⊢ 
+    move ≡ FinSetcarr X.
 (********************************** machine ***********************************)
 
 record TM (sig:FinSet): Type[1] ≝ 
 { states : FinSet;
-  trans : states × (option sig) → states × (option (sig × move));
+  trans : states × (option sig) → states × (option sig) × move;
   start: states;
   halt : states → bool
 }.
 
-definition tape_move_left ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
-  match lt with
-  [ nil ⇒ leftof sig c rt
-  | cons c0 lt0 ⇒ midtape sig lt0 c0 (c::rt) ].
+definition tape_move_left ≝ λsig:FinSet.λt:tape sig.
+  match t with 
+  [ niltape ⇒ niltape sig
+  | leftof _ _ ⇒ t
+  | rightof a ls ⇒ midtape sig ls a [ ]
+  | midtape ls a rs ⇒ 
+    match ls with 
+    [ nil ⇒ leftof sig a rs
+    | cons a0 ls0 ⇒ midtape sig ls0 a0 (a::rs)
+    ]
+  ]. 
+  
+definition tape_move_right ≝ λsig:FinSet.λt:tape sig.
+  match t with 
+  [ niltape ⇒ niltape sig
+  | rightof _ _ ⇒ t
+  | leftof a rs ⇒ midtape sig [ ] a rs
+  | midtape ls a rs ⇒ 
+    match rs with 
+    [ nil ⇒ rightof sig a ls
+    | cons a0 rs0 ⇒ midtape sig (a::ls) a0 rs0
+    ]
+  ]. 
   
-definition tape_move_right ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
-  match rt with
-  [ nil ⇒ rightof sig c lt
-  | cons c0 rt0 ⇒ midtape sig (c::lt) c0 rt0 ].
+definition tape_write ≝ λsig.λt: tape sig.λs:option sig.
+  match s with 
+  [ None ⇒ t
+  | Some s0 ⇒ midtape ? (left ? t) s0 (right ? t)
+  ].
 
-definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
+definition tape_move ≝ λsig.λt: tape sig.λm:move.
   match m with
-  [ None ⇒ t
-  | Some m' ⇒ 
-    let 〈s,m1〉 ≝ m' in 
-    match m1 with
-      [ R ⇒ tape_move_right ? (left ? t) s (right ? t)
-      | L ⇒ tape_move_left ? (left ? t) s (right ? t)
-      | N ⇒ midtape ? (left ? t) s (right ? t)
-      ] ].
+    [ R ⇒ tape_move_right ? t
+    | L ⇒ tape_move_left ? t
+    | N ⇒ t
+    ].
+
+definition tape_move_mono ≝ 
+  λsig,t,mv.
+  tape_move sig (tape_write sig t (\fst mv)) (\snd mv).
 
 record config (sig,states:FinSet): Type[0] ≝ 
 { cstate : states;
@@ -111,9 +195,19 @@ qed.
 
 definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
   let current_char ≝ current ? (ctape ?? c) in
-  let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
-  mk_config ?? news (tape_move sig (ctape ?? c) mv).
+  let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
+  mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) mv).
 
+(*
+lemma step_eq : ∀sig,M,c. 
+  let current_char ≝ current ? (ctape ?? c) in
+  let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
+  step sig M c = 
+    mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) mv).
+#sig #M #c  
+ whd in match (tape_move_mono sig ??);
+*)
+  
 (******************************** loop ****************************************)
 let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
   match n with 
@@ -329,6 +423,13 @@ cases (HR1 intape HP) -HR1 #k * #outc * #Hloop #HR1
 @(ex_intro ?? k) @(ex_intro ?? outc) % /2/
 qed.
 
+lemma GRealize_to_GRealize_2 : ∀alpha,M,P1,P2,R1,R2.
+  P2 ⊆ P1 → R1 ⊆ R2 → GRealize alpha M P1 R1 → GRealize alpha M P2 R2.
+#alpha #M #P1 #P2 #R1 #R2 #Himpl1 #Himpl2 #H1 #intape #HP
+cases (H1 intape (Himpl1 … HP)) -H1 #k * #outc * #Hloop #H1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
 lemma acc_Realize_to_acc_Realize: ∀sig,M.∀q:states sig M.∀R1,R2,R3,R4. 
   R1 ⊆ R3 → R2 ⊆ R4 → M ⊨ [q:R1,R2] → M ⊨ [q:R3,R4].
 #alpha #M #q #R1 #R2 #R3 #R4 #Hsub13 #Hsub24 #HRa #intape
@@ -360,7 +461,7 @@ definition start_nop : initN 1 ≝ mk_Sig ?? 0 (le_n … 1).
 
 definition nop ≝ 
   λalpha:FinSet.mk_TM alpha nop_states
-  (λp.let 〈q,a〉 ≝ p in 〈q,None ?〉)
+  (λp.let 〈q,a〉 ≝ p in 〈q,None ?,N〉)
   start_nop (λ_.true).
   
 definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
@@ -383,9 +484,9 @@ definition seq_trans ≝ λsig. λM1,M2 : TM sig.
 λp. let 〈s,a〉 ≝ p in
   match s with 
   [ inl s1 ⇒ 
-      if halt sig M1 s1 then 〈inr … (start sig M2), None ?〉
-      else let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in 〈inl … news1,m〉
-  | inr s2 ⇒ let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in 〈inr … news2,m〉
+      if halt sig M1 s1 then 〈inr … (start sig M2), None ?,N〉
+      else let 〈news1,newa,m〉 ≝ trans sig M1 〈s1,a〉 in 〈inl … news1,newa,m〉
+  | inr s2 ⇒ let 〈news2,newa,m〉 ≝ trans sig M2 〈s2,a〉 in 〈inr … news2,newa,m〉
   ].
  
 definition seq ≝ λsig. λM1,M2 : TM sig. 
@@ -425,19 +526,19 @@ lemma p_halt_liftL : ∀sig,S1,S2,halt,c.
 #sig #S1 #S2 #halt #c cases c #s #t %
 qed.
 
-lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,newa,move.
   halt ? M1 s = false → 
-  trans sig M1 〈s,a〉 = 〈news,move〉 → 
-  trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
-#sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
+  trans sig M1 〈s,a〉 = 〈news,newa,move〉 → 
+  trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,newa,move〉.
+#sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #newa #move
 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 qed.
 
-lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,newa,move.
   halt ? M2 s = false → 
-  trans sig M2 〈s,a〉 = 〈news,move〉 → 
-  trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
-#sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #move
+  trans sig M2 〈s,a〉 = 〈news,newa,move〉 → 
+  trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,newa,move〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #newa #move
 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 qed.
 
@@ -448,7 +549,7 @@ lemma step_seq_liftR : ∀sig,M1,M2,c0.
 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
   lapply (refl ? (trans ?? 〈s,current sig t〉))
   cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
-  #s0 #m0 cases t
+  * #s0 #a0 #m0 cases t
   [ #Heq #Hhalt
   | 2,3: #s1 #l1 #Heq #Hhalt 
   |#ls #s1 #rs #Heq #Hhalt ]
@@ -463,7 +564,7 @@ lemma step_seq_liftL : ∀sig,M1,M2,c0.
 #sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
   lapply (refl ? (trans ?? 〈s,current sig t〉))
   cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
-  #s0 #m0 cases t
+  * #s0 #a0 #m0 cases t
   [ #Heq #Hhalt
   | 2,3: #s1 #l1 #Heq #Hhalt 
   |#ls #s1 #rs #Heq #Hhalt ]
@@ -473,7 +574,7 @@ qed.
 
 lemma trans_liftL_true : ∀sig,M1,M2,s,a.
   halt ? M1 s = true → 
-  trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?〉.
+  trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?,N〉.
 #sig #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
 qed.
 
@@ -575,3 +676,57 @@ theorem sem_seq_app_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2,R3.
 #k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
 % [@Hloop |@Hsub @Houtc]
 qed.
+
+theorem acc_sem_seq : ∀sig.∀M1,M2:TM sig.∀R1,Rtrue,Rfalse,acc.
+  M1 ⊨ R1 → M2 ⊨ [ acc: Rtrue, Rfalse ] → 
+  M1 · M2 ⊨ [ inr … acc: R1 ∘ Rtrue, R1 ∘ Rfalse ].
+#sig #M1 #M2 #R1 #Rtrue #Rfalse #acc #HR1 #HR2 #t 
+cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
+cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * * #Hloop2 
+#HMtrue #HMfalse
+@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
+% [ %
+[@(loop_merge …
+   (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
+   (step sig M1) (step sig (seq sig M1 M2)) 
+   (λc.halt sig M1 (cstate … c)) 
+   (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
+  [ * *
+   [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+   | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+  || #c0 #Hhalt <step_seq_liftL //
+  | #x <p_halt_liftL %
+  |6:cases outc1 #s1 #t1 %
+  |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2) 
+    [ * #s2 #t2 %
+    | #c0 #Hhalt <step_seq_liftR // ]
+  |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
+   generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
+   >(trans_liftL_true sig M1 M2 ??) 
+    [ whd in ⊢ (??%?); whd in ⊢ (???%); //
+    | @(loop_Some ?????? Hloop10) ]
+ ]
+| >(config_expand … outc2) in ⊢ (%→?); whd in ⊢ (??%?→?); 
+  #Hqtrue destruct (Hqtrue)
+  @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R /2/ ]
+| >(config_expand … outc2) in ⊢ (%→?); whd in ⊢ (?(??%?)→?); #Hqfalse
+  @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+  % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R @HMfalse
+  @(not_to_not … Hqfalse) //
+]
+qed.
+
+lemma acc_sem_seq_app : ∀sig.∀M1,M2:TM sig.∀R1,Rtrue,Rfalse,R2,R3,acc.
+  M1 ⊨ R1 → M2 ⊨ [acc: Rtrue, Rfalse] → 
+    (∀t1,t2,t3. R1 t1 t3 → Rtrue t3 t2 → R2 t1 t2) → 
+    (∀t1,t2,t3. R1 t1 t3 → Rfalse t3 t2 → R3 t1 t2) → 
+    M1 · M2 ⊨ [inr … acc : R2, R3].    
+#sig #M1 #M2 #R1 #Rtrue #Rfalse #R2 #R3 #acc
+#HR1 #HRacc #Hsub1 #Hsub2 
+#t cases (acc_sem_seq … HR1 HRacc t)
+#k * #outc * * #Hloop #Houtc1 #Houtc2 @(ex_intro … k) @(ex_intro … outc)
+% [% [@Hloop
+     |#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
+  |#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]
+qed.