X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fturing%2Fmono.ma;h=0fcbfbe480d5d6cc8db8389c8141daa0dca8cde0;hb=bd5d6160029247d8c4e3f8cec82f7acd7199d7d5;hp=8b3450b3cd8a530e80fba654c9a7cf53a31dd3d8;hpb=482f39fc4f8e1b9cdca50cb0e072bdece36b271a;p=helm.git

diff --git a/matita/matita/lib/turing/mono.ma b/matita/matita/lib/turing/mono.ma
index 8b3450b3c..0fcbfbe48 100644
--- a/matita/matita/lib/turing/mono.ma
+++ b/matita/matita/lib/turing/mono.ma
@@ -47,6 +47,19 @@ definition mk_tape :
       | cons r0 rs0 ⇒ leftof ? r0 rs0 ]
     | cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
 
+lemma current_to_midtape: ∀sig,t,c. current sig t = Some ? c →
+  ∃ls,rs. t = midtape ? ls c rs.
+#sig *
+  [#c whd in ⊢ ((??%?)→?); #Hfalse destruct
+  |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
+  |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
+  |#ls #a #rs #c whd in ⊢ ((??%?)→?); #H destruct 
+   @(ex_intro … ls) @(ex_intro … rs) //
+  ]
+qed.
+
+(*********************************** moves ************************************)
+
 inductive move : Type[0] ≝
   | L : move | R : move | N : move.
 
@@ -84,7 +97,18 @@ record config (sig,states:FinSet): Type[0] ≝
 { cstate : states;
   ctape: tape sig
 }.
+
+lemma config_expand: ∀sig,Q,c. 
+  c = mk_config sig Q (cstate ?? c) (ctape ?? c).
+#sig #Q * // 
+qed.
   
+lemma config_eq : ∀sig,M,c1,c2.
+  cstate sig M c1 = cstate sig M c2 → 
+    ctape sig M c1 = ctape sig M c2 →  c1 = c2.
+#sig #M1 * #s1 #t1 * #s2 #t2 //
+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
@@ -172,10 +196,41 @@ lemma loop_eq : ∀sig,f,q,i,j,a,x,y.
 ]
 qed.
 
+lemma loop_p_true : 
+  ∀A,k,f,p,a.p a = true → loop A (S k) f p a = Some ? a.
+#A #k #f #p #a #Ha normalize >Ha %
+qed.
+
+lemma loop_Some : 
+  ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
+#A #k #f #p elim k 
+  [#a #b normalize #Hfalse destruct
+  |#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
+    [ >Hpa normalize #H1 destruct // | >Hpa normalize @IH ]
+  ]
+qed. 
+
+lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
+  (∀x.hlift (lift x) = h x) → 
+  (∀x.h x = false → lift (f x) = g (lift x)) → 
+  loop A k f h c1 = Some ? c2 → 
+  loop B k g hlift (lift c1) = Some ? (lift … c2).
+#A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
+generalize in match c1; elim k
+[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
+|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
+ cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0 normalize
+ [ #Heq destruct (Heq) % | <Hhlift // @IH ]
+qed.
+
 (************************** Realizability *************************************)
 definition loopM ≝ λsig,M,i,cin.
   loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
 
+lemma loopM_unfold : ∀sig,M,i,cin.
+  loopM sig M i cin = loop ? i (step sig M) (λc.halt sig M (cstate ?? c)) cin.
+// qed.
+
 definition initc ≝ λsig.λM:TM sig.λt.
   mk_config sig (states sig M) (start sig M) t.
 
@@ -189,15 +244,23 @@ definition WRealize ≝ λsig.λM:TM sig.λR:relation (tape sig).
 
 definition Terminate ≝ λsig.λM:TM sig.λt. ∃i,outc.
   loopM sig M i (initc sig M t) = Some ? outc.
+  
+notation "M \vDash R" non associative with precedence 45 for @{ 'models $M $R}.
+interpretation "realizability" 'models M R = (Realize ? M R).
+
+notation "M \VDash R" non associative with precedence 45 for @{ 'wmodels $M $R}.
+interpretation "weak realizability" 'wmodels M R = (WRealize ? M R).
+
+interpretation "termination" 'fintersects M t = (Terminate ? M t).
 
 lemma WRealize_to_Realize : ∀sig.∀M: TM sig.∀R.
-  (∀t.Terminate sig M t) → WRealize sig M R → Realize sig M R.
+  (∀t.M ↓ t) → M ⊫ R → M ⊨ R.
 #sig #M #R #HT #HW #t cases (HT … t) #i * #outc #Hloop 
 @(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
 qed.
 
-theorem Realize_to_WRealize : ∀sig,M,R.
-  Realize sig M R → WRealize sig M R.
+theorem Realize_to_WRealize : ∀sig.∀M:TM sig.∀R.
+  M ⊨ R → M ⊫ R.
 #sig #M #R #H1 #inc #i #outc #Hloop 
 cases (H1 inc) #k * #outc1 * #Hloop1 #HR >(loop_eq … Hloop Hloop1) //
 qed.
@@ -207,6 +270,44 @@ definition accRealize ≝ λsig.λM:TM sig.λacc:states sig M.λRtrue,Rfalse.
   loopM sig M i (initc sig M t) = Some ? outc ∧
     (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧ 
     (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
+    
+notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}.
+interpretation "conditional realizability" 'cmodels M q R1 R2 = (accRealize ? M q R1 R2).
+
+(******************************** monotonicity ********************************)
+lemma Realize_to_Realize : ∀alpha,M,R1,R2.
+  R1 ⊆ R2 → Realize alpha M R1 → Realize alpha M R2.
+#alpha #M #R1 #R2 #Himpl #HR1 #intape
+cases (HR1 intape) -HR1 #k * #outc * #Hloop #HR1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
+lemma WRealize_to_WRealize: ∀sig,M,R1,R2.
+  R1 ⊆ R2 → WRealize sig M R1 → WRealize ? M R2.
+#alpha #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
+@Hsub @(HR1 … i) @Hloop
+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
+cases (HRa intape) -HRa #k * #outc * * #Hloop #HRtrue #HRfalse 
+@(ex_intro ?? k) @(ex_intro ?? outc) % 
+  [ % [@Hloop] #Hq @Hsub13 @HRtrue // | #Hq @Hsub24 @HRfalse //]
+qed.
+
+(**************************** A canonical relation ****************************)
+
+definition R_TM ≝ λsig.λM:TM sig.λq.λt1,t2.
+∃i,outc.
+  loopM ? M i (mk_config ?? q t1) = Some ? outc ∧ 
+  t2 = (ctape ?? outc).
+  
+lemma R_TM_to_R: ∀sig,M,R. ∀t1,t2. 
+  M ⊫ R → R_TM ? M (start sig M) t1 t2 → R t1 t2.
+#sig #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
+#Hloop #Ht2 >Ht2 @(HMR … Hloop)
+qed.
 
 (******************************** NOP Machine *********************************)
 
@@ -224,11 +325,17 @@ definition nop ≝
 definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
 
 lemma sem_nop :
-  ∀alpha.Realize alpha (nop alpha) (R_nop alpha).
+  ∀alpha.nop alpha ⊨ R_nop alpha.
 #alpha #intape @(ex_intro ?? 1) 
 @(ex_intro … (mk_config ?? start_nop intape)) % % 
 qed.
 
+lemma nop_single_state: ∀sig.∀q1,q2:states ? (nop sig). q1 = q2.
+normalize #sig * #n #ltn1 * #m #ltm1 
+generalize in match ltn1; generalize in match ltm1;
+<(le_n_O_to_eq … (le_S_S_to_le … ltn1)) <(le_n_O_to_eq … (le_S_S_to_le … ltm1)) 
+// qed.
+
 (************************** Sequential Composition ****************************)
 
 definition seq_trans ≝ λsig. λM1,M2 : TM sig. 
@@ -236,12 +343,8 @@ definition seq_trans ≝ λsig. λM1,M2 : TM sig.
   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〉
+      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〉
   ].
  
 definition seq ≝ λsig. λM1,M2 : TM sig. 
@@ -249,13 +352,11 @@ definition seq ≝ λsig. λM1,M2 : TM sig.
     (FinSum (states sig M1) (states sig M2))
     (seq_trans sig M1 M2) 
     (inl … (start sig M1))
-    (λs.match s with
+    (λs.match s with 
       [ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]). 
 
-definition Rcomp ≝ λA.λR1,R2:relation A.λa1,a2.
-  ∃am.R1 a1 am ∧ R2 am a2.
-  
-interpretation "relation composition" 'compose R1 R2 = (Rcomp ? R1 R2).
+notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}.
+interpretation "sequential composition" 'middot a b = (seq ? a b).
 
 definition lift_confL ≝ 
   λsig,S1,S2,c.match c with 
@@ -283,7 +384,7 @@ lemma p_halt_liftL : ∀sig,S1,S2,halt,c.
 #sig #S1 #S2 #halt #c cases c #s #t %
 qed.
 
-lemma trans_liftL : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,move.
   halt ? M1 s = false → 
   trans sig M1 〈s,a〉 = 〈news,move〉 → 
   trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
@@ -291,7 +392,7 @@ lemma trans_liftL : ∀sig,M1,M2,s,a,news,move.
 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 qed.
 
-lemma trans_liftR : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,move.
   halt ? M2 s = false → 
   trans sig M2 〈s,a〉 = 〈news,move〉 → 
   trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
@@ -299,14 +400,7 @@ lemma trans_liftR : ∀sig,M1,M2,s,a,news,move.
 #Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
 qed.
 
-lemma config_eq : 
-  ∀sig,M,c1,c2.
-  cstate sig M c1 = cstate sig M c2 → 
-  ctape sig M c1 = ctape sig M c2 →  c1 = c2.
-#sig #M1 * #s1 #t1 * #s2 #t2 //
-qed.
-
-lemma step_lift_confR : ∀sig,M1,M2,c0.
+lemma step_seq_liftR : ∀sig,M1,M2,c0.
  halt ? M2 (cstate ?? c0) = false → 
  step sig (seq sig M1 M2) (lift_confR sig (states ? M1) (states ? M2) c0) =
  lift_confR sig (states ? M1) (states ? M2) (step sig M2 c0).
@@ -317,13 +411,11 @@ lemma step_lift_confR : ∀sig,M1,M2,c0.
   [ #Heq #Hhalt
   | 2,3: #s1 #l1 #Heq #Hhalt 
   |#ls #s1 #rs #Heq #Hhalt ]
-  whd in ⊢ (???(????%)); >Heq
-  whd in ⊢ (???%);
-  whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
-  >(trans_liftR … Heq) //
+  whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftR … Heq) //
 qed.
 
-lemma step_lift_confL : ∀sig,M1,M2,c0.
+lemma step_seq_liftL : ∀sig,M1,M2,c0.
  halt ? M1 (cstate ?? c0) = false → 
  step sig (seq sig M1 M2) (lift_confL sig (states ? M1) (states ? M2) c0) =
  lift_confL sig ?? (step sig M1 c0).
@@ -334,84 +426,14 @@ lemma step_lift_confL : ∀sig,M1,M2,c0.
   [ #Heq #Hhalt
   | 2,3: #s1 #l1 #Heq #Hhalt 
   |#ls #s1 #rs #Heq #Hhalt ]
-  whd in ⊢ (???(????%)); >Heq
-  whd in ⊢ (???%);
-  whd in ⊢ (??(???%)?); whd in ⊢ (??%?);
-  >(trans_liftL … Heq) //
+  whd in ⊢ (???(????%)); >Heq whd in ⊢ (???%);
+  whd in ⊢ (??(???%)?); whd in ⊢ (??%?); >(trans_seq_liftL … Heq) //
 qed.
 
-lemma loop_lift : ∀A,B,k,lift,f,g,h,hlift,c1,c2.
-  (∀x.hlift (lift x) = h x) → 
-  (∀x.h x = false → lift (f x) = g (lift x)) → 
-  loop A k f h c1 = Some ? c2 → 
-  loop B k g hlift (lift c1) = Some ? (lift … c2).
-#A #B #k #lift #f #g #h #hlift #c1 #c2 #Hfg #Hhlift
-generalize in match c1; elim k
-[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
- cases (true_or_false (h c0)) #Hc0 >Hfg >Hc0
- [ normalize #Heq destruct (Heq) %
- | normalize <Hhlift // @IH ]
-qed.
-
-(* 
-lemma loop_liftL : ∀sig,k,M1,M2,c1,c2.
-  loop ? k (step sig M1) (λc.halt sig M1 (cstate ?? c)) c1 = Some ? c2 →
-    loop ? k (step sig (seq sig M1 M2)) 
-      (λc.halt_liftL ?? (halt sig M1) (cstate ?? c)) (lift_confL … c1) = 
-    Some ? (lift_confL … c2).
-#sig #k #M1 #M2 #c1 #c2 generalize in match c1;
-elim k
-[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
- cases (true_or_false (halt ?? (cstate sig (states ? M1) c0))) #Hc0 >Hc0
- [ >(?: halt_liftL ?? (halt sig M1) (cstate sig ? (lift_confL … c0)) = true)
-   [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
-   | <Hc0 cases c0 // ]
- | >(?: halt_liftL ?? (halt sig M1) (cstate ?? (lift_confL … c0)) = false)
-   [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
-    @step_lift_confL //
-   | <Hc0 cases c0 // ]
-qed.
-
-lemma loop_liftR : ∀sig,k,M1,M2,c1,c2.
-  loop ? k (step sig M2) (λc.halt sig M2 (cstate ?? c)) c1 = Some ? c2 →
-    loop ? k (step sig (seq sig M1 M2)) 
-      (λc.halt sig (seq sig M1 M2) (cstate ?? c)) (lift_confR … c1) = 
-    Some ? (lift_confR … c2).
-#sig #k #M1 #M2 #c1 #c2 generalize in match c1;
-elim k
-[#c0 normalize in ⊢ (??%? → ?); #Hfalse destruct (Hfalse)
-|#k0 #IH #c0 whd in ⊢ (??%? → ??%?);
- cases (true_or_false (halt ?? (cstate sig ? c0))) #Hc0 >Hc0
- [ >(?: halt ? (seq sig M1 M2) (cstate sig ? (lift_confR … c0)) = true)
-   [ whd in ⊢ (??%? → ??%?); #Hc2 destruct (Hc2) %
-   | <Hc0 cases c0 // ]
- | >(?: halt ? (seq sig M1 M2) (cstate sig ? (lift_confR … c0)) = false)
-   [whd in ⊢ (??%? → ??%?); #Hc2 <(IH ? Hc2) @eq_f
-    @step_lift_confR //
-   | <Hc0 cases c0 // ]
- ]
-qed.  
-
-*)
-    
-lemma loop_Some : 
-  ∀A,k,f,p,a,b.loop A k f p a = Some ? b → p b = true.
-#A #k #f #p elim k 
-[#a #b normalize #Hfalse destruct
-|#k0 #IH #a #b whd in ⊢ (??%? → ?); cases (true_or_false (p a)) #Hpa
- [ >Hpa normalize #H1 destruct //
- | >Hpa normalize @IH
- ]
-]
-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 ?〉.
-#sig #M1 #M2 #s #a
-#Hhalt whd in ⊢ (??%?); >Hhalt %
+#sig #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
 qed.
 
 lemma eq_ctape_lift_conf_L : ∀sig,S1,S2,outc.
@@ -424,9 +446,8 @@ lemma eq_ctape_lift_conf_R : ∀sig,S1,S2,outc.
 #sig #S1 #S2 #outc cases outc #s #t %
 qed.
 
-theorem sem_seq: ∀sig,M1,M2,R1,R2.
-  Realize sig M1 R1 → Realize sig M2 R2 → 
-    Realize sig (seq sig M1 M2) (R1 ∘ R2).
+theorem sem_seq: ∀sig.∀M1,M2:TM sig.∀R1,R2.
+  M1 ⊨ R1 → M2 ⊨ R2 → M1 · M2 ⊨ R1 ∘ R2.
 #sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t 
 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
 cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
@@ -440,12 +461,12 @@ cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
   [ * *
    [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
    | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
-  || #c0 #Hhalt <step_lift_confL //
+  || #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_lift_confR // ]
+    | #c0 #Hhalt <step_seq_liftR // ]
   |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
    generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10 
    >(trans_liftL_true sig M1 M2 ??) 
@@ -458,3 +479,10 @@ cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
 ]
 qed.
 
+theorem sem_seq_app: ∀sig.∀M1,M2:TM sig.∀R1,R2,R3.
+  M1 ⊨ R1 → M2 ⊨ R2 → R1 ∘ R2 ⊆ R3 → M1 · M2 ⊨ R3.
+#sig #M1 #M2 #R1 #R2 #R3 #HR1 #HR2 #Hsub
+#t cases (sem_seq … HR1 HR2 t)
+#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
+% [@Hloop |@Hsub @Houtc]
+qed.