- csx_cnx_vector.ma completed

[helm.git] / matita / matita / lib / re / moves.ma

diff --git a/matita/matita/lib/re/moves.ma b/matita/matita/lib/re/moves.ma
index 1372b49760ee677eea226d73a3a70392bc869e86..c8c3b3478fc865cf53bcd034110e8b2a05fcc9f4 100644 (file)
--- a/matita/matita/lib/re/moves.ma
+++ b/matita/matita/lib/re/moves.ma
@@ -30,10 +30,10 @@ lifted operators of the previous section:
 
 let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre S ≝
  match E with
-  [ pz ⇒ 〈 `∅, false 〉
-  | pe ⇒ 〈 ϵ, false 〉
-  | ps y ⇒ 〈 `y, false 〉
-  | pp y ⇒ 〈 `y, x == y 〉
+  [ pz ⇒ 〈 pz ?, false 〉
+  | pe ⇒ 〈 pe ? , false 〉
+  | ps y ⇒ 〈 ps ? y, false 〉
+  | pp y ⇒ 〈 ps ? y, x == y 〉
   | po e1 e2 ⇒ (move ? x e1) ⊕ (move ? x e2) 
   | pc e1 e2 ⇒ (move ? x e1) ⊙ (move ? x e2)
   | pk e ⇒ (move ? x e)^⊛ ].
@@ -92,8 +92,7 @@ theorem move_ok:
    (* rhs = a::w1∈\sem{i1}\sem{|i2|} ∨ a::w∈\sem{i2} *)
    @iff_trans[||@(iff_or_l … (HI2 w))] 
    (* rhs = a::w1∈\sem{i1}\sem{|i2|} ∨ w∈\sem{move S a i2} *)
-   @iff_or_r 
-   check deriv_middot 
+   @iff_or_r
    (* we are left to prove that 
      w∈\sem{move S a i1}·\sem{|i2|} ↔ a::w∈\sem{i1}\sem{|i2|}
      we use deriv_middot on the rhs *)
@@ -110,7 +109,7 @@ theorem move_ok:
   ]
 qed.
     
-notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
+notation > "x ↦* E" non associative with precedence 65 for @{moves ? $x $E}.
 let rec moves (S : DeqSet) w e on w : pre S ≝
  match w with
   [ nil ⇒ e
@@ -145,7 +144,6 @@ theorem decidable_sem: ∀S:DeqSet.∀w: word S. ∀e:pre S.
 #S #w elim w 
  [* #i #b >moves_empty cases b % /2/
  |#a #w1 #Hind #e >moves_cons
- check not_epsilon_sem
   @iff_trans [||@iff_sym @not_epsilon_sem]
   @iff_trans [||@move_ok] @Hind
  ]
@@ -211,7 +209,7 @@ e1 and e2 are equivalent iff for any word w the states reachable
 through w are cofinal. *)
 
 theorem equiv_sem: ∀S:DeqSet.∀e1,e2:pre S. 
-  \sem{e1} =1 \sem{e2} ↔ ∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
+  \sem{e1} ≐ \sem{e2} ↔ ∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
 #S #e1 #e2 % 
 [#same_sem #w 
   cut (∀b1,b2. iff (b1 = true) (b2 = true) → (b1 = b2)) 
@@ -244,7 +242,7 @@ occurring the given regular expressions. *)
 
 lemma equiv_sem_occ: ∀S.∀e1,e2:pre S.
 (∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
-→ \sem{e1}=1\sem{e2}.
+→ \sem{e1}≐\sem{e2}.
 #S #e1 #e2 #H @(proj2 … (equiv_sem …)) @occ_enough #w @H 
 qed.
 
@@ -273,7 +271,7 @@ definition is_bisim ≝ λS:DeqSet.λl:list ?.λalpha:list S.
 (* Using lemma equiv_sem_occ it is easy to prove the following result: *)
 
 lemma bisim_to_sem: ∀S:DeqSet.∀l:list ?.∀e1,e2: pre S. 
-  is_bisim S l (occ S e1 e2) → memb ? 〈e1,e2〉 l = true → \sem{e1}=1\sem{e2}.
+  is_bisim S l (occ S e1 e2) → memb ? 〈e1,e2〉 l = true → \sem{e1}≐\sem{e2}.
 #S #l #e1 #e2 #Hbisim #Hmemb @equiv_sem_occ 
 #w #Hsub @(proj1 … (Hbisim 〈moves S w e1,moves S w e2〉 ?))
 lapply Hsub @(list_elim_left … w) [//]
@@ -428,7 +426,7 @@ uniqueb ? l = true ∧
 definition disjoint ≝ λS:DeqSet.λl1,l2.
   ∀p:S. memb S p l1 = true →  memb S p l2 = false.
         
-lemma bisim_correct: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} → 
+lemma bisim_correct: ∀S.∀e1,e2:pre S.\sem{e1}≐\sem{e2} → 
  ∀l,n.∀frontier,visited:list ((pre S)×(pre S)).
  |space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→
  all_reachable S e1 e2 visited →  
@@ -544,7 +542,7 @@ definition equiv ≝ λSig.λre1,re2:re Sig.
   (bisim ? sig n [〈e1,e2〉] []).
 
 theorem euqiv_sem : ∀Sig.∀e1,e2:re Sig.
-   \fst (equiv ? e1 e2) = true ↔ \sem{e1} =1 \sem{e2}.
+   \fst (equiv ? e1 e2) = true ↔ \sem{e1} ≐ \sem{e2}.
 #Sig #re1 #re2 %
   [#H @eqP_trans [|@eqP_sym @re_embedding] @eqP_trans [||@re_embedding]
    cut (equiv ? re1 re2 = 〈true,\snd (equiv ? re1 re2)〉)
@@ -610,10 +608,75 @@ normalize // qed.
 definition exp10 ≝ a·a·a·a·a·a·a·a·a·a·a·a·(a^* ).
 definition exp11 ≝ (a·a·a·a·a + a·a·a·a·a·a·a)^*.
 
-example ex2 : \fst (equiv ? (exp10+exp11) exp10) = true.
+example ex2 : \fst (equiv ? (exp10+exp11) exp11) = false.
 normalize // qed.
 
+definition exp12 ≝
+  (a·a·a·a·a·a·a·a)·(a·a·a·a·a·a·a·a)·(a·a·a·a·a·a·a·a)·(a^* ).
+  
+example ex3 : \fst (equiv ? (exp12+exp11) exp11) = true.
+normalize // qed.
 
+let rec raw (n:nat) ≝
+  match n with
+  [ O ⇒ a
+  | S p ⇒ a · (raw p)
+  ].
+
+let rec alln (n:nat) ≝
+  match n with
+  [O ⇒ ϵ
+  |S m ⇒ raw m + alln m
+  ].
 
+definition testA ≝ λx,y,z,b.
+  let e1 ≝ raw x in
+  let e2 ≝ raw y in
+  let e3 ≝ (raw z) · a^* in
+  let e4 ≝ (e1 + e2)^* in
+  \fst (equiv ? (e3+e4) e4) = b.
+  
+example ex4 : testA 2 4 7 true.
+normalize // qed.
+
+example ex5 : testA 3 4 10 false.
+normalize // qed.
+
+example ex6 : testA 3 4 11 true.
+normalize // qed.
+
+example ex7 : testA 4 5 18 false.
+normalize // qed.
+
+example ex8 : testA 4 5 19 true.
+normalize // qed.
+
+example ex9 : testA 4 6 22 false.
+normalize // qed.
 
-\v
\ No newline at end of file
+example ex10 : testA 4 6 23 true.
+normalize // qed.
+
+definition testB ≝ λn,b.
+  \fst (equiv ? ((alln n)·((raw n)^* )) a^* ) = b.
+
+example ex11 : testB 6 true.
+normalize // qed.
+
+example ex12 : testB 8 true.
+normalize // qed.
+
+example ex13 : testB 10 true.
+normalize // qed.
+
+example ex14 : testB 12 true.
+normalize // qed.
+
+example ex15 : testB 14 true.
+normalize // qed.
+
+example ex16 : testB 16 true.
+normalize // qed.
+
+example ex17 : testB 18 true.
+normalize // qed.