- we set up the support for the "bt-reduction" of Automath literature

[helm.git] / weblib / tutorial / chapter9.ma

diff --git a/weblib/tutorial/chapter9.ma b/weblib/tutorial/chapter9.ma
index c68c8f1a0e978c2188fd53ef313c7a3c0e355501..51fe4a40293090b2945a284dbfca9ce71deccadf 100644 (file)
--- a/weblib/tutorial/chapter9.ma
+++ b/weblib/tutorial/chapter9.ma
@@ -11,26 +11,26 @@ lifted operators of the previous section:
 
 include "tutorial/chapter8.ma".
 
 
 include "tutorial/chapter8.ma".
 

-let rec move (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (x:S) (E: \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on E : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S ≝
+\ 5img class="anchor" src="icons/tick.png" id="move"\ 6let rec move (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (x:S) (E: \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on E : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S ≝

  match E with
  match E with

-  [ pz ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 S, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
-  | pe ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
-  | ps y ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
-  | pp y ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y, x \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6= y 〉
+  [ pz ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 S, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+  | pe ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+  | ps y ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+  | pp y ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y, x \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6

   | po e1 e2 ⇒ (move ? x e1) \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 (move ? x e2) 
   | pc e1 e2 ⇒ (move ? x e1) \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 (move ? x e2)
   | po e1 e2 ⇒ (move ? x e1) \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 (move ? x e2) 
   | pc e1 e2 ⇒ (move ? x e1) \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 (move ? x e2)

-  | pk e ⇒ (move ? x e)\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6⊛ ].
+  | pk e ⇒ (move ? x e)\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6 ].

   
   

-lemma move_plus: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="move_plus"\ 6lemma move_plus: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.

   \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i1) \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i2).
 // qed.
 
   \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i1) \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i2).
 // qed.
 

-lemma move_cat: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="move_cat"\ 6lemma move_cat: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.

   \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i1) \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i2).
 // qed.
 
   \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i1) \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i2).
 // qed.
 

-lemma move_star: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
-  \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i)\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6⊛.
+\ 5img class="anchor" src="icons/tick.png" id="move_star"\ 6lemma move_star: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+  \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i)\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6.

 // qed.
 
 (*
 // qed.
 
 (*


@@ -43,31 +43,31 @@ For a, we have two possible positions (all other points gets erased); the innerm
 point stops in front of the final b, while the other one broadcast inside (b^*a + b)b, 
 so
  
 point stops in front of the final b, while the other one broadcast inside (b^*a + b)b, 
 so
  

-      move((•a + ϵ)((•b)*•a + •b)b,a) = 〈(a + ϵ)((•b)^*•a + •b)•b, false〉
+      move((•a + ϵ)((•b)*•a + •b)b,a) = 〈(a + ϵ)((•b)^*•a + •b)•b, false〉

 
 For b, we have two positions too. The innermost point stops in front of the final b too, 
 while the other point reaches the end of b* and must go back through b*a:  
     
 
 For b, we have two positions too. The innermost point stops in front of the final b too, 
 while the other point reaches the end of b* and must go back through b*a:  
     

-      move((•a + ϵ)((•b)*•a + •b)b ,b) = 〈(a +  ϵ)((•b)*•a + b)•b, false〉
+      move((•a + ϵ)((•b)*•a + •b)b ,b) = 〈(a +  ϵ)((•b)*•a + b)•b, false〉

 
 *)
 
 
 *)
 

-definition pmove ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λx:S.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).
+\ 5img class="anchor" src="icons/tick.png" id="pmove"\ 6definition pmove ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λx:S.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).

 
 

-lemma pmove_def : ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b. 
-  \ 5a href="cic:/matita/tutorial/chapter9/pmove.def(7)"\ 6pmove\ 5/a\ 6 ? x \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i.
+\ 5img class="anchor" src="icons/tick.png" id="pmove_def"\ 6lemma pmove_def : ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b. 
+  \ 5a href="cic:/matita/tutorial/chapter9/pmove.def(7)"\ 6pmove\ 5/a\ 6 ? x \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i.

 // qed.
 
 // qed.
 

-lemma eq_to_eq_hd: ∀A.∀l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀a,b. 
-  a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l2 → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b.
+\ 5img class="anchor" src="icons/tick.png" id="eq_to_eq_hd"\ 6lemma eq_to_eq_hd: ∀A.∀l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀a,b. 
+  a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l2 → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b.

 #A #l1 #l2 #a #b #H destruct //
 qed. 
 
 (* Obviously, a move does not change the carrier of the item, as one can easily 
 prove by induction on the item. *)
 
 #A #l1 #l2 #a #b #H destruct //
 qed. 
 
 (* Obviously, a move does not change the carrier of the item, as one can easily 
 prove by induction on the item. *)
 

-lemma same_kernel: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
-  \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a i)| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|.
+\ 5img class="anchor" src="icons/tick.png" id="same_kernel"\ 6lemma same_kernel: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+  \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a i)\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6.

 #S #a #i elim i //
   [#i1 #i2 #H1 #H2 >\ 5a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"\ 6move_cat\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter8/erase_odot.def(7)"\ 6erase_odot\ 5/a\ 6 //
   |#i1 #i2 #H1 #H2 >\ 5a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"\ 6move_plus\ 5/a\ 6 whd in ⊢ (??%%); // 
 #S #a #i elim i //
   [#i1 #i2 #H1 #H2 >\ 5a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"\ 6move_cat\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter8/erase_odot.def(7)"\ 6erase_odot\ 5/a\ 6 //
   |#i1 #i2 #H1 #H2 >\ 5a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"\ 6move_plus\ 5/a\ 6 whd in ⊢ (??%%); // 


@@ -77,14 +77,14 @@ qed.
 (* Here is our first, major result, stating the correctness of the
 move operation. The proof is a simple induction on i. *)
 
 (* Here is our first, major result, stating the correctness of the
 move operation. The proof is a simple induction on i. *)
 

-theorem move_ok:
+\ 5img class="anchor" src="icons/tick.png" id="move_ok"\ 6theorem move_ok:

  ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. 
  ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. 

-   \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a i} w \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w).
+   \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a i\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w).

 #S #a #i elim i 
   [normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
   |normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
   |normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
 #S #a #i elim i 
   [normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
   |normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
   |normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/

-  |normalize #x #w cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x)) #H >H normalize
+  |normalize #x #w cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x)) #H >H normalize

     [>(\P H) % [* // #bot @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 //| #H1 destruct /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
     |% [@\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 |#H1 cases (\Pf H) #H2 @H2 destruct //]
     ]
     [>(\P H) % [* // #bot @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 //| #H1 destruct /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
     |% [@\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 |#H1 cases (\Pf H) #H2 @H2 destruct //]
     ]


@@ -107,47 +107,47 @@ qed.
 
 notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
 
 
 notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
 

-let rec moves (S : DeqSet) w e on w : pre S ≝
+\ 5img class="anchor" src="icons/tick.png" id="moves"\ 6let rec moves (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) w e on w : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S ≝

  match w with
   [ nil ⇒ e
  match w with
   [ nil ⇒ e

-  | cons x w' ⇒ w' ↦* (move S x (\fst e))]. 
+  | cons x w' ⇒ w' ↦* (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e))]. 

 
 

-lemma moves_empty: ∀S:DeqSet.∀e:pre S. 
-  moves ? [ ] e = e.
+\ 5img class="anchor" src="icons/tick.png" id="moves_empty"\ 6lemma moves_empty: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. 
+  \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e.

 // qed.
 
 // qed.
 

-lemma moves_cons: ∀S:DeqSet.∀a:S.∀w.∀e:pre S. 
-  moves ? (a::w)  e = moves ? w (move S a (\fst e)).
+\ 5img class="anchor" src="icons/tick.png" id="moves_cons"\ 6lemma moves_cons: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀w.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. 
+  \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w)  e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e)).

 // qed.
 
 // qed.
 

-lemma moves_left : ∀S,a,w,e. 
-  moves S (w@[a]) e = move S a (\fst (moves S w e)). 
-#S #a #w elim w // #x #tl #Hind #e >moves_cons >moves_cons //
+\ 5img class="anchor" src="icons/tick.png" id="moves_left"\ 6lemma moves_left : ∀S,a,w,e. 
+  \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S (w\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6)) e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e)). 
+#S #a #w elim w // #x #tl #Hind #e >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6 //

 qed.
 
 qed.
 

-lemma not_epsilon_sem: ∀S:DeqSet.∀a:S.∀w: word S. ∀e:pre S. 
-  iff ((a::w) ∈ e) ((a::w) ∈ \fst e).
+\ 5img class="anchor" src="icons/tick.png" id="not_epsilon_sem"\ 6lemma not_epsilon_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. ∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. 
+  \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 ((a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w) \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e) ((a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w) \ 5a title="in_pl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).

 #S #a #w * #i #b cases b normalize 
 #S #a #w * #i #b cases b normalize 

-  [% /2/ * // #H destruct |% normalize /2/]
+  [% /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ * // #H destruct |% normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/]

 qed.
 
 qed.
 

-lemma same_kernel_moves: ∀S:DeqSet.∀w.∀e:pre S.
-  |\fst (moves ? w e)| = |\fst e|.
+\ 5img class="anchor" src="icons/tick.png" id="same_kernel_moves"\ 6lemma same_kernel_moves: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀w.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+  \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e)\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6.

 #S #w elim w //
 qed.
 
 #S #w elim w //
 qed.
 

-theorem decidable_sem: ∀S:DeqSet.∀w: word S. ∀e:pre S. 
-   (\snd (moves ? w e) = true) ↔ \sem{e} w.
+\ 5img class="anchor" src="icons/tick.png" id="decidable_sem"\ 6theorem decidable_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. ∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. 
+   (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w.

 #S #w elim w 
 #S #w elim w 

- [* #i #b >moves_empty cases b % /2/
- |#a #w1 #Hind #e >moves_cons
-  @iff_trans [||@iff_sym @not_epsilon_sem]
-  @iff_trans [||@move_ok] @Hind
+ [* #i #b >\ 5a href="cic:/matita/tutorial/chapter9/moves_empty.def(8)"\ 6moves_empty\ 5/a\ 6 cases b % /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter7/true_to_epsilon.def(9)"\ 6true_to_epsilon\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/\ 5span class="error" title="error location"\ 6\ 5/span\ 6 #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |#a #w1 #Hind #e >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6
+  @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6 [||@\ 5a href="cic:/matita/basics/logic/iff_sym.def(2)"\ 6iff_sym\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter9/not_epsilon_sem.def(9)"\ 6not_epsilon_sem\ 5/a\ 6]
+  @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6 [||@\ 5a href="cic:/matita/tutorial/chapter9/move_ok.def(14)"\ 6move_ok\ 5/a\ 6] @Hind

  ]
 qed.
 
 (* It is now clear that we can build a DFA D_e for e by taking pre as states, 
  ]
 qed.
 
 (* It is now clear that we can build a DFA D_e for e by taking pre as states, 

-and move as transition function; the initial state is •(e) and a state 〈i,b〉 is 
+and move as transition function; the initial state is •(e) and a state 〈i,b〉 is 

 final if and only if b is true. The fact that states in D_e are finite is obvious: 
 in fact, their cardinality is at most 2^{n+1} where n is the number of symbols in 
 e. This is one of the advantages of pointed regular expressions w.r.t. derivatives, 
 final if and only if b is true. The fact that states in D_e are finite is obvious: 
 in fact, their cardinality is at most 2^{n+1} where n is the number of symbols in 
 e. This is one of the advantages of pointed regular expressions w.r.t. derivatives, 


@@ -158,11 +158,11 @@ Let us discuss a couple of examples.
 \ 5b\ 6Example\ 5/b\ 6. 
 Below is the DFA associated with the regular expression (ac+bc)*.
 
 \ 5b\ 6Example\ 5/b\ 6. 
 Below is the DFA associated with the regular expression (ac+bc)*.
 

-\ 5img src="acUbc.gif" alt="some_text"/\ 6  
+\ 5img src="http://www.cs.unibo.it/~asperti/FIGURES/acUbc.gif" alt="DFA for (ac+bc)"\ 6  

 
 The graphical description of the automaton is the traditional one, with nodes for 
 states and labelled arcs for transitions. Unreachable states are not shown.
 
 The graphical description of the automaton is the traditional one, with nodes for 
 states and labelled arcs for transitions. Unreachable states are not shown.

-Final states are emphasized by a double circle: since a state 〈e,b〉 is final if and 
+Final states are emphasized by a double circle: since a state 〈e,b〉 is final if and 

 only if b is true, we may just label nodes with the item.
 The automaton is not minimal: it is easy to see that the two states corresponding to 
 the items (a•c +bc)* and (ac+b•c)* are equivalent (a way to prove it is to observe 
 only if b is true, we may just label nodes with the item.
 The automaton is not minimal: it is easy to see that the two states corresponding to 
 the items (a•c +bc)* and (ac+b•c)* are equivalent (a way to prove it is to observe 


@@ -176,60 +176,75 @@ Let us consider a more complex case.
 \ 5b\ 6Example\ 5/b\ 6. 
 Starting form the regular expression (a+ϵ)(b*a + b)b, we obtain the following automaton.
 
 \ 5b\ 6Example\ 5/b\ 6. 
 Starting form the regular expression (a+ϵ)(b*a + b)b, we obtain the following automaton.
 

-\ 5img src="automaton.pdf" alt="some_text"/\ 6 
+\ 5img src="http://www.cs.unibo.it/~asperti/FIGURES/automaton.gif" alt="DFA for (a+ϵ)(b*a + b)b"\ 6 

 
 Remarkably, this DFA is minimal, testifying the small number of states produced by our 
 
 Remarkably, this DFA is minimal, testifying the small number of states produced by our 

-technique (the pair ofstates $6-8$ and $7-9$ differ for the fact that $6$ and $7$ 
-are final, while $8$ and $9$ are not). 
-*) 
+technique (the pair of states 6-8 and 7-9 differ for the fact that 6 and 7 
+are final, while 8 and 9 are not). 

 
 

-(************************ pit state ***************************)
-definition pit_pre ≝ λS.λi.〈blank S (|i|), false〉. 

 
 

-let rec occur (S: DeqSet) (i: re S) on i ≝  
+\ 5h2\ 6Move to pit\ 5/h2\ 6. 
+
+We conclude this chapter with a few properties of the move opertions in relation
+with the pit state. *)
+
+\ 5img class="anchor" src="icons/tick.png" id="pit_pre"\ 6definition pit_pre ≝ λS.λi.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6), \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6. 
+
+(* The following function compute the list of characters occurring in a given
+item i. *)
+
+\ 5img class="anchor" src="icons/tick.png" id="occur"\ 6let rec occur (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (i: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on i ≝  

   match i with
   match i with

-  [ z ⇒ [ ]
-  | e ⇒ [ ]
-  | s y ⇒ [y]
-  | o e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2) 
-  | c e1 e2 ⇒ unique_append ? (occur S e1) (occur S e2) 
+  [ z ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6
+  | e ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6
+  | s y ⇒ y\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6
+  | o e1 e2 ⇒ \ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (occur S e1) (occur S e2) 
+  | c e1 e2 ⇒ \ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (occur S e1) (occur S e2) 

   | k e ⇒ occur S e].
 
   | k e ⇒ occur S e].
 

-lemma not_occur_to_pit: ∀S,a.∀i:pitem S. memb S a (occur S (|i|)) ≠ true →
-  move S a i  = pit_pre S i.
+(* If a symbol a does not occur in i, then move(i,a) gets to the
+pit state. *)
+
+\ 5img class="anchor" src="icons/tick.png" id="not_occur_to_pit"\ 6lemma not_occur_to_pit: ∀S,a.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6)) \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+  \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a i  \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i.

 #S #a #i elim i //
 #S #a #i elim i //

-  [#x normalize cases (a==x) normalize // #H @False_ind /2/
-  |#i1 #i2 #Hind1 #Hind2 #H >move_cat 
-   >Hind1 [2:@(not_to_not … H) #H1 @sublist_unique_append_l1 //]
-   >Hind2 [2:@(not_to_not … H) #H1 @sublist_unique_append_l2 //] //
-  |#i1 #i2 #Hind1 #Hind2 #H >move_plus 
-   >Hind1 [2:@(not_to_not … H) #H1 @sublist_unique_append_l1 //]
-   >Hind2 [2:@(not_to_not … H) #H1 @sublist_unique_append_l2 //] //
-  |#i #Hind #H >move_star >Hind // 
+  [#x normalize cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x) normalize // #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+  |#i1 #i2 #Hind1 #Hind2 #H >\ 5a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"\ 6move_cat\ 5/a\ 6 
+   >Hind1 [2:@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"\ 6sublist_unique_append_l1\ 5/a\ 6 //]
+   >Hind2 [2:@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l2.def(6)"\ 6sublist_unique_append_l2\ 5/a\ 6 //] //
+  |#i1 #i2 #Hind1 #Hind2 #H >\ 5a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"\ 6move_plus\ 5/a\ 6 
+   >Hind1 [2:@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"\ 6sublist_unique_append_l1\ 5/a\ 6 //]
+   >Hind2 [2:@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l2.def(6)"\ 6sublist_unique_append_l2\ 5/a\ 6 //] //
+  |#i #Hind #H >\ 5a href="cic:/matita/tutorial/chapter9/move_star.def(7)"\ 6move_star\ 5/a\ 6 >Hind // 

   ]
 qed.
 
   ]
 qed.
 

-lemma move_pit: ∀S,a,i. move S a (\fst (pit_pre S i)) = pit_pre S i.
+(* We cannot escape form the pit state. *)
+
+\ 5img class="anchor" src="icons/tick.png" id="move_pit"\ 6lemma move_pit: ∀S,a,i. \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i.

 #S #a #i elim i //
 #S #a #i elim i //

-  [#i1 #i2 #Hind1 #Hind2 >move_cat >Hind1 >Hind2 // 
-  |#i1 #i2 #Hind1 #Hind2 >move_plus >Hind1 >Hind2 // 
-  |#i #Hind >move_star >Hind //
+  [#i1 #i2 #Hind1 #Hind2 >\ 5a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"\ 6move_cat\ 5/a\ 6 >Hind1 >Hind2 // 
+  |#i1 #i2 #Hind1 #Hind2 >\ 5a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"\ 6move_plus\ 5/a\ 6 >Hind1 >Hind2 // 
+  |#i #Hind >\ 5a href="cic:/matita/tutorial/chapter9/move_star.def(7)"\ 6move_star\ 5/a\ 6 >Hind //

   ]
 qed. 
 
   ]
 qed. 
 

-lemma moves_pit: ∀S,w,i. moves S w (pit_pre S i) = pit_pre S i.
+\ 5img class="anchor" src="icons/tick.png" id="moves_pit"\ 6lemma moves_pit: ∀S,w,i. \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i.

 #S #w #i elim w // 
 qed. 
  
 #S #w #i elim w // 
 qed. 
  

-lemma to_pit: ∀S,w,e. ¬ sublist S w (occur S (|\fst e|)) →
- moves S w e = pit_pre S (\fst e).
+(* If any character in w does not occur in i, then moves(i,w) gets
+to the pit state. *)
+
+\ 5img class="anchor" src="icons/tick.png" id="to_pit"\ 6lemma to_pit: ∀S,w,e. \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6)) →
+ \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).

 #S #w elim w
 #S #w elim w

-  [#e * #H @False_ind @H normalize #a #abs @False_ind /2/
-  |#a #tl #Hind #e #H cases (true_or_false (memb S a (occur S (|\fst e|))))
-    [#Htrue >moves_cons whd in ⊢ (???%); <(same_kernel … a) 
-     @Hind >same_kernel @(not_to_not … H) #H1 #b #memb cases (orb_true_l … memb)
+  [#e * #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @H normalize #a #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+  |#a #tl #Hind #e #H cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6))))
+    [#Htrue >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6 whd in ⊢ (???%); <(\ 5a href="cic:/matita/tutorial/chapter9/same_kernel.def(8)"\ 6same_kernel\ 5/a\ 6 … a) 
+     @Hind >\ 5a href="cic:/matita/tutorial/chapter9/same_kernel.def(8)"\ 6same_kernel\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 #b #memb cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … memb)

       [#H2 >(\P H2) // |#H2 @H1 //]
       [#H2 >(\P H2) // |#H2 @H1 //]

-    |#Hfalse >moves_cons >not_occur_to_pit // >Hfalse /2/ 
+    |#Hfalse >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter9/not_occur_to_pit.def(8)"\ 6not_occur_to_pit\ 5/a\ 6 // >Hfalse /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/bool/eqnot_to_noteq.def(4)"\ 6eqnot_to_noteq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 

     ]
   ]
     ]
   ]

-qed.
\ No newline at end of file
+qed.