milestone in basic_2 with additions in static_2

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

diff --git a/weblib/tutorial/chapter9.ma b/weblib/tutorial/chapter9.ma
index 147bad1b9dd656b11c23d0c783dadcca5b89c9e6..ba759d47cf34a8c20c42c5db01f8d8e42f86cdfc 100644 (file)
--- a/weblib/tutorial/chapter9.ma
+++ b/weblib/tutorial/chapter9.ma
@@ -13,13 +13,13 @@ 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 ≝
  match E with
 
 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 ≝
  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 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.
   \ 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).
   
 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.
   \ 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).


@@ -30,7 +30,7 @@ lemma move_cat: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6D
 // 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.
 // 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⊛.
+  \ 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.
 
 (*


@@ -43,23 +43,23 @@ 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).
 
 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. 
 
 *)
 
 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).
 
 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.
+  \ 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.
 
 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. 
 // 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.
+  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. 
 
 #A #l1 #l2 #a #b #H destruct //
 qed. 
 


@@ -67,7 +67,7 @@ qed.
 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.
 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|.
+  \ 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 ⊢ (??%%); // 


@@ -79,12 +79,12 @@ move operation. The proof is a simple induction on i. *)
 
 theorem 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. 
 
 theorem 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. 

-   \ 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 //]
     ]


@@ -113,33 +113,33 @@ let rec moves (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6Deq
   | 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:\ 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. 
   | 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:\ 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 ] e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e.
+  \ 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.
 
 lemma 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. 
 // qed.
 
 lemma 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:w)  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)).
+  \ 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.
 
 lemma moves_left : ∀S,a,w,e. 
 // qed.
 
 lemma 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="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)). 
+  \ 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.
 
 lemma 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. 
 #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.
 
 lemma 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:w) \ 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:w) \ 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).
+  \ 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 
   [% /\ 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.
 
 lemma 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.
 #S #a #w * #i #b cases b normalize 
   [% /\ 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.
 
 lemma 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="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\ 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.
 
 theorem 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. 
 #S #w elim w //
 qed.
 
 theorem 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} w.
+   (\ 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 >\ 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/
+ [* #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/ #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
  |#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


@@ -147,7 +147,7 @@ theorem decidable_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,
 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, 


@@ -162,7 +162,7 @@ Below is the DFA associated with the regular expression (ac+bc)*.
 
 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 


@@ -188,16 +188,16 @@ are final, while 8 and 9 are not).
 We conclude this chapter with a few properties of the move opertions in relation
 with the pit state. *)
 
 We conclude this chapter with a few properties of the move opertions in relation
 with the pit state. *)
 

-definition 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 href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉. 
+definition 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. *)
 
 let 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
 
 (* The following function compute the list of characters occurring in a given
 item i. *)
 
 let 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

-  [ z ⇒ \ 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 ]
-  | s y ⇒ y\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
+  [ 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].
   | 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].


@@ -205,10 +205,10 @@ let rec occur (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqS
 (* If a symbol a does not occur in i, then move(i,a) gets to the
 pit state. *)
 
 (* If a symbol a does not occur in i, then move(i,a) gets to the
 pit state. *)
 

-lemma 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="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 →
+lemma 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 //
   \ 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 //

-  [#x normalize cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x) 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/
+  [#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_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 //] //


@@ -236,11 +236,11 @@ qed.
 (* If any character in w does not occur in i, then moves(i,w) gets
 to the pit state. *)
 
 (* If any character in w does not occur in i, then moves(i,w) gets
 to the pit state. *)
 

-lemma 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|)) →
+lemma 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
   [#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/
  \ 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
   [#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|))))
+  |#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 //]
     [#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 //]