commit by user andrea

diff --git a/weblib/While/syntax.ma b/weblib/While/syntax.ma
new file mode 100644 (file)
index 0000000..b55d89d
--- /dev/null
+++ b/weblib/While/syntax.ma
@@ -0,0 +1,58 @@
+(* While Syntax *)
+
+include "basics/logic.ma".
+include "basics/types.ma".
+include "basics/bool.ma".
+include "arithmetics/nat.ma".
+
+(* Variables *)
+
+inductive Var : Type[0] ≝
+  Id : nat → Var.
+ 
+(* Arithmetic expressions *)
+
+inductive Aexp : Type[0] ≝
+  | Const : \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Aexp
+  | Aid : \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6Var → Aexp
+  | Add : Aexp → Aexp → Aexp
+  | Sub : Aexp → Aexp → Aexp
+  | Mul : Aexp → Aexp → Aexp
+  | Div : Aexp → Aexp → Aexp
+  | Mod : Aexp → Aexp → Aexp
+ .
+
+interpretation "Aexp plus" 'plus x y = (Add x y).
+interpretation "Aexp minus" 'plus x y = (Sub x y).
+interpretation "Aexp times" 'times x y = (Mul x y).
+
+(* Boolean expressions *)
+inductive Bexp : Type[0] ≝
+  | BFalse : Bexp
+  | BTtrue : Bexp
+  | BNot : Bexp → Bexp
+  | BAnd : Bexp → Bexp → Bexp
+  | BOr : Bexp → Bexp → Bexp
+  | Eq : \ 5a href="cic:/matita/While/syntax/Aexp.ind(1,0,0)"\ 6Aexp\ 5/a\ 6 → \ 5a href="cic:/matita/While/syntax/Aexp.ind(1,0,0)"\ 6Aexp\ 5/a\ 6 → Bexp
+  | LE : \ 5a href="cic:/matita/While/syntax/Aexp.ind(1,0,0)"\ 6Aexp\ 5/a\ 6 → \ 5a href="cic:/matita/While/syntax/Aexp.ind(1,0,0)"\ 6Aexp\ 5/a\ 6 → Bexp
+.
+
+interpretation "Bexp not" 'not x = (BNot x).
+interpretation "Bexp and" 'and x y = (BAnd x y).
+interpretation "Bexp or" 'or x y = (BOr x y).
+interpretation "Bexp 'less or equal to'" 'leq x y = (LE x y).
+
+(* Commands *)
+inductive Cmd : Type[0] ≝
+  | Skip : Cmd
+  | Assign : Var → \ 5a href="cic:/matita/While/syntax/Aexp.ind(1,0,0)"\ 6Aexp\ 5/a\ 6 → Cmd
+  | Seq : Cmd → Cmd → Cmd
+  | If : \ 5a href="cic:/matita/While/syntax/Bexp.ind(1,0,0)"\ 6Bexp\ 5/a\ 6 → Cmd → Cmd → Cmd
+  | While : \ 5a href="cic:/matita/While/syntax/Bexp.ind(1,0,0)"\ 6Bexp\ 5/a\ 6 → Cmd → Cmd
+.
+
+notation "hvbox(c1 break ; c2)"
+  left associative with precedence 47 for @{Seq $c1 $c2}.
+
+notation "hvbox(var break <- val)"
+  left associative with precedence 48 for @{Assign $var $val}.
\ No newline at end of file

diff --git a/weblib/prova.ma b/weblib/prova.ma
new file mode 100644 (file)
index 0000000..5fe5dce
--- /dev/null
+++ b/weblib/prova.ma
@@ -0,0 +1 @@
+(* new script *)
\ No newline at end of file

diff --git a/weblib/tutorial/chapter9.ma b/weblib/tutorial/chapter9.ma
index 51fe4a40293090b2945a284dbfca9ce71deccadf..ba759d47cf34a8c20c42c5db01f8d8e42f86cdfc 100644 (file)
--- a/weblib/tutorial/chapter9.ma
+++ b/weblib/tutorial/chapter9.ma
@@ -11,9 +11,9 @@ lifted operators of the previous section:
 
 include "tutorial/chapter8.ma".
 
-\ 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 ≝
+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 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+  [ 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
@@ -21,15 +21,15 @@ include "tutorial/chapter8.ma".
   | 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\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6 ].
   
-\ 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.
+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).
 // qed.
 
-\ 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.
+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.
   \ 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.
 
-\ 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.
+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="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.
 
@@ -52,13 +52,13 @@ while the other point reaches the end of b* and must go back through b*a:
 
 *)
 
-\ 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).
+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_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. 
+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="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.
 
-\ 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. 
+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\ 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. 
@@ -66,7 +66,7 @@ qed.
 (* Obviously, a move does not change the carrier of the item, as one can easily 
 prove by induction on the item. *)
 
-\ 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.
+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="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 //
@@ -77,7 +77,7 @@ qed.
 (* Here is our first, major result, stating the correctness of the
 move operation. The proof is a simple induction on i. *)
 
-\ 5img class="anchor" src="icons/tick.png" id="move_ok"\ 6theorem move_ok:
+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\ 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 
@@ -107,39 +107,39 @@ qed.
 
 notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
 
-\ 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 ≝
+let 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
   | 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))]. 
 
-\ 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. 
+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 \ 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.
 
-\ 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. 
+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\ 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.
 
-\ 5img class="anchor" src="icons/tick.png" id="moves_left"\ 6lemma moves_left : ∀S,a,w,e. 
+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="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.
 
-\ 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. 
+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\ 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.
 
-\ 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.
+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="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.
 
-\ 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. 
+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\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 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
@@ -188,12 +188,12 @@ 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. *)
 
-\ 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. 
+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. *)
 
-\ 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 ≝  
+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 \ 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
@@ -205,7 +205,7 @@ item 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 →
+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 //
   [#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/
@@ -221,7 +221,7 @@ qed.
 
 (* 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.
+lemma 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 //
   [#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 // 
@@ -229,14 +229,14 @@ qed.
   ]
 qed. 
 
-\ 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.
+lemma 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. 
  
 (* 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)) →
+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/
@@ -247,4 +247,4 @@ to the pit state. *)
     |#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.
+qed.
\ No newline at end of file