X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter1.ma;h=4cf145d60f53f72c70b08846444700cd9b1afc7e;hb=a82d87a4c2ff9f518385fba7357dafb632a22882;hp=c2523f38c803f8472106b853b6d6e4e1d4d0d565;hpb=3476afc7e71ac34beec061b94cf712e5ff85802f;p=helm.git

diff --git a/weblib/tutorial/chapter1.ma b/weblib/tutorial/chapter1.ma
index c2523f38c..4cf145d60 100644
--- a/weblib/tutorial/chapter1.ma
+++ b/weblib/tutorial/chapter1.ma
@@ -1,35 +1,57 @@
-
-(*
-In a system like Matita's very few things are built-in: not even booleans or 
-logical connectives. The main philosophy od the system is to let you define your 
-own data-types and functions using a powerful computational mechanism based on 
-the declaration of inductive types. 
-
-Let us start this tutorial with a simple example based on the following well known
-problem.
-
-h2 class="section"The goat, the wolf and the cabbage/h2div class="paragraph"/divA farmer need to tranfer a goat, a wolf and a cabbage across a river, but there is
-only one place available on his boat. Furthermore, the goat will eat the cabbage if 
-they are left alone on the same bank, and similarly the wolf will eat the goat. 
-The problem consists in bringing all three item safely across the river. 
-
-Our first data type defines the two banks of the river, which we will call east and 
-west. It is a simple example of enumerated type, defined by explicitly declaring all 
-its elements. The type itself is called "bank".
-Before giving its definition we "include" the file "logic.ma" that contains a a few 
-preliminary notions not worth discussing here.
+(* 
+h1Matita Interactive Tutorial/h1
+This is an interactive tutorial. To let you interact on line with the system, 
+you must first of all register yourself.
+
+Before starting, let us briefly explain the meaning of the menu buttons. 
+With the Advance and Retract buttons you respectively perform and undo single 
+computational steps. Each step consists in reading a user command, and processing
+it. The part of the user input file (called script) already executed by the 
+system will be colored, and will not be editable any more. The advance bottom 
+will also automatically advance the focus of the window, but you can inspect the 
+whole file using the scroll bars, as usual. Comments are skipped.
+Try to advance and retract a few steps, to get the feeling of the system. You can 
+also come back here by using the top button, that takes you at the beginning of 
+a file. The play button is meant to process the script up to a position 
+previously selected by the user; the bottom button execute the whole script. 
+That's it: we arespan style="font-family: Verdana,sans-serif;" /spannow ready to start.
+
+h2 class="section"Data types, functions and theorems/h2
+Matita is both a programming language and a theorem proving environment:
+you define datatypes and programs, and then prove properties on them.
+Very few things are built-in: not even booleans or logical connectives
+(but you may of course use libraries, as in normal programming languages). 
+The main philosophy of the system is to let you define your own data-types 
+and functions using a powerful computational mechanism based on the 
+declaration of inductive types. 
+
+Let us start this tutorial with a simple example based on the following well 
+known problem.
+
+bThe goat, the wolf and the cabbage/b
+A farmer need to transfer a goat, a wolf and a cabbage across a river, but there
+is only one place available on his boat. Furthermore, the goat will eat the 
+cabbage if they are left alone on the same bank, and similarly the wolf will eat
+the goat. The problem consists in bringing all three items safely across the 
+river. 
+
+Our first data type defines the two banks of the river, which will be named east
+and west. It is a simple example of enumerated type, defined by explicitly 
+declaring all its elements. The type itself is called "bank".
+Before giving its definition we "include" the file "logic.ma" that contains a 
+few preliminary notions not worth discussing for the moment.
 *)
 
 include "basics/logic.ma".
 
-inductive bank: Type[0] ≝
+img class="anchor" src="icons/tick.png" id="bank"inductive bank: Type[0] ≝
 | east : bank 
 | west : bank.
 
-(* We can now define a simple function computing, for each bank of the river,
-the opposite one. *)
+(* We can now define a simple function computing, for each bank of the river, the
+opposite one. *)
 
-definition opposite ≝ λs.
+img class="anchor" src="icons/tick.png" id="opposite"definition opposite ≝ λs.
 match s with
   [ east ⇒ a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a
   | west ⇒ a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a
@@ -39,9 +61,11 @@ match s with
 us state the property that the opposite bank of east is west; every lemma needs a 
 name for further reference, and we call it "east_to_west". *)
  
-lemma east_to_west : a href="cic:/matita/tutorial/chapter1/opposite.def(1)"opposite/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a.
+img class="anchor" src="icons/tick.png" id="east_to_west"lemma east_to_west : a href="cic:/matita/tutorial/chapter1/opposite.def(1)"opposite/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a.
 
-(* If you stop the execution here you will see a new window on the  right side
+(* 
+h2 class="section"The goal window/h2
+If you stop the execution here you will see a new window on the  right side
 of the screen: it is the goal window, providing a sequent like representation of
 the form
 
@@ -69,170 +93,286 @@ You may use the retract bottom to undo the step, if you wish.
 The new goal west = west is trivial: it is just a consequence of reflexivity.
 Such trivial steps can be just closed in Matita by typing a double slash. 
 We complete the proof by the qed command, that instructs the system to store the
-lemma performing some bookkeeping operations. 
+lemma performing some book-keeping operations. 
 *)
 
 // qed.
 
-lemma west_to_east : change_side west = east.
+(* In exactly the same way, we can prove that the opposite side of west is east.
+In this case, we avoid the unnecessary simplification step: // will take care of 
+it. *) 
+
+img class="anchor" src="icons/tick.png" id="west_to_east"lemma west_to_east : a href="cic:/matita/tutorial/chapter1/opposite.def(1)"opposite/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a.
 // qed.
 
-lemma change_twice : ∀x. change_side (change_side x) = x.
-* //
-qed.
+(*
+h2 class="section"Introduction/h2
+A slightly more complex problem consists in proving that opposite is idempotent *)
 
-inductive item: Type[0] ≝
-| goat : item
-| wolf : item
-| cabbage: item
-| boat : item.
+img class="anchor" src="icons/tick.png" id="idempotent_opposite"lemma idempotent_opposite : ∀x. a href="cic:/matita/tutorial/chapter1/opposite.def(1)"opposite/a (a href="cic:/matita/tutorial/chapter1/opposite.def(1)"opposite/a x) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a x.
 
-(* definition state ≝ item → side. *)
+(* we start the proof moving x from the conclusion into the context, that is a 
+(backward) introduction step. Matita syntax for an introduction step is simply 
+the sharp character followed by the name of the item to be moved into the 
+context. This also allows us to rename the item, if needed: for instance if we 
+wish to rename x into b (since it is a bank), we just type #b. *)
 
-record state : Type[0] ≝
-  {goat_pos : side;
-   wolf_pos : side;
-   cabbage_pos: side;
-   boat_pos : side}.
+#b
 
-definition state_of_item ≝ λs,i.
-match i with
-[ goat ⇒  goat_pos s
-| wolf ⇒  wolf_pos s
-| cabbage ⇒  cabbage_pos s
-| boat ⇒  boat_pos s].
+(* See the effect of the execution in the goal window on the right: b has been 
+added to the context (replacing x in the conclusion); moreover its implicit type 
+"bank" has been made explicit. 
+*)
 
 (*
-definition is_movable ≝ λs,i.
-  state_of_item s i = state_of_item s boat.
- 
-record movable_item (s:state) : Type[0] ≝
-  {elem : item;
-   mov : is_movable s elem}. *)
- 
-definition start :state ≝ 
-  mk_state east east east east.
-
-(* slow
-inductive move : state → state → Type[0] ≝
-| move_goat: ∀g,w,c.
-  move (mk_state (change_side g) w c (change_side g)) (mk_state g w c g) 
-| move_wolf: ∀g,w,c.
-  move (mk_state g (change_side w) c (change_side w)) (mk_state g w c w)
-| move_cabbage: ∀g,w,c.
-  move (mk_state g w (change_side c) (change_side c)) (mk_state g w c c)
-| move_boat: ∀g,w,c,b.
-  move (mk_state g w c (change_side b)) (mk_state g w c b).
+h2 class="section"Case analysis/h2
+But how are we supposed to proceed, now? Simplification cannot help us, since b
+is a variable: just try to call normalize and you will see that it has no effect.
+The point is that we must proceed by cases according to the possible values of b,
+namely east and west. To this aim, you must invoke the cases command, followed by
+the name of the hypothesis (more generally, an arbitrary expression) that must be
+the object of the case analysis (in our case, b).
+*)
+
+cases b
+
+(* Executing the previous command has the effect of opening two subgoals, 
+corresponding to the two cases b=east and b=west: you may switch from one to the 
+other by using the hyperlinks over the top of the goal window. 
+Both goals can be closed by trivial computations, so we may use // as usual.
+If we had to treat each subgoal in a different way, we should focus on each of 
+them in turn, in a way that will be described at the end of this section.
+*)
+
+// qed.
+
+(* 
+h2 class="section"Predicates/h2
+Instead of working with functions, it is sometimes convenient to work with
+predicates. For instance, instead of defining a function computing the opposite 
+bank, we could declare a predicate stating when two banks are opposite to each 
+other. Only two cases are possible, leading naturally to the following 
+definition:
 *)
 
-(*  this is working well *)
-inductive move : state → state → Type[0] ≝
-| move_goat: ∀g,w,c.
-  move (mk_state g w c g) (mk_state (change_side g) w c (change_side g)) 
-| move_wolf: ∀g,w,c.
-  move (mk_state g w c w) (mk_state g (change_side w) c (change_side w))
-| move_cabbage: ∀g,w,c.
-  move (mk_state g w c c) (mk_state g w (change_side c) (change_side c))
-| move_boat: ∀g,w,c,b.
-  move (mk_state g w c b) (mk_state g w c (change_side b)).
+img class="anchor" src="icons/tick.png" id="opp"inductive opp : a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"bank/a → a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"bank/a → Prop ≝ 
+| east_west : opp a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a
+| west_east : opp a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a.
 
+(* In precisely the same way as "bank" is the smallest type containing east and
+west, opp is the smallest predicate containing the two sub-cases east_west and
+weast_east. If you have some familiarity with Prolog, you may look at opp as the
+predicate defined by the two clauses - in this case, the two facts - ast_west and
+west_east.
+
+Between opp and opposite we have the following relation:
+    opp a b iff a = opposite b
+Let us prove it, starting from the left to right implication, first *)
+
+img class="anchor" src="icons/tick.png" id="opp_to_opposite"lemma opp_to_opposite: ∀a,b. a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"opp/a a b → a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter1/opposite.def(1)"opposite/a b.
  
-(* this is working 
-inductive move : state → state → Type[0] ≝
-| move_goat: ∀g,g1,w,c. change_side g = g1 → 
-  move (mk_state g w c g) (mk_state g1 w c g1) 
-| move_wolf: ∀g,w,w1,c. change_side w = w1 → 
-  move (mk_state g w c w) (mk_state g w1 c w1)
-| move_cabbage: ∀g,w,c,c1. change_side c = c1 → 
-  move (mk_state g w c c) (mk_state g w c1 c1)
-| move_boat: ∀g,w,c,b,b1. change_side b = b1 →
-  move (mk_state g w c b) (mk_state g w c b1) 
-.*)
+(* We start the proof introducing a, b and the hypothesis opp a b, that we
+call oppab. *)
+#a #b #oppab
+
+(* Now we proceed by cases on the possible proofs of (opp a b), that is on the 
+possible shapes of oppab. By definition, there are only two possibilities, 
+namely east_west or west_east. Both subcases are trivial, and can be closed by
+automation *)
+
+cases oppab // qed.
+
+(* 
+h2 class="section"Rewriting/h2
+Let us come to the opposite direction. *)
+
+img class="anchor" src="icons/tick.png" id="opposite_to_opp"lemma opposite_to_opp: ∀a,b. a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter1/opposite.def(1)"opposite/a b → a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"opp/a a b.
+
+(* As usual, we start introducing a, b and the hypothesis (a = opposite b), 
+that we call eqa. *)
+
+#a #b #eqa
+
+(* The right way to proceed, now, is by rewriting a into (opposite b). We do
+this by typing ">eqa". If we wished to rewrite in the opposite direction, namely
+opposite b into a, we would have typed "<eqa". *)
+
+>eqa
+
+(* We conclude the proof by cases on b. *)
+
+cases b // qed.
 
 (*
-inductive move : state →state → Type[0] ≝
-| move_goat: ∀s.∀h:is_movable s goat. move s (mk_state (change_side (goat_pos s)) (wolf_pos s) 
-  (cabbage_pos s) (change_side (boat_pos s)))
-| move_wolf: ∀s.∀h:is_movable s wolf. move s (mk_state (goat_pos s) (change_side (wolf_pos s))
-  (cabbage_pos s) (change_side (boat_pos s)))
-| move_cabbage: ∀s.∀h:is_movable s cabbage. move s (mk_state (goat_pos s) (wolf_pos s) 
-  (change_side (cabbage_pos s)) (change_side (boat_pos s)))
-| move_boat: ∀s.move s (mk_state (goat_pos s) (wolf_pos s) 
-  (cabbage_pos s) (change_side (boat_pos s))).
+h2 class="section"Records/h2
+It is time to proceed with our formalization of the farmer's problem. 
+A state of the system is defined by the position of four items: the goat, the 
+wolf, the cabbage, and the boat. The simplest way to declare such a data type
+is to use a record.
 *)
 
-(*
-definition legal_state ≝ λs.
-  goat_pos s = boat_pos s ∨  
-  (goat_pos s = change_side (cabbage_pos s) ∧ 
-   goat_pos s = change_side (wolf_pos s)). *)
-
-inductive legal_state : state → Prop ≝
-| with_boat : ∀g,w,c.legal_state (mk_state g w c g)
-| opposite_side : ∀g,b. 
-    legal_state (mk_state g (change_side g) (change_side g) b).
-  
-inductive reachable : state → state → Prop ≝
-| one : ∀s1,s2.move s1 s2 → reachable s1 s2
-| more : ∀s1,s2,s3. reachable s1 s2 → legal_state s2 → move s2 s3 →
-  reachable s1 s3. 
-
-definition end ≝ mk_state west west west west.
-definition second ≝ mk_state west east east west.
-definition third ≝ mk_state west east east east.
-definition fourth ≝ mk_state west west east west.
-definition fifth ≝ mk_state east west east east.
-definition sixth ≝ mk_state east west west west.
-definition seventh ≝ mk_state east west west east.
-
-lemma problem: reachable start end.
-normalize /8/ qed. 
-
-lemma problem: reachable start end.
-normalize 
-@more [4: @move_goat || 3: //]
-@more [4: @move_boat || 3: //]  
-@more [4: @move_wolf || 3: //]
-@more [4: @move_goat || 3: //]
-@more [4: @move_cabbage || 3: //]
-@more [4: @move_boat || 3: //]
-@one //
-qed.
+img class="anchor" src="icons/tick.png" id="state"record state : Type[0] ≝
+  {goat_pos : a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"bank/a;
+   wolf_pos : a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"bank/a;
+   cabbage_pos: a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"bank/a;
+   boat_pos : a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"bank/a}.
+
+(* When you define a record named foo, the system automatically defines a record 
+constructor named mk_foo. To construct a new record you pass as arguments to 
+mk_foo the values of the record fields *)
+
+img class="anchor" src="icons/tick.png" id="start"definition start ≝ a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a.
+img class="anchor" src="icons/tick.png" id="end"definition end ≝ a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a.
+
+(* We must now define the possible moves. A natural way to do it is in the form 
+of a relation (a binary predicate) over states. *)
+
+img class="anchor" src="icons/tick.png" id="move"inductive move : a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"state/a → a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"state/a → Prop ≝
+| move_goat: ∀g,g1,w,c. a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"opp/a g g1 → move (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a g w c g) (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a g1 w c g1)
+  (* We can move the goat from a bank g to the opposite bank g1 if and only if the
+     boat is on the same bank g of the goat and we move the boat along with it. *)
+| move_wolf: ∀g,w,w1,c. a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"opp/a w w1 → move (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a g w c w) (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a g w1 c w1)
+| move_cabbage: ∀g,w,c,c1.a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"opp/a c c1 → move (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a g w c c) (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a g w c1 c1)
+| move_boat: ∀g,w,c,b,b1. a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"opp/a b b1 → move (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a g w c b) (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a g w c b1).
+
+(* A state is safe if either the goat is on the same bank of the boat, or both 
+the wolf and the cabbage are on the opposite bank of the goat. *)
+
+img class="anchor" src="icons/tick.png" id="safe_state"inductive safe_state : a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"state/a → Prop ≝
+| with_boat : ∀g,w,c.safe_state (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a g w c g)
+| opposite_side : ∀g,g1,b.a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"opp/a g g1 → safe_state (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a g g1 g1 b).
+
+(* Finally, a state y is reachable from x if either there is a single move 
+leading from x to y, or there is a safe state z such that z is reachable from x 
+and there is a move leading from z to y *)
+
+img class="anchor" src="icons/tick.png" id="reachable"inductive reachable : a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"state/a → a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"state/a → Prop ≝
+| one : ∀x,y.a href="cic:/matita/tutorial/chapter1/move.ind(1,0,0)"move/a x y → reachable x y
+| more : ∀x,z,y. span style="text-decoration: underline;"/spanreachable x z → a href="cic:/matita/tutorial/chapter1/safe_state.ind(1,0,0)"safe_state/a z → span style="text-decoration: underline;"/spana href="cic:/matita/tutorial/chapter1/move.ind(1,0,0)"move/a z y → reachable x y.
+
+(* 
+h2 class="section"Automation/h2
+Remarkably, Matita is now able to solve the problem by itslef, provided
+you allow automation to exploit more resources. The command /n/ is similar to
+//, where n is a measure of this complexity: in particular it is a bound to
+the depth of the expected proof tree (more precisely, to the number of nested
+applicative nodes). In this case, there is a solution in six moves, and we
+need a few more applications to handle reachability, and side conditions. 
+The magic number to let automation work is, in this case, 9.  *)
+
+img class="anchor" src="icons/tick.png" id="problem"lemma problem: a href="cic:/matita/tutorial/chapter1/reachable.ind(1,0,0)"reachable/a a href="cic:/matita/tutorial/chapter1/start.def(1)"start/a a href="cic:/matita/tutorial/chapter1/end.def(1)"end/a.
+normalize /span class="autotactic"9span class="autotrace" trace a href="cic:/matita/tutorial/chapter1/reachable.con(0,1,0)"one/a, a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"more/a, a href="cic:/matita/tutorial/chapter1/safe_state.con(0,1,0)"with_boat/a, a href="cic:/matita/tutorial/chapter1/safe_state.con(0,2,0)"opposite_side/a, a href="cic:/matita/tutorial/chapter1/move.con(0,1,0)"move_goat/a, a href="cic:/matita/tutorial/chapter1/move.con(0,2,0)"move_wolf/a, a href="cic:/matita/tutorial/chapter1/move.con(0,3,0)"move_cabbage/a, a href="cic:/matita/tutorial/chapter1/move.con(0,4,0)"move_boat/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"east_west/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"west_east/a/span/span/ qed. 
+
+(* 
+h2 class="section"Application/h2
+Let us now try to derive the proof in a more interactive way. Of course, we
+expect to need several moves to transfer all items from a bank to the other, so 
+we should start our proof by applying "more". Matita syntax for invoking the 
+application of a property named foo is to write "@foo". In general, the philosophy 
+of Matita is to describe each proof of a property P as a structured collection of 
+objects involved in the proof, prefixed by simple modalities (#,<,@,...) explaining 
+the way it is actually used (e.g. for introduction, rewriting, in an applicative 
+step, and so on).
+*)
+
+img class="anchor" src="icons/tick.png" id="problem1"lemma problem1: a href="cic:/matita/tutorial/chapter1/reachable.ind(1,0,0)"reachable/a a href="cic:/matita/tutorial/chapter1/start.def(1)"start/a a href="cic:/matita/tutorial/chapter1/end.def(1)"end/a.
+normalize @a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"more/a
+
+(* 
+h2 class="section"Focusing/h2
+After performing the previous application, we have four open subgoals:
+
+  X : STATE
+  Y : reachable [east,east,east,east] X
+  W : safe_state X
+  Z : move X [west,west,west,west]
+ 
+That is, we must guess a state X, such that this is reachable from start, it is 
+safe, and there is a move leading from X to end. All goals are active, that is
+emphasized by the fact that they are all red. Any command typed by the user is
+normally applied in parallel to all active goals, but clearly we must proceed 
+here is a different way for each of them. The way to do it, is by structuring
+the script using the following syntax: [...|... |...|...] where we typically have
+as many cells inside the brackets as the number of the active subgoals. The
+interesting point is that we can associate with the three symbol "[", "|" and
+"]" a small-step semantics that allow to execute them individually. In particular
+
+- the operator "[" opens a new focusing section for the currently active goals,
+  and focus on the first of them
+- the operator "|" shift the focus to the next goal
+- the operator "]" close the focusing section, falling back to the previous level
+  and adding to it all remaining goals not yet closed
+
+Let us see the effect of the "[" on our proof:
+*)
 
+  [  
+
+(* As you see, only the first goal has now the focus on. Moreover, all goals got
+a progressive numerical label, to help designating them, if needed. 
+We can now proceed in several possible ways. The most straightforward way is to 
+provide the intermediate state, that is [east,west,west,east]. We can do it, by 
+just applying this term. *)
+
+   @(a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a) 
+
+(* This application closes the goal; at present, no goal has the focus on.
+In order to act on the next goal, we must focus on it using the "|" operator. In
+this case, we would like to skip the next goal, and focus on the trivial third 
+subgoal: a simple way to do it, is by retyping "|". The proof that 
+[east,west,west,east] is safe is trivial and can be done with //.*)
+
+  || //
 
 (*
-definition move_item ≝ λs.λi:movable_item s.
-match (elem ? i) with
-[ goat ⇒ mk_state (change_side (goat_pos s)) (wolf_pos s) 
-  (cabbage_pos s) (change_side (boat_pos s))
-| wolf ⇒ mk_state (goat_pos s) (change_side (wolf_pos s)) 
-  (cabbage_pos s) (change_side (boat_pos s))
-| cabbage ⇒ mk_state (goat_pos s) (wolf_pos s) 
-  (change_side (cabbage_pos s)) (change_side (boat_pos s))
-| boat ⇒ mk_state (goat_pos s) (wolf_pos s) 
-  (cabbage_pos s) (change_side (boat_pos s))
-].
-
-definition legal_state ≝ λs. 
-  cabbage_pos s = goat_pos s ∨ 
-  goat_pos s = wolf_pos s → goat_pos s = boat_pos s.
-  
-definition legal_step ≝ λs1,s2. 
-  ∃i.move_item s1 i = s2. 
-
-inductive reachable : state → state → Prop ≝
-| nil : ∀s.reachable s s
-| step : ∀s1,s2,s3. legal_step s1 s2 → legal_state s2 → reachable s2 s3 →
-  reachable s1 s3. 
-
-definition second ≝ mk_state west east east west.
-definition end ≝ mk_state west west west west.
-
-lemma goat_move: ∀s.legal_step s (mk_state (change_side (goat_pos s)) (wolf_pos s) 
-  (cabbage_pos s) (change_side (boat_pos s))).
-
-lemma problem: reachable start second.
-@step [| @ex_intro  // [| @move_goat ] // | //]
-
-  [|| @move_goat //| //
-  |normalize
\ No newline at end of file
+We then advance to the next goal, namely the fact that there is a move from 
+[east,west,west,east] to [west,west,west,west]; this is trivial too, but it 
+requires /2/ since move_goat opens an additional subgoal. By applying "]" we
+refocus on the skipped goal, going back to a situation similar to the one we
+started with. *)
+
+  | /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter1/move.con(0,1,0)"move_goat/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"east_west/a/span/span/ ] 
+
+(* 
+h2 class="section"Implicit arguments/h2
+Let us perform the next step, namely moving back the boat, in a sligtly 
+different way. The more operation expects as second argument the new 
+intermediate state, hence instead of applying more we can apply this term
+already instatated on the next intermediate state. As first argument, we
+type a question mark that stands for an implicit argument to be guessed by
+the system. *)
+
+@(a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"more/a ? (a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"mk_state/a a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a))
+
+(* We now get three independent subgoals, all actives, and two of them are 
+trivial. Wespan style="font-family: Verdana,sans-serif;" /spancan just apply automation to all of them, and it will close the two
+trivial goals. *)
+
+/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter1/safe_state.con(0,2,0)"opposite_side/a, a href="cic:/matita/tutorial/chapter1/move.con(0,4,0)"move_boat/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"east_west/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"west_east/a/span/span/
+
+(* Let us come to the next step, that consists in moving the wolf. Suppose that 
+instead of specifying the next intermediate state, we prefer to specify the next 
+move. In the spirit of the previous example, we can do it in the following way 
+*)
+
+@(a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"more/a … (a href="cic:/matita/tutorial/chapter1/move.con(0,2,0)"move_wolf/a … ))
+
+(* The dots stand here for an arbitrary number of implicit arguments, to be 
+guessed by the system. 
+Unfortunately, the previous move is not enough to fully instantiate the new 
+intermediate state: a bank B remains unknown. Automation cannot help here,
+since all goals depend from this bank and automation refuses to close some
+subgoals instantiating other subgoals remaining open (the instantiation could
+be arbitrary). The simplest way to proceed is to focus on the bank, that is
+the fourth subgoal, and explicitly instatiate it. Instead of repeatedly using "|",
+we can perform focusing by typing "4:" as described by the following command. *)
+
+[4: @a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a] /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter1/safe_state.con(0,1,0)"with_boat/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"east_west/a/span/span/
+
+(* Alternatively, we can directly instantiate the bank into the move. Let
+us complete the proof in this, very readable way. *)
+
+@(a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"more/a … (a href="cic:/matita/tutorial/chapter1/move.con(0,1,0)"move_goat/a a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a … )) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter1/safe_state.con(0,1,0)"with_boat/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"west_east/a/span/span/
+@(a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"more/a … (a href="cic:/matita/tutorial/chapter1/move.con(0,3,0)"move_cabbage/a ?? a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"east/a … )) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter1/safe_state.con(0,2,0)"opposite_side/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"east_west/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"west_east/a/span/span/
+@(a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"more/a … (a href="cic:/matita/tutorial/chapter1/move.con(0,4,0)"move_boat/a ??? a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"west/a … )) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter1/safe_state.con(0,1,0)"with_boat/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"west_east/a/span/span/
+@a href="cic:/matita/tutorial/chapter1/reachable.con(0,1,0)"one/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter1/move.con(0,1,0)"move_goat/a, a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"east_west/a/span/span/ qed.
\ No newline at end of file