X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter1.ma;h=ef764ceae2aa5bff3b8e1375c6a9c84290d498c4;hb=f799225a710e609a59e398eb8d6d45dad7c62f61;hp=c46004f24d106a7f03eea1812ebc5620ee7d65a9;hpb=10b08809fd7b78ff21f020911b7ec383eb4f1f0d;p=helm.git

diff --git a/weblib/tutorial/chapter1.ma b/weblib/tutorial/chapter1.ma
index c46004f24..ef764ceae 100644
--- a/weblib/tutorial/chapter1.ma
+++ b/weblib/tutorial/chapter1.ma
@@ -53,17 +53,19 @@ opposite one. *)
 
 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
+  [ east ⇒ west
+  | west ⇒ east
   ].
 
 (* Functions are live entities, and can be actually computed. To check this, let
 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.
+lemma east_to_west : opposite east = west.
 
-(* 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
 
@@ -100,14 +102,14 @@ lemma performing some book-keeping operations.
 In this case, we avoid the unnecessary simplification step: // will take care of 
 it. *) 
 
-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.
+lemma west_to_east : opposite west = east.
 // qed.
 
 (*
-h2 class="section"Case analysis/h2
+h2 class="section"Introduction/h2
 A slightly more complex problem consists in proving that opposite is idempotent *)
 
-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.
+lemma idempotent_opposite : ∀x. opposite (opposite x) = x.
 
 (* 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 
@@ -120,7 +122,10 @@ wish to rename x into b (since it is a bank), we just type #b. *)
 (* 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. 
+*)
 
+(*
+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,
@@ -150,9 +155,9 @@ other. Only two cases are possible, leading naturally to the following
 definition:
 *)
 
-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.
+inductive opp : bank → bank → Prop ≝ 
+| east_west : opp east west
+| west_east : opp west east.
 
 (* 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
@@ -164,7 +169,7 @@ 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 *)
 
-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.
+lemma opp_to_opposite: ∀a,b. opp a b → a = opposite b.
  
 (* We start the proof introducing a, b and the hypothesis opp a b, that we
 call oppab. *)
@@ -181,7 +186,7 @@ cases oppab // qed.
 h2 class="section"Rewriting/h2
 Let us come to the opposite direction. *)
 
-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.
+lemma opposite_to_opp: ∀a,b. a = opposite b → opp a b.
 
 (* As usual, we start introducing a, b and the hypothesis (a = opposite b), 
 that we call eqa. *)
@@ -201,49 +206,49 @@ cases b // qed.
 (*
 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 item: the goat, the 
+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.
 *)
 
 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}.
+  {goat_pos : bank;
+   wolf_pos : bank;
+   cabbage_pos: bank;
+   boat_pos : bank}.
 
 (* 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 *)
 
-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.
-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.
+definition start ≝ mk_state east east east east.
+definition end ≝ mk_state west west west west.
 
 (* 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. *)
 
-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)
+inductive move : state → state → Prop ≝
+| move_goat: ∀g,g1,w,c. opp g g1 → move (mk_state g w c g) (mk_state 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).
+| move_wolf: ∀g,w,w1,c. opp w w1 → move (mk_state g w c w) (mk_state g w1 c w1)
+| move_cabbage: ∀g,w,c,c1.opp c c1 → move (mk_state g w c c) (mk_state g w c1 c1)
+| move_boat: ∀g,w,c,b,b1. opp b b1 → move (mk_state g w c b) (mk_state 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. *)
 
-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).
+inductive safe_state : state → Prop ≝
+| with_boat : ∀g,w,c.safe_state (mk_state g w c g)
+| opposite_side : ∀g,g1,b.opp g g1 → safe_state (mk_state 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 *)
 
-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.
+inductive reachable : state → state → Prop ≝
+| one : ∀x,y.move x y → reachable x y
+| more : ∀x,z,y. span style="text-decoration: underline;"/spanreachable x z → safe_state z → span style="text-decoration: underline;"/spanmove z y → reachable x y.
 
 (* 
 h2 class="section"Automation/h2
@@ -255,8 +260,8 @@ 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.  *)
 
-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. 
+lemma problem: reachable start end.
+normalize /span class="autotactic"9span class="autotrace" trace one, more, with_boat, opposite_side, move_goat, move_wolf, move_cabbage, move_boat, east_west, west_east/span/span/ qed. 
 
 (* 
 h2 class="section"Application/h2
@@ -265,12 +270,13 @@ 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 
-results required for proving P, prefixed by simple modalities (<,@,...) explaining 
-the way the result is used (e.g. for rewriting, in an applicative step, and so on).
+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).
 *)
 
-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
+lemma problem1: reachable start end.
+normalize @more
 
 (* 
 h2 class="section"Focusing/h2
@@ -308,7 +314,7 @@ 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) 
+   @(mk_state east west west east) 
 
 (* 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
@@ -325,7 +331,7 @@ 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/ ] 
+  | /span class="autotactic"2span class="autotrace" trace move_goat, east_west/span/span/ ] 
 
 (* 
 h2 class="section"Implicit arguments/h2
@@ -336,20 +342,20 @@ 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))
+@(more ? (mk_state east west west west))
 
 (* 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/
+/span class="autotactic"2span class="autotrace" trace opposite_side, move_boat, east_west, west_east/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 … ))
+@(more … (move_wolf … ))
 
 (* The dots stand here for an arbitrary number of implicit arguments, to be 
 guessed by the system. 
@@ -361,12 +367,12 @@ 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/
+[4: @east] /span class="autotactic"2span class="autotrace" trace with_boat, east_west/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
+@(more … (move_goat west … )) /span class="autotactic"2span class="autotrace" trace with_boat, west_east/span/span/
+@(more … (move_cabbage ?? east … )) /span class="autotactic"2span class="autotrace" trace opposite_side, east_west, west_east/span/span/
+@(more … (move_boat ??? west … )) /span class="autotactic"2span class="autotrace" trace with_boat, west_east/span/span/
+@one /span class="autotactic"2span class="autotrace" trace move_goat, east_west/span/span/ qed.
\ No newline at end of file