X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter1.ma;h=4cf145d60f53f72c70b08846444700cd9b1afc7e;hb=528f8ea107f689d07d060e1d31ba32bf65b4e6ba;hp=c46004f24d106a7f03eea1812ebc5620ee7d65a9;hpb=10b08809fd7b78ff21f020911b7ec383eb4f1f0d;p=helm.git

diff --git a/weblib/tutorial/chapter1.ma b/weblib/tutorial/chapter1.ma
index c46004f24..4cf145d60 100644
--- a/weblib/tutorial/chapter1.ma
+++ b/weblib/tutorial/chapter1.ma
@@ -44,14 +44,14 @@ 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. *)
 
-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
@@ -61,9 +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
 
@@ -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.
+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.
 
 (*
-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.
+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.
 
 (* 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,
@@ -150,7 +155,7 @@ 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 ≝ 
+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.
 
@@ -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.
+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.
  
 (* 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.
+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. *)
@@ -201,12 +206,12 @@ 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] ≝
+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;
@@ -216,13 +221,13 @@ record state : Type[0] ≝
 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.
+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. *)
 
-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 ≝
+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. *)
@@ -233,7 +238,7 @@ inductive move : a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"state
 (* 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 ≝
+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).
 
@@ -241,7 +246,7 @@ inductive safe_state : a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"
 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 ≝
+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.
 
@@ -255,7 +260,7 @@ 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.
+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. 
 
 (* 
@@ -265,11 +270,12 @@ 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.
+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
 
 (*