update in basic_2

[helm.git] / weblib / tutorial / chapter1.ma

diff --git a/weblib/tutorial/chapter1.ma b/weblib/tutorial/chapter1.ma
index c46004f24d106a7f03eea1812ebc5620ee7d65a9..4cf145d60f53f72c70b08846444700cd9b1afc7e 100644 (file)
--- 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] ≝
+\ 5img class="anchor" src="icons/tick.png" id="bank"\ 6inductive 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.
+\ 5img class="anchor" src="icons/tick.png" id="opposite"\ 6definition opposite ≝ λs.
 match s with
   [ east ⇒ \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6
   | west ⇒ \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6
@@ -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 : \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="east_to_west"\ 6lemma east_to_west : \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6.
 
-(* If you stop the execution here you will see a new window on the  right side
+(* 
+\ 5h2 class="section"\ 6The goal window\ 5/h2\ 6
+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 : \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="west_to_east"\ 6lemma west_to_east : \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6.
 // qed.
 
 (*
-\ 5h2 class="section"\ 6Case analysis\ 5/h2\ 6
+\ 5h2 class="section"\ 6Introduction\ 5/h2\ 6
 A slightly more complex problem consists in proving that opposite is idempotent *)
 
-lemma idempotent_opposite : ∀x. \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 x) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x.
+\ 5img class="anchor" src="icons/tick.png" id="idempotent_opposite"\ 6lemma idempotent_opposite : ∀x. \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 x) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 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. 
+*)
 
+(*
+\ 5h2 class="section"\ 6Case analysis\ 5/h2\ 6
 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 : \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6 → Prop ≝ 
+\ 5img class="anchor" src="icons/tick.png" id="opp"\ 6inductive opp : \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6 → Prop ≝ 
 | east_west : opp \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6
 | west_east : opp \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6.
 
@@ -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. \ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 a b → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 b.
+\ 5img class="anchor" src="icons/tick.png" id="opp_to_opposite"\ 6lemma opp_to_opposite: ∀a,b. \ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 a b → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 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.
 \ 5h2 class="section"\ 6Rewriting\ 5/h2\ 6
 Let us come to the opposite direction. *)
 
-lemma opposite_to_opp: ∀a,b. a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 b → \ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 a b.
+\ 5img class="anchor" src="icons/tick.png" id="opposite_to_opp"\ 6lemma opposite_to_opp: ∀a,b. a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 b → \ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 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.
 (*
 \ 5h2 class="section"\ 6Records\ 5/h2\ 6
 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] ≝
+\ 5img class="anchor" src="icons/tick.png" id="state"\ 6record state : Type[0] ≝
   {goat_pos : \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6;
    wolf_pos : \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6;
    cabbage_pos: \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6;
@@ -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 ≝ \ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6.
-definition end ≝ \ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="start"\ 6definition start ≝ \ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="end"\ 6definition end ≝ \ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6.
 
 (* 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 : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="move"\ 6inductive move : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → Prop ≝
 | move_goat: ∀g,g1,w,c. \ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 g g1 → move (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g w c g) (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 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 : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5
 (* 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 : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="safe_state"\ 6inductive safe_state : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → Prop ≝
 | with_boat : ∀g,w,c.safe_state (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g w c g)
 | opposite_side : ∀g,g1,b.\ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 g g1 → safe_state (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g g1 g1 b).
 
@@ -241,7 +246,7 @@ inductive safe_state : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6
 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 : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="reachable"\ 6inductive reachable : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → Prop ≝
 | one : ∀x,y.\ 5a href="cic:/matita/tutorial/chapter1/move.ind(1,0,0)"\ 6move\ 5/a\ 6 x y → reachable x y
 | more : ∀x,z,y. \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6reachable x z → \ 5a href="cic:/matita/tutorial/chapter1/safe_state.ind(1,0,0)"\ 6safe_state\ 5/a\ 6 z → \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter1/move.ind(1,0,0)"\ 6move\ 5/a\ 6 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: \ 5a href="cic:/matita/tutorial/chapter1/reachable.ind(1,0,0)"\ 6reachable\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/start.def(1)"\ 6start\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/end.def(1)"\ 6end\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="problem"\ 6lemma problem: \ 5a href="cic:/matita/tutorial/chapter1/reachable.ind(1,0,0)"\ 6reachable\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/start.def(1)"\ 6start\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/end.def(1)"\ 6end\ 5/a\ 6.
 normalize /\ 5span class="autotactic"\ 69\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,1,0)"\ 6one\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"\ 6more\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/safe_state.con(0,1,0)"\ 6with_boat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/safe_state.con(0,2,0)"\ 6opposite_side\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,1,0)"\ 6move_goat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,2,0)"\ 6move_wolf\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,3,0)"\ 6move_cabbage\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,4,0)"\ 6move_boat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"\ 6east_west\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"\ 6west_east\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ 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: \ 5a href="cic:/matita/tutorial/chapter1/reachable.ind(1,0,0)"\ 6reachable\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/start.def(1)"\ 6start\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/end.def(1)"\ 6end\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="problem1"\ 6lemma problem1: \ 5a href="cic:/matita/tutorial/chapter1/reachable.ind(1,0,0)"\ 6reachable\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/start.def(1)"\ 6start\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/end.def(1)"\ 6end\ 5/a\ 6.
 normalize @\ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"\ 6more\ 5/a\ 6
 
 (*