-
-(*
-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.
-
-\ 5h2 class="section"\ 6The goat, the wolf and the cabbage\ 5/h2\ 6\ 5div class="paragraph"\ 6\ 5/div\ 6A 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".
-
+(*
+\ 5h1\ 6Matita Interactive Tutorial\ 5/h1\ 6
+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 are\ 5span style="font-family: Verdana,sans-serif;"\ 6 \ 5/span\ 6now ready to start.
+
+\ 5h2 class="section"\ 6Data types, functions and theorems\ 5/h2\ 6
+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.
+
+\ 5b\ 6The goat, the wolf and the cabbage\ 5/b\ 6
+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/pts.ma".
+include "basics/logic.ma".
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.
match s with
| west ⇒ east
].
-lemma east_to_west : change_side east = west.
+(* 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 : opposite east = west.
+
+(*
+\ 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
+
+B1
+B2
+....
+Bk
+-----------------------
+A
+
+for each open goal remaining to be solved. A is the conclusion of the goal and
+B1, ..., Bk is its context, that is the set of current hypothesis and type
+declarations. In this case, we have only one goal, and its context is empty.
+The proof proceeds by typing commands to the system. In this case, we
+want to evaluate the function, that can be done by invoking the "normalize"
+command:
+*)
+
+normalize
+
+(* By executing it - just type the advance command - you will see that the goal
+has changed to west = west, by reducing the subexpression (opposite east).
+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 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. *)
+
+lemma west_to_east : opposite west = east.
// qed.
-lemma change_twice : ∀x. change_side (change_side x) = x.
-* //
-qed.
+(*
+\ 5h2 class="section"\ 6Introduction\ 5/h2\ 6
+A slightly more complex problem consists in proving that opposite is idempotent *)
-inductive item: Type[0] ≝
-| goat : item
-| wolf : item
-| cabbage: item
-| boat : item.
+lemma idempotent_opposite : ∀x. opposite (opposite x) = 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}.
-
-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].
+#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.
+*)
(*
-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).
+\ 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,
+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.
+
+(*
+\ 5h2 class="section"\ 6Predicates\ 5/h2\ 6
+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)).
+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
+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 *)
+
+lemma opp_to_opposite: ∀a,b. opp a b → a = opposite 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.
+
+(*
+\ 5h2 class="section"\ 6Rewriting\ 5/h2\ 6
+Let us come to the opposite direction. *)
+
+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. *)
+
+#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))).
+\ 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 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.
+record state : Type[0] ≝
+ {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 ≝ mk_state east east east east.
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.
+(* 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 : 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. 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 : 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 : state → state → Prop ≝
+| one : ∀x,y.move x y → reachable x y
+| more : ∀x,z,y. \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6reachable x z → safe_state z → \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6move z y → reachable x y.
+
+(*
+\ 5h2 class="section"\ 6Automation\ 5/h2\ 6
+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. *)
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.
+normalize /\ 5span class="autotactic"\ 69\ 5span class="autotrace"\ 6 trace one, more, with_boat, opposite_side, move_goat, move_wolf, move_cabbage, move_boat, east_west, west_east\ 5/span\ 6\ 5/span\ 6/ qed.
+
+(*
+\ 5h2 class="section"\ 6Application\ 5/h2\ 6
+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).
+*)
+
+lemma problem1: reachable start end.
+normalize @more
+
+(*
+\ 5h2 class="section"\ 6Focusing\ 5/h2\ 6
+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. *)
+
+ @(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
+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.
+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. *)
+
+ | /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace move_goat, east_west\ 5/span\ 6\ 5/span\ 6/ ]
+
+(*
+\ 5h2 class="section"\ 6Implicit arguments\ 5/h2\ 6
+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. *)
+
+@(more ? (mk_state east west west west))
+
+(* We now get three independent subgoals, all actives, and two of them are
+trivial. We\ 5span style="font-family: Verdana,sans-serif;"\ 6 \ 5/span\ 6can just apply automation to all of them, and it will close the two
+trivial goals. *)
+
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace opposite_side, move_boat, east_west, west_east\ 5/span\ 6\ 5/span\ 6/
+
+(* 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
+*)
-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.
+@(more … (move_wolf … ))
-definition second ≝ mk_state west east east west.
-definition end ≝ mk_state west west west west.
+(* 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. *)
-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))).
+[4: @east] /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace with_boat, east_west\ 5/span\ 6\ 5/span\ 6/
-lemma problem: reachable start second.
-@step [| @ex_intro // [| @move_goat ] // | //]
+(* Alternatively, we can directly instantiate the bank into the move. Let
+us complete the proof in this, very readable way. *)
- [|| @move_goat //| //
- |normalize
\ No newline at end of file
+@(more … (move_goat west … )) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace with_boat, west_east\ 5/span\ 6\ 5/span\ 6/
+@(more … (move_cabbage ?? east … )) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace opposite_side, east_west, west_east\ 5/span\ 6\ 5/span\ 6/
+@(more … (move_boat ??? west … )) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace with_boat, west_east\ 5/span\ 6\ 5/span\ 6/
+@one /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace move_goat, east_west\ 5/span\ 6\ 5/span\ 6/ qed.
\ No newline at end of file
projects / helm.git / blobdiff