(**` * Basics: Functional Programming in Coq *)
(* REMINDER:
#####################################################
### PLEASE DO NOT DISTRIBUTE SOLUTIONS PUBLICLY ###
#####################################################
(See the [Preface] for why.)
*)
(* [Admitted] is Coq's "escape hatch" that says accept this definition
without proof. We use it to mark the 'holes' in the development
that should be completed as part of your homework exercises. In
practice, [Admitted] is useful when you're incrementally developing
large proofs. *)
Definition admit {T: Type} : T. Admitted.
(* ################################################################# *)
(** * Introduction *)
(** The functional programming style brings programming closer to
simple, everyday mathematics: If a procedure or method has no side
effects, then (ignoring efficiency) all we need to understand
about it is how it maps inputs to outputs -- that is, we can think
of it as just a concrete method for computing a mathematical
function. This is one sense of the word "functional" in
"functional programming." The direct connection between programs
and simple mathematical objects supports both formal correctness
proofs and sound informal reasoning about program behavior.
The other sense in which functional programming is "functional" is
that it emphasizes the use of functions (or methods) as
_first-class_ values -- i.e., values that can be passed as
arguments to other functions, returned as results, included in
data structures, etc. The recognition that functions can be
treated as data in this way enables a host of useful and powerful
idioms.
Other common features of functional languages include _algebraic
data types_ and _pattern matching_, which make it easy to
construct and manipulate rich data structures, and sophisticated
_polymorphic type systems_ supporting abstraction and code reuse.
Coq shares all of these features.
The first half of this chapter introduces the most essential
elements of Coq's functional programming language. The second
half introduces some basic _tactics_ that can be used to prove
simple properties of Coq programs. *)
(* ################################################################# *)
(** * Enumerated Types *)
(** One unusual aspect of Coq is that its set of built-in
features is _extremely_ small. For example, instead of providing
the usual palette of atomic data types (booleans, integers,
strings, etc.), Coq offers a powerful mechanism for defining new
data types from scratch, from which all these familiar types arise
as instances.
Naturally, the Coq distribution comes with an extensive standard
library providing definitions of booleans, numbers, and many
common data structures like lists and hash tables. But there is
nothing magic or primitive about these library definitions. To
illustrate this, we will explicitly recapitulate all the
definitions we need in this course, rather than just getting them
implicitly from the library.
To see how this definition mechanism works, let's start with a
very simple example. *)
(* ================================================================= *)
(** ** Days of the Week *)
(** The following declaration tells Coq that we are defining
a new set of data values -- a _type_. *)
Inductive day : Type :=
| monday : day
| tuesday : day
| wednesday : day
| thursday : day
| friday : day
| saturday : day
| sunday : day.
(** The type is called [day], and its members are [monday],
[tuesday], etc. The second and following lines of the definition
can be read "[monday] is a [day], [tuesday] is a [day], etc."
Having defined [day], we can write functions that operate on
days. *)
Definition next_weekday (d:day) : day :=
match d with
| monday => tuesday
| tuesday => wednesday
| wednesday => thursday
| thursday => friday
| friday => monday
| saturday => monday
| sunday => monday
end.
(** One thing to note is that the argument and return types of
this function are explicitly declared. Like most functional
programming languages, Coq can often figure out these types for
itself when they are not given explicitly -- i.e., it performs
_type inference_ -- but we'll include them to make reading
easier. *)
(** Having defined a function, we should check that it works on
some examples. There are actually three different ways to do this
in Coq.
First, we can use the command [Compute] to evaluate a compound
expression involving [next_weekday]. *)
Compute (next_weekday friday).
(* ==> monday : day *)
Compute (next_weekday (next_weekday saturday)).
(* ==> tuesday : day *)
(** (We show Coq's responses in comments, but, if you have a
computer handy, this would be an excellent moment to fire up the
Coq interpreter under your favorite IDE -- either CoqIde or Proof
General -- and try this for yourself. Load this file, [Basics.v],
from the book's accompanying Coq sources, find the above example,
submit it to Coq, and observe the result.)
Second, we can record what we _expect_ the result to be in the
form of a Coq example: *)
Example test_next_weekday:
(next_weekday (next_weekday saturday)) = tuesday.
(** This declaration does two things: it makes an
assertion (that the second weekday after [saturday] is [tuesday]),
and it gives the assertion a name that can be used to refer to it
later.
Having made the assertion, we can also ask Coq to verify it, like
this: *)
Proof. simpl. reflexivity. Qed.
(** The details are not important for now (we'll come back to
them in a bit), but essentially this can be read as "The assertion
we've just made can be proved by observing that both sides of the
equality evaluate to the same thing, after some simplification."
Third, we can ask Coq to _extract_, from our [Definition], a
program in some other, more conventional, programming
language (OCaml, Scheme, or Haskell) with a high-performance
compiler. This facility is very interesting, since it gives us a
way to construct _fully certified_ programs in mainstream
languages. Indeed, this is one of the main uses for which Coq was
developed. We'll come back to this topic in later chapters. *)
(* ================================================================= *)
(** ** Booleans *)
(** In a similar way, we can define the standard type [bool] of
booleans, with members [true] and [false]. *)
Inductive bool : Type :=
| true : bool
| false : bool.
(** Although we are rolling our own booleans here for the sake
of building up everything from scratch, Coq does, of course,
provide a default implementation of the booleans in its standard
library, together with a multitude of useful functions and
lemmas. (Take a look at [Coq.Init.Datatypes] in the Coq library
documentation if you're interested.) Whenever possible, we'll
name our own definitions and theorems so that they exactly
coincide with the ones in the standard library.
Functions over booleans can be defined in the same way as
above: *)
Definition negb (b:bool) : bool :=
match b with
| true => false
| false => true
end.
Definition andb (b1:bool) (b2:bool) : bool :=
match b1 with
| true => b2
| false => false
end.
Definition orb (b1:bool) (b2:bool) : bool :=
match b1 with
| true => true
| false => b2
end.
(** The last two illustrate Coq's syntax for multi-argument
function definitions. The corresponding multi-argument
application syntax is illustrated by the following four "unit
tests," which constitute a complete specification -- a truth
table -- for the [orb] function: *)
Example test_orb1: (orb true false) = true.
Proof. simpl. reflexivity. Qed.
Example test_orb2: (orb false false) = false.
Proof. simpl. reflexivity. Qed.
Example test_orb3: (orb false true) = true.
Proof. simpl. reflexivity. Qed.
Example test_orb4: (orb true true) = true.
Proof. simpl. reflexivity. Qed.
(** We can also introduce some familiar syntax for the boolean
operations we have just defined. The [Infix] command defines new,
infix notation for an existing definition. *)
Infix "&&" := andb.
Infix "||" := orb.
Example test_orb5: false || false || true = true.
Proof. simpl. reflexivity. Qed.
(** _A note on notation_: In [.v] files, we use square brackets to
delimit fragments of Coq code within comments; this convention,
also used by the [coqdoc] documentation tool, keeps them visually
separate from the surrounding text. In the html version of the
files, these pieces of text appear in a [different font].
The special phrases [Admitted] and [admit] can be used as a
placeholder for an incomplete definition or proof. We'll use them
in exercises, to indicate the parts that we're leaving for you --
i.e., your job is to replace [admit] or [Admitted] with real
definitions or proofs. *)
(** **** Exercise: 1 star (nandb) *)
(** Remove [admit] and complete the definition of the following
function; then make sure that the [Example] assertions below can
each be verified by Coq. (Remove "[Admitted.]" and fill in each
proof, following the model of the [orb] tests above.) The function
should return [true] if either or both of its inputs are
[false]. *)
Definition nandb (b1:bool) (b2:bool) : bool
(* REPLACE THIS LINE WITH := _your_definition_ . *) . Admitted.
Example test_nandb1: (nandb true false) = true.
(* FILL IN HERE *) Admitted.
Example test_nandb2: (nandb false false) = true.
(* FILL IN HERE *) Admitted.
Example test_nandb3: (nandb false true) = true.
(* FILL IN HERE *) Admitted.
Example test_nandb4: (nandb true true) = false.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 1 star (andb3) *)
(** Do the same for the [andb3] function below. This function should
return [true] when all of its inputs are [true], and [false]
otherwise. *)
Definition andb3 (b1:bool) (b2:bool) (b3:bool) : bool
(* REPLACE THIS LINE WITH := _your_definition_ . *) . Admitted.
Example test_andb31: (andb3 true true true) = true.
(* FILL IN HERE *) Admitted.
Example test_andb32: (andb3 false true true) = false.
(* FILL IN HERE *) Admitted.
Example test_andb33: (andb3 true false true) = false.
(* FILL IN HERE *) Admitted.
Example test_andb34: (andb3 true true false) = false.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ================================================================= *)
(** ** Function Types *)
(** Every expression in Coq has a type, describing what sort of
thing it computes. The [Check] command asks Coq to print the type
of an expression. *)
(** For example, the type of [negb true] is [bool]. *)
Check true.
(* ===> true : bool *)
Check (negb true).
(* ===> negb true : bool *)
(** Functions like [negb] itself are also data values, just like
[true] and [false]. Their types are called _function types_, and
they are written with arrows. *)
Check negb.
(* ===> negb : bool -> bool *)
(** The type of [negb], written [bool -> bool] and pronounced
"[bool] arrow [bool]," can be read, "Given an input of type
[bool], this function produces an output of type [bool]."
Similarly, the type of [andb], written [bool -> bool -> bool], can
be read, "Given two inputs, both of type [bool], this function
produces an output of type [bool]." *)
(* ================================================================= *)
(** ** Modules *)
(** Coq provides a _module system_, to aid in organizing large
developments. In this course we won't need most of its features,
but one is useful: If we enclose a collection of declarations
between [Module X] and [End X] markers, then, in the remainder of
the file after the [End], these definitions are referred to by
names like [X.foo] instead of just [foo]. Here, we use this
feature to introduce the definition of the type [nat] in an inner
module so that it does not interfere with the one from the
standard library, which comes with a bit of special notational
magic. *)
Module Playground1.
(* ================================================================= *)
(** ** Numbers *)
(** The types we have defined so far are examples of "enumerated
types": their definitions explicitly enumerate a finite set of
elements. A more interesting way of defining a type is to give a
collection of _inductive rules_ describing its elements. For
example, we can define the natural numbers as follows: *)
Inductive nat : Type :=
| O : nat
| S : nat -> nat.
(** The clauses of this definition can be read:
- [O] is a natural number (note that this is the letter "[O],"
not the numeral "[0]").
- [S] is a "constructor" that takes a natural number and yields
another one -- that is, if [n] is a natural number, then [S n]
is too.
Let's look at this in a little more detail.
Every inductively defined set ([day], [nat], [bool], etc.) is
actually a set of _expressions_. The definition of [nat] says how
expressions in the set [nat] can be constructed:
- the expression [O] belongs to the set [nat];
- if [n] is an expression belonging to the set [nat], then [S n]
is also an expression belonging to the set [nat]; and
- expressions formed in these two ways are the only ones belonging
to the set [nat].
The same rules apply for our definitions of [day] and [bool]. The
annotations we used for their constructors are analogous to the
one for the [O] constructor, indicating that they don't take any
arguments.
These three conditions are the precise force of the [Inductive]
declaration. They imply that the expression [O], the expression
[S O], the expression [S (S O)], the expression [S (S (S O))], and
so on all belong to the set [nat], while other expressions like
[true], [andb true false], and [S (S false)] do not.
We can write simple functions that pattern match on natural
numbers just as we did above -- for example, the predecessor
function: *)
Definition pred (n : nat) : nat :=
match n with
| O => O
| S n' => n'
end.
(** The second branch can be read: "if [n] has the form [S n']
for some [n'], then return [n']." *)
End Playground1.
Definition minustwo (n : nat) : nat :=
match n with
| O => O
| S O => O
| S (S n') => n'
end.
(** Because natural numbers are such a pervasive form of data,
Coq provides a tiny bit of built-in magic for parsing and printing
them: ordinary arabic numerals can be used as an alternative to
the "unary" notation defined by the constructors [S] and [O]. Coq
prints numbers in arabic form by default: *)
Check (S (S (S (S O)))).
(* ===> 4 : nat *)
Compute (minustwo 4).
(* ===> 2 : nat *)
(** The constructor [S] has the type [nat -> nat], just like the
functions [minustwo] and [pred]: *)
Check S.
Check pred.
Check minustwo.
(** These are all things that can be applied to a number to yield a
number. However, there is a fundamental difference between the
first one and the other two: functions like [pred] and [minustwo]
come with _computation rules_ -- e.g., the definition of [pred]
says that [pred 2] can be simplified to [1] -- while the
definition of [S] has no such behavior attached. Although it is
like a function in the sense that it can be applied to an
argument, it does not _do_ anything at all!
For most function definitions over numbers, just pattern matching
is not enough: we also need recursion. For example, to check that
a number [n] is even, we may need to recursively check whether
[n-2] is even. To write such functions, we use the keyword
[Fixpoint]. *)
Fixpoint evenb (n:nat) : bool :=
match n with
| O => true
| S O => false
| S (S n') => evenb n'
end.
(** We can define [oddb] by a similar [Fixpoint] declaration, but here
is a simpler definition that is a bit easier to work with: *)
Definition oddb (n:nat) : bool := negb (evenb n).
Example test_oddb1: oddb 1 = true.
Proof. simpl. reflexivity. Qed.
Example test_oddb2: oddb 4 = false.
Proof. simpl. reflexivity. Qed.
(** (You will notice if you step through these proofs that
[simpl] actually has no effect on the goal -- all of the work is
done by [reflexivity]. We'll see more about why that is shortly.)
Naturally, we can also define multi-argument functions by
recursion. *)
Module Playground2.
Fixpoint plus (n : nat) (m : nat) : nat :=
match n with
| O => m
| S n' => S (plus n' m)
end.
(** Adding three to two now gives us five, as we'd expect. *)
Compute (plus 3 2).
(** The simplification that Coq performs to reach this conclusion can
be visualized as follows: *)
(* [plus (S (S (S O))) (S (S O))]
==> [S (plus (S (S O)) (S (S O)))]
by the second clause of the [match]
==> [S (S (plus (S O) (S (S O))))]
by the second clause of the [match]
==> [S (S (S (plus O (S (S O)))))]
by the second clause of the [match]
==> [S (S (S (S (S O))))]
by the first clause of the [match]
*)
(** As a notational convenience, if two or more arguments have
the same type, they can be written together. In the following
definition, [(n m : nat)] means just the same as if we had written
[(n : nat) (m : nat)]. *)
Fixpoint mult (n m : nat) : nat :=
match n with
| O => O
| S n' => plus m (mult n' m)
end.
Example test_mult1: (mult 3 3) = 9.
Proof. simpl. reflexivity. Qed.
(** You can match two expressions at once by putting a comma
between them: *)
Fixpoint minus (n m:nat) : nat :=
match n, m with
| O , _ => O
| S _ , O => n
| S n', S m' => minus n' m'
end.
(** The _ in the first line is a _wildcard pattern_. Writing _ in a
pattern is the same as writing some variable that doesn't get used
on the right-hand side. This avoids the need to invent a bogus
variable name. *)
End Playground2.
Fixpoint exp (base power : nat) : nat :=
match power with
| O => S O
| S p => mult base (exp base p)
end.
(** **** Exercise: 1 star (factorial) *)
(** Recall the standard mathematical factorial function:
factorial(0) = 1
factorial(n) = n * factorial(n-1) (if n>0)
Translate this into Coq. *)
Fixpoint factorial (n:nat) : nat
(* REPLACE THIS LINE WITH := _your_definition_ . *) . Admitted.
Example test_factorial1: (factorial 3) = 6.
(* FILL IN HERE *) Admitted.
Example test_factorial2: (factorial 5) = (mult 10 12).
(* FILL IN HERE *) Admitted.
(** [] *)
(** We can make numerical expressions a little easier to read and
write by introducing _notations_ for addition, multiplication, and
subtraction. *)
Notation "x + y" := (plus x y)
(at level 50, left associativity)
: nat_scope.
Notation "x - y" := (minus x y)
(at level 50, left associativity)
: nat_scope.
Notation "x * y" := (mult x y)
(at level 40, left associativity)
: nat_scope.
Check ((0 + 1) + 1).
(** (The [level], [associativity], and [nat_scope] annotations
control how these notations are treated by Coq's parser. The
details are not important, but interested readers can refer to the
optional "More on Notation" section at the end of this chapter.)
Note that these do not change the definitions we've already made:
they are simply instructions to the Coq parser to accept [x + y]
in place of [plus x y] and, conversely, to the Coq pretty-printer
to display [plus x y] as [x + y].
When we say that Coq comes with nothing built-in, we really mean
it: even equality testing for numbers is a user-defined
operation! *)
(** The [beq_nat] function tests [nat]ural numbers for [eq]uality,
yielding a [b]oolean. Note the use of nested [match]es (we could
also have used a simultaneous match, as we did in [minus].) *)
Fixpoint beq_nat (n m : nat) : bool :=
match n with
| O => match m with
| O => true
| S m' => false
end
| S n' => match m with
| O => false
| S m' => beq_nat n' m'
end
end.
(** The [leb] function tests whether its first argument is less than or
equal to its second argument, yielding a boolean. *)
Fixpoint leb (n m : nat) : bool :=
match n with
| O => true
| S n' =>
match m with
| O => false
| S m' => leb n' m'
end
end.
Example test_leb1: (leb 2 2) = true.
Proof. simpl. reflexivity. Qed.
Example test_leb2: (leb 2 4) = true.
Proof. simpl. reflexivity. Qed.
Example test_leb3: (leb 4 2) = false.
Proof. simpl. reflexivity. Qed.
(** **** Exercise: 1 star (blt_nat) *)
(** The [blt_nat] function tests [nat]ural numbers for [l]ess-[t]han,
yielding a [b]oolean. Instead of making up a new [Fixpoint] for
this one, define it in terms of a previously defined function. *)
Definition blt_nat (n m : nat) : bool
(* REPLACE THIS LINE WITH := _your_definition_ . *) . Admitted.
Example test_blt_nat1: (blt_nat 2 2) = false.
(* FILL IN HERE *) Admitted.
Example test_blt_nat2: (blt_nat 2 4) = true.
(* FILL IN HERE *) Admitted.
Example test_blt_nat3: (blt_nat 4 2) = false.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ################################################################# *)
(** * Proof by Simplification *)
(** Now that we've defined a few datatypes and functions, let's
turn to stating and proving properties of their behavior.
Actually, we've already started doing this: each [Example] in the
previous sections makes a precise claim about the behavior of some
function on some particular inputs. The proofs of these claims
were always the same: use [simpl] to simplify both sides of the
equation, then use [reflexivity] to check that both sides contain
identical values.
The same sort of "proof by simplification" can be used to prove
more interesting properties as well. For example, the fact that
[0] is a "neutral element" for [+] on the left can be proved just
by observing that [0 + n] reduces to [n] no matter what [n] is, a
fact that can be read directly off the definition of [plus].*)
Theorem plus_O_n : forall n : nat, 0 + n = n.
Proof.
intros n. simpl. reflexivity. Qed.
(** (You may notice that the above statement looks different in
the [.v] file in your IDE than it does in the HTML rendition in
your browser, if you are viewing both. In [.v] files, we write the
[forall] universal quantifier using the reserved identifier
"forall." When the [.v] files are converted to HTML, this gets
transformed into an upside-down-A symbol.) *)
(** This is a good place to mention that [reflexivity] is a bit
more powerful than we have admitted. In the examples we have seen,
the calls to [simpl] were actually not needed, because
[reflexivity] can perform some simplification automatically when
checking that two sides are equal; [simpl] was just added so that
we could see the intermediate state -- after simplification but
before finishing the proof. Here is a shorter proof of the
theorem: *)
Theorem plus_O_n' : forall n : nat, 0 + n = n.
Proof.
intros n. reflexivity. Qed.
(** Moreover, it will be useful later to know that [reflexivity]
does somewhat _more_ simplification than [simpl] does -- for
example, it tries "unfolding" defined terms, replacing them with
their right-hand sides. The reason for this difference is that,
if reflexivity succeeds, the whole goal is finished and we don't
need to look at whatever expanded expressions [reflexivity] has
created by all this simplification and unfolding; by contrast,
[simpl] is used in situations where we may have to read and
understand the new goal that it creates, so we would not want it
blindly expanding definitions and leaving the goal in a messy
state. *)
(** The form of the theorem we just stated and its proof are
almost exactly the same as the simpler examples we saw earlier;
there are just a few differences.
First, we've used the keyword [Theorem] instead of [Example].
This difference is purely a matter of style; the keywords
[Example] and [Theorem] (and a few others, including [Lemma],
[Fact], and [Remark]) mean exactly the same thing to Coq.
Second, we've added the quantifier [forall n:nat], so that our
theorem talks about _all_ natural numbers [n]. In order to prove
theorems of this form, we need to to be able to reason by
_assuming_ the existence of an arbitrary natural number [n]. This
is achieved in the proof by [intros n], which moves the quantifier
from the goal to a _context_ of current assumptions. In effect, we
start the proof by saying "Suppose [n] is some arbitrary
number..."
The keywords [intros], [simpl], and [reflexivity] are examples of
_tactics_. A tactic is a command that is used between [Proof] and
[Qed] to guide the process of checking some claim we are making.
We will see several more tactics in the rest of this chapter and
yet more in future chapters.
Other similar theorems can be proved with the same pattern. *)
Theorem plus_1_l : forall n:nat, 1 + n = S n.
Proof.
intros n. reflexivity. Qed.
Theorem mult_0_l : forall n:nat, 0 * n = 0.
Proof.
intros n. reflexivity. Qed.
(** The [_l] suffix in the names of these theorems is
pronounced "on the left." *)
(** It is worth stepping through these proofs to observe how the
context and the goal change. *)
(** You may want to add calls to [simpl] before [reflexivity] to
see the simplifications that Coq performs on the terms before
checking that they are equal.
Although simplification is powerful enough to prove some fairly
general facts, there are many statements that cannot be handled by
simplification alone. For instance, we cannot use it to prove
that [0] is also a neutral element for [+] _on the right_. *)
Theorem plus_n_O : forall n, n = n + 0.
Proof.
intros n. simpl. (* Doesn't do anything! *)
(** (Can you explain why this happens? Step through both proofs
with Coq and notice how the goal and context change.)
When stuck in the middle of a proof, we can use the [Abort]
command to give up on it for the moment. *)
Abort.
(** The next chapter will introduce _induction_, a powerful
technique that can be used for proving this goal. For the moment,
though, let's look at a few more simple tactics. *)
(* ################################################################# *)
(** * Proof by Rewriting *)
(** This theorem is a bit more interesting than the others we've
seen: *)
Theorem plus_id_example : forall n m:nat,
n = m ->
n + n = m + m.
(** Instead of making a universal claim about all numbers [n] and [m],
it talks about a more specialized property that only holds when [n
= m]. The arrow symbol is pronounced "implies."
As before, we need to be able to reason by assuming the existence
of some numbers [n] and [m]. We also need to assume the hypothesis
[n = m]. The [intros] tactic will serve to move all three of these
from the goal into assumptions in the current context.
Since [n] and [m] are arbitrary numbers, we can't just use
simplification to prove this theorem. Instead, we prove it by
observing that, if we are assuming [n = m], then we can replace
[n] with [m] in the goal statement and obtain an equality with the
same expression on both sides. The tactic that tells Coq to
perform this replacement is called [rewrite]. *)
Proof.
(* move both quantifiers into the context: *)
intros n m.
(* move the hypothesis into the context: *)
intros H.
(* rewrite the goal using the hypothesis: *)
rewrite -> H.
reflexivity. Qed.
(** The first line of the proof moves the universally quantified
variables [n] and [m] into the context. The second moves the
hypothesis [n = m] into the context and gives it the name [H].
The third tells Coq to rewrite the current goal ([n + n = m + m])
by replacing the left side of the equality hypothesis [H] with the
right side.
(The arrow symbol in the [rewrite] has nothing to do with
implication: it tells Coq to apply the rewrite from left to right.
To rewrite from right to left, you can use [rewrite <-]. Try
making this change in the above proof and see what difference it
makes.) *)
(** **** Exercise: 1 star (plus_id_exercise) *)
(** Remove "[Admitted.]" and fill in the proof. *)
Theorem plus_id_exercise : forall n m o : nat,
n = m -> m = o -> n + m = m + o.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** The [Admitted] command tells Coq that we want to skip trying
to prove this theorem and just accept it as a given. This can be
useful for developing longer proofs, since we can state subsidiary
lemmas that we believe will be useful for making some larger
argument, use [Admitted] to accept them on faith for the moment,
and continue working on the main argument until we are sure it
makes sense; then we can go back and fill in the proofs we
skipped. Be careful, though: every time you say [Admitted] (or
[admit]) you are leaving a door open for total nonsense to enter
Coq's nice, rigorous, formally checked world! *)
(** We can also use the [rewrite] tactic with a previously proved
theorem instead of a hypothesis from the context. If the statement
of the previously proved theorem involves quantified variables,
as in the example below, Coq tries to instantiate them
by matching with the current goal. *)
Theorem mult_0_plus : forall n m : nat,
(0 + n) * m = n * m.
Proof.
intros n m.
rewrite -> plus_O_n.
reflexivity. Qed.
(** **** Exercise: 2 stars (mult_S_1) *)
Theorem mult_S_1 : forall n m : nat,
m = S n ->
m * (1 + n) = m * m.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ################################################################# *)
(** * Proof by Case Analysis *)
(** Of course, not everything can be proved by simple
calculation and rewriting: In general, unknown, hypothetical
values (arbitrary numbers, booleans, lists, etc.) can block
simplification. For example, if we try to prove the following
fact using the [simpl] tactic as above, we get stuck. *)
Theorem plus_1_neq_0_firsttry : forall n : nat,
beq_nat (n + 1) 0 = false.
Proof.
intros n.
simpl. (* does nothing! *)
Abort.
(** The reason for this is that the definitions of both
[beq_nat] and [+] begin by performing a [match] on their first
argument. But here, the first argument to [+] is the unknown
number [n] and the argument to [beq_nat] is the compound
expression [n + 1]; neither can be simplified.
To make progress, we need to consider the possible forms of [n]
separately. If [n] is [O], then we can calculate the final result
of [beq_nat (n + 1) 0] and check that it is, indeed, [false]. And
if [n = S n'] for some [n'], then, although we don't know exactly
what number [n + 1] yields, we can calculate that, at least, it
will begin with one [S], and this is enough to calculate that,
again, [beq_nat (n + 1) 0] will yield [false].
The tactic that tells Coq to consider, separately, the cases where
[n = O] and where [n = S n'] is called [destruct]. *)
Theorem plus_1_neq_0 : forall n : nat,
beq_nat (n + 1) 0 = false.
Proof.
intros n. destruct n as [| n'].
- reflexivity.
- reflexivity. Qed.
(** The [destruct] generates _two_ subgoals, which we must then
prove, separately, in order to get Coq to accept the theorem. The
annotation "[as [| n']]" is called an _intro pattern_. It tells
Coq what variable names to introduce in each subgoal. In general,
what goes between the square brackets is a _list of lists_ of
names, separated by [|]. In this case, the first component is
empty, since the [O] constructor is nullary (it doesn't have any
arguments). The second component gives a single name, [n'], since
[S] is a unary constructor.
The [-] signs on the second and third lines are called _bullets_,
and they mark the parts of the proof that correspond to each
generated subgoal. The proof script that comes after a bullet is
the entire proof for a subgoal. In this example, each of the
subgoals is easily proved by a single use of [reflexivity], which
itself performs some simplification -- e.g., the first one
simplifies [beq_nat (S n' + 1) 0] to [false] by first rewriting
[(S n' + 1)] to [S (n' + 1)], then unfolding [beq_nat], and then
simplifying the [match].
Marking cases with bullets is entirely optional: if bullets are
not present, Coq simply asks you to prove each subgoal in
sequence, one at a time. But it is a good idea to use bullets.
For one thing, they make the structure of a proof apparent, making
it more readable. Also, bullets instruct Coq to ensure that a
subgoal is complete before trying to verify the next one,
preventing proofs for different subgoals from getting mixed
up. These issues become especially important in large
developments, where fragile proofs lead to long debugging
sessions.
There are no hard and fast rules for how proofs should be
formatted in Coq -- in particular, where lines should be broken
and how sections of the proof should be indented to indicate their
nested structure. However, if the places where multiple subgoals
are generated are marked with explicit bullets at the beginning of
lines, then the proof will be readable almost no matter what
choices are made about other aspects of layout.
This is also a good place to mention one other piece of somewhat
obvious advice about line lengths. Beginning Coq users sometimes
tend to the extremes, either writing each tactic on its own line
or writing entire proofs on one line. Good style lies somewhere
in the middle. One reasonable convention is to limit yourself to
80-character lines.
The [destruct] tactic can be used with any inductively defined
datatype. For example, we use it next to prove that boolean
negation is involutive -- i.e., that negation is its own
inverse. *)
Theorem negb_involutive : forall b : bool,
negb (negb b) = b.
Proof.
intros b. destruct b.
- reflexivity.
- reflexivity. Qed.
(** Note that the [destruct] here has no [as] clause because
none of the subcases of the [destruct] need to bind any variables,
so there is no need to specify any names. (We could also have
written [as [|]], or [as []].) In fact, we can omit the [as]
clause from _any_ [destruct] and Coq will fill in variable names
automatically. This is generally considered bad style, since Coq
often makes confusing choices of names when left to its own
devices.
It is sometimes useful to invoke [destruct] inside a subgoal,
generating yet more proof obligations. In this case, we use
different kinds of bullets to mark goals on different "levels."
For example: *)
Theorem andb_commutative : forall b c, andb b c = andb c b.
Proof.
intros b c. destruct b.
- destruct c.
+ reflexivity.
+ reflexivity.
- destruct c.
+ reflexivity.
+ reflexivity.
Qed.
(** Each pair of calls to [reflexivity] corresponds to the
subgoals that were generated after the execution of the [destruct
c] line right above it. Besides [-] and [+], Coq proofs can also
use [*] (asterisk) as a third kind of bullet. If we ever encounter
a proof that generates more than three levels of subgoals, we can
also enclose individual subgoals in curly braces ([{ ... }]): *)
Theorem andb_commutative' : forall b c, andb b c = andb c b.
Proof.
intros b c. destruct b.
{ destruct c.
{ reflexivity. }
{ reflexivity. } }
{ destruct c.
{ reflexivity. }
{ reflexivity. } }
Qed.
(** Since curly braces mark both the beginning and the end of a
proof, they can be used for multiple subgoal levels, as this
example shows. Furthermore, curly braces allow us to reuse the
same bullet shapes at multiple levels in a proof: *)
Theorem andb3_exchange :
forall b c d, andb (andb b c) d = andb (andb b d) c.
Proof.
intros b c d. destruct b.
- destruct c.
{ destruct d.
- reflexivity.
- reflexivity. }
{ destruct d.
- reflexivity.
- reflexivity. }
- destruct c.
{ destruct d.
- reflexivity.
- reflexivity. }
{ destruct d.
- reflexivity.
- reflexivity. }
Qed.
(** Before closing the chapter, let's mention one final
convenience. As you may have noticed, many proofs perform case
analysis on a variable right after introducing it:
intros x y. destruct y as [|y].
This pattern is so common that Coq provides a shorthand for it: we
can perform case analysis on a variable when introducing it by
using an intro pattern instead of a variable name. For instance,
here is a shorter proof of the [plus_1_neq_0] theorem above. *)
Theorem plus_1_neq_0' : forall n : nat,
beq_nat (n + 1) 0 = false.
Proof.
intros [|n].
- reflexivity.
- reflexivity. Qed.
(** If there are no arguments to name, we can just write [[]]. *)
Theorem andb_commutative'' :
forall b c, andb b c = andb c b.
Proof.
intros [] [].
- reflexivity.
- reflexivity.
- reflexivity.
- reflexivity.
Qed.
(** **** Exercise: 2 stars (andb_true_elim2) *)
(** Prove the following claim, marking cases (and subcases) with
bullets when you use [destruct]. *)
Theorem andb_true_elim2 : forall b c : bool,
andb b c = true -> c = true.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 1 star (zero_nbeq_plus_1) *)
Theorem zero_nbeq_plus_1 : forall n : nat,
beq_nat 0 (n + 1) = false.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ================================================================= *)
(** ** More on Notation (Optional) *)
(** (In general, sections marked Optional are not needed to follow the
rest of the book, except possibly other Optional sections. On a
first reading, you might want to skim these sections so that you
know what's there for future reference.)
Recall the notation definitions for infix plus and times: *)
Notation "x + y" := (plus x y)
(at level 50, left associativity)
: nat_scope.
Notation "x * y" := (mult x y)
(at level 40, left associativity)
: nat_scope.
(** For each notation symbol in Coq, we can specify its _precedence
level_ and its _associativity_. The precedence level [n] is
specified by writing [at level n]; this helps Coq parse compound
expressions. The associativity setting helps to disambiguate
expressions containing multiple occurrences of the same
symbol. For example, the parameters specified above for [+] and
[*] say that the expression [1+2*3*4] is shorthand for
[(1+((2*3)*4))]. Coq uses precedence levels from 0 to 100, and
_left_, _right_, or _no_ associativity. We will see more examples
of this later, e.g., in the [Lists]
chapter.
Each notation symbol is also associated with a _notation scope_.
Coq tries to guess what scope is meant from context, so when it
sees [S(O*O)] it guesses [nat_scope], but when it sees the
cartesian product (tuple) type [bool*bool] it guesses
[type_scope]. Occasionally, it is necessary to help it out with
percent-notation by writing [(x*y)%nat], and sometimes in what Coq
prints it will use [%nat] to indicate what scope a notation is in.
Notation scopes also apply to numeral notation ([3], [4], [5],
etc.), so you may sometimes see [0%nat], which means [O] (the
natural number [0] that we're using in this chapter), or [0%Z],
which means the Integer zero (which comes from a different part of
the standard library). *)
(* ================================================================= *)
(** ** Fixpoints and Structural Recursion (Optional) *)
(** Here is a copy of the definition of addition: *)
Fixpoint plus' (n : nat) (m : nat) : nat :=
match n with
| O => m
| S n' => S (plus' n' m)
end.
(** When Coq checks this definition, it notes that [plus'] is
"decreasing on 1st argument." What this means is that we are
performing a _structural recursion_ over the argument [n] -- i.e.,
that we make recursive calls only on strictly smaller values of
[n]. This implies that all calls to [plus'] will eventually
terminate. Coq demands that some argument of _every_ [Fixpoint]
definition is "decreasing."
This requirement is a fundamental feature of Coq's design: In
particular, it guarantees that every function that can be defined
in Coq will terminate on all inputs. However, because Coq's
"decreasing analysis" is not very sophisticated, it is sometimes
necessary to write functions in slightly unnatural ways. *)
(** **** Exercise: 2 stars, optional (decreasing) *)
(** To get a concrete sense of this, find a way to write a sensible
[Fixpoint] definition (of a simple function on numbers, say) that
_does_ terminate on all inputs, but that Coq will reject because
of this restriction. *)
(* FILL IN HERE *)
(** [] *)
(* ################################################################# *)
(** * More Exercises *)
(** **** Exercise: 2 stars (boolean_functions) *)
(** Use the tactics you have learned so far to prove the following
theorem about boolean functions. *)
Theorem identity_fn_applied_twice :
forall (f : bool -> bool),
(forall (x : bool), f x = x) ->
forall (b : bool), f (f b) = b.
Proof.
(* FILL IN HERE *) Admitted.
(** Now state and prove a theorem [negation_fn_applied_twice] similar
to the previous one but where the second hypothesis says that the
function [f] has the property that [f x = negb x].*)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 2 stars (andb_eq_orb) *)
(** Prove the following theorem. (You may want to first prove a
subsidiary lemma or two. Alternatively, remember that you do
not have to introduce all hypotheses at the same time.) *)
Theorem andb_eq_orb :
forall (b c : bool),
(andb b c = orb b c) ->
b = c.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars (binary) *)
(** Consider a different, more efficient representation of natural
numbers using a binary rather than unary system. That is, instead
of saying that each natural number is either zero or the successor
of a natural number, we can say that each binary number is either
- zero,
- twice a binary number, or
- one more than twice a binary number.
(a) First, write an inductive definition of the type [bin]
corresponding to this description of binary numbers.
(Hint: Recall that the definition of [nat] from class,
Inductive nat : Type :=
| O : nat
| S : nat -> nat.
says nothing about what [O] and [S] "mean." It just says "[O] is
in the set called [nat], and if [n] is in the set then so is [S
n]." The interpretation of [O] as zero and [S] as successor/plus
one comes from the way that we _use_ [nat] values, by writing
functions to do things with them, proving things about them, and
so on. Your definition of [bin] should be correspondingly simple;
it is the functions you will write next that will give it
mathematical meaning.)
(b) Next, write an increment function [incr] for binary numbers,
and a function [bin_to_nat] to convert binary numbers to unary numbers.
(c) Write five unit tests [test_bin_incr1], [test_bin_incr2], etc.
for your increment and binary-to-unary functions. Notice that
incrementing a binary number and then converting it to unary
should yield the same result as first converting it to unary and
then incrementing.
*)
(* FILL IN HERE *)
(** [] *)
(** $Date: 2016-07-13 12:41:41 -0400 (Wed, 13 Jul 2016) $ *)