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(* REMINDER:
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### PLEASE DO NOT DISTRIBUTE SOLUTIONS PUBLICLY ###
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(See the [Preface] for why.)
*)
(* ################################################################# *)
The functional programming style is founded on simple, everyday
mathematical intuition: 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 gives rise to a host of useful and powerful
programming 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 offers all of these features.
The first half of this chapter introduces the most essential
elements of Coq's functional programming language, called
Gallina. The second half introduces some basic tactics that
can be used to prove properties of Coq programs.
(* ################################################################# *)
(* ================================================================= *)
One notable 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, with all these familiar types as
instances.
Naturally, the Coq distribution comes preloaded 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. The following declaration tells Coq that
we are defining a new set of data values -- a type.
Inductive day : Type :=
| monday
| tuesday
| wednesday
| thursday
| friday
| saturday
| sunday.
The type is called day, and its members are monday,
tuesday, 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 can do type
inference -- but we'll generally 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.
(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 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:
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:
next_weekday (next_weekday saturday) = tuesdayreflexivity. Qed.tuesday = tuesday
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 go from proved-correct algorithms written in Gallina to
efficient machine code. (Of course, we are trusting the
correctness of the OCaml/Haskell/Scheme compiler, and of Coq's
extraction facility itself, but this is still a big step forward
from the way most software is developed today.) Indeed, this is
one of the main uses for which Coq was developed. We'll come back
to this topic in later chapters.
(* ================================================================= *)
If you are using Software Foundations in a course, your
instructor may use automatic scripts to help grade your homework
assignments. In order for these scripts to work correctly (so
that you get full credit for your work!), please be careful to
follow these rules:
You will also notice that each chapter (like Basics.v) is
accompanied by a test script (BasicsTest.v) that automatically
calculates points for the finished homework problems in the
chapter. These scripts are mostly for the auto-grading
infrastructure that your instructor may use to help process
assignments, but you may also like to use them to double-check
that your file is well formatted before handing it in. In a
terminal window either type make BasicsTest.vo or do the
following:
coqc -Q . LF Basics.v
coqc -Q . LF BasicsTest.v
There is no need to hand in BasicsTest.v itself (or Preface.v).
If your class is using the Canvas system to hand in assignments:
- The grading scripts work by extracting marked regions of the .v files that you submit. It is therefore important that you do not alter the "markup" that delimits exercises: the Exercise header, the name of the exercise, the "empty square bracket" marker at the end, etc. Please leave this markup exactly as you find it.
- Do not delete exercises. If you skip an exercise (e.g., because it is marked Optional, or because you can't solve it), it is OK to leave a partial proof in your .v file, but in this case please make sure it ends with Admitted (not, for example Abort).
- It is fine to use additional definitions (of helper functions, useful lemmas, etc.) in your solutions. You can put these between the exercise header and the theorem you are asked to prove.
- If you submit multiple versions of the assignment, you may notice that they are given different names. This is fine: The most recent submission is the one that will be graded.
- To hand in multiple files at the same time (if more than one chapter is assigned in the same week), you need to make a single submission with all the files at once using the button "Add another file" just above the comment box.
(* ================================================================= *)
In a similar way, we can define the standard type bool of
booleans, with members true and false.
Inductive bool : Type :=
| true
| false.
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, 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 of these illustrate Coq's syntax for
multi-argument function definitions. The corresponding
multi-argument application syntax is illustrated by the following
"unit tests," which constitute a complete specification -- a truth
table -- for the orb function:
orb true false = trueorb true false = truereflexivity. Qed.true = trueorb false false = falseorb false false = falsereflexivity. Qed.false = falseorb false true = trueorb false true = truereflexivity. Qed.true = trueorb true true = trueorb true true = truereflexivity. Qed.true = true
We can also introduce some familiar syntax for the boolean
operations we have just defined. The Notation command defines a new
symbolic notation for an existing definition.
Notation "x && y" := (andb x y). Notation "x || y" := (orb x y).false || false || true = truefalse || false || true = truereflexivity. Qed.true = true
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 command Admitted can be used as a placeholder for an
incomplete proof. We'll use it in exercises, to indicate the
parts that we're leaving for you -- i.e., your job is to replace
Admitteds with real proofs.
Exercise: 1 star, standard (nandb)
Admitted.b1, b2:boolbool(* FILL IN HERE *) Admitted.nandb true false = true(* FILL IN HERE *) Admitted.nandb false false = true(* FILL IN HERE *) Admitted.nandb false true = true(* FILL IN HERE *) Admitted.nandb true true = false
☐
Exercise: 1 star, standard (andb3)
Admitted.b1, b2, b3:boolbool(* FILL IN HERE *) Admitted.andb3 true true true = true(* FILL IN HERE *) Admitted.andb3 false true true = false(* FILL IN HERE *) Admitted.andb3 true false true = false(* FILL IN HERE *) Admitted.andb3 true true false = false
☐
(* ================================================================= *)
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.
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.
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."
(* ================================================================= *)
The types we have defined so far are examples of "enumerated
types": their definitions explicitly enumerate a finite set of
elements, each of which is just a bare constructor. Here is a
more interesting type definition, where one of the constructors
takes an argument:
Inductive rgb : Type := | red | green | blue. Inductive color : Type := | black | white | primary (p : rgb).
Let's look at this in a little more detail.
Every inductively defined type (day, bool, rgb, color,
etc.) contains a set of constructor expressions built from
constructors like red, primary, true, false, monday,
etc.
The definitions of rgb and color say how expressions in the
sets rgb and color can be built:
- red, green, and blue are the constructors of rgb;
- black, white, and primary are the constructors of color;
- the expression red belongs to the set rgb, as do the expressions green and blue;
- the expressions black and white belong to the set color;
- if p is an expression belonging to the set rgb, then primary p (pronounced "the constructor primary applied to the argument p") is an expression belonging to the set color; and
- expressions formed in these ways are the only ones belonging to the sets rgb and color.
We can define functions on colors using pattern matching just as
we have done for day and bool.
Definition monochrome (c : color) : bool :=
match c with
| black => true
| white => true
| primary q => false
end.
Since the primary constructor takes an argument, a pattern
matching primary should include either a variable (as above --
note that we can choose its name freely) or a constant of
appropriate type (as below).
Definition isred (c : color) : bool :=
match c with
| black => false
| white => false
| primary red => true
| primary _ => false
end.
The pattern primary _ here is shorthand for "primary applied
to any rgb constructor except red." (The wildcard pattern _
has the same effect as the dummy pattern variable p in the
definition of monochrome.)
(* ================================================================= *)
A single constructor with multiple parameters can be used
to create a tuple type. As an example, consider representing
the four bits in a nybble (half a byte). We first define
a datatype bit that resembles bool (using the
constructors B0 and B1 for the two possible bit values),
and then define the datatype nybble, which is essentially
a tuple of four bits.
Inductive bit : Type := | B0 | B1. Inductive nybble : Type := | bits (b0 b1 b2 b3 : bit).
The bits constructor acts as a wrapper for its contents.
Unwrapping can be done by pattern-matching, as in the all_zero
function which tests a nybble to see if all its bits are O.
Note that we are using underscore (_) as a wildcard pattern to
avoid inventing variable names that will not be used.
Definition all_zero (nb : nybble) : bool := match nb with | (bits B0 B0 B0 B0) => true | (bits _ _ _ _) => false end.(* ================================================================= *)
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. We will 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 we want to use in the rest because it
comes with a tiny bit of convenient special notation).
Module NatPlayground. (* ================================================================= *)
The types we have defined so far, "enumerated types" such as
day, bool, and bit, and tuple types such as nybble built
from them, share the property that each type has a finite set of
values. The natural numbers are an infinite set, and we need to
represent all of them in a datatype with a finite number of
constructors. There are many representations of numbers to choose
from. We are most familiar with decimal notation (base 10), using
the digits 0 through 9, for example, to form the number 123. You
may have encountered hexadecimal notation (base 16), in which the
same number is represented as 7B, or octal (base 8), where it is
173, or binary (base 2), where it is 1111011. Using an enumerated
type to represent digits, we could use any of these to represent
natural numbers. There are circumstances where each of these
choices can be useful.
Binary is valuable in computer hardware because it can in turn be
represented with two voltage levels, resulting in simple
circuitry. Analogously, we wish here to choose a representation
that makes proofs simpler.
Indeed, there is a representation of numbers that is even simpler
than binary, namely unary (base 1), in which only a single digit
is used (as one might do while counting days in prison by scratching
on the walls). To represent unary with a Coq datatype, we use
two constructors. The capital-letter O constructor represents zero.
When the S constructor is applied to the representation of the
natural number n, the result is the representation of n+1.
(S stands for "successor", or "scratch" if one is in prison.)
Here is the complete datatype definition.
Inductive nat : Type :=
| O
| S (n : nat).
With this definition, 0 is represented by O, 1 by S O,
2 by S (S O), and so on.
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 can be put in front of a natural number to yield another one -- if n is a natural number, then S n is too.
Again, let's look at this in a little more detail. The definition
of nat says how expressions in the set nat can be built:
- O and S are constructors;
- 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, bool,
color, etc.
The above 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 built
from data constructors, like true, andb true false, S (S
false), and O (O (O S)) do not.
A critical point here is that what we've done so far is just to
define a representation of numbers: a way of writing them down.
The names O and S are arbitrary, and at this point they have
no special meaning -- they are just two different marks that we
can use to write down numbers (together with a rule that says any
nat will be written as some string of S marks followed by an
O). If we like, we can write essentially the same definition
this way:
Inductive nat' : Type :=
| stop
| tick (foo : nat').
The interpretation of these marks comes from how we use them to
compute.
We can do this by writing functions that pattern match on
representations of natural numbers just as we did above with
booleans and days -- for example, here is 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 NatPlayground.
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 decimal numerals can be used as an alternative to
the "unary" notation defined by the constructors S and O. Coq
prints numbers in decimal form by default:
Definition minustwo (n : nat) : nat := match n with | O => O | S O => O | S (S n') => n' end.
The constructor S has the type nat → nat, just like
pred and functions like 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! It is just a way of
writing down numbers. (Think about standard decimal numerals: the
numeral 1 is not a computation; it's a piece of data. When we
write 111 to mean the number one hundred and eleven, we are
using 1, three times, to write down a concrete representation of
a number.)
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:
Definition oddb (n:nat) : bool := negb (evenb n).oddb 1 = trueoddb 1 = truereflexivity. Qed.oddb 1 = trueoddb 4 = falseoddb 4 = falsereflexivity. Qed.oddb 4 = false
(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 NatPlayground2. 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.
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.mult 3 3 = 9mult 3 3 = 9reflexivity. Qed.9 = 9
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. End NatPlayground2. Fixpoint exp (base power : nat) : nat := match power with | O => S O | S p => mult base (exp base p) end.
Exercise: 1 star, standard (factorial)
Admitted.factorial:nat -> natn:natnat(* FILL IN HERE *) Admitted.factorial 3 = 6(* FILL IN HERE *) Admitted.factorial 5 = 10 * 12
☐
Again, we can make numerical expressions 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.
(The level, associativity, and nat_scope annotations
control how these notations are treated by Coq's parser. The
details are not important for our purposes, but interested readers
can refer to the "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 almost nothing built-in, we really
mean it: even equality testing is a user-defined operation!
Here is a function eqb, which tests natural numbers for
equality, yielding a boolean. Note the use of nested
matches (we could also have used a simultaneous match, as we did
in minus.)
Fixpoint eqb (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' => eqb n' m'
end
end.
Similarly, 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.leb 2 2 = trueleb 2 2 = truereflexivity. Qed.true = trueleb 2 4 = trueleb 2 4 = truereflexivity. Qed.true = trueleb 4 2 = falseleb 4 2 = falsereflexivity. Qed.false = false
Since we'll be using these (especially eqb) a lot, let's give
them infix notations.
Notation "x =? y" := (eqb x y) (at level 70) : nat_scope. Notation "x <=? y" := (leb x y) (at level 70) : nat_scope.(4 <=? 2) = false(4 <=? 2) = falsereflexivity. Qed.false = false
Exercise: 1 star, standard (ltb)
Admitted. Notation "x <? y" := (ltb x y) (at level 70) : nat_scope.n, m:natbool(* FILL IN HERE *) Admitted.(2 <? 2) = false(* FILL IN HERE *) Admitted.(2 <? 4) = true(* FILL IN HERE *) Admitted.(4 <? 2) = false
☐
(* ################################################################# *)
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.
forall n : nat, 0 + n = nforall n : nat, 0 + n = nn:nat0 + n = nreflexivity. Qed.n:natn = n
(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
∀ 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:
forall n : nat, 0 + n = nforall n : nat, 0 + n = nreflexivity. Qed.n:nat0 + n = n
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 mostly a matter of style; the keywords
Example and Theorem (and a few others, including Lemma,
Fact, and Remark) mean pretty much the same thing to Coq.
Second, we've added the quantifier ∀ n:nat, so that our
theorem talks about all natural numbers n. Informally, to
prove theorems of this form, we generally start by saying "Suppose
n is some number..." Formally, this is achieved in the proof by
intros n, which moves n from the quantifier in the goal to a
context of current assumptions.
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.
forall n : nat, 1 + n = S nforall n : nat, 1 + n = S nreflexivity. Qed.n:nat1 + n = S nforall n : nat, 0 * n = 0forall n : nat, 0 * n = 0reflexivity. Qed.n:nat0 * n = 0
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.
(* ################################################################# *)
This theorem is a bit more interesting than the others we've
seen:
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 we are given such
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.
(* move both quantifiers into the context: *)forall n m : nat, n = m -> n + n = m + m(* move the hypothesis into the context: *)n, m:natn = m -> n + n = m + m(* rewrite the goal using the hypothesis: *)n, m:natH:n = mn + n = m + mreflexivity. Qed.n, m:natH:n = mm + m = m + m
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.)
forall n m o : nat, n = m -> m = o -> n + m = m + o(* FILL IN HERE *) Admitted.forall n m o : nat, n = m -> m = o -> n + m = m + o
☐
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 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.
forall n m : nat, (0 + n) * m = n * mforall n m : nat, (0 + n) * m = n * mn, m:nat(0 + n) * m = n * mreflexivity. Qed.n, m:natn * m = n * m
forall n m : nat, m = S n -> m * (1 + n) = m * m(* FILL IN HERE *) Admitted. (* (N.b. This proof can actually be completed with tactics other than [rewrite], but please do use [rewrite] for the sake of the exercise.) [] *) (* ################################################################# *)forall n m : nat, m = S n -> m * (1 + n) = m * m
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. (We then
use the Abort command to give up on it for the moment.)
forall n : nat, (n + 1 =? 0) = falseforall n : nat, (n + 1 =? 0) = falsen:nat(n + 1 =? 0) = falseAbort.n:nat(n + 1 =? 0) = false
The reason for this is that the definitions of both
eqb 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 eqb 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 (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, (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.
forall n : nat, (n + 1 =? 0) = falseforall n : nat, (n + 1 =? 0) = falsen:nat(n + 1 =? 0) = falsen:natE:n = 0(0 + 1 =? 0) = falsen, n':natE:n = S n'(S n' + 1 =? 0) = falsereflexivity.n:natE:n = 0(0 + 1 =? 0) = falsereflexivity. Qed.n, n':natE:n = S n'(S n' + 1 =? 0) = false
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.
In each subgoal, Coq remembers the assumption about n that is
relevant for this subgoal -- either n = 0 or n = S n' for some
n'. The eqn:E annotation tells destruct to give the name E to
this equation. (Leaving off the eqn:E annotation causes Coq to
elide these assumptions in the subgoals. This slightly
streamlines proofs where the assumptions are not explicitly used,
but it is better practice to keep them for the sake of
documentation, as they can help keep you oriented when working
with the subgoals.)
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 second one
simplifies (S n' + 1) =? 0 to false by first rewriting (S n'
+ 1) to S (n' + 1), then unfolding eqb, 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.
forall b : bool, negb (negb b) = bforall b : bool, negb (negb b) = bb:boolnegb (negb b) = bb:boolE:b = truenegb (negb true) = trueb:boolE:b = falsenegb (negb false) = falsereflexivity.b:boolE:b = truenegb (negb true) = truereflexivity. Qed.b:boolE:b = falsenegb (negb false) = false
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:
forall b c : bool, b && c = c && bforall b c : bool, b && c = c && bb, c:boolb && c = c && bb, c:boolEb:b = truetrue && c = c && trueb, c:boolEb:b = falsefalse && c = c && falseb, c:boolEb:b = truetrue && c = c && trueb, c:boolEb:b = trueEc:c = truetrue && true = true && trueb, c:boolEb:b = trueEc:c = falsetrue && false = false && truereflexivity.b, c:boolEb:b = trueEc:c = truetrue && true = true && truereflexivity.b, c:boolEb:b = trueEc:c = falsetrue && false = false && trueb, c:boolEb:b = falsefalse && c = c && falseb, c:boolEb:b = falseEc:c = truefalse && true = true && falseb, c:boolEb:b = falseEc:c = falsefalse && false = false && falsereflexivity.b, c:boolEb:b = falseEc:c = truefalse && true = true && falsereflexivity. Qed.b, c:boolEb:b = falseEc:c = falsefalse && false = false && false
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 +, we can use × (asterisk) as a third kind of
bullet. We can also enclose sub-proofs in curly braces, which is
useful in case we ever encounter a proof that generates more than
three levels of subgoals:
forall b c : bool, b && c = c && bforall b c : bool, b && c = c && bb, c:boolb && c = c && bb, c:boolEb:b = truetrue && c = c && trueb, c:boolEb:b = falsefalse && c = c && falseb, c:boolEb:b = truetrue && c = c && trueb, c:boolEb:b = trueEc:c = truetrue && true = true && trueb, c:boolEb:b = trueEc:c = falsetrue && false = false && truereflexivity.b, c:boolEb:b = trueEc:c = truetrue && true = true && trueb, c:boolEb:b = trueEc:c = falsetrue && false = false && truereflexivity. }b, c:boolEb:b = trueEc:c = falsetrue && false = false && trueb, c:boolEb:b = falsefalse && c = c && falseb, c:boolEb:b = falsefalse && c = c && falseb, c:boolEb:b = falseEc:c = truefalse && true = true && falseb, c:boolEb:b = falseEc:c = falsefalse && false = false && falsereflexivity.b, c:boolEb:b = falseEc:c = truefalse && true = true && falseb, c:boolEb:b = falseEc:c = falsefalse && false = false && falsereflexivity. } } Qed.b, c:boolEb:b = falseEc:c = falsefalse && false = false && false
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:
forall b c d : bool, b && c && d = b && d && cforall b c d : bool, b && c && d = b && d && cb, c, d:boolb && c && d = b && d && cb, c, d:boolEb:b = truetrue && c && d = true && d && cb, c, d:boolEb:b = falsefalse && c && d = false && d && cb, c, d:boolEb:b = truetrue && c && d = true && d && cb, c, d:boolEb:b = trueEc:c = truetrue && true && d = true && d && trueb, c, d:boolEb:b = trueEc:c = falsetrue && false && d = true && d && falseb, c, d:boolEb:b = trueEc:c = truetrue && true && d = true && d && trueb, c, d:boolEb:b = trueEc:c = trueEd:d = truetrue && true && true = true && true && trueb, c, d:boolEb:b = trueEc:c = trueEd:d = falsetrue && true && false = true && false && truereflexivity.b, c, d:boolEb:b = trueEc:c = trueEd:d = truetrue && true && true = true && true && truereflexivity.b, c, d:boolEb:b = trueEc:c = trueEd:d = falsetrue && true && false = true && false && trueb, c, d:boolEb:b = trueEc:c = falsetrue && false && d = true && d && falseb, c, d:boolEb:b = trueEc:c = falsetrue && false && d = true && d && falseb, c, d:boolEb:b = trueEc:c = falseEd:d = truetrue && false && true = true && true && falseb, c, d:boolEb:b = trueEc:c = falseEd:d = falsetrue && false && false = true && false && falsereflexivity.b, c, d:boolEb:b = trueEc:c = falseEd:d = truetrue && false && true = true && true && falsereflexivity. }b, c, d:boolEb:b = trueEc:c = falseEd:d = falsetrue && false && false = true && false && falseb, c, d:boolEb:b = falsefalse && c && d = false && d && cb, c, d:boolEb:b = falseEc:c = truefalse && true && d = false && d && trueb, c, d:boolEb:b = falseEc:c = falsefalse && false && d = false && d && falseb, c, d:boolEb:b = falseEc:c = truefalse && true && d = false && d && trueb, c, d:boolEb:b = falseEc:c = trueEd:d = truefalse && true && true = false && true && trueb, c, d:boolEb:b = falseEc:c = trueEd:d = falsefalse && true && false = false && false && truereflexivity.b, c, d:boolEb:b = falseEc:c = trueEd:d = truefalse && true && true = false && true && truereflexivity.b, c, d:boolEb:b = falseEc:c = trueEd:d = falsefalse && true && false = false && false && trueb, c, d:boolEb:b = falseEc:c = falsefalse && false && d = false && d && falseb, c, d:boolEb:b = falseEc:c = falsefalse && false && d = false && d && falseb, c, d:boolEb:b = falseEc:c = falseEd:d = truefalse && false && true = false && true && falseb, c, d:boolEb:b = falseEc:c = falseEd:d = falsefalse && false && false = false && false && falsereflexivity.b, c, d:boolEb:b = falseEc:c = falseEd:d = truefalse && false && true = false && true && falsereflexivity. } Qed.b, c, d:boolEb:b = falseEc:c = falseEd:d = falsefalse && false && false = false && false && false
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 eqn:E.
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. (You'll also note one downside of this shorthand: we lose
the equation recording the assumption we are making in each
subgoal, which we previously got from the eqn:E annotation.)
forall n : nat, (n + 1 =? 0) = falseforall n : nat, (n + 1 =? 0) = false(0 + 1 =? 0) = falsen:nat(S n + 1 =? 0) = falsereflexivity.(0 + 1 =? 0) = falsereflexivity. Qed.n:nat(S n + 1 =? 0) = false
If there are no arguments to name, we can just write [].
forall b c : bool, b && c = c && bforall b c : bool, b && c = c && btrue && true = true && truetrue && false = false && truefalse && true = true && falsefalse && false = false && falsereflexivity.true && true = true && truereflexivity.true && false = false && truereflexivity.false && true = true && falsereflexivity. Qed.false && false = false && false
Exercise: 2 stars, standard (andb_true_elim2)
forall b c : bool, b && c = true -> c = true(* FILL IN HERE *) Admitted.forall b c : bool, b && c = true -> c = true
☐
forall n : nat, (0 =? n + 1) = false(* FILL IN HERE *) Admitted.forall n : nat, (0 =? n + 1) = false
☐
(* ================================================================= *)
(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 (which we'll see in
later chapters) 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).
Pro tip: Coq's notation mechanism is not especially powerful.
Don't expect too much from it!
(* ================================================================= *)
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, standard, optional (decreasing)
(* FILL IN HERE
[] *)
(* ################################################################# *)
Each SF chapter comes with a tester file (e.g. BasicsTest.v),
containing scripts that check most of the exercises. You can run
make BasicsTest.vo in a terminal and check its output to make
sure you didn't miss anything.
Exercise: 1 star, standard (indentity_fn_applied_twice)
forall f : bool -> bool, (forall x : bool, f x = x) -> forall b : bool, f (f b) = b(* FILL IN HERE *) Admitted.forall f : bool -> bool, (forall x : bool, f x = x) -> forall b : bool, f (f b) = b
☐
Exercise: 1 star, standard (negation_fn_applied_twice)
(* FILL IN HERE *) (* The [Import] statement on the next line tells Coq to use the standard library String module. We'll use strings more in later chapters, but for the moment we just need syntax for literal strings for the grader comments. *) From Coq Require Export String. (* Do not modify the following line: *) Definition manual_grade_for_negation_fn_applied_twice : option (nat*string) := None.
☐
Exercise: 3 stars, standard, optional (andb_eq_orb)
forall b c : bool, b && c = b || c -> b = c(* FILL IN HERE *) Admitted.forall b c : bool, b && c = b || c -> b = c
☐
Exercise: 3 stars, standard (binary)
Inductive bin : Type :=
| Z
| A (n : bin)
| B (n : bin).
(a) Complete the definitions below of an increment function incr
for binary numbers, and a function bin_to_nat to convert
binary numbers to unary numbers.
Admitted.incr:bin -> binm:binbinAdmitted.bin_to_nat:bin -> natm:binnat
(b) Write five unit tests test_bin_incr1, test_bin_incr2, etc.
for your increment and binary-to-unary functions. (A "unit
test" in Coq is a specific Example that can be proved with
just reflexivity, as we've done for several of our
definitions.) 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 *) (* Do not modify the following line: *) Definition manual_grade_for_binary : option (nat*string) := None.
☐
(* Wed Jan 9 12:02:44 EST 2019 *)