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(** * Lists: Working with Structured Data *)
From LF Require Export Induction.
Module NatList.
(* ################################################################# *)
(** * Pairs of Numbers *)
(** In an [Inductive] type definition, each constructor can take
any number of arguments -- none (as with [true] and [O]), one (as
with [S]), or more than one (as with [nybble] and the following): *)
Inductive natprod : Type :=
| pair (n1 n2 : nat).
(** This declaration can be read: "The one and only way to
construct a pair of numbers is by applying the constructor [pair]
to two arguments of type [nat]." *)
Check (pair 3 5) : natprod.
(** Functions for extracting the first and second components of a pair
can then be defined by pattern matching. *)
Definition fst (p : natprod) : nat :=
match p with
| pair x y => x
end.
Definition snd (p : natprod) : nat :=
match p with
| pair x y => y
end.
Compute (fst (pair 3 5)).
(* ===> 3 *)
(** Since pairs will be used heavily in what follows, it will be
convenient to write them with the standard mathematical notation
[(x,y)] instead of [pair x y]. We can tell Rocq to allow this with
a [Notation] declaration. *)
Notation "( x , y )" := (pair x y).
(** The new notation can be used both in expressions and in pattern
matches. *)
Compute (fst (3,5)).
Definition fst' (p : natprod) : nat :=
match p with
| (x,y) => x
end.
Definition snd' (p : natprod) : nat :=
match p with
| (x,y) => y
end.
Definition swap_pair (p : natprod) : natprod :=
match p with
| (x,y) => (y,x)
end.
(** Note that pattern-matching on a pair (with parentheses: [(x, y)])
is not to be confused with the "multiple pattern" syntax (with no
parentheses: [x, y]) that we have seen previously. The above
examples illustrate pattern matching on a pair with elements [x]
and [y], whereas, for example, the definition of [minus] in
[Basics] performs pattern matching on the values [n] and [m]:
Fixpoint minus (n m : nat) : nat :=
match n, m with
| O , _ => O
| S _ , O => n
| S n', S m' => minus n' m'
end.
The distinction is minor, but it is worth understanding that they
are not the same. For instance, the following definitions are
ill-formed:
(* Can't match on a pair with multiple patterns: *)
Definition bad_fst (p : natprod) : nat :=
match p with
| x, y => x
end.
(* Can't match on multiple values with pair patterns: *)
Definition bad_minus (n m : nat) : nat :=
match n, m with
| (O , _ ) => O
| (S _ , O ) => n
| (S n', S m') => bad_minus n' m'
end.
*)
(** If we state properties of pairs in a slightly peculiar way, we can
sometimes complete their proofs with just reflexivity and its
built-in simplification: *)
Theorem surjective_pairing' : forall (n m : nat),
(n,m) = (fst (n,m), snd (n,m)).
Proof.
reflexivity. Qed.
(** But just [reflexivity] is not enough if we state the lemma in a more
natural way: *)
Theorem surjective_pairing_stuck : forall (p : natprod),
p = (fst p, snd p).
Proof.
simpl. (* Doesn't reduce anything! *)
Abort.
(** Instead, we need to expose the structure of [p] so that
[simpl] can perform the pattern match in [fst] and [snd]. We can
do this with [destruct]. *)
Theorem surjective_pairing : forall (p : natprod),
p = (fst p, snd p).
Proof.
intros p. destruct p as [n m]. simpl. reflexivity. Qed.
(** Notice that, by contrast with the behavior of [destruct] on
[nat]s, where it generates two subgoals, [destruct] generates just
one subgoal here. That's because [natprod]s can only be
constructed in one way. *)
(** **** Exercise: 1 star, standard (snd_fst_is_swap) *)
Theorem snd_fst_is_swap : forall (p : natprod),
(snd p, fst p) = swap_pair p.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 1 star, standard, optional (fst_swap_is_snd) *)
Theorem fst_swap_is_snd : forall (p : natprod),
fst (swap_pair p) = snd p.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ################################################################# *)
(** * Lists of Numbers *)
(** Generalizing the definition of pairs, we can describe the
type of _lists_ of numbers like this: "A list is either the empty
list or else a pair of a number and another list." *)
Inductive natlist : Type :=
| nil
| cons (n : nat) (l : natlist).
(** For example, here is a three-element list: *)
Definition mylist := cons 1 (cons 2 (cons 3 nil)).
(** As with pairs, it is convenient to write lists in familiar
notation. The following declarations allow us to use [::] as an
infix [cons] operator and square brackets as an "outfix" notation
for constructing lists. *)
Notation "x :: l" := (cons x l)
(at level 60, right associativity).
Notation "[ ]" := nil.
Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..).
(** It is not necessary to understand the details of these
declarations, but here is roughly what's going on in case you are
interested. The "[right associativity]" annotation tells Rocq how to
parenthesize expressions involving multiple uses of [::] so that,
for example, the next three declarations mean exactly the same
thing: *)
Definition mylist1 := 1 :: (2 :: (3 :: nil)).
Definition mylist2 := 1 :: 2 :: 3 :: nil.
Definition mylist3 := [1;2;3].
(** The "[at level 60]" part tells Rocq how to parenthesize
expressions that involve both [::] and some other infix operator.
For example, since we defined [+] as infix notation for the [plus]
function at level 50,
Notation "x + y" := (plus x y) (at level 50, left associativity).
the [+] operator will bind tighter than [::], so [1 + 2 :: [3]]
will be parsed, as we'd expect, as [(1 + 2) :: [3]] rather than [1
+ (2 :: [3])].
(Expressions like "[1 + 2 :: [3]]" can be a little confusing when
you read them in a [.v] file. The inner brackets, around 3,
indicate a list, but the outer brackets, which are invisible in
the HTML rendering, are there to instruct the "rocq doc" tool that
the bracketed part should be displayed as Rocq code rather than
running text.)
The second and third [Notation] declarations above introduce the
standard square-bracket notation for lists; the right-hand side of
the third one illustrates Rocq's syntax for declaring n-ary
notations and translating them to nested sequences of binary
constructors.
Again, don't worry if some of these parsing details are puzzling:
all the notations you'll need in this course will be defined for
you. *)
(* ----------------------------------------------------------------- *)
(** *** Repeat *)
(** Next let's look at several functions for constructing and
manipulating lists. First is the [repeat] function, which takes a
number [n] and a [count] and returns a list of length [count] in
which every element is [n]. *)
Fixpoint repeat (n count : nat) : natlist :=
match count with
| O => nil
| S count' => n :: (repeat n count')
end.
(* ----------------------------------------------------------------- *)
(** *** Length *)
(** The [length] function calculates the length of a list. *)
Fixpoint length (l:natlist) : nat :=
match l with
| nil => O
| h :: t => S (length t)
end.
(* ----------------------------------------------------------------- *)
(** *** Append *)
(** The [app] function appends (concatenates) two lists. *)
Fixpoint app (l1 l2 : natlist) : natlist :=
match l1 with
| nil => l2
| h :: t => h :: (app t l2)
end.
(** Since [app] will be used extensively, it is again convenient
to have an infix operator for it. *)
Notation "x ++ y" := (app x y)
(right associativity, at level 60).
Example test_app1: [1;2;3] ++ [4;5] = [1;2;3;4;5].
Proof. reflexivity. Qed.
Example test_app2: nil ++ [4;5] = [4;5].
Proof. reflexivity. Qed.
Example test_app3: [1;2;3] ++ nil = [1;2;3].
Proof. reflexivity. Qed.
(* ----------------------------------------------------------------- *)
(** *** Head and Tail *)
(** Here are two more handy functions for working with lists.
The [hd] function returns the first element (the "head") of the
list, while [tl] returns everything but the first element (the
"tail"). Since the empty list has no first element, we pass
a default value to be returned in that case. *)
Definition hd (default : nat) (l : natlist) : nat :=
match l with
| nil => default
| h :: t => h
end.
Definition tl (l : natlist) : natlist :=
match l with
| nil => nil
| h :: t => t
end.
Example test_hd1: hd 0 [1;2;3] = 1.
Proof. reflexivity. Qed.
Example test_hd2: hd 0 [] = 0.
Proof. reflexivity. Qed.
Example test_tl: tl [1;2;3] = [2;3].
Proof. reflexivity. Qed.
(* ----------------------------------------------------------------- *)
(** *** Exercises *)
(** **** Exercise: 2 stars, standard, especially useful (list_funs)
Complete the definitions of [nonzeros], [oddmembers], and
[countoddmembers] below. Have a look at the tests to understand
what these functions should do. *)
Fixpoint nonzeros (l:natlist) : natlist
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_nonzeros:
nonzeros [0;1;0;2;3;0;0] = [1;2;3].
(* FILL IN HERE *) Admitted.
Fixpoint oddmembers (l:natlist) : natlist
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_oddmembers:
oddmembers [0;1;0;2;3;0;0] = [1;3].
(* FILL IN HERE *) Admitted.
(** For the next problem, [countoddmembers], we're giving you a header
that uses the keyword [Definition] instead of [Fixpoint]. The
point of stating the question this way is to encourage you to
implement the function by using already-defined functions, rather
than writing your own recursive definition. *)
Definition countoddmembers (l:natlist) : nat
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_countoddmembers1:
countoddmembers [1;0;3;1;4;5] = 4.
(* FILL IN HERE *) Admitted.
Example test_countoddmembers2:
countoddmembers [0;2;4] = 0.
(* FILL IN HERE *) Admitted.
Example test_countoddmembers3:
countoddmembers nil = 0.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, advanced (alternate)
Complete the following definition of [alternate], which
interleaves two lists into one, alternating between elements taken
from the first list and elements from the second. See the tests
below for more specific examples.
Hint: there are natural ways of writing [alternate] that fail to
satisfy Rocq's requirement that all [Fixpoint] definitions be
_structurally recursive_, as mentioned in [Basics]. If you
encounter this difficulty, consider pattern matching against both
lists at the same time with the "multiple pattern" syntax we've
seen before. *)
Fixpoint alternate (l1 l2 : natlist) : natlist
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_alternate1:
alternate [1;2;3] [4;5;6] = [1;4;2;5;3;6].
(* FILL IN HERE *) Admitted.
Example test_alternate2:
alternate [1] [4;5;6] = [1;4;5;6].
(* FILL IN HERE *) Admitted.
Example test_alternate3:
alternate [1;2;3] [4] = [1;4;2;3].
(* FILL IN HERE *) Admitted.
Example test_alternate4:
alternate [] [20;30] = [20;30].
(* FILL IN HERE *) Admitted.
(** [] *)
(* ----------------------------------------------------------------- *)
(** *** Bags via Lists *)
(** A [bag] (or [multiset]) is like a set, except that each element
can appear multiple times rather than just once. One way of
representating a bag of numbers is as a list. *)
Definition bag := natlist.
(** **** Exercise: 3 stars, standard, especially useful (bag_functions)
Complete the following definitions for the functions [count],
[sum], [add], and [member] for bags. *)
Fixpoint count (v : nat) (s : bag) : nat
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
(** All these proofs can be completed with [reflexivity]. *)
Example test_count1: count 1 [1;2;3;1;4;1] = 3.
(* FILL IN HERE *) Admitted.
Example test_count2: count 6 [1;2;3;1;4;1] = 0.
(* FILL IN HERE *) Admitted.
(** Multiset [sum] is similar to set [union]: [sum a b] contains all
the elements of [a] and those of [b]. (Mathematicians usually
define [union] on multisets a little bit differently -- using max
instead of sum -- which is why we don't call this operation
[union].)
We've deliberately given you a header that does not give explicit
names to the arguments. Implement [sum] in terms of an
already-defined function, without changing the header. *)
Definition sum : bag -> bag -> bag
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_sum1: count 1 (sum [1;2;3] [1;4;1]) = 3.
(* FILL IN HERE *) Admitted.
Definition add (v : nat) (s : bag) : bag
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_add1: count 1 (add 1 [1;4;1]) = 3.
(* FILL IN HERE *) Admitted.
Example test_add2: count 5 (add 1 [1;4;1]) = 0.
(* FILL IN HERE *) Admitted.
Fixpoint member (v : nat) (s : bag) : bool
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_member1: member 1 [1;4;1] = true.
(* FILL IN HERE *) Admitted.
Example test_member2: member 2 [1;4;1] = false.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, standard, optional (bag_more_functions)
Here are some more [bag] functions for you to practice with. *)
(** When [remove_one] is applied to a bag without the number to
remove, it should return the same bag unchanged. (This exercise
is optional, but students following the advanced track will need
to fill in the definition of [remove_one] for a later
exercise.) *)
Fixpoint remove_one (v : nat) (s : bag) : bag
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_remove_one1:
count 5 (remove_one 5 [2;1;5;4;1]) = 0.
(* FILL IN HERE *) Admitted.
Example test_remove_one2:
count 5 (remove_one 5 [2;1;4;1]) = 0.
(* FILL IN HERE *) Admitted.
Example test_remove_one3:
count 4 (remove_one 5 [2;1;4;5;1;4]) = 2.
(* FILL IN HERE *) Admitted.
Example test_remove_one4:
count 5 (remove_one 5 [2;1;5;4;5;1;4]) = 1.
(* FILL IN HERE *) Admitted.
Fixpoint remove_all (v:nat) (s:bag) : bag
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_remove_all1: count 5 (remove_all 5 [2;1;5;4;1]) = 0.
(* FILL IN HERE *) Admitted.
Example test_remove_all2: count 5 (remove_all 5 [2;1;4;1]) = 0.
(* FILL IN HERE *) Admitted.
Example test_remove_all3: count 4 (remove_all 5 [2;1;4;5;1;4]) = 2.
(* FILL IN HERE *) Admitted.
Example test_remove_all4: count 5 (remove_all 5 [2;1;5;4;5;1;4;5;1;4]) = 0.
(* FILL IN HERE *) Admitted.
Fixpoint included (s1 : bag) (s2 : bag) : bool
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_included1: included [1;2] [2;1;4;1] = true.
(* FILL IN HERE *) Admitted.
Example test_included2: included [1;2;2] [2;1;4;1] = false.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 2 stars, standard, optional (add_inc_count)
Adding a value to a bag should increase the value's count by one.
State this as a theorem and prove it in Rocq. *)
(*
Theorem add_inc_count : ...
Proof.
...
Qed.
*)
(* Do not modify the following line: *)
Definition manual_grade_for_add_inc_count : option (nat*string) := None.
(** [] *)
(* ################################################################# *)
(** * Reasoning About Lists *)
(** As with numbers, simple facts about list-processing
functions can sometimes be proved entirely by simplification. For
example, just [reflexivity] is enough for this theorem... *)
Theorem nil_app : forall l : natlist,
[] ++ l = l.
Proof. reflexivity. Qed.
(** ...because the [[]] is substituted into the "scrutinee" (the
expression whose value is being "scrutinized" by the match) in the
definition of [app], allowing the match itself to be simplified. *)
(** Also, as with numbers, it is sometimes helpful to perform case
analysis on the possible shapes -- empty or non-empty -- of an unknown
list. *)
Theorem tl_length_pred : forall l:natlist,
pred (length l) = length (tl l).
Proof.
intros l. destruct l as [| n l'].
- (* l = nil *)
reflexivity.
- (* l = cons n l' *)
reflexivity. Qed.
(** Here, the [nil] case works because we've chosen to define
[tl nil = nil]. Notice that the [as] annotation on the [destruct]
tactic here introduces two names, [n] and [l'], corresponding to
the fact that the [cons] constructor for lists takes two
arguments (the head and tail of the list it is constructing). *)
(** Usually, though, interesting theorems about lists require
induction for their proofs. We'll see how to do this next. *)
(** (Micro-Sermon: As we get deeper into this material, simply
_reading_ proof scripts will not help you very much. Rather, it
is important to step through the details of each one using Rocq and
think about what each step achieves. Otherwise it is more or less
guaranteed that the exercises will make no sense when you get to
them. 'Nuff said.) *)
(* ================================================================= *)
(** ** Induction on Lists *)
(** Proofs by induction over datatypes like [natlist] are a
little less familiar than standard natural number induction, but
the idea is equally simple. Each [Inductive] declaration defines
a set of data values that can be built up using the declared
constructors. For example, a boolean can be either [true] or
[false]; a number can be either [O] or else [S] applied to another
number; and a list can be either [nil] or else [cons] applied to a
number and a list. Moreover, applications of the declared
constructors to one another are the _only_ possible shapes that
elements of an inductively defined set can have.
This last fact directly gives rise to a way of reasoning about
inductively defined sets: a number is either [O] or else it is [S]
applied to some _smaller_ number; a list is either [nil] or else
it is [cons] applied to some number and some _smaller_ list;
etc. Thus, if we have in mind some proposition [P] that mentions a
list [l] and we want to argue that [P] holds for _all_ lists, we
can reason as follows:
- First, show that [P] is true of [l] when [l] is [nil].
- Then show that [P] is true of [l] when [l] is [cons n l'] for
some number [n] and some smaller list [l'], assuming that [P]
is true for [l'].
Since larger lists can always be broken down into smaller ones,
eventually reaching [nil], these two arguments together establish
the truth of [P] for all lists [l].
Here's a concrete example: *)
Theorem app_assoc : forall l1 l2 l3 : natlist,
(l1 ++ l2) ++ l3 = l1 ++ (l2 ++ l3).
Proof.
intros l1 l2 l3. induction l1 as [| n l1' IHl1'].
- (* l1 = nil *)
reflexivity.
- (* l1 = cons n l1' *)
simpl. rewrite -> IHl1'. reflexivity. Qed.
(** Notice that, as we saw with induction on natural numbers,
the [as...] clause provided to the [induction] tactic gives a name
to the induction hypothesis corresponding to the smaller list
[l1'] in the [cons] case.
Once again, this Rocq proof is not especially illuminating as a
static document -- it is easy to see what's going on if you are
reading the proof in an interactive Rocq session and you can see
the current goal and context at each point, but this state is not
visible in the written-down parts of the Rocq proof. So a
natural-language proof -- one written for human readers -- would
include more explicit signposts; in particular, it helps the
reader stay oriented to remind them exactly what the induction
hypothesis is in the second case. *)
(** For comparison, here is an informal proof of the same theorem. *)
(** _Theorem_: For all lists [l1], [l2], and [l3],
[(l1 ++ l2) ++ l3 = l1 ++ (l2 ++ l3)].
_Proof_: By induction on [l1].
- First, suppose [l1 = []]. We must show
([] ++ l2) ++ l3 = [] ++ (l2 ++ l3),
which follows directly from the definition of [++].
- Next, suppose [l1 = n::l1'], with
(l1' ++ l2) ++ l3 = l1' ++ (l2 ++ l3)
(the induction hypothesis). We must show
((n :: l1') ++ l2) ++ l3 = (n :: l1') ++ (l2 ++ l3).
By the definition of [++], this follows from
n :: ((l1' ++ l2) ++ l3) = n :: (l1' ++ (l2 ++ l3)),
which is immediate from the induction hypothesis. [] *)
(* ----------------------------------------------------------------- *)
(** *** Generalizing Statements *)
(** In some situations, it is necessary to generalize a
statement in order to prove it by induction. Intuitively, the
reason is that a more general statement also yields a more general
(stronger) inductive hypothesis. If you find yourself stuck in a
proof, it may help to step back and see whether you can prove a
stronger statement. *)
Theorem repeat_double_firsttry : forall c n: nat,
repeat n c ++ repeat n c = repeat n (c + c).
Proof.
intros c. induction c as [| c' IHc'].
- (* c = 0 *)
intros n. simpl. reflexivity.
- (* c = S c' *)
intros n. simpl.
(* Now we seem to be stuck. The IH cannot be used to
rewrite [repeat n (c' + S c')]: it only works
for [repeat n (c' + c')]. If the IH were more liberal here
(e.g., if it worked for an arbitrary second summand),
the proof would go through. *)
Abort.
(** To get a more general inductive hypothesis, we can generalize
the statement as follows: *)
Theorem repeat_plus: forall c1 c2 n: nat,
repeat n c1 ++ repeat n c2 = repeat n (c1 + c2).
Proof.
intros c1 c2 n.
induction c1 as [| c1' IHc1'].
- simpl. reflexivity.
- simpl.
rewrite <- IHc1'.
reflexivity.
Qed.
(* ----------------------------------------------------------------- *)
(** *** Reversing a List *)
(** For a slightly more involved example of inductive proof over
lists, suppose we use [app] to define a list-reversing function
[rev]: *)
Fixpoint rev (l:natlist) : natlist :=
match l with
| nil => nil
| h :: t => rev t ++ [h]
end.
Example test_rev1: rev [1;2;3] = [3;2;1].
Proof. reflexivity. Qed.
Example test_rev2: rev nil = nil.
Proof. reflexivity. Qed.
(** For something a bit more challenging, let's prove that
reversing a list does not change its length. Our first attempt
gets stuck in the successor case... *)
Theorem rev_length_firsttry : forall l : natlist,
length (rev l) = length l.
Proof.
intros l. induction l as [| n l' IHl'].
- (* l = nil *)
reflexivity.
- (* l = n :: l' *)
(* This is the tricky case. Let's begin as usual
by simplifying. *)
simpl.
(* Now we seem to be stuck: the goal is an equality
involving [++], but we don't have any useful equations
in either the immediate context or in the global
environment! We can make a little progress by using
the IH to rewrite the goal... *)
rewrite <- IHl'.
(* ... but now we can't go any further. *)
Abort.
(** A first attempt to make progress would be to prove exactly
the statement that we are missing at this point. But this attempt
will fail because the inductive hypothesis is not general enough. *)
Theorem app_rev_length_S_firsttry: forall l n,
length (rev l ++ [n]) = S (length (rev l)).
Proof.
intros l. induction l as [| m l' IHl'].
- (* l = [] *)
intros n. simpl. reflexivity.
- (* l = m:: l' *)
intros n. simpl.
(* IHl' not applicable. *)
Abort.
(** It turns out that the above lemma is more specific than it
needs to be. We can strengthen the lemma to work not only on reversed
lists but on general lists. *)
Theorem app_length_S: forall l n,
length (l ++ [n]) = S (length l).
Proof.
intros l n. induction l as [| m l' IHl'].
- (* l = [] *)
simpl. reflexivity.
- (* l = m:: l' *)
simpl.
rewrite IHl'.
reflexivity.
Qed.
(** Now we can complete the original proof. *)
Theorem rev_length : forall l : natlist,
length (rev l) = length l.
Proof.
intros l. induction l as [| n l' IHl'].
- (* l = nil *)
reflexivity.
- (* l = cons *)
simpl.
rewrite -> app_length_S.
rewrite -> IHl'.
reflexivity.
Qed.
(** Note that the [app_length_S] lemma we proved above is pretty
narrow, requiring that the second list contains only a single element.
We can prove a more general version for any two lists. *)
Theorem app_length : forall l1 l2 : natlist,
length (l1 ++ l2) = (length l1) + (length l2).
Proof.
(* WORKED IN CLASS *)
intros l1 l2. induction l1 as [| n l1' IHl1'].
- (* l1 = nil *)
reflexivity.
- (* l1 = cons *)
simpl. rewrite -> IHl1'. reflexivity. Qed.
(** For comparison, here are informal proofs of these two theorems:
_Theorem_: For all lists [l1] and [l2],
[length (l1 ++ l2) = length l1 + length l2].
_Proof_: By induction on [l1].
- First, suppose [l1 = []]. We must show
length ([] ++ l2) = length [] + length l2,
which follows directly from the definitions of [length],
[++], and [plus].
- Next, suppose [l1 = n::l1'], with
length (l1' ++ l2) = length l1' + length l2.
We must show
length ((n::l1') ++ l2) = length (n::l1') + length l2.
This follows directly from the definitions of [length] and [++]
together with the induction hypothesis. [] *)
(** _Theorem_: For all lists [l], [length (rev l) = length l].
_Proof_: By induction on [l].
- First, suppose [l = []]. We must show
length (rev []) = length [],
which follows directly from the definitions of [length]
and [rev].
- Next, suppose [l = n::l'], with
length (rev l') = length l'.
We must show
length (rev (n :: l')) = length (n :: l').
By the definition of [rev], this follows from
length ((rev l') ++ [n]) = S (length l')
which, by the previous lemma, is the same as
length (rev l') + length [n] = S (length l').
This follows directly from the induction hypothesis and the
definition of [length]. [] *)
(** The style of these proofs is rather longwinded and pedantic.
After reading a couple like this, we might find it easier to
follow proofs that give fewer details (which we can easily work
out in our own minds or on scratch paper if necessary) and just
highlight the non-obvious steps. In this more compressed style,
the above proof might look like this: *)
(** _Theorem_: For all lists [l], [length (rev l) = length l].
_Proof_: First observe, by a straightforward induction on [l],
that [length (l ++ [n]) = S (length l)] for any [l]. The main
property then follows by another induction on [l], using the
observation together with the induction hypothesis in the case
where [l = n'::l']. [] *)
(** Which style is preferable in a given situation depends on
the sophistication of the expected audience and how similar the
proof at hand is to ones that they will already be familiar with.
The more pedantic style is a good default for our present purposes
because we're trying to be ultra-clear about the details. *)
(* ================================================================= *)
(** ** [Search] *)
(** We've seen that proofs can make use of other theorems we've
already proved, e.g., using [rewrite]. But in order to refer to a
theorem, we need to know its name! Indeed, it is often hard even
to remember what theorems have been proven, much less what they
are called.
Rocq's [Search] command is quite helpful with this.
Let's say you've forgotten the name of a theorem about [rev]. The
command [Search rev] will cause Rocq to display a list of all
theorems involving [rev]. *)
Search rev.
(** Or say you've forgotten the name of the theorem showing that plus
is commutative. You can use a pattern to search for all theorems
involving the equality of two additions. *)
Search (_ + _ = _ + _).
(** You'll see a lot of results there, nearly all of them from the
standard library. To restrict the results, you can search inside
a particular module: *)
Search (_ + _ = _ + _) inside Induction.
(** You can also make the search more precise by using variables in
the search pattern instead of wildcards: *)
Search (?x + ?y = ?y + ?x).
(** (The question mark in front of the variable is needed to indicate
that it is a variable in the search pattern, rather than a defined
identifier that is expected to be in scope currently.) *)
(** Keep [Search] in mind as you do the following exercises and
throughout the rest of the book; it can save you a lot of time!
Your IDE likely has its own functionality to help with searching.
For example, in VSRocq, you can open a tab for performing searches
with [Command-Control-K]. *)
(* ================================================================= *)
(** ** List Exercises, Part 1 *)
(** **** Exercise: 3 stars, standard (list_exercises)
More practice with lists: *)
Theorem app_nil_r : forall l : natlist,
l ++ [] = l.
Proof.
(* FILL IN HERE *) Admitted.
Theorem rev_app_distr: forall l1 l2 : natlist,
rev (l1 ++ l2) = rev l2 ++ rev l1.
Proof.
(* FILL IN HERE *) Admitted.
(** An _involution_ is a function that is its own inverse. That is,
applying the function twice yield the original input. *)
Theorem rev_involutive : forall l : natlist,
rev (rev l) = l.
Proof.
(* FILL IN HERE *) Admitted.
(** There is a short solution to the next one. If you find yourself
getting tangled up, step back and try to look for a simpler
way. *)
Theorem app_assoc4 : forall l1 l2 l3 l4 : natlist,
l1 ++ (l2 ++ (l3 ++ l4)) = ((l1 ++ l2) ++ l3) ++ l4.
Proof.
(* FILL IN HERE *) Admitted.
(** An exercise about your implementation of [nonzeros]: *)
Lemma nonzeros_app : forall l1 l2 : natlist,
nonzeros (l1 ++ l2) = (nonzeros l1) ++ (nonzeros l2).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 2 stars, standard (eqblist)
Fill in the definition of [eqblist], which compares
lists of numbers for equality. Prove that [eqblist l l]
yields [true] for every list [l]. *)
Fixpoint eqblist (l1 l2 : natlist) : bool
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_eqblist1 :
(eqblist nil nil = true).
(* FILL IN HERE *) Admitted.
Example test_eqblist2 :
eqblist [1;2;3] [1;2;3] = true.
(* FILL IN HERE *) Admitted.
Example test_eqblist3 :
eqblist [1;2;3] [1;2;4] = false.
(* FILL IN HERE *) Admitted.
Theorem eqblist_refl : forall l:natlist,
true = eqblist l l.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ================================================================= *)
(** ** List Exercises, Part 2 *)
(** Here are a couple of little theorems to prove about your
definitions about bags above. *)
(** **** Exercise: 1 star, standard (count_member_nonzero) *)
Theorem count_member_nonzero : forall (s : bag),
1 <=? (count 1 (1 :: s)) = true.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** The following lemma about [leb] might help you in the next
exercise (it will also be useful in later chapters). *)
Theorem leb_n_Sn : forall n,
n <=? (S n) = true.
Proof.
intros n. induction n as [| n' IHn'].
- (* 0 *)
simpl. reflexivity.
- (* S n' *)
simpl. rewrite IHn'. reflexivity. Qed.
(** Before doing the next exercise, make sure you've filled in the
definition of [remove_one] above. *)
(** **** Exercise: 3 stars, advanced (remove_does_not_increase_count) *)
Theorem remove_does_not_increase_count: forall (s : bag),
(count 0 (remove_one 0 s)) <=? (count 0 s) = true.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, standard, optional (bag_count_sum)
Write down an interesting theorem [bag_count_sum] about bags
involving the functions [count] and [sum], and prove it using
Rocq. (You may find that the difficulty of the proof depends on
how you defined [count]!
Hint: If you defined [count] using [=?] you may find it useful
to know that [destruct] works on arbitrary expressions, not just
simple identifiers.)
*)
(* FILL IN HERE
[] *)
(** **** Exercise: 3 stars, advanced (involution_injective) *)
(** Prove that every involution is injective.
Involutions were defined above in [rev_involutive]. An _injective_
function is one-to-one: it maps distinct inputs to distinct
outputs, without any collisions. *)
Theorem involution_injective : forall (f : nat -> nat),
(forall n : nat, n = f (f n)) -> (forall n1 n2 : nat, f n1 = f n2 -> n1 = n2).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 2 stars, advanced (rev_injective)
Prove that [rev] is injective. Do not prove this by induction --
that would be hard. Instead, re-use the same proof technique that
you used for [involution_injective]. (But: Don't try to use that
exercise directly as a lemma: the types are not the same!) *)
Theorem rev_injective : forall (l1 l2 : natlist),
rev l1 = rev l2 -> l1 = l2.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ################################################################# *)
(** * Options *)
(** Suppose we want to write a function that returns the [n]th
element of some list. If we give it type [nat -> natlist -> nat],
then we'll have to choose some number to return when the list is
too short... *)
Fixpoint nth_bad (l:natlist) (n:nat) : nat :=
match l with
| nil => 42
| a :: l' => match n with
| 0 => a
| S n' => nth_bad l' n'
end
end.
(** This solution is not so good: If [nth_bad] returns [42], we
don't know whether that value actually appears in the input
or whether we gave bad arguments. A better alternative is to change the
return type of [nth_bad] to include an error value as a possible
outcome. We call this new type [natoption]. *)
Inductive natoption : Type :=
| Some (n : nat)
| None.
(* Note that we've capitalized the constructor names [None] and
[Some], following their definition in Rocq's standard library. In
general, constructor (and variable) names can begin with either
capital or lowercase letters. *)
(** We can then change the above definition of [nth_bad] to
return [None] when the list is too short and [Some a] when the
list has enough members and [a] appears at position [n]. We call
this new function [nth_error] to indicate that it may result in an
error.
(As we see here, constructors of inductive definitions are allowed
to be be capitalized.) *)
Fixpoint nth_error (l:natlist) (n:nat) : natoption :=
match l with
| nil => None
| a :: l' => match n with
| O => Some a
| S n' => nth_error l' n'
end
end.
Example test_nth_error1 : nth_error [4;5;6;7] 0 = Some 4.
Proof. reflexivity. Qed.
Example test_nth_error2 : nth_error [4;5;6;7] 3 = Some 7.
Proof. reflexivity. Qed.
Example test_nth_error3 : nth_error [4;5;6;7] 9 = None.
Proof. reflexivity. Qed.
(** (In the HTML version, the boilerplate proofs of these
examples are elided. Click on a box if you want to see the
details.) *)
(** The function below pulls the [nat] out of a [natoption], returning
a supplied default in the [None] case. *)
Definition option_elim (d : nat) (o : natoption) : nat :=
match o with
| Some n' => n'
| None => d
end.
(** **** Exercise: 2 stars, standard (hd_error)
Using the same idea, fix the [hd] function from earlier so we don't
have to pass a default element for the [nil] case. *)
Definition hd_error (l : natlist) : natoption
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_hd_error1 : hd_error [] = None.
(* FILL IN HERE *) Admitted.
Example test_hd_error2 : hd_error [1] = Some 1.
(* FILL IN HERE *) Admitted.
Example test_hd_error3 : hd_error [5;6] = Some 5.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 1 star, standard, optional (option_elim_hd)
This exercise relates your new [hd_error] to the old [hd]. *)
Theorem option_elim_hd : forall (l:natlist) (default:nat),
hd default l = option_elim default (hd_error l).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
End NatList.
(* ################################################################# *)
(** * Partial Maps *)
(** As a final illustration of how data structures can be defined in
Rocq, here is a simple _partial map_ data type, analogous to the
map or dictionary data structures found in most programming
languages. *)
(** First, we define a new inductive datatype [id] to serve as the
"keys" of our partial maps. *)
Inductive id : Type :=
| Id (n : nat).
(** Internally, an [id] is just a number. Introducing a separate type
by wrapping each nat with the tag [Id] makes definitions more
readable and gives us flexibility to change representations later
if we want to. *)
(** We'll also need an equality test for [id]s: *)
Definition eqb_id (x1 x2 : id) :=
match x1, x2 with
| Id n1, Id n2 => n1 =? n2
end.
(** **** Exercise: 1 star, standard (eqb_id_refl) *)
Theorem eqb_id_refl : forall x, eqb_id x x = true.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** Now we define the type of partial maps: *)
Module PartialMap.
Export NatList. (* make the definitions from NatList available here *)
Inductive partial_map : Type :=
| empty
| record (i : id) (v : nat) (m : partial_map).
(** This declaration can be read: "There are two ways to construct a
[partial_map]: either using the constructor [empty] to represent an
empty partial map, or applying the constructor [record] to
a key, a value, and an existing [partial_map] to construct a
[partial_map] with an additional key-to-value mapping." *)
(** The [update] function overrides the entry for a given key in a
partial map by shadowing it with a new one (or simply adds a new
entry if the given key is not already present). *)
Definition update (d : partial_map)
(x : id) (value : nat)
: partial_map :=
record x value d.
(** Last, the [find] function searches a [partial_map] for a given
key. It returns [None] if the key was not found and [Some val] if
the key was associated with [val]. If the same key is mapped to
multiple values, [find] will return the first one it
encounters. *)
Fixpoint find (x : id) (d : partial_map) : natoption :=
match d with
| empty => None
| record y v d' => if eqb_id x y
then Some v
else find x d'
end.
(** **** Exercise: 1 star, standard (update_eq) *)
Theorem update_eq :
forall (d : partial_map) (x : id) (v: nat),
find x (update d x v) = Some v.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 1 star, standard (update_neq) *)
Theorem update_neq :
forall (d : partial_map) (x y : id) (o: nat),
eqb_id x y = false -> find x (update d y o) = find x d.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
End PartialMap.
(* 2026-01-07 13:17 *)
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