(** * IndProp: Inductively Defined Propositions *)
Require Export Logic.
(* ################################################################# *)
(** * Inductively Defined Propositions *)
(** In the [Logic] chapter, we looked at several ways of writing
propositions, including conjunction, disjunction, and quantifiers.
In this chapter, we bring a new tool into the mix: _inductive
definitions_.
Recall that we have seen two ways of stating that a number [n] is
even: We can say (1) [evenb n = true], or (2) [exists k, n =
double k]. Yet another possibility is to say that [n] is even if
we can establish its evenness from the following rules:
- Rule [ev_0]: The number [0] is even.
- Rule [ev_SS]: If [n] is even, then [S (S n)] is even. *)
(** To illustrate how this definition of evenness works, let's
imagine using it to show that [4] is even. By rule [ev_SS], it
suffices to show that [2] is even. This, in turn, is again
guaranteed by rule [ev_SS], as long as we can show that [0] is
even. But this last fact follows directly from the [ev_0] rule. *)
(** We will see many definitions like this one during the rest
of the course. For purposes of informal discussions, it is
helpful to have a lightweight notation that makes them easy to
read and write. _Inference rules_ are one such notation: *)
(**
------------ (ev_0)
ev 0
ev n
-------------- (ev_SS)
ev (S (S n))
*)
(** Each of the textual rules above is reformatted here as an
inference rule; the intended reading is that, if the _premises_
above the line all hold, then the _conclusion_ below the line
follows. For example, the rule [ev_SS] says that, if [n]
satisfies [ev], then [S (S n)] also does. If a rule has no
premises above the line, then its conclusion holds
unconditionally.
We can represent a proof using these rules by combining rule
applications into a _proof tree_. Here's how we might transcribe
the above proof that [4] is even: *)
(**
------ (ev_0)
ev 0
------ (ev_SS)
ev 2
------ (ev_SS)
ev 4
*)
(** Why call this a "tree" (rather than a "stack", for example)?
Because, in general, inference rules can have multiple premises.
We will see examples of this below. *)
(** Putting all of this together, we can translate the definition of
evenness into a formal Coq definition using an [Inductive]
declaration, where each constructor corresponds to an inference
rule: *)
Inductive ev : nat -> Prop :=
| ev_0 : ev 0
| ev_SS : forall n : nat, ev n -> ev (S (S n)).
(** This definition is different in one crucial respect from
previous uses of [Inductive]: its result is not a [Type], but
rather a function from [nat] to [Prop] -- that is, a property of
numbers. Note that we've already seen other inductive definitions
that result in functions, such as [list], whose type is [Type ->
Type]. What is new here is that, because the [nat] argument of
[ev] appears _unnamed_, to the _right_ of the colon, it is allowed
to take different values in the types of different constructors:
[0] in the type of [ev_0] and [S (S n)] in the type of [ev_SS].
In contrast, the definition of [list] names the [X] parameter
_globally_, to the _left_ of the colon, forcing the result of
[nil] and [cons] to be the same ([list X]). Had we tried to bring
[nat] to the left in defining [ev], we would have seen an error: *)
Fail Inductive wrong_ev (n : nat) : Prop :=
| wrong_ev_0 : wrong_ev 0
| wrong_ev_SS : forall n, wrong_ev n -> wrong_ev (S (S n)).
(* ===> Error: A parameter of an inductive type n is not
allowed to be used as a bound variable in the type
of its constructor. *)
(** ("Parameter" here is Coq jargon for an argument on the left of the
colon in an [Inductive] definition; "index" is used to refer to
arguments on the right of the colon.) *)
(** We can think of the definition of [ev] as defining a Coq property
[ev : nat -> Prop], together with theorems [ev_0 : ev 0] and
[ev_SS : forall n, ev n -> ev (S (S n))]. Such "constructor
theorems" have the same status as proven theorems. In particular,
we can use Coq's [apply] tactic with the rule names to prove [ev]
for particular numbers... *)
Theorem ev_4 : ev 4.
Proof. apply ev_SS. apply ev_SS. apply ev_0. Qed.
(** ... or we can use function application syntax: *)
Theorem ev_4' : ev 4.
Proof. apply (ev_SS 2 (ev_SS 0 ev_0)). Qed.
(** We can also prove theorems that have hypotheses involving [ev]. *)
Theorem ev_plus4 : forall n, ev n -> ev (4 + n).
Proof.
intros n. simpl. intros Hn.
apply ev_SS. apply ev_SS. apply Hn.
Qed.
(** More generally, we can show that any number multiplied by 2 is even: *)
(** **** Exercise: 1 star (ev_double) *)
Theorem ev_double : forall n,
ev (double n).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ################################################################# *)
(** * Using Evidence in Proofs *)
(** Besides _constructing_ evidence that numbers are even, we can also
_reason about_ such evidence.
Introducing [ev] with an [Inductive] declaration tells Coq not
only that the constructors [ev_0] and [ev_SS] are valid ways to
build evidence that some number is even, but also that these two
constructors are the _only_ ways to build evidence that numbers
are even (in the sense of [ev]). *)
(** In other words, if someone gives us evidence [E] for the assertion
[ev n], then we know that [E] must have one of two shapes:
- [E] is [ev_0] (and [n] is [O]), or
- [E] is [ev_SS n' E'] (and [n] is [S (S n')], where [E'] is
evidence for [ev n']). *)
(** This suggests that it should be possible to analyze a hypothesis
of the form [ev n] much as we do inductively defined data
structures; in particular, it should be possible to argue by
_induction_ and _case analysis_ on such evidence. Let's look at a
few examples to see what this means in practice. *)
(* ================================================================= *)
(** ** Inversion on Evidence *)
(** Suppose we are proving some fact involving a number [n], and we
are given [ev n] as a hypothesis. We already know how to perform
case analysis on [n] using the [inversion] tactic, generating
separate subgoals for the case where [n = O] and the case where [n
= S n'] for some [n']. But for some proofs we may instead want to
analyze the evidence that [ev n] _directly_.
By the definition of [ev], there are two cases to consider:
- If the evidence is of the form [ev_0], we know that [n = 0].
- Otherwise, the evidence must have the form [ev_SS n' E'], where
[n = S (S n')] and [E'] is evidence for [ev n']. *)
(** We can perform this kind of reasoning in Coq, again using
the [inversion] tactic. Besides allowing us to reason about
equalities involving constructors, [inversion] provides a
case-analysis principle for inductively defined propositions.
When used in this way, its syntax is similar to [destruct]: We
pass it a list of identifiers separated by [|] characters to name
the arguments to each of the possible constructors. *)
Theorem ev_minus2 : forall n,
ev n -> ev (pred (pred n)).
Proof.
intros n E.
inversion E as [| n' E'].
- (* E = ev_0 *) simpl. apply ev_0.
- (* E = ev_SS n' E' *) simpl. apply E'. Qed.
(** In words, here is how the inversion reasoning works in this proof:
- If the evidence is of the form [ev_0], we know that [n = 0].
Therefore, it suffices to show that [ev (pred (pred 0))] holds.
By the definition of [pred], this is equivalent to showing that
[ev 0] holds, which directly follows from [ev_0].
- Otherwise, the evidence must have the form [ev_SS n' E'], where
[n = S (S n')] and [E'] is evidence for [ev n']. We must then
show that [ev (pred (pred (S (S n'))))] holds, which, after
simplification, follows directly from [E']. *)
(** This particular proof also works if we replace [inversion] by
[destruct]: *)
Theorem ev_minus2' : forall n,
ev n -> ev (pred (pred n)).
Proof.
intros n E.
destruct E as [| n' E'].
- (* E = ev_0 *) simpl. apply ev_0.
- (* E = ev_SS n' E' *) simpl. apply E'. Qed.
(** The difference between the two forms is that [inversion] is more
convenient when used on a hypothesis that consists of an inductive
property applied to a complex expression (as opposed to a single
variable). Here's is a concrete example. Suppose that we wanted
to prove the following variation of [ev_minus2]: *)
Theorem evSS_ev : forall n,
ev (S (S n)) -> ev n.
(** Intuitively, we know that evidence for the hypothesis cannot
consist just of the [ev_0] constructor, since [O] and [S] are
different constructors of the type [nat]; hence, [ev_SS] is the
only case that applies. Unfortunately, [destruct] is not smart
enough to realize this, and it still generates two subgoals. Even
worse, in doing so, it keeps the final goal unchanged, failing to
provide any useful information for completing the proof. *)
Proof.
intros n E.
destruct E as [| n' E'].
- (* E = ev_0. *)
(* We must prove that [n] is even from no assumptions! *)
Abort.
(** What happened, exactly? Calling [destruct] has the effect of
replacing all occurrences of the property argument by the values
that correspond to each constructor. This is enough in the case
of [ev_minus2'] because that argument, [n], is mentioned directly
in the final goal. However, it doesn't help in the case of
[evSS_ev] since the term that gets replaced ([S (S n)]) is not
mentioned anywhere. *)
(** The [inversion] tactic, on the other hand, can detect (1) that the
first case does not apply, and (2) that the [n'] that appears on
the [ev_SS] case must be the same as [n]. This allows us to
complete the proof: *)
Theorem evSS_ev : forall n,
ev (S (S n)) -> ev n.
Proof.
intros n E.
inversion E as [| n' E'].
(* We are in the [E = ev_SS n' E'] case now. *)
apply E'.
Qed.
(** By using [inversion], we can also apply the principle of explosion
to "obviously contradictory" hypotheses involving inductive
properties. For example: *)
Theorem one_not_even : ~ ev 1.
Proof.
intros H. inversion H. Qed.
(** **** Exercise: 1 star (inversion_practice) *)
(** Prove the following results using [inversion]. *)
Theorem SSSSev__even : forall n,
ev (S (S (S (S n)))) -> ev n.
Proof.
(* FILL IN HERE *) Admitted.
Theorem even5_nonsense :
ev 5 -> 2 + 2 = 9.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** The way we've used [inversion] here may seem a bit
mysterious at first. Until now, we've only used [inversion] on
equality propositions, to utilize injectivity of constructors or
to discriminate between different constructors. But we see here
that [inversion] can also be applied to analyzing evidence for
inductively defined propositions.
Here's how [inversion] works in general. Suppose the name [I]
refers to an assumption [P] in the current context, where [P] has
been defined by an [Inductive] declaration. Then, for each of the
constructors of [P], [inversion I] generates a subgoal in which
[I] has been replaced by the exact, specific conditions under
which this constructor could have been used to prove [P]. Some of
these subgoals will be self-contradictory; [inversion] throws
these away. The ones that are left represent the cases that must
be proved to establish the original goal. For those, [inversion]
adds all equations into the proof context that must hold of the
arguments given to [P] (e.g., [S (S n') = n] in the proof of
[evSS_ev]). *)
(** The [ev_double] exercise above shows that our new notion of
evenness is implied by the two earlier ones (since, by
[even_bool_prop] in chapter [Logic], we already know that
those are equivalent to each other). To show that all three
coincide, we just need the following lemma: *)
Lemma ev_even_firsttry : forall n,
ev n -> exists k, n = double k.
Proof.
(* WORKED IN CLASS *)
(** We could try to proceed by case analysis or induction on [n]. But
since [ev] is mentioned in a premise, this strategy would probably
lead to a dead end, as in the previous section. Thus, it seems
better to first try inversion on the evidence for [ev]. Indeed,
the first case can be solved trivially. *)
intros n E. inversion E as [| n' E'].
- (* E = ev_0 *)
exists 0. reflexivity.
- (* E = ev_SS n' E' *) simpl.
(** Unfortunately, the second case is harder. We need to show [exists
k, S (S n') = double k], but the only available assumption is
[E'], which states that [ev n'] holds. Since this isn't directly
useful, it seems that we are stuck and that performing case
analysis on [E] was a waste of time.
If we look more closely at our second goal, however, we can see
that something interesting happened: By performing case analysis
on [E], we were able to reduce the original result to an similar
one that involves a _different_ piece of evidence for [ev]: [E'].
More formally, we can finish our proof by showing that
exists k', n' = double k',
which is the same as the original statement, but with [n'] instead
of [n]. Indeed, it is not difficult to convince Coq that this
intermediate result suffices. *)
assert (I : (exists k', n' = double k') ->
(exists k, S (S n') = double k)).
{ intros [k' Hk']. rewrite Hk'. exists (S k'). reflexivity. }
apply I. (* reduce the original goal to the new one *)
Admitted.
(* ================================================================= *)
(** ** Induction on Evidence *)
(** If this looks familiar, it is no coincidence: We've encountered
similar problems in the [Induction] chapter, when trying to use
case analysis to prove results that required induction. And once
again the solution is... induction!
The behavior of [induction] on evidence is the same as its
behavior on data: It causes Coq to generate one subgoal for each
constructor that could have used to build that evidence, while
providing an induction hypotheses for each recursive occurrence of
the property in question. *)
(** Let's try our current lemma again: *)
Lemma ev_even : forall n,
ev n -> exists k, n = double k.
Proof.
intros n E.
induction E as [|n' E' IH].
- (* E = ev_0 *)
exists 0. reflexivity.
- (* E = ev_SS n' E'
with IH : exists k', n' = double k' *)
destruct IH as [k' Hk'].
rewrite Hk'. exists (S k'). reflexivity.
Qed.
(** Here, we can see that Coq produced an [IH] that corresponds to
[E'], the single recursive occurrence of [ev] in its own
definition. Since [E'] mentions [n'], the induction hypothesis
talks about [n'], as opposed to [n] or some other number. *)
(** The equivalence between the second and third definitions of
evenness now follows. *)
Theorem ev_even_iff : forall n,
ev n <-> exists k, n = double k.
Proof.
intros n. split.
- (* -> *) apply ev_even.
- (* <- *) intros [k Hk]. rewrite Hk. apply ev_double.
Qed.
(** As we will see in later chapters, induction on evidence is a
recurring technique across many areas, and in particular when
formalizing the semantics of programming languages, where many
properties of interest are defined inductively. *)
(** The following exercises provide simple examples of this
technique, to help you familiarize yourself with it. *)
(** **** Exercise: 2 stars (ev_sum) *)
Theorem ev_sum : forall n m, ev n -> ev m -> ev (n + m).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 4 stars, advanced, optional (ev_alternate) *)
(** In general, there may be multiple ways of defining a
property inductively. For example, here's a (slightly contrived)
alternative definition for [ev]: *)
Inductive ev' : nat -> Prop :=
| ev'_0 : ev' 0
| ev'_2 : ev' 2
| ev'_sum : forall n m, ev' n -> ev' m -> ev' (n + m).
(** Prove that this definition is logically equivalent to the old
one. (You may want to look at the previous theorem when you get
to the induction step.) *)
Theorem ev'_ev : forall n, ev' n <-> ev n.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, advanced, recommended (ev_ev__ev) *)
(** Finding the appropriate thing to do induction on is a
bit tricky here: *)
Theorem ev_ev__ev : forall n m,
ev (n+m) -> ev n -> ev m.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, optional (ev_plus_plus) *)
(** This exercise just requires applying existing lemmas. No
induction or even case analysis is needed, though some of the
rewriting may be tedious. *)
Theorem ev_plus_plus : forall n m p,
ev (n+m) -> ev (n+p) -> ev (m+p).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ################################################################# *)
(** * Inductive Relations *)
(** A proposition parameterized by a number (such as [ev])
can be thought of as a _property_ -- i.e., it defines
a subset of [nat], namely those numbers for which the proposition
is provable. In the same way, a two-argument proposition can be
thought of as a _relation_ -- i.e., it defines a set of pairs for
which the proposition is provable. *)
Module Playground.
(** One useful example is the "less than or equal to" relation on
numbers. *)
(** The following definition should be fairly intuitive. It
says that there are two ways to give evidence that one number is
less than or equal to another: either observe that they are the
same number, or give evidence that the first is less than or equal
to the predecessor of the second. *)
Inductive le : nat -> nat -> Prop :=
| le_n : forall n, le n n
| le_S : forall n m, (le n m) -> (le n (S m)).
Notation "m <= n" := (le m n).
(** Proofs of facts about [<=] using the constructors [le_n] and
[le_S] follow the same patterns as proofs about properties, like
[ev] above. We can [apply] the constructors to prove [<=]
goals (e.g., to show that [3<=3] or [3<=6]), and we can use
tactics like [inversion] to extract information from [<=]
hypotheses in the context (e.g., to prove that [(2 <= 1) ->
2+2=5].) *)
(** Here are some sanity checks on the definition. (Notice that,
although these are the same kind of simple "unit tests" as we gave
for the testing functions we wrote in the first few lectures, we
must construct their proofs explicitly -- [simpl] and
[reflexivity] don't do the job, because the proofs aren't just a
matter of simplifying computations.) *)
Theorem test_le1 :
3 <= 3.
Proof.
(* WORKED IN CLASS *)
apply le_n. Qed.
Theorem test_le2 :
3 <= 6.
Proof.
(* WORKED IN CLASS *)
apply le_S. apply le_S. apply le_S. apply le_n. Qed.
Theorem test_le3 :
(2 <= 1) -> 2 + 2 = 5.
Proof.
(* WORKED IN CLASS *)
intros H. inversion H. inversion H2. Qed.
(** The "strictly less than" relation [n < m] can now be defined
in terms of [le]. *)
End Playground.
Definition lt (n m:nat) := le (S n) m.
Notation "m < n" := (lt m n).
(** Here are a few more simple relations on numbers: *)
Inductive square_of : nat -> nat -> Prop :=
sq : forall n:nat, square_of n (n * n).
Inductive next_nat : nat -> nat -> Prop :=
| nn : forall n:nat, next_nat n (S n).
Inductive next_even : nat -> nat -> Prop :=
| ne_1 : forall n, ev (S n) -> next_even n (S n)
| ne_2 : forall n, ev (S (S n)) -> next_even n (S (S n)).
(** **** Exercise: 2 stars, optional (total_relation) *)
(** Define an inductive binary relation [total_relation] that holds
between every pair of natural numbers. *)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 2 stars, optional (empty_relation) *)
(** Define an inductive binary relation [empty_relation] (on numbers)
that never holds. *)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 3 stars, optional (le_exercises) *)
(** Here are a number of facts about the [<=] and [<] relations that
we are going to need later in the course. The proofs make good
practice exercises. *)
Lemma le_trans : forall m n o, m <= n -> n <= o -> m <= o.
Proof.
(* FILL IN HERE *) Admitted.
Theorem O_le_n : forall n,
0 <= n.
Proof.
(* FILL IN HERE *) Admitted.
Theorem n_le_m__Sn_le_Sm : forall n m,
n <= m -> S n <= S m.
Proof.
(* FILL IN HERE *) Admitted.
Theorem Sn_le_Sm__n_le_m : forall n m,
S n <= S m -> n <= m.
Proof.
(* FILL IN HERE *) Admitted.
Theorem le_plus_l : forall a b,
a <= a + b.
Proof.
(* FILL IN HERE *) Admitted.
Theorem plus_lt : forall n1 n2 m,
n1 + n2 < m ->
n1 < m /\ n2 < m.
Proof.
unfold lt.
(* FILL IN HERE *) Admitted.
Theorem lt_S : forall n m,
n < m ->
n < S m.
Proof.
(* FILL IN HERE *) Admitted.
Theorem leb_complete : forall n m,
leb n m = true -> n <= m.
Proof.
(* FILL IN HERE *) Admitted.
(** Hint: The next one may be easiest to prove by induction on [m]. *)
Theorem leb_correct : forall n m,
n <= m ->
leb n m = true.
Proof.
(* FILL IN HERE *) Admitted.
(** Hint: This theorem can easily be proved without using [induction]. *)
Theorem leb_true_trans : forall n m o,
leb n m = true -> leb m o = true -> leb n o = true.
Proof.
(* FILL IN HERE *) Admitted.
(** **** Exercise: 2 stars, optional (leb_iff) *)
Theorem leb_iff : forall n m,
leb n m = true <-> n <= m.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
Module R.
(** **** Exercise: 3 stars, recommendedM (R_provability) *)
(** We can define three-place relations, four-place relations,
etc., in just the same way as binary relations. For example,
consider the following three-place relation on numbers: *)
Inductive R : nat -> nat -> nat -> Prop :=
| c1 : R 0 0 0
| c2 : forall m n o, R m n o -> R (S m) n (S o)
| c3 : forall m n o, R m n o -> R m (S n) (S o)
| c4 : forall m n o, R (S m) (S n) (S (S o)) -> R m n o
| c5 : forall m n o, R m n o -> R n m o.
(** - Which of the following propositions are provable?
- [R 1 1 2]
- [R 2 2 6]
- If we dropped constructor [c5] from the definition of [R],
would the set of provable propositions change? Briefly (1
sentence) explain your answer.
- If we dropped constructor [c4] from the definition of [R],
would the set of provable propositions change? Briefly (1
sentence) explain your answer.
(* FILL IN HERE *)
[]
*)
(** **** Exercise: 3 stars, optional (R_fact) *)
(** The relation [R] above actually encodes a familiar function.
Figure out which function; then state and prove this equivalence
in Coq? *)
Definition fR : nat -> nat -> nat
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Theorem R_equiv_fR : forall m n o, R m n o <-> fR m n = o.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
End R.
(** **** Exercise: 4 stars, advanced (subsequence) *)
(** A list is a _subsequence_ of another list if all of the elements
in the first list occur in the same order in the second list,
possibly with some extra elements in between. For example,
[1;2;3]
is a subsequence of each of the lists
[1;2;3]
[1;1;1;2;2;3]
[1;2;7;3]
[5;6;1;9;9;2;7;3;8]
but it is _not_ a subsequence of any of the lists
[1;2]
[1;3]
[5;6;2;1;7;3;8].
- Define an inductive proposition [subseq] on [list nat] that
captures what it means to be a subsequence. (Hint: You'll need
three cases.)
- Prove [subseq_refl] that subsequence is reflexive, that is,
any list is a subsequence of itself.
- Prove [subseq_app] that for any lists [l1], [l2], and [l3],
if [l1] is a subsequence of [l2], then [l1] is also a subsequence
of [l2 ++ l3].
- (Optional, harder) Prove [subseq_trans] that subsequence is
transitive -- that is, if [l1] is a subsequence of [l2] and [l2]
is a subsequence of [l3], then [l1] is a subsequence of [l3].
Hint: choose your induction carefully! *)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 2 stars, optionalM (R_provability2) *)
(** Suppose we give Coq the following definition:
Inductive R : nat -> list nat -> Prop :=
| c1 : R 0 []
| c2 : forall n l, R n l -> R (S n) (n :: l)
| c3 : forall n l, R (S n) l -> R n l.
Which of the following propositions are provable?
- [R 2 [1;0]]
- [R 1 [1;2;1;0]]
- [R 6 [3;2;1;0]] *)
(** [] *)
(* ################################################################# *)
(** * Case Study: Regular Expressions *)
(** The [ev] property provides a simple example for illustrating
inductive definitions and the basic techniques for reasoning about
them, but it is not terribly exciting -- after all, it is
equivalent to the two non-inductive of evenness that we had
already seen, and does not seem to offer any concrete benefit over
them. To give a better sense of the power of inductive
definitions, we now show how to use them to model a classic
concept in computer science: _regular expressions_. *)
(** Regular expressions are a simple language for describing strings,
defined as follows: *)
Inductive reg_exp (T : Type) : Type :=
| EmptySet : reg_exp T
| EmptyStr : reg_exp T
| Char : T -> reg_exp T
| App : reg_exp T -> reg_exp T -> reg_exp T
| Union : reg_exp T -> reg_exp T -> reg_exp T
| Star : reg_exp T -> reg_exp T.
Arguments EmptySet {T}.
Arguments EmptyStr {T}.
Arguments Char {T} _.
Arguments App {T} _ _.
Arguments Union {T} _ _.
Arguments Star {T} _.
(** Note that this definition is _polymorphic_: Regular
expressions in [reg_exp T] describe strings with characters drawn
from [T] -- that is, lists of elements of [T].
(We depart slightly from standard practice in that we do not
require the type [T] to be finite. This results in a somewhat
different theory of regular expressions, but the difference is not
significant for our purposes.) *)
(** We connect regular expressions and strings via the following
rules, which define when a regular expression _matches_ some
string:
- The expression [EmptySet] does not match any string.
- The expression [EmptyStr] matches the empty string [[]].
- The expression [Char x] matches the one-character string [[x]].
- If [re1] matches [s1], and [re2] matches [s2], then [App re1
re2] matches [s1 ++ s2].
- If at least one of [re1] and [re2] matches [s], then [Union re1
re2] matches [s].
- Finally, if we can write some string [s] as the concatenation of
a sequence of strings [s = s_1 ++ ... ++ s_k], and the
expression [re] matches each one of the strings [s_i], then
[Star re] matches [s].
As a special case, the sequence of strings may be empty, so
[Star re] always matches the empty string [[]] no matter what
[re] is.
We can easily translate this informal definition into an
[Inductive] one as follows: *)
Inductive exp_match {T} : list T -> reg_exp T -> Prop :=
| MEmpty : exp_match [] EmptyStr
| MChar : forall x, exp_match [x] (Char x)
| MApp : forall s1 re1 s2 re2,
exp_match s1 re1 ->
exp_match s2 re2 ->
exp_match (s1 ++ s2) (App re1 re2)
| MUnionL : forall s1 re1 re2,
exp_match s1 re1 ->
exp_match s1 (Union re1 re2)
| MUnionR : forall re1 s2 re2,
exp_match s2 re2 ->
exp_match s2 (Union re1 re2)
| MStar0 : forall re, exp_match [] (Star re)
| MStarApp : forall s1 s2 re,
exp_match s1 re ->
exp_match s2 (Star re) ->
exp_match (s1 ++ s2) (Star re).
(** Again, for readability, we can also display this definition using
inference-rule notation. At the same time, let's introduce a more
readable infix notation. *)
Notation "s =~ re" := (exp_match s re) (at level 80).
(**
---------------- (MEmpty)
[] =~ EmptyStr
--------------- (MChar)
[x] =~ Char x
s1 =~ re1 s2 =~ re2
------------------------- (MApp)
s1 ++ s2 =~ App re1 re2
s1 =~ re1
--------------------- (MUnionL)
s1 =~ Union re1 re2
s2 =~ re2
--------------------- (MUnionR)
s2 =~ Union re1 re2
--------------- (MStar0)
[] =~ Star re
s1 =~ re s2 =~ Star re
--------------------------- (MStarApp)
s1 ++ s2 =~ Star re
*)
(** Notice that these rules are not _quite_ the same as the informal
ones that we gave at the beginning of the section. First, we
don't need to include a rule explicitly stating that no string
matches [EmptySet]; we just don't happen to include any rule that
would have the effect of some string matching [EmptySet]. (Indeed,
the syntax of inductive definitions doesn't even _allow_ us to
give such a "negative rule.")
Second, the informal rules for [Union] and [Star] correspond
to two constructors each: [MUnionL] / [MUnionR], and [MStar0] /
[MStarApp]. The result is logically equivalent to the original
rules but more convenient to use in Coq, since the recursive
occurrences of [exp_match] are given as direct arguments to the
constructors, making it easier to perform induction on evidence.
(The [exp_match_ex1] and [exp_match_ex2] exercises below ask you
to prove that the constructors given in the inductive declaration
and the ones that would arise from a more literal transcription of
the informal rules are indeed equivalent.)
Let's illustrate these rules with a few examples. *)
Example reg_exp_ex1 : [1] =~ Char 1.
Proof.
apply MChar.
Qed.
Example reg_exp_ex2 : [1; 2] =~ App (Char 1) (Char 2).
Proof.
apply (MApp [1] _ [2]).
- apply MChar.
- apply MChar.
Qed.
(** (Notice how the last example applies [MApp] to the strings [[1]]
and [[2]] directly. Since the goal mentions [[1; 2]] instead of
[[1] ++ [2]], Coq wouldn't be able to figure out how to split the
string on its own.)
Using [inversion], we can also show that certain strings do _not_
match a regular expression: *)
Example reg_exp_ex3 : ~ ([1; 2] =~ Char 1).
Proof.
intros H. inversion H.
Qed.
(** We can define helper functions to help write down regular
expressions. The [reg_exp_of_list] function constructs a regular
expression that matches exactly the list that it receives as an
argument: *)
Fixpoint reg_exp_of_list {T} (l : list T) :=
match l with
| [] => EmptyStr
| x :: l' => App (Char x) (reg_exp_of_list l')
end.
Example reg_exp_ex4 : [1; 2; 3] =~ reg_exp_of_list [1; 2; 3].
Proof.
simpl. apply (MApp [1]).
{ apply MChar. }
apply (MApp [2]).
{ apply MChar. }
apply (MApp [3]).
{ apply MChar. }
apply MEmpty.
Qed.
(** We can also prove general facts about [exp_match]. For instance,
the following lemma shows that every string [s] that matches [re]
also matches [Star re]. *)
Lemma MStar1 :
forall T s (re : reg_exp T) ,
s =~ re ->
s =~ Star re.
Proof.
intros T s re H.
rewrite <- (app_nil_r _ s).
apply (MStarApp s [] re).
- apply H.
- apply MStar0.
Qed.
(** (Note the use of [app_nil_r] to change the goal of the theorem to
exactly the same shape expected by [MStarApp].) *)
(** **** Exercise: 3 stars (exp_match_ex1) *)
(** The following lemmas show that the informal matching rules given
at the beginning of the chapter can be obtained from the formal
inductive definition. *)
Lemma empty_is_empty : forall T (s : list T),
~ (s =~ EmptySet).
Proof.
(* FILL IN HERE *) Admitted.
Lemma MUnion' : forall T (s : list T) (re1 re2 : reg_exp T),
s =~ re1 \/ s =~ re2 ->
s =~ Union re1 re2.
Proof.
(* FILL IN HERE *) Admitted.
(** The next lemma is stated in terms of the [fold] function from the
[Poly] chapter: If [ss : list (list T)] represents a sequence of
strings [s1, ..., sn], then [fold app ss []] is the result of
concatenating them all together. *)
Lemma MStar' : forall T (ss : list (list T)) (re : reg_exp T),
(forall s, In s ss -> s =~ re) ->
fold app ss [] =~ Star re.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 4 stars (reg_exp_of_list) *)
(** Prove that [reg_exp_of_list] satisfies the following
specification: *)
Lemma reg_exp_of_list_spec : forall T (s1 s2 : list T),
s1 =~ reg_exp_of_list s2 <-> s1 = s2.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** Since the definition of [exp_match] has a recursive
structure, we might expect that proofs involving regular
expressions will often require induction on evidence. For
example, suppose that we wanted to prove the following intuitive
result: If a regular expression [re] matches some string [s], then
all elements of [s] must occur somewhere in [re]. To state this
theorem, we first define a function [re_chars] that lists all
characters that occur in a regular expression: *)
Fixpoint re_chars {T} (re : reg_exp T) : list T :=
match re with
| EmptySet => []
| EmptyStr => []
| Char x => [x]
| App re1 re2 => re_chars re1 ++ re_chars re2
| Union re1 re2 => re_chars re1 ++ re_chars re2
| Star re => re_chars re
end.
(** We can then phrase our theorem as follows: *)
Theorem in_re_match : forall T (s : list T) (re : reg_exp T) (x : T),
s =~ re ->
In x s ->
In x (re_chars re).
Proof.
intros T s re x Hmatch Hin.
induction Hmatch
as [
|x'
|s1 re1 s2 re2 Hmatch1 IH1 Hmatch2 IH2
|s1 re1 re2 Hmatch IH|re1 s2 re2 Hmatch IH
|re|s1 s2 re Hmatch1 IH1 Hmatch2 IH2].
(* WORKED IN CLASS *)
- (* MEmpty *)
apply Hin.
- (* MChar *)
apply Hin.
- simpl. rewrite in_app_iff in *.
destruct Hin as [Hin | Hin].
+ (* In x s1 *)
left. apply (IH1 Hin).
+ (* In x s2 *)
right. apply (IH2 Hin).
- (* MUnionL *)
simpl. rewrite in_app_iff.
left. apply (IH Hin).
- (* MUnionR *)
simpl. rewrite in_app_iff.
right. apply (IH Hin).
- (* MStar0 *)
destruct Hin.
(** Something interesting happens in the [MStarApp] case. We obtain
_two_ induction hypotheses: One that applies when [x] occurs in
[s1] (which matches [re]), and a second one that applies when [x]
occurs in [s2] (which matches [Star re]). This is a good
illustration of why we need induction on evidence for [exp_match],
as opposed to [re]: The latter would only provide an induction
hypothesis for strings that match [re], which would not allow us
to reason about the case [In x s2]. *)
- (* MStarApp *)
simpl. rewrite in_app_iff in Hin.
destruct Hin as [Hin | Hin].
+ (* In x s1 *)
apply (IH1 Hin).
+ (* In x s2 *)
apply (IH2 Hin).
Qed.
(** **** Exercise: 4 stars (re_not_empty) *)
(** Write a recursive function [re_not_empty] that tests whether a
regular expression matches some string. Prove that your function
is correct. *)
Fixpoint re_not_empty {T : Type} (re : reg_exp T) : bool
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Lemma re_not_empty_correct : forall T (re : reg_exp T),
(exists s, s =~ re) <-> re_not_empty re = true.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ================================================================= *)
(** ** The [remember] Tactic *)
(** One potentially confusing feature of the [induction] tactic is
that it happily lets you try to set up an induction over a term
that isn't sufficiently general. The effect of this is to lose
information (much as [destruct] can do), and leave you unable to
complete the proof. Here's an example: *)
Lemma star_app: forall T (s1 s2 : list T) (re : reg_exp T),
s1 =~ Star re ->
s2 =~ Star re ->
s1 ++ s2 =~ Star re.
Proof.
intros T s1 s2 re H1.
(** Just doing an [inversion] on [H1] won't get us very far in the
recursive cases. (Try it!). So we need induction. Here is a naive
first attempt: *)
induction H1
as [|x'|s1 re1 s2' re2 Hmatch1 IH1 Hmatch2 IH2
|s1 re1 re2 Hmatch IH|re1 s2' re2 Hmatch IH
|re''|s1 s2' re'' Hmatch1 IH1 Hmatch2 IH2].
(** But now, although we get seven cases (as we would expect from the
definition of [exp_match]), we have lost a very important bit of
information from [H1]: the fact that [s1] matched something of the
form [Star re]. This means that we have to give proofs for _all_
seven constructors of this definition, even though all but two of
them ([MStar0] and [MStarApp]) are contradictory. We can still
get the proof to go through for a few constructors, such as
[MEmpty]... *)
- (* MEmpty *)
simpl. intros H. apply H.
(** ... but most cases get stuck. For [MChar], for instance, we
must show that
s2 =~ Char x' -> x' :: s2 =~ Char x',
which is clearly impossible. *)
- (* MChar. Stuck... *)
Abort.
(** The problem is that [induction] over a Prop hypothesis only works
properly with hypotheses that are completely general, i.e., ones
in which all the arguments are variables, as opposed to more
complex expressions, such as [Star re].
(In this respect, [induction] on evidence behaves more like
[destruct] than like [inversion].)
We can solve this problem by generalizing over the problematic
expressions with an explicit equality: *)
Lemma star_app: forall T (s1 s2 : list T) (re re' : reg_exp T),
s1 =~ re' ->
re' = Star re ->
s2 =~ Star re ->
s1 ++ s2 =~ Star re.
(** We can now proceed by performing induction over evidence directly,
because the argument to the first hypothesis is sufficiently
general, which means that we can discharge most cases by inverting
the [re' = Star re] equality in the context.
This idiom is so common that Coq provides a tactic to
automatically generate such equations for us, avoiding thus the
need for changing the statements of our theorems. *)
(** Invoking the tactic [remember e as x] causes Coq to (1) replace
all occurrences of the expression [e] by the variable [x], and (2)
add an equation [x = e] to the context. Here's how we can use it
to show the above result: *)
Abort.
Lemma star_app: forall T (s1 s2 : list T) (re : reg_exp T),
s1 =~ Star re ->
s2 =~ Star re ->
s1 ++ s2 =~ Star re.
Proof.
intros T s1 s2 re H1.
remember (Star re) as re'.
(** We now have [Heqre' : re' = Star re]. *)
generalize dependent s2.
induction H1
as [|x'|s1 re1 s2' re2 Hmatch1 IH1 Hmatch2 IH2
|s1 re1 re2 Hmatch IH|re1 s2' re2 Hmatch IH
|re''|s1 s2' re'' Hmatch1 IH1 Hmatch2 IH2].
(** The [Heqre'] is contradictory in most cases, which allows us to
conclude immediately. *)
- (* MEmpty *) inversion Heqre'.
- (* MChar *) inversion Heqre'.
- (* MApp *) inversion Heqre'.
- (* MUnionL *) inversion Heqre'.
- (* MUnionR *) inversion Heqre'.
(** The interesting cases are those that correspond to [Star]. Note
that the induction hypothesis [IH2] on the [MStarApp] case
mentions an additional premise [Star re'' = Star re'], which
results from the equality generated by [remember]. *)
- (* MStar0 *)
inversion Heqre'. intros s H. apply H.
- (* MStarApp *)
inversion Heqre'. rewrite H0 in IH2, Hmatch1.
intros s2 H1. rewrite <- app_assoc.
apply MStarApp.
+ apply Hmatch1.
+ apply IH2.
* reflexivity.
* apply H1.
Qed.
(** **** Exercise: 4 stars (exp_match_ex2) *)
(** The [MStar''] lemma below (combined with its converse, the
[MStar'] exercise above), shows that our definition of [exp_match]
for [Star] is equivalent to the informal one given previously. *)
Lemma MStar'' : forall T (s : list T) (re : reg_exp T),
s =~ Star re ->
exists ss : list (list T),
s = fold app ss []
/\ forall s', In s' ss -> s' =~ re.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 5 stars, advanced (pumping) *)
(** One of the first really interesting theorems in the theory of
regular expressions is the so-called _pumping lemma_, which
states, informally, that any sufficiently long string [s] matching
a regular expression [re] can be "pumped" by repeating some middle
section of [s] an arbitrary number of times to produce a new
string also matching [re].
To begin, we need to define "sufficiently long." Since we are
working in a constructive logic, we actually need to be able to
calculate, for each regular expression [re], the minimum length
for strings [s] to guarantee "pumpability." *)
Module Pumping.
Fixpoint pumping_constant {T} (re : reg_exp T) : nat :=
match re with
| EmptySet => 0
| EmptyStr => 1
| Char _ => 2
| App re1 re2 =>
pumping_constant re1 + pumping_constant re2
| Union re1 re2 =>
pumping_constant re1 + pumping_constant re2
| Star _ => 1
end.
(** Next, it is useful to define an auxiliary function that repeats a
string (appends it to itself) some number of times. *)
Fixpoint napp {T} (n : nat) (l : list T) : list T :=
match n with
| 0 => []
| S n' => l ++ napp n' l
end.
Lemma napp_plus: forall T (n m : nat) (l : list T),
napp (n + m) l = napp n l ++ napp m l.
Proof.
intros T n m l.
induction n as [|n IHn].
- reflexivity.
- simpl. rewrite IHn, app_assoc. reflexivity.
Qed.
(** Now, the pumping lemma itself says that, if [s =~ re] and if the
length of [s] is at least the pumping constant of [re], then [s]
can be split into three substrings [s1 ++ s2 ++ s3] in such a way
that [s2] can be repeated any number of times and the result, when
combined with [s1] and [s3] will still match [re]. Since [s2] is
also guaranteed not to be the empty string, this gives us
a (constructive!) way to generate strings matching [re] that are
as long as we like. *)
Lemma pumping : forall T (re : reg_exp T) s,
s =~ re ->
pumping_constant re <= length s ->
exists s1 s2 s3,
s = s1 ++ s2 ++ s3 /\
s2 <> [] /\
forall m, s1 ++ napp m s2 ++ s3 =~ re.
(** To streamline the proof (which you are to fill in), the [omega]
tactic, which is enabled by the following [Require], is helpful in
several places for automatically completing tedious low-level
arguments involving equalities or inequalities over natural
numbers. We'll return to [omega] in a later chapter, but feel
free to experiment with it now if you like. The first case of the
induction gives an example of how it is used. *)
Require Import Coq.omega.Omega.
Proof.
intros T re s Hmatch.
induction Hmatch
as [ | x | s1 re1 s2 re2 Hmatch1 IH1 Hmatch2 IH2
| s1 re1 re2 Hmatch IH | re1 s2 re2 Hmatch IH
| re | s1 s2 re Hmatch1 IH1 Hmatch2 IH2 ].
- (* MEmpty *)
simpl. omega.
(* FILL IN HERE *) Admitted.
End Pumping.
(** [] *)
(* ################################################################# *)
(** * Case Study: Improving Reflection *)
(** We've seen in the [Logic] chapter that we often need to
relate boolean computations to statements in [Prop]. But
performing this conversion in the way we did it there can result
in tedious proof scripts. Consider the proof of the following
theorem: *)
Theorem filter_not_empty_In : forall n l,
filter (beq_nat n) l <> [] ->
In n l.
Proof.
intros n l. induction l as [|m l' IHl'].
- (* l = [] *)
simpl. intros H. apply H. reflexivity.
- (* l = m :: l' *)
simpl. destruct (beq_nat n m) eqn:H.
+ (* beq_nat n m = true *)
intros _. rewrite beq_nat_true_iff in H. rewrite H.
left. reflexivity.
+ (* beq_nat n m = false *)
intros H'. right. apply IHl'. apply H'.
Qed.
(** In the first branch after [destruct], we explicitly apply
the [beq_nat_true_iff] lemma to the equation generated by
destructing [beq_nat n m], to convert the assumption [beq_nat n m
= true] into the assumption [n = m]; then we had to [rewrite]
using this assumption to complete the case.
We can streamline this by defining an inductive proposition that
yields a better case-analysis principle for [beq_nat n m].
Instead of generating an equation such as [beq_nat n m = true],
which is generally not directly useful, this principle gives us
right away the assumption we really need: [n = m].
We'll actually define something a bit more general, which can be
used with arbitrary properties (and not just equalities): *)
Module FirstTry.
Inductive reflect : Prop -> bool -> Prop :=
| ReflectT : forall (P:Prop), P -> reflect P true
| ReflectF : forall (P:Prop), ~ P -> reflect P false.
(** Before explaining this, let's rearrange it a little: Since the
types of both [ReflectT] and [ReflectF] begin with
[forall (P:Prop)], we can make the definition a bit more readable
and easier to work with by making [P] a parameter of the whole
Inductive declaration. *)
End FirstTry.
Inductive reflect (P : Prop) : bool -> Prop :=
| ReflectT : P -> reflect P true
| ReflectF : ~ P -> reflect P false.
(** The [reflect] property takes two arguments: a proposition
[P] and a boolean [b]. Intuitively, it states that the property
[P] is _reflected_ in (i.e., equivalent to) the boolean [b]: [P]
holds if and only if [b = true]. To see this, notice that, by
definition, the only way we can produce evidence that [reflect P
true] holds is by showing that [P] is true and using the
[ReflectT] constructor. If we invert this statement, this means
that it should be possible to extract evidence for [P] from a
proof of [reflect P true]. Conversely, the only way to show
[reflect P false] is by combining evidence for [~ P] with the
[ReflectF] constructor.
It is easy to formalize this intuition and show that the two
statements are indeed equivalent: *)
Theorem iff_reflect : forall P b, (P <-> b = true) -> reflect P b.
Proof.
(* WORKED IN CLASS *)
intros P b H. destruct b.
- apply ReflectT. rewrite H. reflexivity.
- apply ReflectF. rewrite H. intros H'. inversion H'.
Qed.
(** **** Exercise: 2 stars, recommended (reflect_iff) *)
Theorem reflect_iff : forall P b, reflect P b -> (P <-> b = true).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** The advantage of [reflect] over the normal "if and only if"
connective is that, by destructing a hypothesis or lemma of the
form [reflect P b], we can perform case analysis on [b] while at
the same time generating appropriate hypothesis in the two
branches ([P] in the first subgoal and [~ P] in the second). *)
Lemma beq_natP : forall n m, reflect (n = m) (beq_nat n m).
Proof.
intros n m.
apply iff_reflect. rewrite beq_nat_true_iff. reflexivity.
Qed.
(** The new proof of [filter_not_empty_In] now goes as follows.
Notice how the calls to [destruct] and [apply] are combined into a
single call to [destruct]. *)
(** (To see this clearly, look at the two proofs of
[filter_not_empty_In] with Coq and observe the differences in
proof state at the beginning of the first case of the
[destruct].) *)
Theorem filter_not_empty_In' : forall n l,
filter (beq_nat n) l <> [] ->
In n l.
Proof.
intros n l. induction l as [|m l' IHl'].
- (* l = [] *)
simpl. intros H. apply H. reflexivity.
- (* l = m :: l' *)
simpl. destruct (beq_natP n m) as [H | H].
+ (* n = m *)
intros _. rewrite H. left. reflexivity.
+ (* n <> m *)
intros H'. right. apply IHl'. apply H'.
Qed.
(** **** Exercise: 3 stars, recommended (beq_natP_practice) *)
(** Use [beq_natP] as above to prove the following: *)
Fixpoint count n l :=
match l with
[] => 0
| m :: l' => (if beq_nat n m then 1 else 0) + count n l'
end.
Theorem beq_natP_practice : forall n l,
count n l = 0 -> ~(In n l).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** This technique gives us only a small gain in convenience for
the proofs we've seen here, but using [reflect] consistently often
leads to noticeably shorter and clearer scripts as proofs get
larger. We'll see many more examples in later chapters.
The use of the [reflect] property was popularized by _SSReflect_,
a Coq library that has been used to formalize important results in
mathematics, including as the 4-color theorem and the
Feit-Thompson theorem. The name SSReflect stands for _small-scale
reflection_, i.e., the pervasive use of reflection to simplify
small proof steps with boolean computations. *)
(* ################################################################# *)
(** * Additional Exercises *)
(** **** Exercise: 3 stars, recommended (nostutter) *)
(** Formulating inductive definitions of properties is an important
skill you'll need in this course. Try to solve this exercise
without any help at all.
We say that a list "stutters" if it repeats the same element
consecutively. The property "[nostutter mylist]" means that
[mylist] does not stutter. Formulate an inductive definition for
[nostutter]. (This is different from the [NoDup] property in the
exercise above; the sequence [1;4;1] repeats but does not
stutter.) *)
Inductive nostutter {X:Type} : list X -> Prop :=
(* FILL IN HERE *)
.
(** Make sure each of these tests succeeds, but feel free to change
the suggested proof (in comments) if the given one doesn't work
for you. Your definition might be different from ours and still
be correct, in which case the examples might need a different
proof. (You'll notice that the suggested proofs use a number of
tactics we haven't talked about, to make them more robust to
different possible ways of defining [nostutter]. You can probably
just uncomment and use them as-is, but you can also prove each
example with more basic tactics.) *)
Example test_nostutter_1: nostutter [3;1;4;1;5;6].
(* FILL IN HERE *) Admitted.
(*
Proof. repeat constructor; apply beq_nat_false_iff; auto.
Qed.
*)
Example test_nostutter_2: nostutter (@nil nat).
(* FILL IN HERE *) Admitted.
(*
Proof. repeat constructor; apply beq_nat_false_iff; auto.
Qed.
*)
Example test_nostutter_3: nostutter [5].
(* FILL IN HERE *) Admitted.
(*
Proof. repeat constructor; apply beq_nat_false; auto. Qed.
*)
Example test_nostutter_4: not (nostutter [3;1;1;4]).
(* FILL IN HERE *) Admitted.
(*
Proof. intro.
repeat match goal with
h: nostutter _ |- _ => inversion h; clear h; subst
end.
contradiction H1; auto. Qed.
*)
(** [] *)
(** **** Exercise: 4 stars, advanced (filter_challenge) *)
(** Let's prove that our definition of [filter] from the [Poly]
chapter matches an abstract specification. Here is the
specification, written out informally in English:
A list [l] is an "in-order merge" of [l1] and [l2] if it contains
all the same elements as [l1] and [l2], in the same order as [l1]
and [l2], but possibly interleaved. For example,
[1;4;6;2;3]
is an in-order merge of
[1;6;2]
and
[4;3].
Now, suppose we have a set [X], a function [test: X->bool], and a
list [l] of type [list X]. Suppose further that [l] is an
in-order merge of two lists, [l1] and [l2], such that every item
in [l1] satisfies [test] and no item in [l2] satisfies test. Then
[filter test l = l1].
Translate this specification into a Coq theorem and prove
it. (You'll need to begin by defining what it means for one list
to be a merge of two others. Do this with an inductive relation,
not a [Fixpoint].) *)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 5 stars, advanced, optional (filter_challenge_2) *)
(** A different way to characterize the behavior of [filter] goes like
this: Among all subsequences of [l] with the property that [test]
evaluates to [true] on all their members, [filter test l] is the
longest. Formalize this claim and prove it. *)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 4 stars, optional (palindromes) *)
(** A palindrome is a sequence that reads the same backwards as
forwards.
- Define an inductive proposition [pal] on [list X] that
captures what it means to be a palindrome. (Hint: You'll need
three cases. Your definition should be based on the structure
of the list; just having a single constructor like
c : forall l, l = rev l -> pal l
may seem obvious, but will not work very well.)
- Prove ([pal_app_rev]) that
forall l, pal (l ++ rev l).
- Prove ([pal_rev] that)
forall l, pal l -> l = rev l.
*)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 5 stars, optional (palindrome_converse) *)
(** Again, the converse direction is significantly more difficult, due
to the lack of evidence. Using your definition of [pal] from the
previous exercise, prove that
forall l, l = rev l -> pal l.
*)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 4 stars, advanced, optional (NoDup) *)
(** Recall the definition of the [In] property from the [Logic]
chapter, which asserts that a value [x] appears at least once in a
list [l]: *)
(* Fixpoint In (A : Type) (x : A) (l : list A) : Prop :=
match l with
| [] => False
| x' :: l' => x' = x \/ In A x l'
end *)
(** Your first task is to use [In] to define a proposition [disjoint X
l1 l2], which should be provable exactly when [l1] and [l2] are
lists (with elements of type X) that have no elements in
common. *)
(* FILL IN HERE *)
(** Next, use [In] to define an inductive proposition [NoDup X
l], which should be provable exactly when [l] is a list (with
elements of type [X]) where every member is different from every
other. For example, [NoDup nat [1;2;3;4]] and [NoDup
bool []] should be provable, while [NoDup nat [1;2;1]] and
[NoDup bool [true;true]] should not be. *)
(* FILL IN HERE *)
(** Finally, state and prove one or more interesting theorems relating
[disjoint], [NoDup] and [++] (list append). *)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 4 stars, advanced, optional (pigeonhole principle) *)
(** The _pigeonhole principle_ states a basic fact about counting: if
we distribute more than [n] items into [n] pigeonholes, some
pigeonhole must contain at least two items. As often happens, this
apparently trivial fact about numbers requires non-trivial
machinery to prove, but we now have enough... *)
(** First prove an easy useful lemma. *)
Lemma in_split : forall (X:Type) (x:X) (l:list X),
In x l ->
exists l1 l2, l = l1 ++ x :: l2.
Proof.
(* FILL IN HERE *) Admitted.
(** Now define a property [repeats] such that [repeats X l] asserts
that [l] contains at least one repeated element (of type [X]). *)
Inductive repeats {X:Type} : list X -> Prop :=
(* FILL IN HERE *)
.
(** Now, here's a way to formalize the pigeonhole principle. Suppose
list [l2] represents a list of pigeonhole labels, and list [l1]
represents the labels assigned to a list of items. If there are
more items than labels, at least two items must have the same
label -- i.e., list [l1] must contain repeats.
This proof is much easier if you use the [excluded_middle]
hypothesis to show that [In] is decidable, i.e., [forall x l, (In x
l) \/ ~ (In x l)]. However, it is also possible to make the proof
go through _without_ assuming that [In] is decidable; if you
manage to do this, you will not need the [excluded_middle]
hypothesis. *)
Theorem pigeonhole_principle: forall (X:Type) (l1 l2:list X),
excluded_middle ->
(forall x, In x l1 -> In x l2) ->
length l2 < length l1 ->
repeats l1.
Proof.
intros X l1. induction l1 as [|x l1' IHl1'].
(* FILL IN HERE *) Admitted.
(** [] *)
(** $Date: 2016-10-11 14:39:27 -0400 (Tue, 11 Oct 2016) $ *)