(** * Prop: Propositions and Evidence *)
Require Export Logic.
(* ####################################################### *)
(** * Inductively Defined Propositions *)
(** In chapter [Basics] we defined a _function_ [evenb] that tests a
number for evenness, yielding [true] if so. We can use this
function to define the _proposition_ that some number [n] is
even: *)
Definition even (n:nat) : Prop :=
evenb n = true.
(** That is, we can define "[n] is even" to mean "the function [evenb]
returns [true] when applied to [n]."
Note that here we have given a name
to a proposition using a [Definition], just as we have
given names to expressions of other sorts. This isn't a fundamentally
new kind of proposition; it is still just an equality. *)
(** Another alternative is to define the concept of evenness
directly. Instead of going via the [evenb] function ("a number is
even if a certain computation yields [true]"), we can say what the
concept of evenness means by giving two different ways of
presenting _evidence_ that a number is even. *)
Inductive ev : nat -> Prop :=
| ev_0 : ev O
| ev_SS : forall n:nat, ev n -> ev (S (S n)).
(** The first line declares that [ev] is a proposition -- or,
more formally, a family of propositions "indexed by" natural
numbers. (That is, for each number [n], the claim that "[n] is
even" is a proposition.) Such a family of propositions is
often called a _property_ of numbers.
The last two lines declare the two ways to give evidence that a
number [m] is even. First, [0] is even, and [ev_0] is evidence
for this. Second, if [m = S (S n)] for some [n] and we can give
evidence [e] that [n] is even, then [m] is also even, and [ev_SS n
e] is the evidence.
*)
(** **** Exercise: 1 star (double_even) *)
Theorem double_even : forall n,
ev (double n).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ##################################################### *)
(** For [ev], we had already defined [even] as a function (returning a
boolean), and then defined an inductive relation that agreed with
it. However, we don't necessarily need to think about propositions
first as boolean functions, we can start off with the inductive
definition.
*)
(** As another example of an inductively defined proposition, let's
define a simple property of natural numbers -- we'll call it
"[beautiful]." *)
(** Informally, a number is [beautiful] if it is [0], [3], [5], or the
sum of two [beautiful] numbers.
More pedantically, we can define [beautiful] numbers by giving four
rules:
- Rule [b_0]: The number [0] is [beautiful].
- Rule [b_3]: The number [3] is [beautiful].
- Rule [b_5]: The number [5] is [beautiful].
- Rule [b_sum]: If [n] and [m] are both [beautiful], then so is
their sum. *)
(** We will see many definitions like this one during the rest
of the course, and 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: *)
(**
----------- (b_0)
beautiful 0
------------ (b_3)
beautiful 3
------------ (b_5)
beautiful 5
beautiful n beautiful m
--------------------------- (b_sum)
beautiful (n+m)
*)
(** *** *)
(** 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 [b_sum] says that, if [n] and [m]
are both [beautiful] numbers, then it follows that [n+m] is
[beautiful] too. If a rule has no premises above the line, then
its conclusion holds unconditionally.
These rules _define_ the property [beautiful]. That is, if we
want to convince someone that some particular number is [beautiful],
our argument must be based on these rules. For a simple example,
suppose we claim that the number [5] is [beautiful]. To support
this claim, we just need to point out that rule [b_5] says so.
Or, if we want to claim that [8] is [beautiful], we can support our
claim by first observing that [3] and [5] are both [beautiful] (by
rules [b_3] and [b_5]) and then pointing out that their sum, [8],
is therefore [beautiful] by rule [b_sum]. This argument can be
expressed graphically with the following _proof tree_: *)
(**
----------- (b_3) ----------- (b_5)
beautiful 3 beautiful 5
------------------------------- (b_sum)
beautiful 8
*)
(** *** *)
(**
Of course, there are other ways of using these rules to argue that
[8] is [beautiful], for instance:
----------- (b_5) ----------- (b_3)
beautiful 5 beautiful 3
------------------------------- (b_sum)
beautiful 8
*)
(** **** Exercise: 1 star (varieties_of_beauty) *)
(** How many different ways are there to show that [8] is [beautiful]? *)
(* FILL IN HERE *)
(** [] *)
(* ####################################################### *)
(** ** Constructing Evidence *)
(** In Coq, we can express the definition of [beautiful] as
follows: *)
Inductive beautiful : nat -> Prop :=
b_0 : beautiful 0
| b_3 : beautiful 3
| b_5 : beautiful 5
| b_sum : forall n m, beautiful n -> beautiful m -> beautiful (n+m).
(** *** *)
(**
The rules introduced this way have the same status as proven
theorems; that is, they are true axiomatically.
So we can use Coq's [apply] tactic with the rule names to prove
that particular numbers are [beautiful]. *)
Theorem three_is_beautiful: beautiful 3.
Proof.
(* This simply follows from the rule [b_3]. *)
apply b_3.
Qed.
Theorem eight_is_beautiful: beautiful 8.
Proof.
(* First we use the rule [b_sum], telling Coq how to
instantiate [n] and [m]. *)
apply b_sum with (n:=3) (m:=5).
(* To solve the subgoals generated by [b_sum], we must provide
evidence of [beautiful 3] and [beautiful 5]. Fortunately we
have rules for both. *)
apply b_3.
apply b_5.
Qed.
(** *** *)
(** As you would expect, we can also prove theorems that have
hypotheses about [beautiful]. *)
Theorem beautiful_plus_eight: forall n, beautiful n -> beautiful (8+n).
Proof.
intros n B.
apply b_sum with (n:=8) (m:=n).
apply eight_is_beautiful.
apply B.
Qed.
(** **** Exercise: 2 stars (b_times2) *)
Theorem b_times2: forall n, beautiful n -> beautiful (2*n).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars (b_timesm) *)
Theorem b_timesm: forall n m, beautiful n -> beautiful (m*n).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ####################################################### *)
(** * Using Evidence in Proofs *)
(** ** Induction over Evidence *)
(** Besides _constructing_ evidence that numbers are beautiful, we can
also _reason about_ such evidence. *)
(** The fact that we introduced [beautiful] with an [Inductive]
declaration tells Coq not only that the constructors [b_0], [b_3],
[b_5] and [b_sum] are ways to build evidence, but also that these
four constructors are the _only_ ways to build evidence that
numbers are beautiful. *)
(** In other words, if someone gives us evidence [E] for the assertion
[beautiful n], then we know that [E] must have one of four shapes:
- [E] is [b_0] (and [n] is [O]),
- [E] is [b_3] (and [n] is [3]),
- [E] is [b_5] (and [n] is [5]), or
- [E] is [b_sum n1 n2 E1 E2] (and [n] is [n1+n2], where [E1] is
evidence that [n1] is beautiful and [E2] is evidence that [n2]
is beautiful). *)
(** *** *)
(** This permits us to _analyze_ any hypothesis of the form [beautiful
n] to see how it was constructed, using the tactics we already
know. In particular, we can use the [induction] tactic that we
have already seen for reasoning about inductively defined _data_
to reason about inductively defined _evidence_.
To illustrate this, let's define another property of numbers: *)
Inductive gorgeous : nat -> Prop :=
g_0 : gorgeous 0
| g_plus3 : forall n, gorgeous n -> gorgeous (3+n)
| g_plus5 : forall n, gorgeous n -> gorgeous (5+n).
(** **** Exercise: 1 star (gorgeous_tree) *)
(** Write out the definition of [gorgeous] numbers using inference rule
notation.
(* FILL IN HERE *)
[]
*)
(** **** Exercise: 1 star (gorgeous_plus13) *)
Theorem gorgeous_plus13: forall n,
gorgeous n -> gorgeous (13+n).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** *** *)
(** It seems intuitively obvious that, although [gorgeous] and
[beautiful] are presented using slightly different rules, they are
actually the same property in the sense that they are true of the
same numbers. Indeed, we can prove this. *)
Theorem gorgeous__beautiful_FAILED : forall n,
gorgeous n -> beautiful n.
Proof.
intros. induction n as [| n'].
Case "n = 0". apply b_0.
Case "n = S n'". (* We are stuck! *)
Abort.
(** The problem here is that doing induction on [n] doesn't yield a
useful induction hypothesis. Knowing how the property we are
interested in behaves on the predecessor of [n] doesn't help us
prove that it holds for [n]. Instead, we would like to be able to
have induction hypotheses that mention other numbers, such as [n -
3] and [n - 5]. This is given precisely by the shape of the
constructors for [gorgeous]. *)
(** *** *)
(** Let's see what happens if we try to prove this by induction on the evidence [H]
instead of on [n]. *)
Theorem gorgeous__beautiful : forall n,
gorgeous n -> beautiful n.
Proof.
intros n H.
induction H as [|n'|n'].
Case "g_0".
apply b_0.
Case "g_plus3".
apply b_sum. apply b_3.
apply IHgorgeous.
Case "g_plus5".
apply b_sum. apply b_5. apply IHgorgeous.
Qed.
(* These exercises also require the use of induction on the evidence. *)
(** **** Exercise: 2 stars (gorgeous_sum) *)
Theorem gorgeous_sum : forall n m,
gorgeous n -> gorgeous m -> gorgeous (n + m).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, advanced (beautiful__gorgeous) *)
Theorem beautiful__gorgeous : forall n, beautiful n -> gorgeous n.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, optional (g_times2) *)
(** Prove the [g_times2] theorem below without using [gorgeous__beautiful].
You might find the following helper lemma useful. *)
Lemma helper_g_times2 : forall x y z, x + (z + y) = z + x + y.
Proof.
(* FILL IN HERE *) Admitted.
Theorem g_times2: forall n, gorgeous n -> gorgeous (2*n).
Proof.
intros n H. simpl.
induction H.
(* FILL IN HERE *) Admitted.
(** [] *)
(** Here is a proof that the inductive definition of evenness implies
the computational one. *)
Theorem ev__even : forall n,
ev n -> even n.
Proof.
intros n E. induction E as [| n' E'].
Case "E = ev_0".
unfold even. reflexivity.
Case "E = ev_SS n' E'".
unfold even. apply IHE'.
Qed.
(** **** Exercise: 1 star (ev__even) *)
(** Could this proof also be carried out by induction on [n] instead
of [E]? If not, why not? *)
(* FILL IN HERE *)
(** [] *)
(** Intuitively, the induction principle [ev n] evidence [ev n] is
similar to induction on [n], but restricts our attention to only
those numbers for which evidence [ev n] could be generated. *)
(** **** Exercise: 1 star (l_fails) *)
(** The following proof attempt will not succeed.
Theorem l : forall n,
ev n.
Proof.
intros n. induction n.
Case "O". simpl. apply ev_0.
Case "S".
...
Intuitively, we expect the proof to fail because not every
number is even. However, what exactly causes the proof to fail?
(* FILL IN HERE *)
*)
(** [] *)
(** Here's another exercise requiring induction on evidence. *)
(** **** Exercise: 2 stars (ev_sum) *)
Theorem ev_sum : forall n m,
ev n -> ev m -> ev (n+m).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ####################################################### *)
(** ** Inversion on Evidence *)
(** Having evidence for a proposition is useful while proving, because we
can _look_ at that evidence for more information. For example, consider
proving that, if [n] is even, then [pred (pred n)] is
too. In this case, we don't need to do an inductive proof. Instead
the [inversion] tactic provides all of the information that we need.
*)
Theorem ev_minus2: forall n, ev n -> ev (pred (pred n)).
Proof.
intros n E.
inversion E as [| n' E'].
Case "E = ev_0". simpl. apply ev_0.
Case "E = ev_SS n' E'". simpl. apply E'. Qed.
(** **** Exercise: 1 star, optional (ev_minus2_n) *)
(** What happens if we try to use [destruct] on [n] instead of [inversion] on [E]? *)
(* FILL IN HERE *)
(** [] *)
(** *** *)
(** Here is another example, in which [inversion] helps narrow down to
the relevant cases. *)
Theorem SSev__even : forall n,
ev (S (S n)) -> ev n.
Proof.
intros n E.
inversion E as [| n' E'].
apply E'. Qed.
(** ** The Inversion Tactic Revisited *)
(** These uses of [inversion] 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.
(You might also expect that [destruct] would be a more suitable
tactic to use here. Indeed, it is possible to use [destruct], but
it often throws away useful information, and the [eqn:] qualifier
doesn't help much in this case.)
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.
In this particular case, the [inversion] analyzed the construction
[ev (S (S n))], determined that this could only have been
constructed using [ev_SS], and generated a new subgoal with the
arguments of that constructor as new hypotheses. (It also
produced an auxiliary equality, which happens to be useless here.)
We'll begin exploring this more general behavior of inversion in
what follows. *)
(** **** Exercise: 1 star (inversion_practice) *)
Theorem SSSSev__even : forall n,
ev (S (S (S (S n)))) -> ev n.
Proof.
(* FILL IN HERE *) Admitted.
(** The [inversion] tactic can also be used to derive goals by showing
the absurdity of a hypothesis. *)
Theorem even5_nonsense :
ev 5 -> 2 + 2 = 9.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, advanced (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) *)
(** Here's an exercise that just requires applying existing lemmas. No
induction or even case analysis is needed, but 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.
(** [] *)
(* ####################################################### *)
(** * Discussion and Variations *)
(** ** Computational vs. Inductive Definitions *)
(** We have seen that the proposition "[n] is even" can be
phrased in two different ways -- indirectly, via a boolean testing
function [evenb], or directly, by inductively describing what
constitutes evidence for evenness. These two ways of defining
evenness are about equally easy to state and work with. Which we
choose is basically a question of taste.
However, for many other properties of interest, the direct
inductive definition is preferable, since writing a testing
function may be awkward or even impossible.
One such property is [beautiful]. This is a perfectly sensible
definition of a set of numbers, but we cannot translate its
definition directly into a Coq Fixpoint (or into a recursive
function in any other common programming language). We might be
able to find a clever way of testing this property using a
[Fixpoint] (indeed, it is not too hard to find one in this case),
but in general this could require arbitrarily deep thinking. In
fact, if the property we are interested in is uncomputable, then
we cannot define it as a [Fixpoint] no matter how hard we try,
because Coq requires that all [Fixpoint]s correspond to
terminating computations.
On the other hand, writing an inductive definition of what it
means to give evidence for the property [beautiful] is
straightforward. *)
(* ####################################################### *)
(** ** Parameterized Data Structures *)
(** So far, we have only looked at propositions about natural numbers. However,
we can define inductive predicates about any type of data. For example,
suppose we would like to characterize lists of _even_ length. We can
do that with the following definition. *)
Inductive ev_list {X:Type} : list X -> Prop :=
| el_nil : ev_list []
| el_cc : forall x y l, ev_list l -> ev_list (x :: y :: l).
(** Of course, this proposition is equivalent to just saying that the
length of the list is even. *)
Lemma ev_list__ev_length: forall X (l : list X), ev_list l -> ev (length l).
Proof.
intros X l H. induction H.
Case "el_nil". simpl. apply ev_0.
Case "el_cc". simpl. apply ev_SS. apply IHev_list.
Qed.
(** However, because evidence for [ev] contains less information than
evidence for [ev_list], the converse direction must be stated very
carefully. *)
Lemma ev_length__ev_list: forall X n, ev n -> forall (l : list X), n = length l -> ev_list l.
Proof.
intros X n H.
induction H.
Case "ev_0". intros l H. destruct l.
SCase "[]". apply el_nil.
SCase "x::l". inversion H.
Case "ev_SS". intros l H2. destruct l.
SCase "[]". inversion H2. destruct l.
SCase "[x]". inversion H2.
SCase "x :: x0 :: l". apply el_cc. apply IHev. inversion H2. reflexivity.
Qed.
(** **** Exercise: 4 stars (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
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 *)
(** [] *)
(* Again, the converse direction is much more difficult, due to the
lack of evidence. *)
(** **** Exercise: 5 stars, optional (palindrome_converse) *)
(** Using your definition of [pal] from the previous exercise, prove
that
forall l, l = rev l -> pal l.
*)
(* FILL IN HERE *)
(** [] *)
(* ####################################################### *)
(** ** Relations *)
(** A proposition parameterized by a number (such as [ev] or
[beautiful]) 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 LeModule.
(** 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] in chapter [Prop]. 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 LeModule.
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 (total_relation) *)
(** Define an inductive binary relation [total_relation] that holds
between every pair of natural numbers. *)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 2 stars (empty_relation) *)
(** Define an inductive binary relation [empty_relation] (on numbers)
that never holds. *)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 2 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 ble_nat_true : forall n m,
ble_nat n m = true -> n <= m.
Proof.
(* FILL IN HERE *) Admitted.
Theorem le_ble_nat : forall n m,
n <= m ->
ble_nat n m = true.
Proof.
(* Hint: This may be easiest to prove by induction on [m]. *)
(* FILL IN HERE *) Admitted.
Theorem ble_nat_true_trans : forall n m o,
ble_nat n m = true -> ble_nat m o = true -> ble_nat n o = true.
Proof.
(* Hint: This theorem can be easily proved without using [induction]. *)
(* FILL IN HERE *) Admitted.
(** **** Exercise: 2 stars, optional (ble_nat_false) *)
Theorem ble_nat_false : forall n m,
ble_nat n m = false -> ~(n <= m).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars (R_provability2) *)
Module R.
(** 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) *)
(** Relation [R] actually encodes a familiar function. State and prove two
theorems that formally connects the relation and the function.
That is, if [R m n o] is true, what can we say about [m],
[n], and [o], and vice versa?
*)
(* FILL IN HERE *)
(** [] *)
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, optional (R_provability) *)
(** 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]]
*)
(** [] *)
(* ##################################################### *)
(** * Programming with Propositions *)
(** As we have seen, a _proposition_ is a statement expressing a factual claim,
like "two plus two equals four." In Coq, propositions are written
as expressions of type [Prop]. . *)
Check (2 + 2 = 4).
(* ===> 2 + 2 = 4 : Prop *)
Check (ble_nat 3 2 = false).
(* ===> ble_nat 3 2 = false : Prop *)
Check (beautiful 8).
(* ===> beautiful 8 : Prop *)
(** *** *)
(** Both provable and unprovable claims are perfectly good
propositions. Simply _being_ a proposition is one thing; being
_provable_ is something else! *)
Check (2 + 2 = 5).
(* ===> 2 + 2 = 5 : Prop *)
Check (beautiful 4).
(* ===> beautiful 4 : Prop *)
(** Both [2 + 2 = 4] and [2 + 2 = 5] are legal expressions
of type [Prop]. *)
(** *** *)
(** We've mainly seen one place that propositions can appear in Coq: in
[Theorem] (and [Lemma] and [Example]) declarations. *)
Theorem plus_2_2_is_4 :
2 + 2 = 4.
Proof. reflexivity. Qed.
(** But they can be used in many other ways. For example, we have also seen that
we can give a name to a proposition using a [Definition], just as we have
given names to expressions of other sorts. *)
Definition plus_fact : Prop := 2 + 2 = 4.
Check plus_fact.
(* ===> plus_fact : Prop *)
(** We can later use this name in any situation where a proposition is
expected -- for example, as the claim in a [Theorem] declaration. *)
Theorem plus_fact_is_true :
plus_fact.
Proof. reflexivity. Qed.
(** *** *)
(** We've seen several ways of constructing propositions.
- We can define a new proposition primitively using [Inductive].
- Given two expressions [e1] and [e2] of the same type, we can
form the proposition [e1 = e2], which states that their
values are equal.
- We can combine propositions using implication and
quantification. *)
(** *** *)
(** We have also seen _parameterized propositions_, such as [even] and
[beautiful]. *)
Check (even 4).
(* ===> even 4 : Prop *)
Check (even 3).
(* ===> even 3 : Prop *)
Check even.
(* ===> even : nat -> Prop *)
(** *** *)
(** The type of [even], i.e., [nat->Prop], can be pronounced in
three equivalent ways: (1) "[even] is a _function_ from numbers to
propositions," (2) "[even] is a _family_ of propositions, indexed
by a number [n]," or (3) "[even] is a _property_ of numbers." *)
(** Propositions -- including parameterized propositions -- are
first-class citizens in Coq. For example, we can define functions
from numbers to propositions... *)
Definition between (n m o: nat) : Prop :=
andb (ble_nat n o) (ble_nat o m) = true.
(** ... and then partially apply them: *)
Definition teen : nat->Prop := between 13 19.
(** We can even pass propositions -- including parameterized
propositions -- as arguments to functions: *)
Definition true_for_zero (P:nat->Prop) : Prop :=
P 0.
(** *** *)
(** Here are two more examples of passing parameterized propositions
as arguments to a function.
The first function, [true_for_all_numbers], takes a proposition
[P] as argument and builds the proposition that [P] is true for
all natural numbers. *)
Definition true_for_all_numbers (P:nat->Prop) : Prop :=
forall n, P n.
(** The second, [preserved_by_S], takes [P] and builds the proposition
that, if [P] is true for some natural number [n'], then it is also
true by the successor of [n'] -- i.e. that [P] is _preserved by
successor_: *)
Definition preserved_by_S (P:nat->Prop) : Prop :=
forall n', P n' -> P (S n').
(** *** *)
(** Finally, we can put these ingredients together to define
a proposition stating that induction is valid for natural numbers: *)
Definition natural_number_induction_valid : Prop :=
forall (P:nat->Prop),
true_for_zero P ->
preserved_by_S P ->
true_for_all_numbers P.
(** **** Exercise: 3 stars (combine_odd_even) *)
(** Complete the definition of the [combine_odd_even] function
below. It takes as arguments two properties of numbers [Podd] and
[Peven]. As its result, it should return a new property [P] such
that [P n] is equivalent to [Podd n] when [n] is odd, and
equivalent to [Peven n] otherwise. *)
Definition combine_odd_even (Podd Peven : nat -> Prop) : nat -> Prop :=
(* FILL IN HERE *) admit.
(** To test your definition, see whether you can prove the following
facts: *)
Theorem combine_odd_even_intro :
forall (Podd Peven : nat -> Prop) (n : nat),
(oddb n = true -> Podd n) ->
(oddb n = false -> Peven n) ->
combine_odd_even Podd Peven n.
Proof.
(* FILL IN HERE *) Admitted.
Theorem combine_odd_even_elim_odd :
forall (Podd Peven : nat -> Prop) (n : nat),
combine_odd_even Podd Peven n ->
oddb n = true ->
Podd n.
Proof.
(* FILL IN HERE *) Admitted.
Theorem combine_odd_even_elim_even :
forall (Podd Peven : nat -> Prop) (n : nat),
combine_odd_even Podd Peven n ->
oddb n = false ->
Peven n.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ##################################################### *)
(** One more quick digression, for adventurous souls: if we can define
parameterized propositions using [Definition], then can we also
define them using [Fixpoint]? Of course we can! However, this
kind of "recursive parameterization" doesn't correspond to
anything very familiar from everyday mathematics. The following
exercise gives a slightly contrived example. *)
(** **** Exercise: 4 stars, optional (true_upto_n__true_everywhere) *)
(** Define a recursive function
[true_upto_n__true_everywhere] that makes
[true_upto_n_example] work. *)
(*
Fixpoint true_upto_n__true_everywhere
(* FILL IN HERE *)
Example true_upto_n_example :
(true_upto_n__true_everywhere 3 (fun n => even n))
= (even 3 -> even 2 -> even 1 -> forall m : nat, even m).
Proof. reflexivity. Qed.
*)
(** [] *)
(** $Date: 2014-12-31 11:17:56 -0500 (Wed, 31 Dec 2014) $ *)