Induction.v

(** * Induction: Proof by Induction *)

(* ################################################################# *)
(** * Separate Compilation *)

(** Before getting started on this chapter, we need to import
    all of our definitions from the previous chapter: *)

From LF Require Export Basics.

(** For this [Require Export] command to work, Coq needs to be
    able to find a compiled version of [Basics.v], called [Basics.vo],
    in a directory associated with the prefix [LF].  This file is
    analogous to the [.class] files compiled from [.java] source files
    and the [.o] files compiled from [.c] files.

    First create a file named [_CoqProject] containing the following
    line (if you obtained the whole volume "Logical Foundations" as a
    single archive, a [_CoqProject] should already exist and you can
    skip this step):

      -Q . LF

    This maps the current directory ("[.]", which contains [Basics.v],
    [Induction.v], etc.) to the prefix (or "logical directory")
    "[LF]".  Proof General and CoqIDE read [_CoqProject]
    automatically, so they know to where to look for the file
    [Basics.vo] corresponding to the library [LF.Basics].

    Once [_CoqProject] is thus created, there are various ways to
    build [Basics.vo]:

     - In Proof General or CoqIDE, the compilation should happen
       automatically when you submit the [Require] line above to PG.

     - For VSCode users, open the terminal pane at the bottom 
       and then use the command line instructions below. 
       (If you downloaded the project setup .tgz file, just doing `make`
       should build all the code.)

     - If you want to compile from the command line, generate a
       [Makefile] using the [coq_makefile] utility, which comes
       installed with Coq (if you obtained the whole volume as a
       single archive, a [Makefile] should already exist and you can
       skip this step):

         coq_makefile -f _CoqProject *.v -o Makefile

       Note: You should rerun that command whenever you add or remove
       Coq files to the directory.

       Now you can compile [Basics.v] by running [make] with the
       corresponding [.vo] file as a target:

         make Basics.vo

       All files in the directory can be compiled by giving no
       arguments:

         make

       Under the hood, [make] uses the Coq compiler, [coqc].  You can
       also run [coqc] directly:

         coqc -Q . LF Basics.v

       But [make] also calculates dependencies between source files to
       compile them in the right order, so [make] should generally be
       preferred over explicit [coqc].

    If you have trouble (e.g., if you get complaints about missing
    identifiers later in the file), it may be because the "load path"
    for Coq is not set up correctly.  The [Print LoadPath.] command
    may be helpful in sorting out such issues.

    In particular, if you see a message like

        Compiled library Foo makes inconsistent assumptions over
        library Bar

    check whether you have multiple installations of Coq on your
    machine.  It may be that commands (like [coqc]) that you execute
    in a terminal window are getting a different version of Coq than
    commands executed by Proof General or CoqIDE.

    - Another common reason is that the library [Bar] was modified and
      recompiled without also recompiling [Foo] which depends on it.
      Recompile [Foo], or everything if too many files are
      affected.  (Using the third solution above: [make clean; make].)

    One more tip for CoqIDE users: If you see messages like [Error:
    Unable to locate library Basics], a likely reason is
    inconsistencies between compiling things _within CoqIDE_ vs _using
    [coqc] from the command line_.  This typically happens when there
    are two incompatible versions of [coqc] installed on your
    system (one associated with CoqIDE, and one associated with [coqc]
    from the terminal).  The workaround for this situation is
    compiling using CoqIDE only (i.e. choosing "make" from the menu),
    and avoiding using [coqc] directly at all. *)

(* ################################################################# *)
(** * Proof by Induction *)

(** We can prove that [0] is a neutral element for [+] on the _left_
    using just [reflexivity].  But the proof that it is also a neutral
    element on the _right_ ... *)

Theorem add_0_r_firsttry : forall n:nat,
  n + 0 = n.

(** ... can't be done in the same simple way.  Just applying
  [reflexivity] doesn't work, since the [n] in [n + 0] is an arbitrary
  unknown number, so the [match] in the definition of [+] can't be
  simplified.  *)

Proof.
  intros n.
  simpl. (* Does nothing! *)
Abort.

(** And reasoning by cases using [destruct n] doesn't get us much
    further: the branch of the case analysis where we assume [n = 0]
    goes through fine, but in the branch where [n = S n'] for some [n'] we
    get stuck in exactly the same way. *)

Theorem add_0_r_secondtry : forall n:nat,
  n + 0 = n.
Proof.
  intros n. destruct n as [| n'] eqn:E.
  - (* n = 0 *)
    reflexivity. (* so far so good... *)
  - (* n = S n' *)
    simpl.       (* ...but here we are stuck again *)
Abort.

(** We could use [destruct n'] to get one step further, but,
    since [n] can be arbitrarily large, we'll never get all the there
    if we just go on like this. *)

(** To prove interesting facts about numbers, lists, and other
    inductively defined sets, we often need a more powerful reasoning
    principle: _induction_.

    Recall (from high school, a discrete math course, etc.) the
    _principle of induction over natural numbers_: If [P(n)] is some
    proposition involving a natural number [n] and we want to show
    that [P] holds for all numbers [n], we can reason like this:
         - show that [P(O)] holds;
         - show that, for any [n'], if [P(n')] holds, then so does
           [P(S n')];
         - conclude that [P(n)] holds for all [n].

    In Coq, the steps are the same: we begin with the goal of proving
    [P(n)] for all [n] and break it down (by applying the [induction]
    tactic) into two separate subgoals: one where we must show [P(O)]
    and another where we must show [P(n') -> P(S n')].  Here's how
    this works for the theorem at hand: *)

Theorem add_0_r : forall n:nat, n + 0 = n.
Proof.
  intros n. induction n as [| n' IHn'].
  - (* n = 0 *)    reflexivity.
  - (* n = S n' *) simpl. rewrite -> IHn'. reflexivity.  Qed.

(** Like [destruct], the [induction] tactic takes an [as...]
    clause that specifies the names of the variables to be introduced
    in the subgoals.  Since there are two subgoals, the [as...] clause
    has two parts, separated by [|].  (Strictly speaking, we can omit
    the [as...] clause and Coq will choose names for us.  In practice,
    this is a bad idea, as Coq's automatic choices tend to be
    confusing.)

    In the first subgoal, [n] is replaced by [0].  No new variables
    are introduced (so the first part of the [as...] is empty), and
    the goal becomes [0 = 0 + 0], which follows by simplification.

    In the second subgoal, [n] is replaced by [S n'], and the
    assumption [n' + 0 = n'] is added to the context with the name
    [IHn'] (i.e., the Induction Hypothesis for [n']).  These two names
    are specified in the second part of the [as...] clause.  The goal
    in this case becomes [S n' = (S n') + 0], which simplifies to
    [S n' = S (n' + 0)], which in turn follows from [IHn']. *)

Theorem minus_n_n : forall n,
  minus n n = 0.
Proof.
  (* WORKED IN CLASS *)
  intros n. induction n as [| n' IHn'].
  - (* n = 0 *)
    simpl. reflexivity.
  - (* n = S n' *)
    simpl. rewrite -> IHn'. reflexivity.  Qed.

(** (The use of the [intros] tactic in these proofs is actually
    redundant.  When applied to a goal that contains quantified
    variables, the [induction] tactic will automatically move them
    into the context as needed.) *)

(** **** Exercise: 2 stars, standard, especially useful (basic_induction)

    Prove the following using induction. You might need previously
    proven results. *)

Theorem mul_0_r : forall n:nat,
  n * 0 = 0.
Proof.
  induction n as [| n' IHn'].
  - reflexivity.
  - simpl. rewrite -> IHn'. reflexivity.
Qed.  

Theorem plus_n_Sm : forall n m : nat,
  S (n + m) = n + (S m).
Proof.
  intros. induction n as [| n' IHn'].
  - simpl. reflexivity.
  - simpl. rewrite -> IHn'. reflexivity.
Qed.   

Theorem add_comm : forall n m : nat,
  n + m = m + n.
Proof.
  intros. induction n as [| n' IHn'].
  - simpl. rewrite -> add_0_r. reflexivity.
  - simpl. rewrite -> IHn'. rewrite -> plus_n_Sm. reflexivity.
Qed.

Theorem add_assoc : forall n m p : nat,
  n + (m + p) = (n + m) + p.
Proof.
  intros. induction n as [| n' IHn'].
  - simpl. reflexivity.
  - simpl. rewrite -> IHn'. reflexivity.
Qed.  
(** [] *)

(** **** Exercise: 2 stars, standard (double_plus)

    Consider the following function, which doubles its argument: *)

Fixpoint double (n:nat) :=
  match n with
  | O => O
  | S n' => S (S (double n'))
  end.

(** Use induction to prove this simple fact about [double]: *)

Lemma double_plus : forall n, double n = n + n .
Proof.
  induction n as [| n' IHn'].
  - simpl. reflexivity.
  - simpl. rewrite -> IHn'. rewrite -> plus_n_Sm. reflexivity.
Qed.
(** [] *)

(** **** Exercise: 2 stars, standard (eqb_refl)

    The following theorem relates the computational equality [=?] on
    [nat] with the definitional equality [=] on [bool]. *)
Theorem eqb_refl : forall n : nat,
  (n =? n) = true.
Proof.
  induction n as [| n' IHn'].
  - reflexivity.
  - simpl. rewrite -> IHn'. reflexivity.
Qed.   
(** [] *)

(** **** Exercise: 2 stars, standard, optional (even_S)

    One inconvenient aspect of our definition of [even n] is the
    recursive call on [n - 2]. This makes proofs about [even n]
    harder when done by induction on [n], since we may need an
    induction hypothesis about [n - 2]. The following lemma gives an
    alternative characterization of [even (S n)] that works better
    with induction: *)

Theorem even_S : forall n : nat,
  even (S n) = negb (even n).
Proof.
  intros. induction n as [| n' IHn'].
  - reflexivity.
  - rewrite -> IHn'. simpl. rewrite -> negb_involutive. reflexivity.
Qed.  
(** [] *)

(** **** Exercise: 1 star, standard, optional (destruct_induction)

    Briefly explain the difference between the tactics [destruct]
    and [induction].

(* FILL IN HERE *)
*)

(** [] *)

(* ################################################################# *)
(** * Proofs Within Proofs *)

(** In Coq, as in informal mathematics, large proofs are often
    broken into a sequence of theorems, with later proofs referring to
    earlier theorems.  But sometimes a proof will involve some
    miscellaneous fact that is too trivial and of too little general
    interest to bother giving it its own top-level name.  In such
    cases, it is convenient to be able to simply state and prove the
    needed "sub-theorem" right at the point where it is used.  The
    [assert] tactic allows us to do this. *)

Theorem mult_0_plus' : forall n m : nat,
  (n + 0 + 0) * m = n * m.
Proof.
  intros n m.
  assert (H: n + 0 + 0 = n).
    { rewrite add_comm. simpl. rewrite add_comm. reflexivity. }
  rewrite -> H.
  reflexivity.  Qed.

(** The [assert] tactic introduces two sub-goals.  The first is
    the assertion itself; by prefixing it with [H:] we name the
    assertion [H].  (We can also name the assertion with [as] just as
    we did above with [destruct] and [induction], i.e., [assert (n + 0
    + 0 = n) as H].)  Note that we surround the proof of this
    assertion with curly braces [{ ... }], both for readability and so
    that, when using Coq interactively, we can see more easily when we
    have finished this sub-proof.  The second goal is the same as the
    one at the point where we invoke [assert] except that, in the
    context, we now have the assumption [H] that [n + 0 + 0 = n].
    That is, [assert] generates one subgoal where we must prove the
    asserted fact and a second subgoal where we can use the asserted
    fact to make progress on whatever we were trying to prove in the
    first place. *)

(** As another example, suppose we want to prove that [(n + m)
    + (p + q) = (m + n) + (p + q)]. The only difference between the
    two sides of the [=] is that the arguments [m] and [n] to the
    first inner [+] are swapped, so it seems we should be able to use
    the commutativity of addition ([add_comm]) to rewrite one into the
    other.  However, the [rewrite] tactic is not very smart about
    _where_ it applies the rewrite.  There are three uses of [+] here,
    and it turns out that doing [rewrite -> add_comm] will affect only
    the _outer_ one... *)

Theorem plus_rearrange_firsttry : forall n m p q : nat,
  (n + m) + (p + q) = (m + n) + (p + q).
Proof.
  intros n m p q.
  (* We just need to swap (n + m) for (m + n)... seems
     like add_comm should do the trick! *)
  rewrite add_comm.
  (* Doesn't work... Coq rewrites the wrong plus! :-( *)
Abort.

(** To use [add_comm] at the point where we need it, we can introduce
    a local lemma stating that [n + m = m + n] (for the _particular_ [m]
    and [n] that we are talking about here), prove this lemma using
    [add_comm], and then use it to do the desired rewrite. *)

Theorem plus_rearrange : forall n m p q : nat,
  (n + m) + (p + q) = (m + n) + (p + q).
Proof.
  intros n m p q.
  assert (H: n + m = m + n).
  { rewrite add_comm. reflexivity. }
  rewrite H. reflexivity.  Qed.

(* ################################################################# *)
(** * Formal vs. Informal Proof *)

(** "_Informal proofs are algorithms; formal proofs are code_." *)

(** What constitutes a successful proof of a mathematical claim?
    The question has challenged philosophers for millennia, but a
    rough and ready definition could be this: A proof of a
    mathematical proposition [P] is a written (or spoken) text that
    instills in the reader or hearer the certainty that [P] is true --
    an unassailable argument for the truth of [P].  That is, a proof
    is an act of communication.

    Acts of communication may involve different sorts of readers.  On
    one hand, the "reader" can be a program like Coq, in which case
    the "belief" that is instilled is that [P] can be mechanically
    derived from a certain set of formal logical rules, and the proof
    is a recipe that guides the program in checking this fact.  Such
    recipes are _formal_ proofs.

    Alternatively, the reader can be a human being, in which case the
    proof will be written in English or some other natural language,
    and will thus necessarily be _informal_.  Here, the criteria for
    success are less clearly specified.  A "valid" proof is one that
    makes the reader believe [P].  But the same proof may be read by
    many different readers, some of whom may be convinced by a
    particular way of phrasing the argument, while others may not be.
    Some readers may be particularly pedantic, inexperienced, or just
    plain thick-headed; the only way to convince them will be to make
    the argument in painstaking detail.  But other readers, more
    familiar in the area, may find all this detail so overwhelming
    that they lose the overall thread; all they want is to be told the
    main ideas, since it is easier for them to fill in the details for
    themselves than to wade through a written presentation of them.
    Ultimately, there is no universal standard, because there is no
    single way of writing an informal proof that is guaranteed to
    convince every conceivable reader.

    In practice, however, mathematicians have developed a rich set of
    conventions and idioms for writing about complex mathematical
    objects that -- at least within a certain community -- make
    communication fairly reliable.  The conventions of this stylized
    form of communication give a fairly clear standard for judging
    proofs good or bad.

    Because we are using Coq in this course, we will be working
    heavily with formal proofs.  But this doesn't mean we can
    completely forget about informal ones!  Formal proofs are useful
    in many ways, but they are _not_ very efficient ways of
    communicating ideas between human beings. *)

(** For example, here is a proof that addition is associative: *)

Theorem add_assoc' : forall n m p : nat,
  n + (m + p) = (n + m) + p.
Proof. intros n m p. induction n as [| n' IHn']. reflexivity.
  simpl. rewrite IHn'. reflexivity.  Qed.

(** Coq is perfectly happy with this.  For a human, however, it
    is difficult to make much sense of it.  We can use comments and
    bullets to show the structure a little more clearly... *)

Theorem add_assoc'' : forall n m p : nat,
  n + (m + p) = (n + m) + p.
Proof.
  intros n m p. induction n as [| n' IHn'].
  - (* n = 0 *)
    reflexivity.
  - (* n = S n' *)
    simpl. rewrite IHn'. reflexivity.   Qed.

(** ... and if you're used to Coq you might be able to step
    through the tactics one after the other in your mind and imagine
    the state of the context and goal stack at each point, but if the
    proof were even a little bit more complicated this would be next
    to impossible.

    A (pedantic) mathematician might write the proof something like
    this: *)

(** - _Theorem_: For any [n], [m] and [p],

      n + (m + p) = (n + m) + p.

    _Proof_: By induction on [n].

    - First, suppose [n = 0].  We must show that

        0 + (m + p) = (0 + m) + p.

      This follows directly from the definition of [+].

    - Next, suppose [n = S n'], where

        n' + (m + p) = (n' + m) + p.

      We must now show that

        (S n') + (m + p) = ((S n') + m) + p.

      By the definition of [+], this follows from

        S (n' + (m + p)) = S ((n' + m) + p),

      which is immediate from the induction hypothesis.  _Qed_. *)

(** The overall form of the proof is basically similar, and of
    course this is no accident: Coq has been designed so that its
    [induction] tactic generates the same sub-goals, in the same
    order, as the bullet points that a mathematician would write.  But
    there are significant differences of detail: the formal proof is
    much more explicit in some ways (e.g., the use of [reflexivity])
    but much less explicit in others (in particular, the "proof state"
    at any given point in the Coq proof is completely implicit,
    whereas the informal proof reminds the reader several times where
    things stand). *)

(** **** Exercise: 2 stars, advanced, especially useful (add_comm_informal)

    Translate your solution for [add_comm] into an informal proof:

    Theorem: Addition is commutative.

    Proof: (* FILL IN HERE *)
*)

(* Do not modify the following line: *)
Definition manual_grade_for_add_comm_informal : option (nat*string) := None.
(** [] *)

(** **** Exercise: 2 stars, standard, optional (eqb_refl_informal)

    Write an informal proof of the following theorem, using the
    informal proof of [add_assoc] as a model.  Don't just
    paraphrase the Coq tactics into English!

    Theorem: [(n =? n) = true] for any [n].

    Proof: (* FILL IN HERE *)
*)

(* Do not modify the following line: *)
Definition manual_grade_for_eqb_refl_informal : option (nat*string) := None.
(** [] *)

(* ################################################################# *)
(** * More Exercises *)

(** **** Exercise: 3 stars, standard, especially useful (mul_comm)

    Use [assert] to help prove [add_shuffle3].  You don't need to
    use induction yet. *)

Theorem add_shuffle3 : forall n m p : nat,
  n + (m + p) = m + (n + p).
Proof.
  intros n m p.
  assert (H1: n + (m + p) = (n + m) + p).
  - rewrite add_assoc. reflexivity.
  - rewrite H1.
    assert (H2: n + m = m + n).
    + rewrite add_comm. reflexivity.
    + rewrite H2. rewrite add_assoc. reflexivity.
Qed.

(** Now prove commutativity of multiplication.  You will probably want
    to look for (or define and prove) a "helper" theorem to be used in
    the proof of this one. Hint: what is [n * (1 + k)]? *)

Theorem mul_comm : forall m n : nat,
  m * n = n * m.
Proof.
  intros.
  induction m as [| m' IHm'].
  - simpl. rewrite <- mult_n_O. reflexivity.
  - simpl.
    assert (H1: n + m' * n = n * (1 + m')).
    + simpl. rewrite <- mult_n_Sm. rewrite add_comm. rewrite IHm'. reflexivity.
    + rewrite <- mult_n_Sm. rewrite IHm'. rewrite add_comm. reflexivity.
Qed.
(** [] *)

(** **** Exercise: 2 stars, standard, optional (plus_leb_compat_l)

    If a hypothesis has the form [H: P -> a = b], then [rewrite H] will
    rewrite [a] to [b] in the goal, and add [P] as a new subgoal. Use
    that in the inductive step of this exercise. *)

Check leb.

Theorem plus_leb_compat_l : forall n m p : nat,
  n <=? m = true -> (p + n) <=? (p + m) = true.
Proof.
  intros n m p. intro H.
  induction p as [| p' IHp'].
  - simpl. rewrite H. reflexivity.
  - simpl. rewrite IHp'. reflexivity.
Qed.  

(** [] *)

(** **** Exercise: 3 stars, standard, optional (more_exercises)

    Take a piece of paper.  For each of the following theorems, first
    _think_ about whether (a) it can be proved using only
    simplification and rewriting, (b) it also requires case
    analysis ([destruct]), or (c) it also requires induction.  Write
    down your prediction.  Then fill in the proof.  (There is no need
    to turn in your piece of paper; this is just to encourage you to
    reflect before you hack!) *)

Theorem leb_refl : forall n:nat,
  (n <=? n) = true.
Proof.
  intros. induction n as [| n' IHn'].
  - simpl. reflexivity.
  - simpl. rewrite IHn'. reflexivity.
Qed.

Theorem zero_neqb_S : forall n:nat,
  0 =? (S n) = false.
Proof.
  intros. simpl. reflexivity.
Qed.

Theorem andb_false_r : forall b : bool,
  andb b false = false.
Proof.
  intros. destruct b.
  - reflexivity.
  - reflexivity.
Qed.

Theorem S_neqb_0 : forall n:nat,
  (S n) =? 0 = false.
Proof.
  intros. simpl. reflexivity.
Qed.

Theorem mult_1_l : forall n:nat, 1 * n = n.
Proof.
  intros. rewrite mul_comm. destruct n.
  - reflexivity.
  - rewrite <- mult_n_Sm. rewrite mul_0_r. simpl. reflexivity.
Qed.     

Theorem all3_spec : forall b c : bool,
  orb
    (andb b c)
    (orb (negb b)
         (negb c))
  = true.
Proof.
  intros. destruct b , c; reflexivity.
Qed.  

Theorem mult_plus_distr_r : forall n m p : nat,
  (n + m) * p = (n * p) + (m * p).
Proof.
  intros. induction n as [| n' IHn'].
  - simpl. reflexivity.
  - simpl. rewrite IHn'. rewrite add_assoc. reflexivity.
Qed.

Theorem mult_assoc : forall n m p : nat,
  n * (m * p) = (n * m) * p.
Proof.
  intros.
  assert (H1: n + m * n = n * (1 + m)).
    { simpl. rewrite <- mult_n_Sm. rewrite mul_comm. rewrite add_comm. reflexivity. }
  induction p as [| p' IHp'].
  - rewrite mul_0_r. rewrite mul_0_r. rewrite mul_0_r. reflexivity.
  - rewrite <- mult_n_Sm. 
    rewrite <- mult_n_Sm. 
    rewrite mul_comm.
    rewrite mult_plus_distr_r.
    assert (H2: n * m = m * n).
      { rewrite mul_comm. reflexivity. }
    rewrite H2. rewrite mul_comm.  rewrite IHp'. rewrite H2. reflexivity.
Qed.
(** [] *)

(** **** Exercise: 2 stars, standard, optional (add_shuffle3')

    The [replace] tactic allows you to specify a particular subterm to
   rewrite and what you want it rewritten to: [replace (t) with (u)]
   replaces (all copies of) expression [t] in the goal by expression
   [u], and generates [t = u] as an additional subgoal. This is often
   useful when a plain [rewrite] acts on the wrong part of the goal.

   Use the [replace] tactic to do a proof of [add_shuffle3'], just like
   [add_shuffle3] but without needing [assert]. *)

Theorem add_shuffle3' : forall n m p : nat,
  n + (m + p) = m + (n + p).
Proof.
  intros. rewrite add_assoc'.
  replace (n + m) with (m + n).
  - rewrite add_assoc'. reflexivity.
  - rewrite add_comm. reflexivity.
Qed.
  
(** [] *)

(* ################################################################# *)
(** * Nat to Bin and Back to Nat *)

(** Recall the [bin] type we defined in [Basics]: *)

Inductive bin : Type :=
  | Z
  | B0 (n : bin)
  | B1 (n : bin)
.
(** Before you start working on the next exercise, replace the stub
    definitions of [incr] and [bin_to_nat], below, with your solution
    from [Basics].  That will make it possible for this file to
    be graded on its own. *)

Fixpoint incr (m:bin) : bin :=
  match m with
  | Z => B1 Z
  | B0 m' => B1 m'
  | B1 m' => B0 (incr m')
  end.

Fixpoint bin_to_nat (m:bin) : nat :=
  match m with
  | Z => 0
  | B0 m' => 2 * bin_to_nat m' 
  | B1 m' => 1 + (2 * bin_to_nat m')
  end.

(** In [Basics], we did some unit testing of [bin_to_nat], but we
    didn't prove its correctness. Now we'll do so. *)

(** **** Exercise: 3 stars, standard, especially useful (binary_commute)

    Prove that the following diagram commutes:

                            incr
              bin ----------------------> bin
               |                           |
    bin_to_nat |                           |  bin_to_nat
               |                           |
               v                           v
              nat ----------------------> nat
                             S

    That is, incrementing a binary number and then converting it to
    a (unary) natural number yields the same result as first converting
    it to a natural number and then incrementing.

    If you want to change your previous definitions of [incr] or [bin_to_nat]
    to make the property easier to prove, feel free to do so! *)

Theorem bin_to_nat_pres_incr : forall b : bin,
  bin_to_nat (incr b) = 1 + bin_to_nat b.
  Proof.
  intros. induction b as [| b' | b'' IHb'].
  - simpl. reflexivity.
  - simpl. reflexivity.
  - simpl. rewrite IHb'. simpl. rewrite add_0_r. 
    rewrite plus_n_Sm. rewrite plus_n_Sm. rewrite plus_n_Sm. 
    reflexivity.
Qed.

(** [] *)

(** **** Exercise: 3 stars, standard (nat_bin_nat) *)

(** Write a function to convert natural numbers to binary numbers. *)

Fixpoint nat_to_bin (n:nat) : bin :=
  match n with
  | 0 => Z
  | S n' => incr ( nat_to_bin n')
  end.

(** Prove that, if we start with any [nat], convert it to [bin], and
    convert it back, we get the same [nat] which we started with.

    Hint: This proof should go through smoothly using the previous
    exercise about [incr] as a lemma. If not, revisit your definitions
    of the functions involved and consider whether they are more
    complicated than necessary: the shape of a proof by induction will
    match the recursive structure of the program being verified, so
    make the recursions as simple as possible. *)

Theorem nat_bin_nat : forall n, bin_to_nat (nat_to_bin n) = n.
Proof.
  intros. induction n as [| n' IHn'].
  - simpl. reflexivity.
  - simpl. rewrite bin_to_nat_pres_incr. simpl. rewrite IHn'. reflexivity.
Qed.

(** [] *)

(* ################################################################# *)
(** * Bin to Nat and Back to Bin (Advanced) *)

(** The opposite direction -- starting with a [bin], converting to [nat],
    then converting back to [bin] -- turns out to be problematic. That
    is, the following theorem does not hold. *)

Theorem bin_nat_bin_fails : forall b, nat_to_bin (bin_to_nat b) = b.
Abort.

(** Let's explore why that theorem fails, and how to prove a modified
    version of it. We'll start with some lemmas that might seem
    unrelated, but will turn out to be relevant. *)

(** **** Exercise: 2 stars, advanced (double_bin) *)

(** Prove this lemma about [double], which we defined earlier in the
    chapter. *)

Lemma double_incr : forall n : nat, double (S n) = S (S (double n)).
Proof.
  intros. simpl. reflexivity.
Qed.

(** Now define a similar doubling function for [bin]. *)

Definition double_bin (b:bin) : bin :=
  match b with
  | Z => Z
  | b' => B0 b'
  end.

(** Check that your function correctly doubles zero. *)

Example double_bin_zero : double_bin Z = Z.
Proof.
  intros. simpl. reflexivity.
Qed.

(** Prove this lemma, which corresponds to [double_incr]. *)

Lemma double_incr_bin : forall b,
    double_bin (incr b) = incr (incr (double_bin b)).
Proof.
  intros.
  induction b as [| b' | b'' Ihb'].
  - simpl. reflexivity.
  - simpl. reflexivity.
  - simpl. reflexivity.
Qed. 

(** [] *)

(** Let's return to our desired theorem: *)

Theorem bin_nat_bin_fails : forall b, nat_to_bin (bin_to_nat b) = b.
Abort.

(** The theorem fails because there are some [bin] such that we won't
    necessarily get back to the _original_ [bin], but instead to an
    "equivalent" [bin].  (We deliberately leave that notion undefined
    here for you to think about.)

    Explain in a comment, below, why this failure occurs. Your
    explanation will not be graded, but it's important that you get it
    clear in your mind before going on to the next part. If you're
    stuck on this, think about alternative implementations of
    [double_bin] that might have failed to satisfy [double_bin_zero]
    yet otherwise seem correct. *)

(* FILL IN HERE *)

(* 
I think the Idea is that binary zero aka "Z" can be written with 
any number of trailing "B0" constructors and still be equivalent to zero.
Similarly, other numbers can have leading "B0"s and still be equivalent to their respective values.
*)

(** To solve that problem, we can introduce a _normalization_ function
    that selects the simplest [bin] out of all the equivalent
    [bin]. Then we can prove that the conversion from [bin] to [nat] and
    back again produces that normalized, simplest [bin]. *)

(** **** Exercise: 4 stars, advanced (bin_nat_bin) *)

(** Define [normalize]. You will need to keep its definition as simple
    as possible for later proofs to go smoothly. Do not use
    [bin_to_nat] or [nat_to_bin], but do use [double_bin].

    Hint: Structure the recursion such that it _always_ reaches the
    end of the [bin] and process each bit only once. Do not try to
    "look ahead" at future bits. *)

Fixpoint normalize (b:bin) : bin :=
  (* B0 Z *)
  match b with
  | Z => Z
  | B0 b' => double_bin (normalize b')
  | B1 b' => B1 (normalize b')
  end.

(** It would be wise to do some [Example] proofs to check that your definition of
    [normalize] works the way you intend before you proceed. They won't be graded,
    but fill them in below. *)

(* FILL IN HERE *)

(** Finally, prove the main theorem. The inductive cases could be a
    bit tricky.

    Hint: Start by trying to prove the main statement, see where you
    get stuck, and see if you can find a lemma -- perhaps requiring
    its own inductive proof -- that will allow the main proof to make
    progress. We have one lemma for the [B0] case (which also makes 
    use of [double_incr_bin]) and another for the [B1] case. *)

Lemma b0_case : forall n : nat, nat_to_bin (double n) = double_bin (nat_to_bin n).
Proof.
  intros.
  induction n as [| n' IHn'].
  - simpl. reflexivity.
  - simpl. rewrite double_incr_bin. rewrite <- IHn'. reflexivity.
Qed. 

Lemma b1_case : forall b : bin, incr (double_bin (b)) = B1 (b).
Proof.
  intros. 
  induction b as [| b' | b'' IHb'].
  - simpl. reflexivity.
  - simpl. reflexivity.
  - simpl. reflexivity.
Qed.  

Theorem bin_nat_bin : forall b, nat_to_bin (bin_to_nat b) = normalize b.
Proof.
  intros. 
  induction b as [| b' | b'' IHb'].
  - simpl. reflexivity.
  - simpl. rewrite add_0_r. rewrite <- double_plus. rewrite <- IHb'. rewrite b0_case. reflexivity.
  - simpl. rewrite add_0_r. rewrite <- double_plus. rewrite b0_case. rewrite IHb'. rewrite b1_case. reflexivity.
Qed.

(** [] *)

(* 2023-08-23 11:29 *)