Tactics.v

(** * Tactics: More Basic Tactics *)

(** This chapter introduces several additional proof strategies
    and tactics that allow us to begin proving more interesting
    properties of functional programs.

    We will see:
    - how to use auxiliary lemmas in both "forward-" and
      "backward-style" proofs;
    - how to reason about data constructors -- in particular, how to
      use the fact that they are injective and disjoint;
    - how to strengthen an induction hypothesis, and when such
      strengthening is required; and
    - more details on how to reason by case analysis. *)

Set Warnings "-notation-overridden,-parsing,-deprecated-hint-without-locality".
From LF Require Export Lists.
From LF Require Export Poly.

(* ################################################################# *)
(** * The [apply] Tactic *)

(** We often encounter situations where the goal to be proved is
    _exactly_ the same as some hypothesis in the context or some
    previously proved lemma. *)

Theorem silly1 : forall (n m : nat),
  n = m ->
  n = m.
Proof.
  intros n m eq.

(** Here, we could finish with "[rewrite -> eq.  reflexivity.]" as we
    have done several times before.  Or we can finish in a single step
    by using [apply]: *)

  apply eq.  Qed.

(** The [apply] tactic also works with _conditional_ hypotheses
    and lemmas: if the statement being applied is an implication, then
    the premises of this implication will be added to the list of
    subgoals needing to be proved.

    [apply] also works with _conditional_ hypotheses: *)

Theorem silly2 : forall (n m o p : nat),
  n = m ->
  (n = m -> [n;o] = [m;p]) ->
  [n;o] = [m;p].
Proof.
  intros n m o p eq1 eq2.
  apply eq2. apply eq1.  Qed.

(** Typically, when we use [apply H], the statement [H] will
    begin with a [forall] that introduces some _universally quantified
    variables_.

    When Coq matches the current goal against the conclusion of [H],
    it will try to find appropriate values for these variables.  For
    example, when we do [apply eq2] in the following proof, the
    universal variable [q] in [eq2] gets instantiated with [n], and
    [r] gets instantiated with [m]. *)

Theorem silly2a : forall (n m : nat),
  (n,n) = (m,m)  ->
  (forall (q r : nat), (q,q) = (r,r) -> [q] = [r]) ->
  [n] = [m].
Proof.
  intros n m eq1 eq2.
  apply eq2. apply eq1.  Qed.

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

    Complete the following proof using only [intros] and [apply]. *)
Theorem silly_ex : forall p,
  (forall n, even n = true -> even (S n) = false) ->
  (forall n, even n = false -> odd n = true) ->
  even p = true ->
  odd (S p) = true.
Proof.
    intros p H1 H2. intro H. apply H2. apply H1. apply H. Qed.
(** [] *)

(** To use the [apply] tactic, the (conclusion of the) fact
    being applied must match the goal exactly (perhaps after
    simplification) -- for example, [apply] will not work if the left
    and right sides of the equality are swapped. *)

Theorem silly3 : forall (n m : nat),
  n = m ->
  m = n.
Proof.
  intros n m H.

  (** Here we cannot use [apply] directly... *)

  Fail apply H.

  (** but we can use the [symmetry] tactic, which switches the left
      and right sides of an equality in the goal. *)

  symmetry. apply H.  Qed.

(** **** Exercise: 2 stars, standard (apply_exercise1)

    You can use [apply] with previously defined theorems, not
    just hypotheses in the context.  Use [Search] to find a
    previously-defined theorem about [rev] from [Lists].  Use
    that theorem as part of your (relatively short) solution to this
    exercise. You do not need [induction]. *)

(* Since I can't invoke it because list nat != natlist, I rewrote the theorem here *)
Lemma rev_injective : forall (l1 l2 : list nat),
  rev l1 = rev l2 -> l1 = l2.
Proof.
    intros l1 l2.
    intro H.
    rewrite <- rev_involutive.
    rewrite <- H.
    rewrite rev_involutive.
    reflexivity.
Qed.

Theorem rev_exercise1 : forall (l l' : list nat),
  l = rev l' ->
  l' = rev l.
Proof.
    intros l l'.
    intros H.
    apply rev_injective.
    rewrite rev_involutive. 
    symmetry. apply H.
    Qed.
(** [] *)

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

    Briefly explain the difference between the tactics [apply] and
    [rewrite].  What are the situations where both can usefully be
    applied? *)

(* 
Rewrite deal with equalities, but apply deals with conditionals. If we want to prove a=b,
and our current goal is b=a, and we have the hypothesis a=b and if a=b->b=a, 
then rewrite gets b=b, whereas applying the hypothesis we get an empty goal.
    [] *)

(* ################################################################# *)
(** * The [apply with] Tactic *)

(** The following silly example uses two rewrites in a row to
    get from [[a;b]] to [[e;f]]. *)

Example trans_eq_example : forall (a b c d e f : nat),
     [a;b] = [c;d] ->
     [c;d] = [e;f] ->
     [a;b] = [e;f].
Proof.
  intros a b c d e f eq1 eq2.
  rewrite -> eq1. rewrite -> eq2. reflexivity.  Qed.

(** Since this is a common pattern, we might like to pull it out as a
    lemma that records, once and for all, the fact that equality is
    transitive. *)

Theorem trans_eq : forall (X:Type) (n m o : X),
  n = m -> m = o -> n = o.
Proof.
  intros X n m o eq1 eq2. rewrite -> eq1. rewrite -> eq2.
  reflexivity.  Qed.

(** Now, we should be able to use [trans_eq] to prove the above
    example.  However, to do this we need a slight refinement of the
    [apply] tactic. *)

Example trans_eq_example' : forall (a b c d e f : nat),
     [a;b] = [c;d] ->
     [c;d] = [e;f] ->
     [a;b] = [e;f].
Proof.
  intros a b c d e f eq1 eq2.

(** If we simply tell Coq [apply trans_eq] at this point, it can
    tell (by matching the goal against the conclusion of the lemma)
    that it should instantiate [X] with [[nat]], [n] with [[a,b]], and
    [o] with [[e,f]].  However, the matching process doesn't determine
    an instantiation for [m]: we have to supply one explicitly by
    adding "[with (m:=[c,d])]" to the invocation of [apply]. *)

  apply trans_eq with (m:=[c;d]).
  apply eq1. apply eq2.   Qed.

(** Actually, the name [m] in the [with] clause is not required,
    since Coq is often smart enough to figure out which variable we
    are instantiating. We could instead simply write [apply trans_eq
    with [c;d]]. *)

(** Coq also has a built-in tactic [transitivity] that
    accomplishes the same purpose as applying [trans_eq]. The tactic
    requires us to state the instantiation we want, just like [apply
    with] does. *)

Example trans_eq_example'' : forall (a b c d e f : nat),
     [a;b] = [c;d] ->
     [c;d] = [e;f] ->
     [a;b] = [e;f].
Proof.
  intros a b c d e f eq1 eq2.
  transitivity [c;d].
  apply eq1. apply eq2.   Qed.

(** **** Exercise: 3 stars, standard, optional (trans_eq_exercise) *)
Example trans_eq_exercise : forall (n m o p : nat),
     m = (minustwo o) ->
     (n + p) = m ->
     (n + p) = (minustwo o).
Proof. intros n m o p H1 H2. transitivity m. apply H2. apply H1. Qed.
(** [] *)

(* ################################################################# *)
(** * The [injection] and [discriminate] Tactics *)

(** Recall the definition of natural numbers:

     Inductive nat : Type :=
       | O
       | S (n : nat).

    It is obvious from this definition that every number has one of
    two forms: either it is the constructor [O] or it is built by
    applying the constructor [S] to another number.  But there is more
    here than meets the eye: implicit in the definition are two
    additional facts:

    - The constructor [S] is _injective_ (or _one-to-one_).  That is,
      if [S n = S m], it must also be that [n = m].

    - The constructors [O] and [S] are _disjoint_.  That is, [O] is not
      equal to [S n] for any [n]. *)

(** Similar principles apply to every inductively defined type:
    all constructors are injective, and the values built from distinct
    constructors are never equal.  For lists, the [cons] constructor
    is injective and the empty list [nil] is different from every
    non-empty list.  For booleans, [true] and [false] are different.
    (Since [true] and [false] take no arguments, their injectivity is
    neither here nor there.)  And so on. *)

(** We can _prove_ the injectivity of [S] by using the [pred] function
    defined in [Basics.v]. *)

Theorem S_injective : forall (n m : nat),
  S n = S m ->
  n = m.
Proof.
  intros n m H1.
  assert (H2: n = pred (S n)). { reflexivity. }
  rewrite H2. rewrite H1. simpl. reflexivity.
Qed.

(** This technique can be generalized to any constructor by
    writing the equivalent of [pred] -- i.e., writing a function that
    "undoes" one application of the constructor.

    As a more convenient alternative, Coq provides a tactic called
    [injection] that allows us to exploit the injectivity of any
    constructor.  Here is an alternate proof of the above theorem
    using [injection]: *)

Theorem S_injective' : forall (n m : nat),
  S n = S m ->
  n = m.
Proof.
  intros n m H.

(** By writing [injection H as Hmn] at this point, we are asking Coq
    to generate all equations that it can infer from [H] using the
    injectivity of constructors (in the present example, the equation
    [n = m]). Each such equation is added as a hypothesis (called
    [Hmn] in this case) into the context. *)

  injection H as Hnm. apply Hnm.
Qed.

(** Here's a more interesting example that shows how [injection] can
    derive multiple equations at once. *)

Theorem injection_ex1 : forall (n m o : nat),
  [n;m] = [o;o] ->
  n = m.
Proof.
  intros n m o H.
  (* WORKED IN CLASS *)
  injection H as H1 H2.
  rewrite H1. rewrite H2. reflexivity.
Qed.

(** **** Exercise: 3 stars, standard (injection_ex3) *)
Example injection_ex3 : forall (X : Type) (x y z : X) (l j : list X),
  x :: y :: l = z :: j ->
  j = z :: l ->
  x = y.
Proof.
    intros X x y z l j. intros H1 H2.
    injection H1 as H H'. rewrite <- H' in H2. injection H2 as H3.
    transitivity z. apply H. symmetry. apply H3.
Qed.
(** [] *)

(** So much for injectivity of constructors.  What about disjointness? *)

(** The principle of disjointness says that two terms beginning
    with different constructors (like [O] and [S], or [true] and [false])
    can never be equal.  This means that, any time we find ourselves
    in a context where we've _assumed_ that two such terms are equal,
    we are justified in concluding anything we want, since the
    assumption is nonsensical. *)

(** The [discriminate] tactic embodies this principle: It is used
    on a hypothesis involving an equality between different
    constructors (e.g., [false = true]), and it solves the current
    goal immediately.  Some examples: *)

Theorem discriminate_ex1 : forall (n m : nat),
  false = true ->
  n = m.
Proof.
  intros n m contra. discriminate contra. Qed.

Theorem discriminate_ex2 : forall (n : nat),
  S n = O ->
  2 + 2 = 5.
Proof.
  intros n contra. discriminate contra. Qed.

(** These examples are instances of a logical principle known as the
    _principle of explosion_, which asserts that a contradictory
    hypothesis entails anything (even manifestly false things!). *)

(** If you find the principle of explosion confusing, remember
    that these proofs are _not_ showing that the conclusion of the
    statement holds.  Rather, they are showing that, _if_ the
    nonsensical situation described by the premise did somehow hold,
    _then_ the nonsensical conclusion would also follow, because we'd
    be living in an inconsistent universe where every statement is
    true.

    We'll explore the principle of explosion in more detail in the
    next chapter. *)

(** **** Exercise: 1 star, standard (discriminate_ex3) *)
Example discriminate_ex3 :
  forall (X : Type) (x y z : X) (l j : list X),
    x :: y :: l = [] ->
    x = z.
Proof. intros X x y z l j H. discriminate H. Qed.
(** [] *)

(** For a more useful example, we can use [discriminate] to make a
    connection between the two different notions of equality ([=] and
    [=?]) that we have seen for natural numbers. *)
Theorem eqb_0_l : forall n,
   0 =? n = true -> n = 0.
Proof.
  intros n.

(** We can proceed by case analysis on [n]. The first case is
    trivial. *)

  destruct n as [| n'] eqn:E.
  - (* n = 0 *)
    intros H. reflexivity.

(** However, the second one doesn't look so simple: assuming [0
    =? (S n') = true], we must show [S n' = 0]!  The way forward is to
    observe that the assumption itself is nonsensical: *)

  - (* n = S n' *)
    simpl.

(** If we use [discriminate] on this hypothesis, Coq confirms
    that the subgoal we are working on is impossible and removes it
    from further consideration. *)

    intros H. discriminate H.
Qed.

(** The injectivity of constructors allows us to reason that
    [forall (n m : nat), S n = S m -> n = m].  The converse of this
    implication is an instance of a more general fact about both
    constructors and functions, which we will find convenient
    below: *)

Theorem f_equal : forall (A B : Type) (f: A -> B) (x y: A),
  x = y -> f x = f y.
Proof. intros A B f x y eq. rewrite eq.  reflexivity.  Qed.

Theorem eq_implies_succ_equal : forall (n m : nat),
  n = m -> S n = S m.
Proof. intros n m H. apply f_equal. apply H. Qed.

(** Indeed, there is also a tactic named `f_equal` that can
    prove such theorems directly.  Given a goal of the form [f a1
    ... an = g b1 ... bn], the tactic [f_equal] will produce subgoals
    of the form [f = g], [a1 = b1], ..., [an = bn]. At the same time,
    any of these subgoals that are simple enough (e.g., immediately
    provable by [reflexivity]) will be automatically discharged by
    [f_equal]. *)

Theorem eq_implies_succ_equal' : forall (n m : nat),
  n = m -> S n = S m.
Proof. intros n m H. f_equal. apply H. Qed.

(* ################################################################# *)
(** * Using Tactics on Hypotheses *)

(** By default, most tactics work on the goal formula and leave
    the context unchanged.  However, most tactics also have a variant
    that performs a similar operation on a statement in the context.

    For example, the tactic "[simpl in H]" performs simplification on
    the hypothesis [H] in the context. *)

Theorem S_inj : forall (n m : nat) (b : bool),
  ((S n) =? (S m)) = b  ->
  (n =? m) = b.
Proof.
  intros n m b H. simpl in H. apply H.  Qed.

(** Similarly, [apply L in H] matches some conditional statement
    [L] (of the form [X -> Y], say) against a hypothesis [H] in the
    context.  However, unlike ordinary [apply] (which rewrites a goal
    matching [Y] into a subgoal [X]), [apply L in H] matches [H]
    against [X] and, if successful, replaces it with [Y].

    In other words, [apply L in H] gives us a form of "forward
    reasoning": given [X -> Y] and a hypothesis matching [X], it
    produces a hypothesis matching [Y].

    By contrast, [apply L] is "backward reasoning": it says that if we
    know [X -> Y] and we are trying to prove [Y], it suffices to prove
    [X].

    Here is a variant of a proof from above, using forward reasoning
    throughout instead of backward reasoning. *)

Theorem silly4 : forall (n m p q : nat),
  (n = m -> p = q) ->
  m = n ->
  q = p.
Proof.
  intros n m p q EQ H.
  symmetry in H. apply EQ in H. symmetry in H.
  apply H.  Qed.

(** Forward reasoning starts from what is _given_ (premises,
    previously proven theorems) and iteratively draws conclusions from
    them until the goal is reached.  Backward reasoning starts from
    the _goal_ and iteratively reasons about what would imply the
    goal, until premises or previously proven theorems are reached.

    The informal proofs seen in math or computer science classes tend
    to use forward reasoning.  By contrast, idiomatic use of Coq
    generally favors backward reasoning, though in some situations the
    forward style can be easier to think about. *)

(* ################################################################# *)
(** * Specializing Hypotheses *)

(** Another handy tactic for fiddling with hypotheses is [specialize].
    It is essentially just a combination of [assert] and [apply], but
    it often provides a pleasingly smooth way to nail down overly
    general assumptions.  It works like this:

    If [H] is a quantified hypothesis in the current context -- i.e.,
    [H : forall (x:T), P] -- then [specialize H with (x := e)] will
    change [H] so that it looks like [[x:=e]P], that is, [P] with [x]
    replaced by [e].

    For example: *)

Theorem specialize_example: forall n,
     (forall m, m*n = 0)
  -> n = 0.
Proof.
  intros n H.
  specialize H with (m := 1).
  simpl in H.
  rewrite add_comm in H.
  simpl in H.
  apply H. Qed.

(** Using [specialize] before [apply] gives us yet another way to
    control where [apply] does its work. *)
Example trans_eq_example''' : forall (a b c d e f : nat),
     [a;b] = [c;d] ->
     [c;d] = [e;f] ->
     [a;b] = [e;f].
Proof.
  intros a b c d e f eq1 eq2.
  specialize trans_eq with (m:=[c;d]) as H.
  apply H.
  apply eq1.
  apply eq2. Qed.
(** Note:
    - We can [specialize] facts in the global context, not just
      local hypotheses.
    - The [as...] clause at the end tells [specialize] how to name
      the new hypothesis in this case. *)

(* ################################################################# *)
(** * Varying the Induction Hypothesis *)

(** Sometimes it is important to control the exact form of the
    induction hypothesis when carrying out inductive proofs in Coq.
    In particular, we may need to be careful about which of the
    assumptions we move (using [intros]) from the goal to the context
    before invoking the [induction] tactic.

    For example, suppose we want to show that [double] is injective --
    i.e., that it maps different arguments to different results:

       Theorem double_injective: forall n m,
         double n = double m ->
         n = m.

    The way we start this proof is a bit delicate: if we begin it with

       intros n. induction n.

    then all is well.  But if we begin it with introducing both
    variables

       intros n m. induction n.

    we get stuck in the middle of the inductive case... *)

Theorem double_injective_FAILED : forall n m,
  double n = double m ->
  n = m.
Proof.
  intros n m. induction n as [| n' IHn'].
  - (* n = O *) simpl. intros eq. destruct m as [| m'] eqn:E.
    + (* m = O *) reflexivity.
    + (* m = S m' *) discriminate eq.
  - (* n = S n' *) intros eq. destruct m as [| m'] eqn:E.
    + (* m = O *) discriminate eq.
    + (* m = S m' *) f_equal.

(** At this point, the induction hypothesis ([IHn']) does _not_ give us
    [n' = m'] -- there is an extra [S] in the way -- so the goal is
    not provable. *)

Abort.

(** What went wrong? *)

(** The problem is that, at the point where we invoke the
    induction hypothesis, we have already introduced [m] into the
    context -- intuitively, we have told Coq, "Let's consider some
    particular [n] and [m]..." and we now have to prove that, if
    [double n = double m] for _these particular_ [n] and [m], then
    [n = m].

    The next tactic, [induction n] says to Coq: We are going to show
    the goal by induction on [n].  That is, we are going to prove, for
    _all_ [n], that the proposition

      - [P n] = "if [double n = double m], then [n = m]"

    holds, by showing

      - [P O]

         (i.e., "if [double O = double m] then [O = m]") and

      - [P n -> P (S n)]

        (i.e., "if [double n = double m] then [n = m]" implies "if
        [double (S n) = double m] then [S n = m]").

    If we look closely at the second statement, it is saying something
    rather strange: that, for a _particular_ [m], if we know

      - "if [double n = double m] then [n = m]"

    then we can prove

       - "if [double (S n) = double m] then [S n = m]".

    To see why this is strange, let's think of a particular [m] --
    say, [5].  The statement is then saying that, if we know

      - [Q] = "if [double n = 10] then [n = 5]"

    then we can prove

      - [R] = "if [double (S n) = 10] then [S n = 5]".

    But knowing [Q] doesn't give us any help at all with proving [R]!
    If we tried to prove [R] from [Q], we would start with something
    like "Suppose [double (S n) = 10]..." but then we'd be stuck:
    knowing that [double (S n)] is [10] tells us nothing helpful about
    whether [double n] is [10] (indeed, it strongly suggests that
    [double n] is _not_ [10]!!), so [Q] is useless. *)

(** Trying to carry out this proof by induction on [n] when [m] is
    already in the context doesn't work because we are then trying to
    prove a statement involving _every_ [n] but just a _particular_
    [m]. *)

(** A successful proof of [double_injective] leaves [m] universally
    quantified in the goal statement at the point where the
    [induction] tactic is invoked on [n]: *)

Theorem double_injective : forall n m,
  double n = double m ->
  n = m.
Proof.
  intros n. induction n as [| n' IHn'].
  - (* n = O *) simpl. intros m eq. destruct m as [| m'] eqn:E.
    + (* m = O *) reflexivity.
    + (* m = S m' *) discriminate eq.

  - (* n = S n' *)

(** Notice that both the goal and the induction hypothesis are
    different this time: the goal asks us to prove something more
    general (i.e., we must prove the statement for _every_ [m]), but
    the IH is correspondingly more flexible, allowing us to choose any
    [m] we like when we apply the IH. *)

    intros m eq.

(** Now we've chosen a particular [m] and introduced the assumption
    that [double n = double m].  Since we are doing a case analysis on
    [n], we also need a case analysis on [m] to keep the two "in sync." *)

    destruct m as [| m'] eqn:E.
    + (* m = O *)

(** The 0 case is trivial: *)

    discriminate eq.
    + (* m = S m' *)
      f_equal.

(** Since we are now in the second branch of the [destruct m], the
    [m'] mentioned in the context is the predecessor of the [m] we
    started out talking about.  Since we are also in the [S] branch of
    the induction, this is perfect: if we instantiate the generic [m]
    in the IH with the current [m'] (this instantiation is performed
    automatically by the [apply] in the next step), then [IHn'] gives
    us exactly what we need to finish the proof. *)

      apply IHn'. simpl in eq. injection eq as goal. apply goal. Qed.

(** The thing to take away from all this is that you need to be
    careful, when using induction, that you are not trying to prove
    something too specific: When proving a property quantified over
    variables [n] and [m] by induction on [n], it is sometimes crucial
    to leave [m] generic. *)

(** The following exercise, which further strengthens the link between
    [=?] and [=], follows the same pattern. *)
(** **** Exercise: 2 stars, standard (eqb_true) *)
Theorem eqb_true : forall n m,
  n =? m = true -> n = m.
Proof.
    induction n as [|n' IH].
    - destruct m as [|m'].
        + reflexivity.
        + simpl. discriminate.
    - destruct m as [|m'].
        + simpl. discriminate.
        + intro H. f_equal. apply IH. simpl in H. apply H. 
Qed.
(** [] *)

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

    Give a careful informal proof of [eqb_true], stating the induction
    hypothesis explicitly and being as explicit as possible about
    quantifiers, everywhere. *)

(* We induct on n.
If n = 0, then we prove "forall m, 0 =? m = true -> 0 = m" by cases:
    If m = 0, then 0 =? 0 and 0 = 0
    If m = S m' for some m', then it is not the case that 0 =? S m', so the conditional is true
If n = S n' for some n', then the inductive hypothesis is "forall k, n' =? k = true -> n' = k".
We split m into two cases:
    If m = 0, then it is not the case that S n' =? 0, so the conditional is trivially true
    If m = S m', then we want to prove S n' =? S m' = true -> S n' = S m'
    Assume S n' =? S m' = true.
    By injectivity of S, S n' = S m' if n' = m'.
    To prove that, we try to prove n' =? m' = true (from the IH where k = m').
    If we simplify our asumption,S n' =? S m' = true just in case that n' =? m' = true.
    So we have proved that n' =? m' = true. *)

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

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

    In addition to being careful about how you use [intros], practice
    using "in" variants in this proof.  (Hint: use [plus_n_Sm].) *)
Theorem plus_n_n_injective : forall n m,
  n + n = m + m ->
  n = m.
Proof.
    induction n as [|n' IH].
    - simpl. destruct m as [|m'].
        + reflexivity.
        + simpl. discriminate.
    - simpl. destruct m as [|m'].
        + simpl. discriminate.
        + intro H. simpl in H.
          rewrite <- plus_n_Sm in H. rewrite <- plus_n_Sm in H.
          f_equal. apply IH. injection H as H. rewrite H. reflexivity.
Qed.
(** [] *)

(** The strategy of doing fewer [intros] before an [induction] to
    obtain a more general IH doesn't always work; sometimes some
    _rearrangement_ of quantified variables is needed.  Suppose, for
    example, that we wanted to prove [double_injective] by induction
    on [m] instead of [n]. *)

Theorem double_injective_take2_FAILED : forall n m,
  double n = double m ->
  n = m.
Proof.
  intros n m. induction m as [| m' IHm'].
  - (* m = O *) simpl. intros eq. destruct n as [| n'] eqn:E.
    + (* n = O *) reflexivity.
    + (* n = S n' *) discriminate eq.
  - (* m = S m' *) intros eq. destruct n as [| n'] eqn:E.
    + (* n = O *) discriminate eq.
    + (* n = S n' *) f_equal.
        (* We are stuck here, just like before. *)
Abort.

(** The problem is that, to do induction on [m], we must first
    introduce [n].  (If we simply say [induction m] without
    introducing anything first, Coq will automatically introduce [n]
    for us!)  *)

(** What can we do about this?  One possibility is to rewrite the
    statement of the lemma so that [m] is quantified before [n].  This
    works, but it's not nice: We don't want to have to twist the
    statements of lemmas to fit the needs of a particular strategy for
    proving them!  Rather we want to state them in the clearest and
    most natural way. *)

(** What we can do instead is to first introduce all the quantified
    variables and then _re-generalize_ one or more of them,
    selectively taking variables out of the context and putting them
    back at the beginning of the goal.  The [generalize dependent]
    tactic does this. *)

Theorem double_injective_take2 : forall n m,
  double n = double m ->
  n = m.
Proof.
  intros n m.
  (* [n] and [m] are both in the context *)
  generalize dependent n.
  (* Now [n] is back in the goal and we can do induction on
     [m] and get a sufficiently general IH. *)
  induction m as [| m' IHm'].
  - (* m = O *) simpl. intros n eq. destruct n as [| n'] eqn:E.
    + (* n = O *) reflexivity.
    + (* n = S n' *) discriminate eq.
  - (* m = S m' *) intros n eq. destruct n as [| n'] eqn:E.
    + (* n = O *) discriminate eq.
    + (* n = S n' *) f_equal.
      apply IHm'. injection eq as goal. apply goal. Qed.

(** Let's look at an informal proof of this theorem.  Note that
    the proposition we prove by induction leaves [n] quantified,
    corresponding to the use of generalize dependent in our formal
    proof.

    _Theorem_: For any nats [n] and [m], if [double n = double m], then
      [n = m].

    _Proof_: Let [m] be a [nat]. We prove by induction on [m] that, for
      any [n], if [double n = double m] then [n = m].

      - First, suppose [m = 0], and suppose [n] is a number such
        that [double n = double m].  We must show that [n = 0].

        Since [m = 0], by the definition of [double] we have [double n =
        0].  There are two cases to consider for [n].  If [n = 0] we are
        done, since [m = 0 = n], as required.  Otherwise, if [n = S n']
        for some [n'], we derive a contradiction: by the definition of
        [double], we can calculate [double n = S (S (double n'))], but
        this contradicts the assumption that [double n = 0].

      - Second, suppose [m = S m'] and that [n] is again a number such
        that [double n = double m].  We must show that [n = S m'], with
        the induction hypothesis that for every number [s], if [double s =
        double m'] then [s = m'].

        By the fact that [m = S m'] and the definition of [double], we
        have [double n = S (S (double m'))].  There are two cases to
        consider for [n].

        If [n = 0], then by definition [double n = 0], a contradiction.

        Thus, we may assume that [n = S n'] for some [n'], and again by
        the definition of [double] we have [S (S (double n')) =
        S (S (double m'))], which implies by injectivity that [double n' =
        double m'].  Instantiating the induction hypothesis with [n'] thus
        allows us to conclude that [n' = m'], and it follows immediately
        that [S n' = S m'].  Since [S n' = n] and [S m' = m], this is just
        what we wanted to show. [] *)

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

    Prove this by induction on [l]. *)

Theorem nth_error_after_last: forall (n : nat) (X : Type) (l : list X),
  length l = n ->
  nth_error l n = None.
Proof.
    intros n X l. generalize dependent n. 
    induction l as [|h t IH].
    - reflexivity.
    - destruct n as [|n'].
        + simpl. discriminate.
        + simpl. intro H. apply IH. injection H as H. apply H.
Qed.
(** [] *)

(* ################################################################# *)
(** * Unfolding Definitions *)

(** It sometimes happens that we need to manually unfold a name that
    has been introduced by a [Definition] so that we can manipulate
    the expression it stands for.

    For example, if we define... *)

Definition square n := n * n.

(** ...and try to prove a simple fact about [square]... *)

Lemma square_mult : forall n m, square (n * m) = square n * square m.
Proof.
  intros n m.
  simpl.

(** ...we appear to be stuck: [simpl] doesn't simplify anything, and
    since we haven't proved any other facts about [square], there is
    nothing we can [apply] or [rewrite] with. *)

(** To make progress, we can manually [unfold] the definition of
    [square]: *)

  unfold square.

(** Now we have plenty to work with: both sides of the equality are
    expressions involving multiplication, and we have lots of facts
    about multiplication at our disposal.  In particular, we know that
    it is commutative and associative, and from these it is not hard
    to finish the proof. *)

  rewrite mult_assoc.
  assert (H : n * m * n = n * n * m).
    { rewrite mul_comm. apply mult_assoc. }
  rewrite H. rewrite mult_assoc. reflexivity.
Qed.

(** At this point, a bit deeper discussion of unfolding and
    simplification is in order.

    We already have observed that tactics like [simpl], [reflexivity],
    and [apply] will often unfold the definitions of functions
    automatically when this allows them to make progress.  For
    example, if we define [foo m] to be the constant [5]... *)

Definition foo (x: nat) := 5.

(** .... then the [simpl] in the following proof (or the
    [reflexivity], if we omit the [simpl]) will unfold [foo m] to
    [(fun x => 5) m] and further simplify this expression to just
    [5]. *)

Fact silly_fact_1 : forall m, foo m + 1 = foo (m + 1) + 1.
Proof.
  intros m.
  simpl.
  reflexivity.
Qed.

(** But this automatic unfolding is somewhat conservative.  For
    example, if we define a slightly more complicated function
    involving a pattern match... *)

Definition bar x :=
  match x with
  | O => 5
  | S _ => 5
  end.

(** ...then the analogous proof will get stuck: *)

Fact silly_fact_2_FAILED : forall m, bar m + 1 = bar (m + 1) + 1.
Proof.
  intros m.
  simpl. (* Does nothing! *)
Abort.

(** The reason that [simpl] doesn't make progress here is that it
    notices that, after tentatively unfolding [bar m], it is left with
    a match whose scrutinee, [m], is a variable, so the [match] cannot
    be simplified further.  It is not smart enough to notice that the
    two branches of the [match] are identical, so it gives up on
    unfolding [bar m] and leaves it alone.

    Similarly, tentatively unfolding [bar (m+1)] leaves a [match]
    whose scrutinee is a function application (that cannot itself be
    simplified, even after unfolding the definition of [+]), so
    [simpl] leaves it alone. *)

(** At this point, there are two ways to make progress.  One is to use
    [destruct m] to break the proof into two cases, each focusing on a
    more concrete choice of [m] ([O] vs [S _]).  In each case, the
    [match] inside of [bar] can now make progress, and the proof is
    easy to complete. *)

Fact silly_fact_2 : forall m, bar m + 1 = bar (m + 1) + 1.
Proof.
  intros m.
  destruct m eqn:E.
  - simpl. reflexivity.
  - simpl. reflexivity.
Qed.

(** This approach works, but it depends on our recognizing that the
    [match] hidden inside [bar] is what was preventing us from making
    progress. *)

(** A more straightforward way forward is to explicitly tell Coq to
    unfold [bar]. *)

Fact silly_fact_2' : forall m, bar m + 1 = bar (m + 1) + 1.
Proof.
  intros m.
  unfold bar.

(** Now it is apparent that we are stuck on the [match] expressions on
    both sides of the [=], and we can use [destruct] to finish the
    proof without thinking so hard. *)

  destruct m eqn:E.
  - reflexivity.
  - reflexivity.
Qed.

(* ################################################################# *)
(** * Using [destruct] on Compound Expressions *)

(** We have seen many examples where [destruct] is used to
    perform case analysis of the value of some variable.  Sometimes we
    need to reason by cases on the result of some _expression_.  We
    can also do this with [destruct].

    Here are some examples: *)

Definition sillyfun (n : nat) : bool :=
  if n =? 3 then false
  else if n =? 5 then false
  else false.

Theorem sillyfun_false : forall (n : nat),
  sillyfun n = false.
Proof.
  intros n. unfold sillyfun.
  destruct (n =? 3) eqn:E1.
    - (* n =? 3 = true *) reflexivity.
    - (* n =? 3 = false *) destruct (n =? 5) eqn:E2.
      + (* n =? 5 = true *) reflexivity.
      + (* n =? 5 = false *) reflexivity.  Qed.

(** After unfolding [sillyfun] in the above proof, we find that
    we are stuck on [if (n =? 3) then ... else ...].  But either
    [n] is equal to [3] or it isn't, so we can use [destruct (eqb
    n 3)] to let us reason about the two cases.

    In general, the [destruct] tactic can be used to perform case
    analysis of the results of arbitrary computations.  If [e] is an
    expression whose type is some inductively defined type [T], then,
    for each constructor [c] of [T], [destruct e] generates a subgoal
    in which all occurrences of [e] (in the goal and in the context)
    are replaced by [c]. *)

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

    Here is an implementation of the [split] function mentioned in
    chapter [Poly]: *)

Fixpoint split {X Y : Type} (l : list (X*Y))
               : (list X) * (list Y) :=
  match l with
  | [] => ([], [])
  | (x, y) :: t =>
      match split t with
      | (lx, ly) => (x :: lx, y :: ly)
      end
  end.

(** Prove that [split] and [combine] are inverses in the following
    sense: *)
Theorem combine_split : forall X Y (l : list (X * Y)) l1 l2,
  split l = (l1, l2) ->
  combine l1 l2 = l.
Proof.
    intros X Y l.
    induction l as [|[x y] t IH].
    - simpl. intros l1 l2 H. injection H as H1 H2.
        rewrite <- H1. reflexivity.
    - simpl. destruct (split t) as [tx ty].
    intros l1 l2 H. injection H as H1 H2. 
    rewrite <- H1. rewrite <- H2. simpl. f_equal.
    apply IH. reflexivity.
Qed.
(** [] *)

(** The [eqn:] part of the [destruct] tactic is optional; although
    we've chosen to include it most of the time, for the sake of
    documentation, it can often be omitted without harm.

    However, when [destruct]ing compound expressions, the information
    recorded by the [eqn:] can actually be critical: if we leave it
    out, then [destruct] can erase information we need to complete a
    proof.

    For example, suppose we define a function [sillyfun1] like
    this: *)

Definition sillyfun1 (n : nat) : bool :=
  if n =? 3 then true
  else if n =? 5 then true
  else false.

(** Now suppose that we want to convince Coq that [sillyfun1 n]
    yields [true] only when [n] is odd.  If we start the proof like
    this (with no [eqn:] on the [destruct])... *)

Theorem sillyfun1_odd_FAILED : forall (n : nat),
  sillyfun1 n = true ->
  odd n = true.
Proof.
  intros n eq. unfold sillyfun1 in eq.
  destruct (n =? 3).
  (* stuck... *)
Abort.

(** ... then we are stuck at this point because the context does
    not contain enough information to prove the goal!  The problem is
    that the substitution performed by [destruct] is quite brutal --
    in this case, it throws away every occurrence of [n =? 3], but we
    need to keep some memory of this expression and how it was
    destructed, because we need to be able to reason that, since we
    are assuming [n =? 3 = true] in this branch of the case analysis,
    it must be that [n = 3], from which it follows that [n] is odd.

    What we want here is to substitute away all existing occurrences
    of [n =? 3], but at the same time add an equation to the context
    that records which case we are in.  This is precisely what the
    [eqn:] qualifier does. *)

Theorem sillyfun1_odd : forall (n : nat),
  sillyfun1 n = true ->
  odd n = true.
Proof.
  intros n eq. unfold sillyfun1 in eq.
  destruct (n =? 3) eqn:Heqe3.
  (** Now we have the same state as at the point where we got
      stuck above, except that the context contains an extra
      equality assumption, which is exactly what we need to
      make progress. *)
    - (* e3 = true *) apply eqb_true in Heqe3.
      rewrite -> Heqe3. reflexivity.
    - (* e3 = false *)
     (** When we come to the second equality test in the body
         of the function we are reasoning about, we can use
         [eqn:] again in the same way, allowing us to finish the
         proof. *)
      destruct (n =? 5) eqn:Heqe5.
        + (* e5 = true *)
          apply eqb_true in Heqe5.
          rewrite -> Heqe5. reflexivity.
        + (* e5 = false *) discriminate eq.  Qed.

(** **** Exercise: 2 stars, standard (destruct_eqn_practice) *)
Theorem bool_fn_applied_thrice :
  forall (f : bool -> bool) (b : bool),
  f (f (f b)) = f b.
Proof.
    intros f b.
    destruct b eqn:B.
    - destruct (f b) eqn:FB.
        + rewrite B in FB. rewrite FB. rewrite FB. rewrite FB. reflexivity.
        + rewrite B in FB. rewrite FB. destruct (f false) eqn:FF.
            * rewrite FB. reflexivity.
            * rewrite FF. reflexivity.
    - destruct (f b) eqn:FB.
        + rewrite B in FB. rewrite FB.  destruct (f true) eqn: FT.
            * rewrite FT. reflexivity.
            * rewrite FB. reflexivity.
        + rewrite B in FB. rewrite FB. rewrite FB. rewrite FB. reflexivity.
Qed.
(** [] *)

(* ################################################################# *)
(** * Review *)

(** We've now seen many of Coq's most fundamental tactics.  We'll
    introduce a few more in the coming chapters, and later on we'll
    see some more powerful _automation_ tactics that make Coq help us
    with low-level details.  But basically we've got what we need to
    get work done.

    Here are the ones we've seen:

      - [intros]: move hypotheses/variables from goal to context

      - [reflexivity]: finish the proof (when the goal looks like [e =
        e])

      - [apply]: prove goal using a hypothesis, lemma, or constructor

      - [apply... in H]: apply a hypothesis, lemma, or constructor to
        a hypothesis in the context (forward reasoning)

      - [apply... with...]: explicitly specify values for variables
        that cannot be determined by pattern matching

      - [simpl]: simplify computations in the goal

      - [simpl in H]: ... or a hypothesis

      - [rewrite]: use an equality hypothesis (or lemma) to rewrite
        the goal

      - [rewrite ... in H]: ... or a hypothesis

      - [symmetry]: changes a goal of the form [t=u] into [u=t]

      - [symmetry in H]: changes a hypothesis of the form [t=u] into
        [u=t]

      - [transitivity y]: prove a goal [x=z] by proving two new subgoals,
        [x=y] and [y=z]

      - [unfold]: replace a defined constant by its right-hand side in
        the goal

      - [unfold... in H]: ... or a hypothesis

      - [destruct... as...]: case analysis on values of inductively
        defined types

      - [destruct... eqn:...]: specify the name of an equation to be
        added to the context, recording the result of the case
        analysis

      - [induction... as...]: induction on values of inductively
        defined types

      - [injection... as...]: reason by injectivity on equalities
        between values of inductively defined types

      - [discriminate]: reason by disjointness of constructors on
        equalities between values of inductively defined types

      - [assert (H: e)] (or [assert (e) as H]): introduce a "local
        lemma" [e] and call it [H]

      - [generalize dependent x]: move the variable [x] (and anything
        else that depends on it) from the context back to an explicit
        hypothesis in the goal formula

      - [f_equal]: change a goal of the form [f x = f y] into [x = y] *)

(* ################################################################# *)
(** * Additional Exercises *)

(** **** Exercise: 3 stars, standard (eqb_sym) *)
Theorem eqb_sym : forall (n m : nat),
  (n =? m) = (m =? n).
Proof.
    induction n as [|n' IH].
    - destruct m as [|m'].
        * reflexivity.
        * reflexivity.
    - destruct m as [|m'].
        * reflexivity.
        * simpl. apply IH.
Qed.
(** [] *)

(** **** Exercise: 3 stars, advanced, optional (eqb_sym_informal)

    Give an informal proof of this lemma that corresponds to your
    formal proof above:

   Theorem: For any [nat]s [n] [m], [(n =? m) = (m =? n)].

   Proof: *)
   (* 
    We induct on n.
    Suppose n = 0. Then either m = 0 or m = S m'.
        If m = 0, then (0 =? 0) = (0 =? 0).
        If m = S m', then (0 =? S m') = false = (S m' = 0).
    Suppose n = S n'. We prove by the same cases:
        If m = 0, then (S n' =? 0) = false  = (S m' = 0).
        If m = S m', then (S n' =? S m') = (S m' =? S n') if (n' =? m') = (m' =? n'), which we have by the IH.
    [] *)

(** **** Exercise: 3 stars, standard, optional (eqb_trans) *)
Theorem eqb_trans : forall n m p,
  n =? m = true ->
  m =? p = true ->
  n =? p = true.
Proof.
    intros n m p.
    intros H1 H2.
    apply eqb_true in H1.
    apply eqb_true in H2.
    rewrite H1. rewrite H2.
    rewrite eqb_refl.
    reflexivity. 
Qed.
(** [] *)

(** **** Exercise: 3 stars, advanced (split_combine)

    We proved, in an exercise above, that [combine] is the inverse of
    [split].  Complete the definition of [split_combine_statement]
    below with a property that states that [split] is the inverse of
    [combine]. Then, prove that the property holds.

    Hint: Take a look at the definition of [combine] in [Poly].
    Your property will need to account for the behavior of [combine]
    in its base cases, which possibly drop some list elements. *)

Definition split_combine_statement : Prop :=
    forall X Y (l1 : list X) (l2 : list Y),
    length l1 = length l2 ->
    split (combine l1 l2) = (l1, l2).
  (* ("[: Prop]" means that we are giving a name to a
     logical proposition here.) *)

Theorem split_combine : split_combine_statement.
Proof.
    intros X Y l1 l2.
    generalize dependent l2.
    induction l1 as [|h1 t1 IH].
    (* induction l1 as [|h1 t1 IH]. *)
    - destruct l2.
        + reflexivity.
        + intro H. simpl in H. discriminate H.
    - intros l2 H. destruct l2.
        + discriminate H.
        + injection H as H. apply IH in H.
          simpl. rewrite H. reflexivity.
Qed.

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

(** **** Exercise: 3 stars, advanced (filter_exercise) *)
Theorem filter_exercise : forall (X : Type) (test : X -> bool)
                                 (x : X) (l lf : list X),
  filter test l = x :: lf ->
  test x = true.
Proof.
    intros X test x l lf.
    intro H.
    induction l as [|h t IH].
    - simpl in H. discriminate H.
    - simpl in H. destruct (test h) eqn:T.
        + injection H as H1 H2. rewrite <- H1. apply T.
        + apply IH. apply H.
Qed.
(** [] *)

(** **** Exercise: 4 stars, advanced, especially useful (forall_exists_challenge)

    Define two recursive [Fixpoints], [forallb] and [existsb].  The
    first checks whether every element in a list satisfies a given
    predicate:

      forallb odd [1;3;5;7;9] = true
      forallb negb [false;false] = true
      forallb even [0;2;4;5] = false
      forallb (eqb 5) [] = true

    The second checks whether there exists an element in the list that
    satisfies a given predicate:

      existsb (eqb 5) [0;2;3;6] = false
      existsb (andb true) [true;true;false] = true
      existsb odd [1;0;0;0;0;3] = true
      existsb even [] = false

    Next, define a _nonrecursive_ version of [existsb] -- call it
    [existsb'] -- using [forallb] and [negb].

    Finally, prove a theorem [existsb_existsb'] stating that
    [existsb'] and [existsb] have the same behavior.
*)

Fixpoint forallb {X : Type} (test : X -> bool) (l : list X) : bool :=
    match l with
    | [] => true
    | h::t => if test h then (forallb test t) else false
    end.

Example test_forallb_1 : forallb odd [1;3;5;7;9] = true.
Proof. reflexivity. Qed.

Example test_forallb_2 : forallb negb [false;false] = true.
Proof. reflexivity. Qed.

Example test_forallb_3 : forallb even [0;2;4;5] = false.
Proof. reflexivity. Qed.

Example test_forallb_4 : forallb (eqb 5) [] = true.
Proof. reflexivity. Qed.

Fixpoint existsb {X : Type} (test : X -> bool) (l : list X) : bool :=
    match l with
    | [] => false
    | h::t => if test h then true else existsb test t
    end.

Example test_existsb_1 : existsb (eqb 5) [0;2;3;6] = false.
Proof. reflexivity. Qed.

Example test_existsb_2 : existsb (andb true) [true;true;false] = true.
Proof. reflexivity. Qed.

Example test_existsb_3 : existsb odd [1;0;0;0;0;3] = true.
Proof. reflexivity. Qed.

Example test_existsb_4 : existsb even [] = false.
Proof. reflexivity. Qed.

Definition existsb' {X : Type} (test : X -> bool) (l : list X) : bool :=
    negb (forallb (fun x => negb (test x)) l).

Theorem existsb_existsb' : forall (X : Type) (test : X -> bool) (l : list X),
  existsb test l = existsb' test l.
Proof. 
    intros X test l.
    induction l as [|h t IH].
    - reflexivity.
    - simpl. destruct (test h) eqn:T.
      + unfold existsb'. simpl. rewrite T. reflexivity.
      + unfold existsb'. simpl. rewrite T. simpl. rewrite IH. reflexivity.
Qed.

(** [] *)

(* 2023-12-29 17:12 *)