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-style" 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".
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 o p : nat),
     n = m  ->
     [n;o] = [n;p] ->
     [n;o] = [m;p].
Proof.
  intros n m o p eq1 eq2.
  rewrite <- eq1.

(** Here, we could finish with "[rewrite -> eq2.  reflexivity.]" as we
    have done several times before.  We can achieve the same effect in
    a single step by using the [apply] tactic instead: *)

  apply eq2.  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. *)

Theorem silly2 : forall (n m o p : nat),
     n = m  ->
     (forall (q r : nat), q = r -> [q;o] = [r;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 binds some _universal 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 without using [simpl]. *)

Theorem silly_ex :
     (forall n, evenb n = true -> oddb (S n) = true) ->
     oddb 3 = true ->
     evenb 4 = true.
Proof.
  intros n H.
  apply H.
Qed.
(** [] *)

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

Theorem silly3_firsttry : forall (n : nat),
     true = (n =? 5)  ->
     (S (S n)) =? 7 = true.
Proof.
  intros n H.

(** Here we cannot use [apply] directly, but we can use the [symmetry]
    tactic, which switches the left and right sides of an equality in
    the goal. *)

  symmetry.
  simpl. (** (This [simpl] is optional, since [apply] will perform
             simplification first, if needed.) *)
  apply H.  Qed.

(** **** Exercise: 3 stars, standard (apply_exercise1)  

    (_Hint_: You can use [apply] with previously defined lemmas, not
    just hypotheses in the context.  Remember that [Search] is
    your friend.) *)

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

(* FILL IN HERE 

    [] *)

(* ################################################################# *)
(** * 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 recording, 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, we usually don't have to include the name [m] in
    the [with] clause; Coq is often smart enough to figure out which
    instantiation we're giving. We could instead write: [apply
    trans_eq with [c;d]]. *)

(** **** Exercise: 3 stars, standard, optional (apply_with_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 eq1 eq2.
  apply trans_eq with m.
  apply eq2. apply eq1.
Qed.
(** [] *)

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

(** Recall the definition of natural numbers:

     Inductive nat : Type :=
       | O : nat
       | S : nat -> 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 (and in our
    informal understanding of how datatype declarations work in other
    programming languages) are two more facts:

    - The constructor [S] is _injective_.  That is, if [S n = S m], it
      must be the case 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 all inductively defined types: all
    constructors are injective, and the values built from distinct
    constructors are never equal.  For lists, the [cons] constructor
    is injective and [nil] is different from every non-empty list.
    For booleans, [true] and [false] are different.  (Since neither
    [true] nor [false] take any arguments, their injectivity is not
    interesting.)  And so on. *)

(** For example, 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. reflexivity.
Qed.

(** This technique can be generalized to any constructor by
    writing the equivalent of [pred] for that constructor -- 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] at this point, we are asking Coq to
    generate all equations that it can infer from [H] using the
    injectivity of constructors. Each such equation is added as a
    premise to the goal. In the present example, adds the premise
    [n = m]. *)

  injection H. intros 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.
  injection H. intros H1 H2.
  rewrite H1. rewrite H2. reflexivity.
Qed.

(** The "[as]" variant of [injection] permits us to choose names for
    the introduced equations rather than letting Coq do it. *)

Theorem injection_ex2 : forall (n m : nat),
  [n] = [m] ->
  n = m.
Proof.
  intros n m H.
  injection H as Hnm. rewrite Hnm.
  reflexivity. Qed.

(** **** Exercise: 1 star, standard (injection_ex3)  *)
Example injection_ex3 : forall (X : Type) (x y z : X) (l j : list X),
  x :: y :: l = z :: j ->
  y :: l = x :: j ->
  x = y.
Proof.
  intros X x y z l j.
  intros H1 H2.
  injection H1. intros H3 H4.
  injection H2. intros H5 H6.
  rewrite -> H6.
  reflexivity.
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
    working in a context where we've _assumed_ that two such terms are
    equal, we are justified in concluding anything we want to (because
    the assumption is nonsensical).

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

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.

(** This is an instance of a logical principle known as the _principle
    of explosion_, which asserts that a contradictory hypothesis
    entails anything, even false things! *)

Theorem discriminate_ex1 : forall (n : nat),
  S n = O ->
  2 + 2 = 5.
Proof.
  intros n contra. discriminate contra. Qed.

Theorem discriminate_ex2 : forall (n m : nat),
  false = true ->
  [n] = [m].
Proof.
  intros n m contra. discriminate contra. Qed.

(** 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 arise,
    then the nonsensical conclusion would follow.  We'll explore the
    principle of explosion of 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.
  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 in a few
    places 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.

(* ################################################################# *)
(** * 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 in
    the hypothesis named [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": from [X -> Y] and a hypothesis matching [X], it
    produces a hypothesis matching [X].  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 silly3' : forall (n : nat),
  (n =? 5 = true -> (S (S n)) =? 7 = true) ->
  true = (n =? 5)  ->
  true = ((S (S n)) =? 7).
Proof.
  intros n 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.

    If you've seen informal proofs before (for example, in a math or
    computer science class), they probably used forward reasoning.  In
    general, idiomatic use of Coq tends to favor backward reasoning,
    but in some situations the forward style can be easier to think
    about.  *)

(** **** Exercise: 3 stars, standard, recommended (plus_n_n_injective)  

    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.
  intros n. induction n as [| n'].
  - intros m H0. destruct m.
    + reflexivity.
    + discriminate.
  - intros m. destruct m.
    + discriminate.
    + intros H0.
      injection H0.
      intros H1.
      rewrite <- plus_n_Sm in H1.
      rewrite <- plus_n_Sm in H1.
      injection H1. intros H2.
      apply IHn' in H2.
      rewrite -> H2.
      reflexivity.
Qed.
(** [] *)

(* ################################################################# *)
(** * 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 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 with

       intros n. induction n.

    all is well.  But if we begin it with

       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'].
  - (* 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' *) apply 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 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: it says 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 about
    whether [double n] is [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 _single_ [m]. *)

(** The successful proof of [double_injective] leaves [m] 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'].
  - (* n = O *) simpl. intros m eq. destruct m as [| m'] eqn:E.
    + (* m = O *) reflexivity.
    + (* m = S m' *) discriminate eq.

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

(** Notice that both the goal and the induction hypothesis are
    different this time: the goal asks us to prove something more
    general (i.e., to 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 *) simpl in eq.

(** The 0 case is trivial: *)

      discriminate eq.

    + (* m = S m' *)
      apply f_equal.

(** At this point, since we are 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'. injection eq as goal. apply goal. Qed.

(** What you should take away from all this is that we need to be
    careful, when using induction, that we are not trying to prove
    something too specific: To prove a property of [n] and [m] by
    induction on [n], it is sometimes important to leave [m]
    generic. *)

(** The following exercise requires the same pattern. *)

(** **** Exercise: 2 stars, standard (eqb_true)  *)
Theorem eqb_true : forall n m,
    n =? m = true -> n = m.
Proof.
  intros n. induction n as [| n'].
  - destruct m as [| m'].
    + simpl. reflexivity.
    + simpl. intros H. discriminate H.
  - destruct m as [| m'].
    + simpl. intros H. discriminate H.
    + simpl. intros H.
      apply IHn' in H. 
      apply f_equal.
      apply H.
Qed.
(** [] *)

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

    Give a careful informal proof of [eqb_true], being as explicit
    as possible about quantifiers. *)

(* FILL IN HERE *)

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

(** The strategy of doing fewer [intros] before an [induction] to
    obtain a more general IH doesn't always work by itself; 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'].
  - (* 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' *) apply f_equal.
        (* Stuck again 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'].
  - (* 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' *) apply 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. [] *)

(** Before we close this section and move on to some exercises,
    let's digress briefly and use [eqb_true] to prove a similar
    property of identifiers that we'll need in later chapters: *)

Theorem eqb_id_true : forall x y,
  eqb_id x y = true -> x = y.
Proof.
  intros [m] [n]. simpl. intros H.
  assert (H' : m = n). { apply eqb_true. apply H. }
  rewrite H'. reflexivity.
Qed.

(** **** Exercise: 3 stars, standard, recommended (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 [| x' l'].
  - simpl. reflexivity.
  - simpl. intros n H. destruct n.
    + simpl. discriminate H.
    + simpl.
      injection H. intros H0.
      apply IHl' in H0.
      apply H0.
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
    its right-hand side.  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 at
    this point, 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 mult_comm. apply mult_assoc. }
  rewrite H. rewrite mult_assoc. reflexivity.
Qed.

(** At this point, some discussion of unfolding and simplification is
    in order.

    You may 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 then 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.

(** However, 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 to make progress 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 too 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.  But
    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, optional (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: *)

Lemma remove_cons : forall X (l1 l2 : list X) (x : X),
    l1 = l2 -> x :: l1 = x :: l2.
Proof.
  intros X l1 l2 x H.
  rewrite <- H.
  reflexivity.
Qed.

Lemma split_cons : forall X Y (l : list (X * Y)),
    split l = (fst (split l), snd (split l)).
Proof.
Abort.

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 [| [a b] l'].
  + intros l1 l2. simpl.
    intros H.
    inversion H.
    reflexivity.
  + intros l1 l2. 
    unfold split.
Abort.

(** [] *)

(** The [eqn:] part of the [destruct] tactic is optional: We've chosen
    to include it most of the time, just for the sake of
    documentation, but many Coq proofs omit it.

    When [destruct]ing compound expressions, however, the information
    recorded by the [eqn:] can actually be critical: if we leave it
    out, then [destruct] can sometimes 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 of the (rather
    obvious) fact 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 ->
     oddb 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 thows 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 [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 occurences 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 ->
     oddb 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:e1.
  + destruct (f b) eqn:e2.
    - apply f_equal.
      rewrite -> e1 in e2.
      rewrite e2.
      apply e2.
    - rewrite -> e1 in e2.
      rewrite e2.
      destruct (f false) eqn:e3.
      * apply e2.
      * apply e3.
  + destruct (f b) eqn:e4.
    - rewrite e1 in e4.
      rewrite e4.
      destruct (f true) eqn:e5.
      * apply e5.
      * apply e4.
    - rewrite e1 in e4.
      rewrite e4.
      destruct (f false) eqn:e6.
      * discriminate e4.
      * apply e6.
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]

      - [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]: 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 *)

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

(** **** Exercise: 3 stars, standard (eqb_sym)  *)
Theorem eqb_sym : forall (n m : nat),
  (n =? m) = (m =? n).
Proof.
  intros n. induction n.
  - intros m. destruct m.
    + reflexivity.
    + reflexivity.
  - intros m. destruct m.
    + reflexivity.
    + simpl. apply IHn.
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: *)
   (* FILL IN HERE 

    [] *)

(** **** 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 H1 H2.
  apply eqb_true in H1.
  apply eqb_true in H2.
  rewrite -> H2 in H1.
  rewrite H1.
  rewrite <- eqb_refl.
  reflexivity.
Qed.
(** [] *)

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

    We proved, in an exercise above, that for all lists of pairs,
    [combine] is the inverse of [split].  How would you formalize the
    statement that [split] is the inverse of [combine]?  When is this
    property true?

    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. (Be sure to leave
    your induction hypothesis general by not doing [intros] on more
    things than necessary.  Hint: what property do you need of [l1]
    and [l2] for [split (combine l1 l2) = (l1,l2)] to be true?) *)

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

Theorem split_combine : split_combine_statement.
Proof.
  intros X Y l1. induction l1 as [| x' l' IHl'].
  - intros l2 H. simpl. inversion H.
    destruct l2.
    + reflexivity.
    + discriminate H1.
Abort.

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

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

    This one is a bit challenging.  Pay attention to the form of your
    induction hypothesis. *)

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.
Abort.
(** [] *)

(** **** Exercise: 4 stars, advanced, recommended (forall_exists_challenge)  

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

      forallb oddb [1;3;5;7;9] = true

      forallb negb [false;false] = true

      forallb evenb [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 oddb [1;0;0;0;0;3] = true

      existsb evenb [] = 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
  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_forallb_1 : forallb oddb [1;3;5;7;9] = true.
Proof. (* FILL IN HERE *) Admitted.

Example test_forallb_2 : forallb negb [false;false] = true.
Proof. (* FILL IN HERE *) Admitted.

Example test_forallb_3 : forallb evenb [0;2;4;5] = false.
Proof. (* FILL IN HERE *) Admitted.

Example test_forallb_4 : forallb (eqb 5) [] = true.
Proof. (* FILL IN HERE *) Admitted.

Fixpoint existsb {X : Type} (test : X -> bool) (l : list X) : bool
  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_existsb_1 : existsb (eqb 5) [0;2;3;6] = false.
Proof. (* FILL IN HERE *) Admitted.

Example test_existsb_2 : existsb (andb true) [true;true;false] = true.
Proof. (* FILL IN HERE *) Admitted.

Example test_existsb_3 : existsb oddb [1;0;0;0;0;3] = true.
Proof. (* FILL IN HERE *) Admitted.

Example test_existsb_4 : existsb evenb [] = false.
Proof. (* FILL IN HERE *) Admitted.

Definition existsb' {X : Type} (test : X -> bool) (l : list X) : bool
  (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Theorem existsb_existsb' : forall (X : Type) (test : X -> bool) (l : list X),
  existsb test l = existsb' test l.
Proof. (* FILL IN HERE *) Admitted.

(** [] *)



(* Wed Jan 9 12:02:44 EST 2019 *)