(** * Induction: Proof by Induction *) (** The next line imports all of our definitions from the previous chapter. *) Require Export Basics. (** For it to work, you need to use [coqc] to compile [Basics.v] into [Basics.vo]. (This is like making a .class file from a .java file, or a .o file from a .c file.) Here are two ways to compile your code: - CoqIDE: Open [Basics.v]. In the "Compile" menu, click on "Compile Buffer". - Command line: Run [coqc Basics.v] *) (* ###################################################################### *) (** * Naming Cases *) (** The fact that there is no explicit command for moving from one branch of a case analysis to the next can make proof scripts rather hard to read. In larger proofs, with nested case analyses, it can even become hard to stay oriented when you're sitting with Coq and stepping through the proof. (Imagine trying to remember that the first five subgoals belong to the inner case analysis and the remaining seven cases are what remains of the outer one...) Disciplined use of indentation and comments can help, but a better way is to use the [Case] tactic. *) (* [Case] is not built into Coq: we need to define it ourselves. There is no need to understand how it works -- you can just skip over the definition to the example that follows. It uses some facilities of Coq that we have not discussed -- the string library (just for the concrete syntax of quoted strings) and the [Ltac] command, which allows us to declare custom tactics. Kudos to Aaron Bohannon for this nice hack! *) Require String. Open Scope string_scope. Ltac move_to_top x := match reverse goal with | H : _ |- _ => try move x after H end. Tactic Notation "assert_eq" ident(x) constr(v) := let H := fresh in assert (x = v) as H by reflexivity; clear H. Tactic Notation "Case_aux" ident(x) constr(name) := first [ set (x := name); move_to_top x | assert_eq x name; move_to_top x | fail 1 "because we are working on a different case" ]. Tactic Notation "Case" constr(name) := Case_aux Case name. Tactic Notation "SCase" constr(name) := Case_aux SCase name. Tactic Notation "SSCase" constr(name) := Case_aux SSCase name. Tactic Notation "SSSCase" constr(name) := Case_aux SSSCase name. Tactic Notation "SSSSCase" constr(name) := Case_aux SSSSCase name. Tactic Notation "SSSSSCase" constr(name) := Case_aux SSSSSCase name. Tactic Notation "SSSSSSCase" constr(name) := Case_aux SSSSSSCase name. Tactic Notation "SSSSSSSCase" constr(name) := Case_aux SSSSSSSCase name. (** Here's an example of how [Case] is used. Step through the following proof and observe how the context changes. *) Theorem andb_true_elim1 : forall b c : bool, andb b c = true -> b = true. Proof. intros b c H. destruct b. Case "b = true". (* <----- here *) reflexivity. Case "b = false". (* <---- and here *) rewrite <- H. reflexivity. Qed. (** [Case] does something very straightforward: It simply adds a string that we choose (tagged with the identifier "Case") to the context for the current goal. When subgoals are generated, this string is carried over into their contexts. When the last of these subgoals is finally proved and the next top-level goal becomes active, this string will no longer appear in the context and we will be able to see that the case where we introduced it is complete. Also, as a sanity check, if we try to execute a new [Case] tactic while the string left by the previous one is still in the context, we get a nice clear error message. For nested case analyses (e.g., when we want to use a [destruct] to solve a goal that has itself been generated by a [destruct]), there is an [SCase] ("subcase") tactic. *) (** **** Exercise: 2 stars (andb_true_elim2) *) (** Prove [andb_true_elim2], marking cases (and subcases) when you use [destruct]. *) Theorem andb_true_elim2 : forall b c : bool, andb b c = true -> c = true. (* Proof. (* FILL IN HERE *) Admitted. (** [] *) *) Proof. intros b c H. destruct c. reflexivity. rewrite <- H. destruct b. reflexivity. reflexivity. Qed. (** There are no hard and fast rules for how proofs should be formatted in Coq -- in particular, where lines should be broken and how sections of the proof should be indented to indicate their nested structure. However, if the places where multiple subgoals are generated are marked with explicit [Case] tactics placed at the beginning of lines, then the proof will be readable almost no matter what choices are made about other aspects of layout. This is a good place to mention one other piece of (possibly obvious) advice about line lengths. Beginning Coq users sometimes tend to the extremes, either writing each tactic on its own line or entire proofs on one line. Good style lies somewhere in the middle. In particular, one reasonable convention is to limit yourself to 80-character lines. Lines longer than this are hard to read and can be inconvenient to display and print. Many editors have features that help enforce this. *) (* ###################################################################### *) (** * Proof by Induction *) (** We proved in the last chapter that [0] is a neutral element for [+] on the left using a simple argument. The fact that it is also a neutral element on the _right_... *) Theorem plus_0_r_firsttry : forall n:nat, n + 0 = n. (** ... cannot be proved in the same simple way. Just applying [reflexivity] doesn't work: 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, but in the branch where [n = S n'] for some [n'] we get stuck in exactly the same way. We could use [destruct n'] to get one step further, but since [n] can be arbitrarily large, if we try to keep on like this we'll never be done. *) Theorem plus_0_r_secondtry : forall n:nat, n + 0 = n. Proof. intros n. destruct n as [| n']. Case "n = 0". reflexivity. (* so far so good... *) Case "n = S n'". simpl. (* ...but here we are stuck again *) Abort. (** To prove such facts -- indeed, to prove most interesting facts about numbers, lists, and other inductively defined sets -- we need a more powerful reasoning principle: _induction_. Recall (from high school) 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 but the order is backwards: 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: first showing [P(O)] and then showing [P(n') -> P(S n')]. Here's how this works for the theorem we are trying to prove at the moment: *) Theorem plus_0_r : forall n:nat, n + 0 = n. Proof. intros n. induction n as [| n']. Case "n = 0". reflexivity. Case "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. In the first branch, [n] is replaced by [0] and the goal becomes [0 + 0 = 0], which follows by simplification. In the second, [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']). The goal in this case becomes [(S n') + 0 = S n'], which simplifies to [S (n' + 0) = S n'], which in turn follows from the induction hypothesis. *) Theorem minus_diag : forall n, minus n n = 0. Proof. (* WORKED IN CLASS *) intros n. induction n as [| n']. Case "n = 0". simpl. reflexivity. Case "n = S n'". simpl. rewrite -> IHn'. reflexivity. Qed. (** **** Exercise: 2 stars (basic_induction) *) (** Prove the following lemmas using induction. You might need previously proven results. *) Theorem mult_0_r : forall n:nat, n * 0 = 0. Proof. (* (* FILL IN HERE *) Admitted.*) intros. induction n. simpl. reflexivity. simpl. apply IHn. Qed. Theorem plus_n_Sm : forall n m : nat, S (n + m) = n + (S m). Proof. (* (* FILL IN HERE *) Admitted.*) intros. induction n. simpl. reflexivity. simpl. rewrite -> IHn. reflexivity. Qed. Theorem plus_comm : forall n m : nat, n + m = m + n. Proof. (* (* FILL IN HERE *) Admitted.*) Lemma plus_distr : forall n m: nat, S (n + m) = n + (S m). Proof. intros. induction n. Case "n = 0". reflexivity. Case "n = S n'". simpl. rewrite -> IHn. reflexivity. Qed. intros. induction n. Case "n=0". simpl. rewrite -> plus_0_r. reflexivity. Case "n =S n". simpl. rewrite ->IHn. rewrite -> plus_distr. reflexivity. Qed. Theorem plus_assoc : forall n m p : nat, n + (m + p) = (n + m) + p. Proof. (* (* FILL IN HERE *) Admitted.*) intros. induction n. reflexivity. simpl. rewrite -> IHn. reflexivity. Qed. (** [] *) (** **** Exercise: 2 stars (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. (* (* FILL IN HERE *) Admitted.*) intros. induction n. Case "n = 0". reflexivity. Case "n = S n'". simpl. rewrite -> IHn. rewrite -> plus_distr. reflexivity. Qed. (** [] *) (* Briefly explain the difference between the tactics apply and rewrite . Are there situations where either one can be applied? Answer: The rewrite tactic is used to apply a known equality to either the goal or a hypothesis in the context, replacing all occurrences of one side by the other. The apply tactic uses a known implication (a hypothesis from the context, a previously proved lemma, o r a constructor) to modify the proof state either backward (if the goal matches the conclusion of the im plication, in which case a subgoal is generated for each premise of the implication) or forward (i f some hypothesis matches the premise of the implication, in which case this hypothesis is replaced b y the conclusion of the implication). If the fact is itself an equality (i.e., an implication with no prem ises), then either tactic can be used. *) (** **** Exercise: 1 star (destruct_induction) *) (** Briefly explain the difference between the tactics [destruct] and [induction]. (* FILL IN HERE *) (* Answer: Answer: Both are used to perform case analysis on an element o f an inductively defined type; induction also generates an induction hypothesis, while destruct does not. *) *) (** [] *) (* ###################################################################### *) (** * Proofs Within Proofs *) (** In Coq, as in informal mathematics, large proofs are very often broken into a sequence of theorems, with later proofs referring to earlier theorems. Occasionally, however, a proof will need 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. For example, our earlier proof of the [mult_0_plus] theorem referred to a previous theorem named [plus_O_n]. We can also use [assert] to state and prove [plus_O_n] in-line: *) Theorem mult_0_plus' : forall n m : nat, (0 + n) * m = n * m. Proof. intros n m. assert (H: 0 + n = n). Case "Proof of assertion". 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]. (Note that we could also name the assertion with [as] just as we did above with [destruct] and [induction], i.e., [assert (0 + n = n) as H]. Also note that we mark the proof of this assertion with a [Case], both for readability and so that, when using Coq interactively, we can see when we're finished proving the assertion by observing when the ["Proof of assertion"] string disappears from the context.) The second goal is the same as the one at the point where we invoke [assert], except that, in the context, we have the assumption [H] that [0 + n = 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. *) (** Actually, [assert] will turn out to be handy in many sorts of situations. For 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 ([plus_comm]) to rewrite one into the other. However, the [rewrite] tactic is a little stupid about _where_ it applies the rewrite. There are three uses of [+] here, and it turns out that doing [rewrite -> plus_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)... it seems like plus_comm should do the trick! *) rewrite -> plus_comm. (* Doesn't work...Coq rewrote the wrong plus! *) Abort. (** To get [plus_comm] to apply at the point where we want 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 [plus_comm], and then use this lemma 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). Case "Proof of assertion". rewrite -> plus_comm. reflexivity. rewrite -> H. reflexivity. Qed. (** **** Exercise: 4 stars (mult_comm) *) (** Use [assert] to help prove this theorem. You shouldn't need to use induction. *) Theorem plus_swap : forall n m p : nat, n + (m + p) = m + (n + p). Proof. (* (* FILL IN HERE *) Admitted.*) intros n m p. rewrite -> plus_assoc. assert(H: m + (n + p) = (m + n) + p). rewrite -> plus_assoc. reflexivity. rewrite -> H. assert(H2: m + n = n + m). rewrite -> plus_comm. reflexivity. rewrite -> H2. reflexivity. Qed. (** Now prove commutativity of multiplication. (You will probably need to define and prove a separate subsidiary theorem to be used in the proof of this one.) You may find that [plus_swap] comes in handy. *) Theorem mult_comm : forall m n : nat, m * n = n * m. Proof. (* FILL IN HERE *) Admitted. (** **** Exercise: 2 stars, optional (evenb_n__oddb_Sn) *) (** Prove the following simple fact: *) Theorem evenb_n__oddb_Sn : forall n : nat, evenb n = negb (evenb (S n)). Proof. (* FILL IN HERE *) Admitted. (** [] *) (* ###################################################################### *) (** * More Exercises *) (** **** Exercise: 3 stars, 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 hacking!) *) Theorem ble_nat_refl : forall n:nat, true = ble_nat n n. Proof. (* FILL IN HERE *) Admitted. Theorem zero_nbeq_S : forall n:nat, beq_nat 0 (S n) = false. Proof. (* FILL IN HERE *) Admitted. Theorem andb_false_r : forall b : bool, andb b false = false. Proof. (* FILL IN HERE *) Admitted. Theorem plus_ble_compat_l : forall n m p : nat, ble_nat n m = true -> ble_nat (p + n) (p + m) = true. Proof. (* FILL IN HERE *) Admitted. Theorem S_nbeq_0 : forall n:nat, beq_nat (S n) 0 = false. Proof. (* FILL IN HERE *) Admitted. Theorem mult_1_l : forall n:nat, 1 * n = n. Proof. (* FILL IN HERE *) Admitted. Theorem all3_spec : forall b c : bool, orb (andb b c) (orb (negb b) (negb c)) = true. Proof. (* FILL IN HERE *) Admitted. Theorem mult_plus_distr_r : forall n m p : nat, (n + m) * p = (n * p) + (m * p). Proof. (* FILL IN HERE *) Admitted. Theorem mult_assoc : forall n m p : nat, n * (m * p) = (n * m) * p. Proof. (* FILL IN HERE *) Admitted. (** [] *) (** **** Exercise: 2 stars, optional (beq_nat_refl) *) (** Prove the following theorem. Putting [true] on the left-hand side of the equality may seem odd, but this is how the theorem is stated in the standard library, so we follow suit. Since rewriting works equally well in either direction, we will have no problem using the theorem no matter which way we state it. *) Theorem beq_nat_refl : forall n : nat, true = beq_nat n n. Proof. (* FILL IN HERE *) Admitted. (** [] *) (** **** Exercise: 2 stars, optional (plus_swap') *) (** The [replace] tactic allows you to specify a particular subterm to rewrite and what you want it rewritten to. More precisely, [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 [plus_swap'], just like [plus_swap] but without needing [assert (n + m = m + n)]. *) Theorem plus_swap' : forall n m p : nat, n + (m + p) = m + (n + p). Proof. (* FILL IN HERE *) Admitted. (** [] *) (** **** Exercise: 3 stars (binary_commute) *) (** Recall the [increment] and [binary-to-unary] functions that you wrote for the [binary] exercise in the [Basics] chapter. Prove that these functions commute -- that is, incrementing a binary number and then converting it to unary yields the same result as first converting it to unary and then incrementing. (Before you start working on this exercise, please copy the definitions from your solution to the [binary] exercise here so that this file can be graded on its own. If you find yourself wanting to change your original definitions to make the property easier to prove, feel free to do so.) *) (* FILL IN HERE *) (** [] *) (** **** Exercise: 5 stars, advanced (binary_inverse) *) (** This exercise is a continuation of the previous exercise about binary numbers. You will need your definitions and theorems from the previous exercise to complete this one. (a) First, write a function to convert natural numbers to binary numbers. Then prove that starting with any natural number, converting to binary, then converting back yields the same natural number you started with. (b) You might naturally think that we should also prove the opposite direction: that starting with a binary number, converting to a natural, and then back to binary yields the same number we started with. However, it is not true! Explain what the problem is. (c) Define a function [normalize] from binary numbers to binary numbers such that for any binary number b, converting to a natural and then back to binary yields [(normalize b)]. Prove it. Again, feel free to change your earlier definitions if this helps here. *) (* FILL IN HERE *) (** [] *) (* ###################################################################### *) (** * Advanced Material *) (** ** Formal vs. Informal Proof *) (** "Informal proofs are algorithms; formal proofs are code." *) (** The question of what, exactly, constitutes a "proof" of a mathematical claim has challenged philosophers for millenia. A rough and ready definition, though, 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. That is, a proof is an act of communication. Now, 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 a simple mechanical check that [P] can be derived from a certain set of formal logical rules, and the proof is a recipe that guides the program in performing this check. 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, thus necessarily _informal_. Here, the criteria for success are less clearly specified. A "good" 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. One reader 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 another reader, 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, because it is easier to fill in the details for themselves. 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, 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 ignore the 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 plus_assoc' : forall n m p : nat, n + (m + p) = (n + m) + p. Proof. intros n m p. induction n as [| n']. reflexivity. simpl. rewrite -> IHn'. reflexivity. Qed. (** Coq is perfectly happy with this as a proof. For a human, however, it is difficult to make much sense of it. If you're used to Coq you can probably 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. Instead, a mathematician might write it 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 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 show (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. [] *) (** The overall form of the proof is basically similar. 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). *) (** Here is a formal proof that shows the structure more clearly: *) Theorem plus_assoc'' : forall n m p : nat, n + (m + p) = (n + m) + p. Proof. intros n m p. induction n as [| n']. Case "n = 0". reflexivity. Case "n = S n'". simpl. rewrite -> IHn'. reflexivity. Qed. (** **** Exercise: 2 stars, advanced (plus_comm_informal) *) (** Translate your solution for [plus_comm] into an informal proof. *) (** Theorem: Addition is commutative. Proof: (* FILL IN HERE *) [] *) (** **** Exercise: 2 stars, optional (beq_nat_refl_informal) *) (** Write an informal proof of the following theorem, using the informal proof of [plus_assoc] as a model. Don't just paraphrase the Coq tactics into English! Theorem: [true = beq_nat n n] for any [n]. Proof: (* FILL IN HERE *) [] *) (* $Date: 2013-07-17 16:19:11 -0400 (Wed, 17 Jul 2013) $ *)