# Data and Functions

## Enumerated Types

In Coq, we can build practically everything from first principles...

## Days of the Week

A datatype definition:
Inductive day : Type :=
| monday
| tuesday
| wednesday
| thursday
| friday
| saturday
| sunday.
A function on days:
Definition next_weekday (d:day) : day :=
match d with
| mondaytuesday
| tuesdaywednesday
| wednesdaythursday
| thursdayfriday
| fridaymonday
| saturdaymonday
| sundaymonday
end.
Simplification:
Compute (next_weekday friday).
(* ==> monday : day *)

Compute (next_weekday (next_weekday saturday)).
(* ==> tuesday : day *)
Second, we can record what we expect the result to be in the form of a Coq example:
Example test_next_weekday:
(next_weekday (next_weekday saturday)) = tuesday.
A proof script giving evidence for the claim:
Proof. simpl. reflexivity. Qed.

## Booleans

Another familiar enumerated type:
Inductive bool : Type :=
| true
| false.
Booleans are also available from Coq's standard library, but in this course we'll define everything from scratch, just to see how it's done.

Definition negb (b:bool) : bool :=
match b with
| truefalse
| falsetrue
end.
Definition andb (b1:bool) (b2:bool) : bool :=
match b1 with
| trueb2
| falsefalse
end.

Definition orb (b1:bool) (b2:bool) : bool :=
match b1 with
| truetrue
| falseb2
end.
Note the syntax for defining multi-argument functions (andb and orb).

Example test_orb1: (orb true false) = true.
Proof. simpl. reflexivity. Qed.
Example test_orb2: (orb false false) = false.
Proof. simpl. reflexivity. Qed.
Example test_orb3: (orb false true) = true.
Proof. simpl. reflexivity. Qed.
Example test_orb4: (orb true true) = true.
Proof. simpl. reflexivity. Qed.
We can define new symbolic notation for an existing definition.
Notation "x && y" := (andb x y).
Notation "x || y" := (orb x y).

Example test_orb5: false || false || true = true.
Proof. simpl. reflexivity. Qed.

#### Exercise: 1 star, standard (nandb)

Remove "Admitted." and complete the definition of the following function; then make sure that the Example assertions below can each be verified by Coq. (I.e., fill in each proof, following the model of the orb tests above.) The function should return true if either or both of its inputs are false.
Definition nandb (b1:bool) (b2:bool) : bool
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_nandb1: (nandb true false) = true.
(* FILL IN HERE *) Admitted.
Example test_nandb2: (nandb false false) = true.
(* FILL IN HERE *) Admitted.
Example test_nandb3: (nandb false true) = true.
(* FILL IN HERE *) Admitted.
Example test_nandb4: (nandb true true) = false.
(* FILL IN HERE *) Admitted.
Most exercises are omitted from the "terse" version of the notes used in lecture.
The "full" version contains both the assigned reading and all the exercises for your homework assignment.

## Types

Every expression in Coq has a type, describing what sort of thing it computes. The Check command asks Coq to print the type of an expression.
Check true.
(* ===> true : bool *)
Check (negb true).
(* ===> negb true : bool *)
Functions like negb itself are also data values, just like true and false. Their types are called function types, and they are written with arrows.
Check negb.
(* ===> negb : bool -> bool *)

## New Types from Old

A more interesting type definition:
Inductive rgb : Type :=
| red
| green
| blue.

Inductive color : Type :=
| black
| white
| primary (p : rgb).
In general, Inductive declarations introduce "inductively defined sets," which consist of all legal combinations of the constructors. Let's look at this in a little more detail.
Every inductively defined type (day, bool, rgb, color, etc.) contains a set of constructor expressions built from constructors like red, primary, true, false, monday, etc. The definitions of rgb and color say how expressions in the sets rgb and color can be built:
• red, green, and blue are the constructors of rgb;
• black, white, and primary are the constructors of color;
• the expression red belongs to the set rgb, as do the expressions green and blue;
• the expressions black and white belong to the set color;
• if p is an expression belonging to the set rgb, then primary p (pronounced "the constructor primary applied to the argument p") is an expression belonging to the set color; and
• expressions formed in these ways are the only ones belonging to the sets rgb and color.
We can define functions on colors using pattern matching just as we have done for day and bool.
Definition monochrome (c : color) : bool :=
match c with
| blacktrue
| whitetrue
| primary qfalse
end.
Since the primary constructor takes an argument, a pattern matching primary should include either a variable (as above — note that we can choose its name freely) or a constant of appropriate type (as below).
Definition isred (c : color) : bool :=
match c with
| blackfalse
| whitefalse
| primary redtrue
| primary _false
end.
The pattern primary _ here is shorthand for "primary applied to any rgb constructor except red." (The wildcard pattern _ has the same effect as the dummy pattern variable p in the definition of monochrome.)

## Tuples

A nybble is half a byte — that is, four bits.
Inductive bit : Type :=
| B0
| B1.

Inductive nybble : Type :=
| bits (b0 b1 b2 b3 : bit).

Check (bits B1 B0 B1 B0).
(* ==> bits B1 B0 B1 B0 : nybble *)
We deconstruct a nybble by pattern-matching.
Definition all_zero (nb : nybble) : bool :=
match nb with
| (bits B0 B0 B0 B0) ⇒ true
| (bits _ _ _ _) ⇒ false
end.

Compute (all_zero (bits B1 B0 B1 B0)).
(* ===> false : bool *)
Compute (all_zero (bits B0 B0 B0 B0)).
(* ===> true : bool *)

## Modules

This Module declaration puts the following definitions in their own namespace (so that they don't conflict with things from the standard library).
Module NatPlayground.

## Numbers

There are many representations possible of natural numbers. You may be familiar with decimal, hexadecimal, octal, and binary. For simplicity in proofs, we choose unary. O represents zero, and S represents the act of adding an additional unary digit, that is, the "successor" operation, which when applied to the representation of n gives the representation of n+1.
Inductive nat : Type :=
| O
| S (n : nat).
With this definition, 0 is represented by O, 1 by S O, 2 by S (S O), and so on.
Again, let's look at this in a little more detail. The definition of nat says how expressions in the set nat can be built:
• O and S are constructors;
• the expression O belongs to the set nat;
• if n is an expression belonging to the set nat, then S n is also an expression belonging to the set nat; and
• expressions formed in these two ways are the only ones belonging to the set nat.
Critical point: this just defines a representation of numbers — a unary notation for writing them down.
• The names O and S are arbitrary.
• They are just two different "marks", with no intrinsic meaning.
We could just as well represent numbers with different marks:
Inductive nat' : Type :=
| stop
| tick (foo : nat').
The interpretation of these marks comes from how we use them to compute.
Definition pred (n : nat) : nat :=
match n with
| OO
| S n'n'
end.

End NatPlayground.
Standard decimal numerals can be used in input expressions, as a shorthand for sequences of applications of S to O, and Coq uses the same shorthand in things it prints:
Check (S (S (S (S O)))).
(* ===> 4 : nat *)

Definition minustwo (n : nat) : nat :=
match n with
| OO
| S OO
| S (S n') ⇒ n'
end.

Compute (minustwo 4).
(* ===> 2 : nat *)
Recursive functions are defined using the Fixpoint keyword.
Fixpoint evenb (n:nat) : bool :=
match n with
| Otrue
| S Ofalse
| S (S n') ⇒ evenb n'
end.
We can define oddb by a similar Fixpoint declaration, but here is a simpler definition:
Definition oddb (n:nat) : bool := negb (evenb n).

Example test_oddb1: oddb 1 = true.
Proof. simpl. reflexivity. Qed.
Example test_oddb2: oddb 4 = false.
Proof. simpl. reflexivity. Qed.
A multi-argument recursive function.
Module NatPlayground2.

Fixpoint plus (n : nat) (m : nat) : nat :=
match n with
| Om
| S n'S (plus n' m)
end.

Compute (plus 3 2).

(*  plus (S (S (S O))) (S (S O))
==> S (plus (S (S O)) (S (S O)))
by the second clause of the match
==> S (S (plus (S O) (S (S O))))
by the second clause of the match
==> S (S (S (plus O (S (S O)))))
by the second clause of the match
==> S (S (S (S (S O))))
by the first clause of the match
*)

Another:
Fixpoint mult (n m : nat) : nat :=
match n with
| OO
| S n'plus m (mult n' m)
end.

Example test_mult1: (mult 3 3) = 9.
Proof. simpl. reflexivity. Qed.
Pattern-matching two values at the same time:
Fixpoint minus (n m:nat) : nat :=
match n, m with
| O , _O
| S _ , On
| S n', S m'minus n' m'
end.

End NatPlayground2.
Again, we can make numerical expressions easier to read and write by introducing notations for addition, multiplication, and subtraction.
Notation "x + y" := (plus x y)
(at level 50, left associativity)
: nat_scope.
Notation "x - y" := (minus x y)
(at level 50, left associativity)
: nat_scope.
Notation "x * y" := (mult x y)
(at level 40, left associativity)
: nat_scope.

Check ((0 + 1) + 1).
When we say that Coq comes with almost nothing built-in, we really mean it: even equality testing is a user-defined operation!
Here is a function eqb, which tests natural numbers for equality, yielding a boolean. Note the use of nested matches (we could also have used a simultaneous match, as we did in minus.)
Fixpoint eqb (n m : nat) : bool :=
match n with
| Omatch m with
| Otrue
| S m'false
end
| S n'match m with
| Ofalse
| S m'eqb n' m'
end
end.
Similarly, the leb function tests whether its first argument is less than or equal to its second argument, yielding a boolean.
Fixpoint leb (n m : nat) : bool :=
match n with
| Otrue
| S n'
match m with
| Ofalse
| S m'leb n' m'
end
end.

Example test_leb1: (leb 2 2) = true.
Proof. simpl. reflexivity. Qed.
Example test_leb2: (leb 2 4) = true.
Proof. simpl. reflexivity. Qed.
Example test_leb3: (leb 4 2) = false.
Proof. simpl. reflexivity. Qed.
Since we'll be using these (especially eqb) a lot, let's give them infix notations.
Notation "x =? y" := (eqb x y) (at level 70) : nat_scope.
Notation "x <=? y" := (leb x y) (at level 70) : nat_scope.

Example test_leb3': (4 <=? 2) = false.
Proof. simpl. reflexivity. Qed.

# Proof by Simplification

A general property of natural numbers:
Theorem plus_O_n : n : nat, 0 + n = n.
Proof.
intros n. simpl. reflexivity. Qed.
The simpl tactic is actually redundant, as reflexivity already does some simplification for us:
Theorem plus_O_n' : n : nat, 0 + n = n.
Proof.
intros n. reflexivity. Qed.
Other similar theorems can be proved with the same pattern.
Theorem plus_1_l : n:nat, 1 + n = S n.
Proof.
intros n. reflexivity. Qed.

Theorem mult_0_l : n:nat, 0 * n = 0.
Proof.
intros n. reflexivity. Qed.

# Proof by Rewriting

A (slightly) more interesting theorem:
Theorem plus_id_example : n m:nat,
n = m
n + n = m + m.

Proof.
(* move both quantifiers into the context: *)
intros n m.
(* move the hypothesis into the context: *)
intros H.
(* rewrite the goal using the hypothesis: *)
rewriteH.
reflexivity. Qed.
The uses of intros name the hypotheses as they are moved to the context. The rewrite needs to know which equality is being used and in which direction to do the replacement.
We can also use the rewrite tactic with a previously proved theorem instead of a hypothesis from the context. If the statement of the previously proved theorem involves quantified variables, as in the example below, Coq tries to instantiate them by matching with the current goal.
Theorem mult_0_plus : n m : nat,
(0 + n) * m = n * m.
Proof.
intros n m.
rewriteplus_O_n.
reflexivity. Qed.

# Proof by Case Analysis

Sometimes simple calculating and rewriting are not enough...
Theorem plus_1_neq_0_firsttry : n : nat,
(n + 1) =? 0 = false.
Proof.
intros n.
simpl. (* does nothing! *)
Abort.
We can use destruct to perform case analysis:
Theorem plus_1_neq_0 : n : nat,
(n + 1) =? 0 = false.
Proof.
intros n. destruct n as [| n'] eqn:E.
- reflexivity.
- reflexivity. Qed.
Note the "bullets" marking the proofs of the two subgoals.
Another example (using booleans):
Theorem negb_involutive : b : bool,
negb (negb b) = b.
Proof.
intros b. destruct b eqn:E.
- reflexivity.
- reflexivity. Qed.
We can have nested subgoals (and we use different "bullets" to mark the inner ones):
Theorem andb_commutative : b c, andb b c = andb c b.
Proof.
intros b c. destruct b eqn:Eb.
- destruct c eqn:Ec.
+ reflexivity.
+ reflexivity.
- destruct c eqn:Ec.
+ reflexivity.
+ reflexivity.
Qed.
Besides - and +, we can use * (asterisk) as a third kind of bullet. We can also enclose sub-proofs in curly braces, which is useful in case we ever encounter a proof that generates more than three levels of subgoals:
Theorem andb_commutative' : b c, andb b c = andb c b.
Proof.
intros b c. destruct b eqn:Eb.
{ destruct c eqn:Ec.
{ reflexivity. }
{ reflexivity. } }
{ destruct c eqn:Ec.
{ reflexivity. }
{ reflexivity. } }
Qed.
intros x y. destruct y as [|yeqn:E.
we can write
intros x [|y].
Theorem plus_1_neq_0' : n : nat,
(n + 1) =? 0 = false.
Proof.
intros [|n].
- reflexivity.
- reflexivity. Qed.
If there are no arguments to name, we can just write [].
Theorem andb_commutative'' :
b c, andb b c = andb c b.
Proof.
intros [] [].
- reflexivity.
- reflexivity.
- reflexivity.
- reflexivity.
Qed.