# Induction in Coq

From: https://www.cs.cornell.edu/courses/cs3110/2018sp/l/22-coq-induction/notes.v
Topics:
• recursive functions
• induction on lists
• induction on natural numbers
• rings and fields
• induction principles

We'll need the list library for these notes.

Require Import List.
Import ListNotations.

## Recursive functions

The List library defines the list append operator, which in Coq is written as infix operator ++, or as prefix function app. In OCaml, the same operator is written @. In OCaml's standard library, you'll recall that append is defined as follows:
```let rec append lst1 lst2 =
match lst1 with
| [] -> lst2
| h::t -> h :: (append t lst2)
```
The Coq equivalent of that would be:

Fixpoint append {A : Type} (lst1 : list A) (lst2 : list A) :=
match lst1 with
| nil => lst2
| h::t => h :: (append t lst2)
end.

The Fixpoint keyword is similar to a let rec definition in OCaml. The braces around A : Type in that definition make the A argument implicit, hence we don't have to provide it at the recursive call to append in the second branch of the pattern match. Without the braces, we'd have to write the function as follows:

Fixpoint append' (A : Type) (lst1 : list A) (lst2 : list A) :=
match lst1 with
| nil => lst2
| h::t => h :: (append' A t lst2)
end.

The actual definition of ++ in the Coq standard library is a little more complicated, but it's essentially the same idea. Here's that definition:
```Definition app (A : Type) : list A -> list A -> list A :=
fix app' lst1 lst2 :=
match lst1 with
| nil => lst2
| h :: t => h :: app' t lst2
end.
```
The Coq fix keyword is similar to a let rec expression in OCaml, but where the body of the expression is implicit: Coq fix f x1 .. xn := e is like OCaml let rec f x1 .. xn = e in f. So in OCaml we could rephrase the above definition as follows:
```let app : 'a list -> 'a list -> 'a list =
let rec app' lst1 lst2 =
match lst1 with
| [] -> lst2
| h :: t -> h :: app' t lst2
in
app'
```
Now that we know how Coq defines ++, let's prove a theorem about it.

Theorem nil_app : forall (A:Type) (lst : list A),
[] ++ lst = lst.

Intuition: appending the empty list to lst immediately returns lst, by the definition of ++.

Proof.
intros A lst.
simpl.
trivial.
Qed.

The second step in that proof simplifies [] ++ lst to lst. That's because of how ++ is defined: it pattern matches against its first argument, which here is [], hence simply returns its second argument.
Next, let's prove that appending nil on the right-hand side also results in lst:

Theorem app_nil : forall (A:Type) (lst : list A),
lst ++ [] = lst.

Proof.
intros A lst. destruct lst as [ | h t].
- trivial.
- simpl. Abort.

When we get to the end of that proof, we are trying to show that h :: (t ++ []) = h :: t. There's no way to make progress on that, because we can't simplify t ++ [] to just t. Of course as humans we know that holds. But to Coq, that's a fact that hasn't yet been proved. Indeed, it is an instance of the theorem we're currently trying to prove!
What's going wrong here is that case analysis is not a sufficiently powerful proof technique for this theorem. We need to be able to recursively apply the theorem we're trying to prove to smaller lists. That's where induction comes into play.

## Induction on lists

Induction is a proof technique that you will have encountered in math classes before---including CS 2800. It is useful when you want to prove that some property holds of all the members of an infinite set, such as the natural numbers, as well as lists, trees, and other data types.
One classic metaphor for induction is falling dominos: if you arrange some dominos such that each domino, when it falls, will knock over the next domino, and if you knock over the first domino, then all the dominos will fall. Another classic metaphor for induction is a ladder: if you can reach the first rung, and if for any given rung the next rung can be reached, then you can reach any rung you wish. (As long as you're not afraid of heights.)
What both of those metaphors have in common is
• a base case, in which something is done first. For the dominos, it's knocking over the first domino; for the ladder, it's climbing the first rung. And,
• an inductive case, in which a step is taken from one thing to the the next. For the dominos, it's one domino knocking over the next; for the ladder, it's literally taking the step from one rung to the next. In both cases, it must actually be possible for the action to occur: if the dominos or the rungs were spaced too far apart, then progress would stop.
A proof by induction likewise has a base case and an inductive case. Here's the structure of a proof by induction on a list:
```Theorem.  For all lists lst, P(lst).

Proof.  By induction on lst.

Case:  lst = nil
Show:  P(nil)

Case:  lst = h::t
IH:    P(t)
Show:  P(h::t)

QED.
```
The base case of a proof by induction on lists is for when the list is empty. The inductive case is when the list is non-empty, hence is the cons of a head to a tail. In the inductive case, we get to assume an inductive hypothesis, which is that the property P holds of the tail.
In the metaphors above, the inductive hypothesis is the assumption that we've already reached some particular domino or rung. From there, the metaphorical work we do in the inductive case of the proof is to show that from that domino or rung, we can reach the next one.
An inductive hypothesis is exactly the kind of assumption we needed to get our proof about appending nil to go through.
Here's how that proof could be written:
```Theorem:  for all lists lst, lst ++ nil = lst.

Proof:  by induction on lst.
P(lst) = lst ++ nil = lst.

Case:  lst = nil
Show:
P(nil)
= nil ++ nil = nil
= nil = nil

Case:  lst = h::t
IH: P(t) = (t ++ nil = t)
Show
P(h::t)
= (h::t) ++ nil = h::t
= h::(t ++ nil) = h::t     // by definition of ++
= h::t = h::t              // by IH

QED
```
In Coq, we could prove that theorem as follows:

Theorem app_nil : forall (A:Type) (lst : list A),
lst ++ nil = lst.
Proof.
intros A lst. induction lst as [ | h t IH].
- simpl. trivial.
- simpl. rewrite -> IH. trivial.
Qed.

The tactics used in that proof correspond exactly to the non-Coq proof above.
The induction tactic is new to us. It initiates a proof by induction on its argument, in this case lst, and provides names for the variables to be used in the cases. There aren't any new variables in the base case, but the inductive case has variables for the head of the list, the tail, and the inductive hypothesis. You could leave out those variables and simply write induction lst., but that leads to a less human-readable proof.
In the inductive case, we use the rewrite -> tactic to rewrite t ++ nil to t. The IH says those terms are equal. That tactic replaces the left-hand side of the equality with the right-hand side, wherever the left-hand side appears in the subgoal. It's also possible to rewrite from right to left with the rewrite <- tactic. If you leave out the arrow, Coq assumes that you mean ->.
Here's another theorem we can prove in exactly the same manner. This theorem shows that append is associative.
```Theorem:  forall lists l1 l2 l3, l1 ++ (l2 ++ l3) = (l1 ++ l2) ++ l3.

Proof: by induction on l1.
P(l1) = l1 ++ (l2 ++ l3) = (l1 ++ l2) ++ l3

Case:  l1 = nil
Show:
P(nil)
= nil ++ (l2 ++ l3) = (nil ++ l2) ++ l3
= l2 ++ l3 = l2 ++ l3   // simplifying ++, twice

Case:  l1 = h::t
IH:  P(t) = t ++ (l2 ++ l3) = (t ++ l2) ++ l3
Show:
P(h::t)
= h::t ++ (l2 ++ l3) = (h::t ++ l2) ++ l3
= h::(t ++ (l2 ++ l3)) = h::((t ++ l2) ++ l3)  // simplifying ++, thrice
= h::((t ++ l2) ++ l3) = h::((t ++ l2) ++ l3)  // by IH

QED
```
In Coq, that proof looks more or less identical to our previous Coq proof about append and nil:

Theorem app_assoc : forall (A:Type) (l1 l2 l3 : list A),
l1 ++ (l2 ++ l3) = (l1 ++ l2) ++ l3.
Proof.
intros A l1 l2 l3.
induction l1 as [ | h t IH].
- trivial.
- simpl. rewrite -> IH. trivial.
Qed.

## Induction on natural numbers

One of the classic theorems proved by induction is that 0 + 1 + ... + n is equal to n * (n+1) / 2. It uses proof by induction on the natural numbers, which are the non-negative integers. The structure of a proof by induction on the naturals is as follows:
```Theorem.  For all natural numbers n, P(n).

Proof.  By induction on n.

Case:  n = 0
Show:  P(0)

Case:  n = k+1
IH:    P(k)
Show:  P(k+1)

QED.
```
The base case is for zero, the smallest natural number. The inductive case assumes that P holds of k, then shows that P holds of k+1.
The induction tactic in Coq works for inductive types---that is, types defined with the Inductive keyword. You might, therefore, suspect that if we're going to do induction over nats, the type nat must be inductively defined. Indeed it is. There is a famous inductive definition of the natural numbers that is credited to Giuseppe Peano (1858-1932). In OCaml, that definition would be:
```type nat = O | S of nat
```
The O constructor (that's the letter capital O) represents zero. The S constructor represents the successor function---that is, the function that adds one to its argument. So:
• 0 is O
• 1 is S O
• 2 is S (S O)
• 3 is S (S (S O))
• etc.
This is a kind of unary representation of the naturals, in which we repeat the symbol S a total of n times to represent the natural number n.
The Coq definition of nat is much the same:

Print nat.

Coq responds with output that is equivalent to the following:
```Inductive nat : Set :=
| O : nat
| S : nat -> nat
```
That is, nat has two constructors, O and S, which are just like the OCaml constructors we examined above. And nat has type Set, meaning that nat is a specification for program computations. (Or, a little more loosely, that nat is a type representing program values.)
Anywhere we write something that looks like an integer literal in Coq, Coq actually understand that as its expansion in the Peano representation defined above. For example, 2 is understood by Coq as just syntactic sugar for S (S O). We can even write computations using those constructors:

Compute S (S O).

Coq responds, though, by reintroducing the syntactic sugar:
```= 2 : nat
```
The Coq standard library defines many functions over nat using those constructors and pattern matching, including addition, subtraction, and multiplication. For example, addition is defined like this:

Fixpoint my_add (a b : nat) : nat :=
match a with
| 0 => b
| S c => S (my_add c b)
end.

Note that we're allowed to use either 0 or O as a pattern, because the former is just syntactic sugar for the latter. The second branch of the pattern match is effectively calling my_add recursively with a-1 as its first argument, since a = S c, meaning that a is the successor of c.
Now that we know how nat is defined inductively, let's try to prove the classic theorem mentioned above about summation. Moreover, let's prove that a program that computes the sum 0 + 1 + ... + n does in fact compute n * (n+1) / 2. First, we need to write that program. In OCaml, we could write the program quite easily:
```let rec sum_to n =
if n=0 then 0
else n + sum_to (n-1)
```
In Coq, it will turn out to be surprisingly tricky...

## Recursive functions, revisited

Here's a first attempt at defining sum_to, which is just a direct translation of the OCaml code into Coq. The Fail keyword before it tells Coq to expect the definition to fail to compile.

Fail Fixpoint sum_to (n:nat) : nat :=
if n = 0 then 0 else n + sum_to (n-1).

Coq responds:
```The command has indeed failed with message:
The term "n = 0" has type "Prop"
which is not a (co-)inductive type.
```
The problem is the the equality operator = returns a proposition (i.e., something we could try to prove), whereas the if expression expects a Boolean (i.e., true or false) as its guard. (Actually if is willing to accept any value of an inductive type with exactly two constructors as its guard, and bool is an example of such a type.)
To fix this problem, we need to use an equality operator that returns a bool, rather than a Prop, when applied to two nats. Such an operator is defined in the Arith library for us:

Require Import Arith.

Locate "=?".
Check Nat.eqb.

Coq responds:
```Nat.eqb : nat -> nat -> bool
```
We can now try to use that operator. Unfortunately, we discover a new problem:

Fail Fixpoint sum_to (n:nat) : nat :=
if n =? 0 then 0 else n + sum_to (n-1).

Coq responds with output that contains the following lines:
```Recursive definition of sum_to is ill-formed.
...
Recursive call to sum_to has principal argument equal to
"n - 1" instead of a subterm of "n".
...
```
Although the error message might be cryptic, you can see that Coq is complaining about the recursive call in the else branch. For some reason, Coq is unhappy about the argument n-1 provided at that call. Coq wants that argument to be a "subterm" of n. The words term and expression are synonymous here, so Coq is saying that it wants the argument to be a subexpression of n. Of course n doesn't have any subexpressions. So why is Coq giving us this error?
Before we can answer that question, let's look at a different recursive function---one that implements an infinite loop:

Fail Fixpoint inf (x:nat) : nat := inf x.

Coq responds very similarly to how it did with sum_to:
```Recursive definition of inf is ill-formed.
...
Recursive call to inf has principal argument equal to
"x" instead of a subterm of "x".
```
The reason Coq rejects inf is that Coq does not permit any infinite loops. That might seem strange, but there's an excellent reason for it. Consider how inf would be defined in OCaml:
```# let rec inf x = inf x
val inf : 'a -> 'b = <fun>
```
Let's look at the type of that function, using what we learned about propositions-as-types in the previous lecture. The type 'a -> 'b corresponds to the proposition A -> B, where A and B could be any propositions. In particular, A could be True and B could be False, leading to the proposition True -> False. That's a proposition that should never be provable: truth does not imply falsehood. And yet, since inf is a program of that type, inf corresponds to a proof of that proposition. So using inf we could actually prove False:
```type void = {nope : 'a . 'a};;
let rec inf x = inf x;;
let ff : void = inf ();;
```
The void type is how we represented False in the previous lecturev. The value ff above corresponds to a proof of False. So infinite loops are able to prove False.
In OCaml we don't mind that phenomenon, because OCaml's purpose is not to be a proof assistant. But in Coq it would be deadly: we should never allow the proof assistant to prove false propositions. Coq therefore wants to prohibit all infinite loops. But that's easier said than done! Recall from CS 2800 that the halting problem is undecidable: we can't write a program that precisely determines whether another program will terminate. Well, the Coq compiler is a program, and it wants to detect which programs terminate and which programs do not---which is exactly what the halting problem says is impossible.
So instead of trying to do something impossible, Coq settles for doing something possible but imprecise, specifically, something that prohibits all non-terminating programs as well as prohibiting some terminating programs. Coq enforces a syntactic restriction on recursive functions: there must always be an argument that is syntactically smaller at every recursive function application. An expression e1 is syntactically smaller than e2 if e1 is a subexpression of e2. For example, 1 is syntactically smaller than 1-x, but n-1 is not syntactically smaller than n. It turns out this restriction is sufficient to guarantee that programs must terminate: eventually, if every call results in something smaller, you must reach something that is small enough that you cannot make a recursive call on it, hence evaluation must terminate. A synonym for "syntactically smaller" is structurally decreasing.
But that does rule out some programs that we as humans know will terminate yet do not meet the syntactic restriction. And sum_to is one of them. Here is the definition we previously tried:

Fail Fixpoint sum_to (n:nat) : nat :=
if n =? 0 then 0 else n + sum_to (n-1).

The recursive call to sum_to has as its argument n-1, which syntactically is actually bigger than the original argument of n. So Coq rejects the program.
To finally succeed in definining sum_to, we can make use of what we know about how nat is defined: since it's an inductive type, we can pattern match on it:

Fixpoint sum_to (n:nat) : nat :=
match n with
| 0 => 0
| S k => n + sum_to k
end.

The second branch of the pattern match recurses on an argument that is both syntactically and arithmetically smaller, just as our definition of my_add did, above. So Coq's syntactic restriction on recursion is satisfied, and the definition is accepted as a program that definitely terminates.

## Inductive proof of the summation formula

Now that we've finally succeeded in defining sum_to, we can prove the classic theorem about summation.
Here's how we would write the proof mathematically:
```Theorem:  for all natural numbers n, sum_to n = n * (n+1) / 2.

Proof:  by induction on n.
P(n) = sum_to n = n * (n+1) / 2

Case:  n=0
Show:
P(0)
= sum_to 0 = 0 * (0+1) / 2
= 0 = 0 * (0+1) / 2         // simplifying sum_to 0
= 0 = 0                     // 0 * x = 0

Case:  n=k+1
IH:  P(k) = sum_to k = k * (k+1) / 2
Show:
P(k+1)
= sum_to (k+1) = (k+1) * (k+1+1) / 2
= k + 1 + sum_to k = (k+1) * (k+1+1) / 2          // simplifying sum_to (k+1)
= k + 1 + k * (k+1) / 2 = (k+1) * (k+1+1) / 2     // using IH
= 2 + 3k + k*k = 2 + 3k + k*k                     // simplifying terms on each side

QED
```
Now let's do the proof in Coq.

Theorem sum_sq : forall n : nat,
sum_to n = n * (n+1) / 2.
Proof.
intros n.
induction n as [ | k IH].
- trivial.
- simpl. rewrite -> IH.
Abort.

The proof is working fine so far, but now we have a complicated algebraic equation we need to prove:
```S (k + k * (k + 1) / 2) = fst (Nat.divmod (k + 1 + k * S (k + 1)) 1 0 0)
```
(divmod is part of how / is implemented in Coq.)
Although we could try to prove that manually using the definitions of all the operators, it would be much nicer to get Coq to find the proof for us. It turns out that Coq does have great support for finding proofs that involve rings: algebraic structures that support addition and multiplication operations. (We'll discuss rings in detail after we finish the current proof.) But we can't use that automation here, because the equation we want to prove also involves division, and rings do not support division operations.
To avoid having to reason about division, we could rewrite the theorem we want to prove: by multiplying both sides by 2, the division goes away:

Theorem sum_sq_no_div : forall n : nat,
2 * sum_to n = n * (n+1).
Proof.
intros n.
induction n as [ | k IH].
- trivial.
- simpl.
Abort.

Now, after the call to simpl, we don't have any division, but we also don't have any expressions that look exactly like the left-hand side of the inductive hypothesis. The problem is that simpl was too agressive in simplifying all the expressions. All we really want is to transform the left-hand side of the subgoal, 2 * sum_to (S k), into an expression that contains the left-hand side of the inductive hypothesis, 2 * sum_to k. Thinking about the definition of sum_to, we ought to be able to transform 2 * sum_to (S k) into 2 * (S k + sum_to k), which equals 2 * (S k) + 2 * sum_to k. That final expression does have the left-hand side of the inductive hypothesis in it, as desired. Let's factor out that reasoning as a separate "helper" theorem. In math, helper theorems are usually called lemmas. The Coq keyword Lemma is synonymous with Theorem.

Lemma sum_helper : forall n : nat,
2 * sum_to (S n) = 2 * (S n) + 2 * sum_to n.
Proof.
intros n. simpl. ring.
Qed.

The proof above simplifies the application of sum_to (S n), then invokes a new tactic called ring. That tactic is able to automatically search for proofs of equations involving addition and multiplication of natural numbers.
Now that we have our helper lemma, we can use it to prove the theorem:

Theorem sum_sq_no_div : forall n : nat,
2 * sum_to n = n * (n+1).
Proof.
intros n.
induction n as [ | k IH].
- trivial.
- rewrite -> sum_helper.
rewrite -> IH.
ring.
Qed.

Once more, after doing the rewriting with the lemma and the inductive hypothesis, we're left with algebraic equations that can be proved simply by invoking the ring tactic.
Finally, we can use sum_sq_no_div to prove the original theorem involving division. To do that, we need to first prove another lemma that shows we can transform a multiplication into a division:

Lemma div_helper : forall a b c : nat,
c <> 0 -> c * a = b -> a = b / c.
Proof.
intros a b c neq eq.
rewrite <- eq.
rewrite Nat.mul_comm.
rewrite Nat.div_mul.
trivial.
assumption.
Qed.

That lemma involves two library theorems, mult_comm and Nat.div_mul. How did we know to use these? Coq can help us search for useful theorems. Right after we rewrite <- eq in the above proof, our subgoal is a = c * a / c. It looks like we ought to be able to cancel the c term on the right-hand side. So we can search for a theorem that would help us do that. The Search command takes wildcards and reports all theorems that match the pattern we supply, for example:

Search (_ * _ / _).

Unfortunately, the search results currently seem to be broken in Visual Studio Code: we get only one matching library theorem, instead of all of them. Doing the search in coqide or coqtop (the Coq REPL), or just browsing through the web documentation, does reveal a useful theorem:
```Nat.div_mul: forall a b : nat, b <> 0 -> a * b / b = a
```
That would let us cancel a term from the numerator and denominator, but it requires the left-hand side of the equality to be of the form a * b / b, whereas we have c * a / b. The problem is that the two sides of the multiplication are reversed. No worries; multiplication is commutative, and there is a library theorem that proves it. Again, we could find that theorem:

Search (_ * _ = _ * _).

One of the results is:
```Nat.mul_comm: forall n m : nat, n * m = m * n
```
Putting those two library theorems to use, we're able to prove the lemma as above.
Finally, we can use that lemma to prove our classic theorem about summation.

Theorem sum_sq : forall n : nat,
sum_to n = n * (n+1) / 2.

Proof.
intros n.
apply div_helper.
- discriminate.
- rewrite sum_sq_no_div. trivial.
Qed.

When we use apply div_helper in that proof, Coq generates two new subgoals---one for each of the propositions c <> 0 and c * a = b in the type of div_helper.
Summary: wow, that was a lot of work to prove that seemingly simple classic theorem! We had to figure out how to code sum_to, and we had to deal with a lot of complications involving algebra. This situation is not uncommon: the theorems that we think of as easy with pencil-and-paper (like arithmetic) turn out to be hard to convince Coq of, whereas the theorems that we think of as challenging with pencil-and-paper (like induction) turn out to be easy.

## Rings and fields

In the proof we just did above about summation, we used a tactic called ring that we said searches for proofs about algebraic equations involving multiplication and addition. Let's look more closely at that tactic.
When we studied OCaml modules, we looked at rings as an example of a mathematical abstraction of addition, multiplication, and negation. Here is an OCaml signature for a ring:
```module type Ring = sig
type t
val zero : t
val one  : t
val add  : t -> t -> t
val mult : t -> t -> t
val neg  : t -> t
end
```
We could implement that signature with a representation type t that is int, or float, or even bool.
The names given in Ring are suggestive of the operations they represent, but to really specify how those operations should behave, we need to write some equations that relate them. Below are the equations that (it turns out) fully specify zero, one, add, and mult. Rather than use those identifiers, we use the more familiar notation of 0, 1, +, and *.
```0 + x = 0
x + y = y + x
x + (y + z) = (x + y) + z

0 * x = 0
1 * x = 1
x * y = y * x
x * (y * z) = (x * y) * z

(x + y) * z = (x * z) + (y * z)
```
Technically these equations specify what is known as a commutative semi-ring. It's a semi-ring because we don't have equations specifying negation yet. It's a commutative semi-ring because the * operation commutes. (The + operation commutes too, but that's always required of a semi-ring.)
The first group of equations specifies how + behaves on its own. The second group specifies how * behaves on its own. The final equation specifies how + and * interact.
If we extend the equations above with the following two, we get a specification for a ring:
```x - y = x + (-y)
x + (-x) = 0
```
It's a remarkable fact from the study of abstract algebra that those equations completely specify a ring. Any theorem you want to prove about addition, multiplication, and negation follows from those equations. We call the equations the axioms that specify a ring.
Rings don't have a division operation. Let's introduce a new operator called inv (short for "inverse"), and let's write 1/x as syntactic sugar for inv x. If we take all the the ring axioms and add the following axiom for inv, we get what is called a field:
```x * 1/x = 1     if x<>0
```
A field is an abstraction of addition, multiplication, negation, and division. Note that OCaml ints do not satisfy the inv axiom above. For example, 2 * (1/2) equals 0 in OCaml, not 1. OCaml floats mostly do satisfy the field axioms, up to the limits of floating-point arithmetic. And in mathematics, the rational numbers and the real numbers are fields.
Coq provides two tactics, ring and field, that automatically search for proofs using the ring and field axioms. The ring tactic was already loaded for us when we wrote Require Import Arith earlier in this file. We can use the ring tactic to easily prove equalities that follow from the ring axioms. Here are two examples.

Theorem plus_comm : forall a b,
a + b = b + a.
Proof.
intros a b. ring.
Qed.

Theorem foil : forall a b c d,
(a + b) * (c + d) = a*c + b*c + a*d + b*d.
Proof.
intros a b c d. ring.
Qed.

Coq infers the types of the variables above to be nat, because the + and * operators are defined on nat.
The proofs that the ring tactic finds can be quite complicated. For example, try looking at the output of the following command. It's so long that we won't put that output in this file!

Print foil.

Of course, ring won't find proofs of equations that don't actually hold. For example, if we had a typo in our statement of foil, then ring would fail.

Theorem broken_foil: forall a b c d,
(a + b) * (c + d) = a*c + b*c + c*d + b*d.
Proof.
intros a b c d. try ring.
Abort.

Here's a theorem that ring, perhaps surprisingly, cannot prove.

Theorem sub_add_1 : forall a : nat, a - 1 + 1 = a.
Proof.
intros a.
try ring.
Abort.

What's going wrong here is that nat is really only a semi-ring, not a ring. That is, nat doesn't satisfy the axioms about negation. Why? Remember that the natural numbers stop at 0; we don't get any negative numbers. So if a is 0 in the above theorem, a-1 actually evaluates to 0 rather than -1.

Compute 0-1.
If we want to reason about the integers instead of the natural numbers, we can use a library called ZArith for that. The name comes from the fact that Z is used in mathematics to denote the integers.

Require Import ZArith.
Open Scope Z_scope.

The Open Scope command causes the ZArith library's scope to be used to resolve names, hence + becomes the operator on Z instead of on nat, as does -, etc.

Compute 0-1.
Now we can prove the theorem from before.

Theorem sub_add_1 : forall a : Z, a - 1 + 1 = a.
Proof.
intros a. ring.
Qed.

Before going on, let's close the Z scope so that the operators go back to working on nat, as usual.

Close Scope Z_scope.

Compute 0-1.
Coq also provides implementations of the rational numbers as a field, as well as the real numbers as a field. To get the field tactic, we first need to load the Field library.

Require Import Field.

The rational numbers are provided in a couple different Coq libraries; the one we'll use here is Qcanon. In mathematics, Q denotes the rational numbers, and canon indicates that the numbers are stored in a canonical form---that is, as simplified fractions. For example, Qcanon would represent 2/4 as 1/2, eliminating the common factor of 2 from the numerator and the denominator. (The QArith library provides rational numbers that are not in canonical form.)

Require Import Qcanon.
Open Scope Qc_scope.

Theorem frac_qc: forall x y z : Qc, z <> 0 -> (x + y) / z = x / z + y /z.
Proof.
intros x y z z_not_0.
field. assumption.
Qed.

Close Scope Qc_scope.

The real numbers are provided in the Reals library. Here's that same theorem again.

Module RealExample.

This code is in its own module for an annoying reason: Reals redefines its own nil, which will interefere with the examples want to give further below in this file with lists.

Require Import Reals.
Open Scope R_scope.

Theorem frac_r : forall x y z, z <> 0 -> (x + y) / z = x / z + y /z.
Proof.
intros x y z z_not_0.
field. assumption.
Qed.

The assumption that z <> 0 was needed in the above theorems to avoid division by zero. If we omitted that assumption, the field tactic would leave us with an unprovable subgoal, as in the proof below.

Theorem frac_r_broken : forall x y z, (x + y) / z = x / z + y /z.
Proof.
intros x y z.
field.
Abort.

Close Scope R_scope.
End RealExample.

## Induction principles

When we studied the Curry-Howard correspondence, we learned that proofs correspond to programs. That correspondence applies to inductive proofs as well, and as it turns out, inductive proofs correspond to recursive programs. Intuitively, that's because an inductive proof involves an inductive hypothesis---which is an instance of the theorem that is being proved, but applied to a smaller value. Likewise, recursive programs involve recursive calls---which are like another instance of the function that is already being evaluated, but on a smaller value.
To get a more concrete idea of what this means, let's look at the proof value (i.e., program) that Coq produces for our original inductive proof in these notes:

Print app_nil.

Coq responds:
```app_nil =
fun (A : Type) (lst : list A) =>
list_ind (fun lst0 : list A => lst0 ++ nil = lst0) eq_refl
(fun (h : A) (t : list A) (IH : t ++ nil = t) =>
eq_ind_r (fun l : list A => h :: l = h :: t) eq_refl IH) lst
: forall (A : Type) (lst : list A), lst ++ nil = lst
```
That's dense, but let's start picking it apart. First, we see that app_nil is a function that takes in two arguments: A and lst. Then it immediately applies another function named list_ind. That function was defined for us in the standard library, and it's what "implements" induction on lists. Let's check it out:

Check list_ind.

Coq responds:
```list_ind
: forall (A : Type) (P : list A -> Prop),
P nil ->
(forall (a : A) (l : list A), P l -> P (a :: l)) ->
forall l : list A, P l
```
We call list_ind the induction principle for lists. It is a proposition that says, intuitively, that induction is a valid reasoning principle for lists. In more detail, it takes these arguments:
• A, which is the type of the list elements.
• P, which is the property to be proved by induction. For example, the property being proved in app_nil is fun (lst: list A) => lst ++ nil = lst.
• P nil, which is a proof that P holds of the empty list. In other words, a proof of the base case.
• A final argument of type forall (a : A) (l : list A), P l -> P (a :: l). This is the proof of the inductive case. It takes an argument a, which is the head of a list, l, which is the tail of a list, and a proof P l that P holds of l. So, P l is the inductive hypothesis. The output is of type P (a :: l), which is a proof that P holds of a::l.
Finally, list_ind returns a value of type forall l : list A, P l, which is a proof that P holds of all lists.
Ok, so that's the type of list_ind: a proposition asserting that if you have a proof of the base case, and a proof of the inductive case, you can assemble those to prove that a property holds of a list. Next, what's the value of list_ind? In other words, what's the proof that list_ind itself is actually a true proposition?

Print list_ind.

Coq responds:
```list_ind =
fun (A : Type) (P : list A -> Prop) => list_rect P
...
```
So list_ind takes in A and P and just immediately applies another function, list_rect, to P. (The name rect is not especially helpful to understand, but alludes to recursion over a type.) Before we look at list_rect's actual implementation, let's look at our own equivalent implementation that is easier to read:

Fixpoint my_list_rect
(A : Type)
(P : list A -> Prop)
(baseCase : P nil)
(inductiveCase : forall (h : A) (t : list A), P t -> P (h::t))
(lst : list A)
: P lst
:=
match lst with
| nil => baseCase
| h::t => inductiveCase h t
(my_list_rect A P baseCase inductiveCase t)
end.

The arguments to my_list_rect are the same as the arguments to list_ind: an element type, a property to be proved, a proof of the base case, and a proof of the inductive case. Then my_list_rect takes an argument lst, which is the list for which we want to prove that P holds. Finally, my_list_rect returns that proof specifically for lst.
The body of my_list_rect constructs the proof that P holds of lst. It does so by matching against lst:
• If lst is empty, then my_list_rect returns the proof of the base case.
• If lst is h::t, then my_list_rect returns the proof of the inductive case. To construct that proof, it applies inductiveCase to h and t as the head and tail. But inductiveCase also requires a final argument, which is the proof that P holds of t. To construct that proof, my_list_rect calls itself recursively on t.
That recursive call is exactly why we said that inductive proofs are recursive programs. The inductive proof needs evidence that the inductive hypothesis holds of the smaller list, and recursing on that smaller list produces the evidence.
It's not immediately obvious, but my_list_rect is almost just fold_right. Here's how we could implement fold_right in Coq, with a slightly different argument order than the same function in OCaml:

Fixpoint my_fold_right
{A B : Type}
(init : B)
(f : A -> B -> B)
(lst : list A)
:=
match lst with
| nil => init
| h::t => f h (my_fold_right init f t)
end.

Now compare the body of my_fold_right with my_list_rect:
```my_fold_right's body:

match lst with
| nil => init
| h::t => f h (my_fold_right init f t)
end.

my_list_rect's body:

match lst with
| nil => baseCase
| h::t => inductiveCase h t (my_list_rect A P baseCase inductiveCase t)
end.
```
Both match against lst. If lst is empty, both return an initial/base-case value. If lst is non-empty, both recurse on the tail, then pass the result of the recursive call to a function (f or inductiveCase) that combines that result with the head. The only essential difference is that f does not take t directly as an input, whereas inductiveCase does.
So there you have it: induction over a list is really just folding over the list, eventually reaching the empty list and returning the proof of the base case for it, then working the way back up the call stack, assembling an ever-larger proof for each element of the list. An inductive proof is a recursive program.
Going back to the actual definition of list_rect, here it is:

Print list_rect.

Coq responds:
```list_rect =
fun (A : Type) (P : list A -> Type) (f : P nil)
(f0 : forall (a : A) (l : list A), P l -> P (a :: l)) =>
fix F (l : list A) : P l :=
match l as l0 return (P l0) with
| nil => f
| y :: l0 => f0 y l0 (F l0)
end
: forall (A : Type) (P : list A -> Type),
P nil ->
(forall (a : A) (l : list A), P l -> P (a :: l)) ->
forall l : list A, P l
```
That uses different syntax, but it ends up defining the same function as my_list_rect.
Whenever you define an inductive type, Coq automatically generates the induction principle and recursive function that implements it for you. For example, we could define our own lists:

Inductive mylist (A:Type) : Type :=
| mynil : mylist A
| mycons : A -> mylist A -> mylist A.

Coq automatically generates mylist_ind for us:

Print mylist_ind.

## Extraction

Coq makes it possible to extract OCaml code (or Haskell or Scheme) from Coq code. That makes it possible for us to
• write Coq code,
• prove the Coq code is correct, and
• extract OCaml code that can be compiled and run more efficiently than the original Coq code.
Let's first prove that a tail recursive factorial is equivalent to the non-tail-recursive one, and then extract the code for the tail recursive factorial.

Fixpoint fact (n:nat) : nat :=
match n with
| 0 => 1
| S k => n * fact k
end.

Fixpoint fact_tail_rec' (n : nat) (acc: nat) : nat :=
match n with
| 0 => acc
| S k => fact_tail_rec' k (acc * n)
end.

Definition fact_tail_rec (n : nat) := fact_tail_rec' n 1.

We need to prove an intermediate lemma about fact_tail_rec' for the proof of our main theorem to go through.

Lemma fact_tail_rec_lem : forall n acc,
fact_tail_rec' n acc = acc * fact_tail_rec' n 1.
Proof.
intro n.
induction n.
- intro acc. simpl. ring.
- intro acc. simpl (fact_tail_rec' (S n) 1). rewrite IHn. simpl. rewrite IHn. ring.
Qed.

In the above proof, the simpl tactic is applied with a specific pattern only on which simplification occurs. This is done so that the subsequent rewrite tactic does not pick the wrong term to rewrite. Try changing simpl (fact_tail_rec' (S n) 1) to simpl and make the proof go through.
Now we are ready to prove our main theorem. The proof involves induction on the input and an application of the lemma fact_tail_rec_lem that we had proved.

Theorem fact_tail_rec_ok : forall n, fact n = fact_tail_rec n.
Proof.
unfold fact_tail_rec.
induction n.
- simpl. trivial.
- simpl. rewrite fact_tail_rec_lem. simpl. rewrite IHn. ring.
Qed.

Let's extract fact_tail_rec as an example.

Require Import Extraction.
Extraction Language OCaml.
Extraction "/tmp/fact.ml" fact_tail_rec.

That produces the following file:
```
type nat =
| O
| S of nat

(** val add : nat -> nat -> nat **)

let rec add n m =
match n with
| O -> m
| S p -> S (add p m)

(** val mul : nat -> nat -> nat **)

let rec mul n m =
match n with
| O -> O
| S p -> add m (mul p m)

(** val fact_tail_rec' : nat -> nat -> nat **)

let rec fact_tail_rec' n acc =
match n with
| O -> acc
| S k -> fact_tail_rec' k (mul acc n)

(** val fact_tail_rec : nat -> nat **)

let fact_tail_rec n =
fact_tail_rec' n (S O)

```
As you can see, Coq has preserved the nat type in this extracted code. Unforunately, computation on natural numbers is not efficient. (Addition requires linear time; multiplication, quadratic!)
We can direct Coq to extract its own nat type to OCaml's int type as follows:

Extract Inductive nat =>
int [ "0" "succ" ] "(fun fO fS n -> if n=0 then fO () else fS (n-1))".
Extract Inlined Constant Init.Nat.mul => "( * )".

The first command says to
• use int instead of nat in the extract code,
• use 0 instead of O and succ instead of S (the succ function is in Pervasives and is fun x -> x + 1), and
• use the provided function to emulate pattern matching over the type.
The second command says to use OCaml's integer ( * ) operator instead of Coq's natural-number multiplication operator.
After issuing those commands, the extraction looks cleaner:

Extraction "/tmp/fact.ml" fact_tail_rec.

```
(** val fact_tail_rec' : int -> int -> int **)

let rec fact_tail_rec' n acc =
(fun fO fS n -> if n=0 then fO () else fS (n-1))
(fun _ -> acc)
(fun k -> fact_tail_rec' k (( * ) acc n))
n

(** val fact_tail_rec : int -> int **)

let fact_tail_rec n =
fact_tail_rec' n (succ 0)

```
There is, however, a tradeoff. The original version we extracted worked (albeit inefficiently) for arbitrarily large numbers without any error. But the second version is subject to integer overflow errors. So the proofs of correctness that we did for fact_tail_rec are no longer completely applicable: they hold only up to the limits of the types we subsituted during extraction.
Do we truly care about the limits of machine arithmetic? Maybe, maybe not. For sake of this little example, we might not. If we were verifying software to control the flight dynamics of a space shuttle, maybe we would. The Coq standard library does contain a module 31-bit integers and operators on them, which we could use if we wanted to precisely model what would happen on a particular architecture.

## Summary

Coq excels as a proof assistant when it comes to proof by induction. Whenever we define an inductive type, Coq generates an induction principle for us automatically. That principle is really a recursive program that knows how to assemble evidence for a proposition, given the constructors of the inductive type. The induction tactic manages the proof for us, automatically figuring out what the base case and the inductive case, and automatically generating the inductive hypothesis.

## Terms and concepts

• append
• base case
• field
• fix
• Fixpoint
• induction
• induction principle
• inductive case
• inductive hypothesis
• lemma
• Peano natural numbers
• Prop vs bool
• ring
• searching for library theorems
• semi-ring
• syntactically smaller restriction on recursive calls

## Tactics

• field
• induction
• rewrite
• ring
• tacticals: try