# Library Cpdt.InductiveTypes

The logical foundation of Coq is the Calculus of Inductive Constructions, or CIC. In a sense, CIC is built from just two relatively straightforward features: function types and inductive types. From this modest foundation, we can prove essentially all of the theorems of math and carry out effectively all program verifications, with enough effort expended. This chapter introduces induction and recursion for functional programming in Coq. Most of our examples reproduce functionality from the Coq standard library, and I have tried to copy the standard library's choices of identifiers, where possible, so many of the definitions here are already available in the default Coq environment.
The last chapter took a deep dive into some of the more advanced Coq features, to highlight the unusual approach that I advocate in this book. However, from this point on, we will rewind and go back to basics, presenting the relevant features of Coq in a more bottom-up manner. A useful first step is a discussion of the differences and relationships between proofs and programs in Coq.

# Proof Terms

Mainstream presentations of mathematics treat proofs as objects that exist outside of the universe of mathematical objects. However, for a variety of reasoning tasks, it is convenient to encode proofs, traditional mathematical objects, and programs within a single formal language. Validity checks on mathematical objects are useful in any setting, to catch typos and other uninteresting errors. The benefits of static typing for programs are widely recognized, and Coq brings those benefits to both mathematical objects and programs via a uniform mechanism. In fact, from this point on, we will not bother to distinguish between programs and mathematical objects. Many mathematical formalisms are most easily encoded in terms of programs.
Proofs are fundamentally different from programs, because any two proofs of a theorem are considered equivalent, from a formal standpoint if not from an engineering standpoint. However, we can use the same type-checking technology to check proofs as we use to validate our programs. This is the Curry-Howard correspondence , an approach for relating proofs and programs. We represent mathematical theorems as types, such that a theorem's proofs are exactly those programs that type-check at the corresponding type.
The last chapter's example already snuck in an instance of Curry-Howard. We used the token -> to stand for both function types and logical implications. One reasonable conclusion upon seeing this might be that some fancy overloading of notations is at work. In fact, functions and implications are precisely identical according to Curry-Howard! That is, they are just two ways of describing the same computational phenomenon.
A short demonstration should explain how this can be. The identity function over the natural numbers is certainly not a controversial program.

Check (fun x : nat => x).
: nat -> nat
Consider this alternate program, which is almost identical to the last one.

Check (fun x : True => x).
: True -> True
The identity program is interpreted as a proof that True, the always-true proposition, implies itself! What we see is that Curry-Howard interprets implications as functions, where an input is a proposition being assumed and an output is a proposition being deduced. This intuition is not too far from a common one for informal theorem proving, where we might already think of an implication proof as a process for transforming a hypothesis into a conclusion.
There are also more primitive proof forms available. For instance, the term I is the single proof of True, applicable in any context.

Check I.
: True
With I, we can prove another simple propositional theorem.

Check (fun _ : False => I).
: False -> True
No proofs of False exist in the top-level context, but the implication-as-function analogy gives us an easy way to, for example, show that False implies itself.

Check (fun x : False => x).
: False -> False
Every one of these example programs whose type looks like a logical formula is a proof term. We use that name for any Gallina term of a logical type, and we will elaborate shortly on what makes a type logical.
In the rest of this chapter, we will introduce different ways of defining types. Every example type can be interpreted alternatively as a type of programs or proofs.
One of the first types we introduce will be bool, with constructors true and false. Newcomers to Coq often wonder about the distinction between True and true and the distinction between False and false. One glib answer is that True and False are types, but true and false are not. A more useful answer is that Coq's metatheory guarantees that any term of type bool evaluates to either true or false. This means that we have an algorithm for answering any question phrased as an expression of type bool. Conversely, most propositions do not evaluate to True or False; the language of inductively defined propositions is much richer than that. We ought to be glad that we have no algorithm for deciding our formalized version of mathematical truth, since otherwise it would be clear that we could not formalize undecidable properties, like almost any interesting property of general-purpose programs.

# Enumerations

Coq inductive types generalize the algebraic datatypes found in Haskell and ML. Confusingly enough, inductive types also generalize generalized algebraic datatypes (GADTs), by adding the possibility for type dependency. Even so, it is worth backing up from the examples of the last chapter and going over basic, algebraic-datatype uses of inductive datatypes, because the chance to prove things about the values of these types adds new wrinkles beyond usual practice in Haskell and ML.
The singleton type unit is an inductive type:

Inductive unit : Set :=
| tt.

This vernacular command defines a new inductive type unit whose only value is tt. We can verify the types of the two identifiers we introduce:

Check unit.
unit : Set

Check tt.
tt : unit
We can prove that unit is a genuine singleton type.

Theorem unit_singleton : forall x : unit, x = tt.

The important thing about an inductive type is, unsurprisingly, that you can do induction over its values, and induction is the key to proving this theorem. We ask to proceed by induction on the variable x.

induction x.

The goal changes to:
tt = tt
...which we can discharge trivially.

reflexivity.
Qed.

It seems kind of odd to write a proof by induction with no inductive hypotheses. We could have arrived at the same result by beginning the proof with:
destruct x.
...which corresponds to "proof by case analysis" in classical math. For non-recursive inductive types, the two tactics will always have identical behavior. Often case analysis is sufficient, even in proofs about recursive types, and it is nice to avoid introducing unneeded induction hypotheses.
What exactly is the induction principle for unit? We can ask Coq:

Check unit_ind.
unit_ind : forall P : unit -> Prop, P tt -> forall u : unit, P u
Every Inductive command defining a type T also defines an induction principle named T_ind. Recall from the last section that our type, operations over it, and principles for reasoning about it all live in the same language and are described by the same type system. The key to telling what is a program and what is a proof lies in the distinction between the type Prop, which appears in our induction principle; and the type Set, which we have seen a few times already.
The convention goes like this: Set is the type of normal types used in programming, and the values of such types are programs. Prop is the type of logical propositions, and the values of such types are proofs. Thus, an induction principle has a type that shows us that it is a function for building proofs.
Specifically, unit_ind quantifies over a predicate P over unit values. If we can present a proof that P holds of tt, then we are rewarded with a proof that P holds for any value u of type unit. In our last proof, the predicate was (fun u : unit => u = tt).
The definition of unit places the type in Set. By replacing Set with Prop, unit with True, and tt with I, we arrive at precisely the definition of True that the Coq standard library employs! The program type unit is the Curry-Howard equivalent of the proposition True. We might make the tongue-in-cheek claim that, while philosophers have expended much ink on the nature of truth, we have now determined that truth is the unit type of functional programming.
We can define an inductive type even simpler than unit:

Inductive Empty_set : Set := .

Empty_set has no elements. We can prove fun theorems about it:

Theorem the_sky_is_falling : forall x : Empty_set, 2 + 2 = 5.
destruct 1.
Qed.

Because Empty_set has no elements, the fact of having an element of this type implies anything. We use destruct 1 instead of destruct x in the proof because unused quantified variables are relegated to being referred to by number. (There is a good reason for this, related to the unity of quantifiers and implication. At least within Coq's logical foundation of constructive logic, which we elaborate on more in the next chapter, an implication is just a quantification over a proof, where the quantified variable is never used. It generally makes more sense to refer to implication hypotheses by number than by name, and Coq treats our quantifier over an unused variable as an implication in determining the proper behavior.)
We can see the induction principle that made this proof so easy:

Check Empty_set_ind.
Empty_set_ind : forall (P : Empty_set -> Prop) (e : P e
In other words, any predicate over values from the empty set holds vacuously of every such element. In the last proof, we chose the predicate (fun _ : Empty_set => 2 + 2 = 5).
We can also apply this get-out-of-jail-free card programmatically. Here is a lazy way of converting values of Empty_set to values of unit:

Definition e2u (e : Empty_set) : unit := match e with end.

We employ match pattern matching as in the last chapter. Since we match on a value whose type has no constructors, there is no need to provide any branches. It turns out that Empty_set is the Curry-Howard equivalent of False. As for why Empty_set starts with a capital letter and not a lowercase letter like unit does, we must refer the reader to the authors of the Coq standard library, to which we try to be faithful.
Moving up the ladder of complexity, we can define the Booleans:

Inductive bool : Set :=
| true
| false.

We can use less vacuous pattern matching to define Boolean negation.

Definition negb (b : bool) : bool :=
match b with
| true => false
| false => true
end.

An alternative definition desugars to the above, thanks to an if notation overloaded to work with any inductive type that has exactly two constructors:

Definition negb' (b : bool) : bool :=
if b then false else true.

We might want to prove that negb is its own inverse operation.

Theorem negb_inverse : forall b : bool, negb (negb b) = b.
destruct b.

After we case-analyze on b, we are left with one subgoal for each constructor of bool.
2 subgoals

============================
negb (negb true) = true

subgoal 2 is

negb (negb false) = false
The first subgoal follows by Coq's rules of computation, so we can dispatch it easily:

reflexivity.

Likewise for the second subgoal, so we can restart the proof and give a very compact justification.

Restart.

destruct b; reflexivity.
Qed.

Another theorem about Booleans illustrates another useful tactic.

Theorem negb_ineq : forall b : bool, negb b <> b.
destruct b; discriminate.
Qed.

The discriminate tactic is used to prove that two values of an inductive type are not equal, whenever the values are formed with different constructors. In this case, the different constructors are true and false.
At this point, it is probably not hard to guess what the underlying induction principle for bool is.

Check bool_ind.
bool_ind : forall P : bool -> Prop, P true -> P false -> forall b : bool, P b
That is, to prove that a property describes all bools, prove that it describes both true and false.
There is no interesting Curry-Howard analogue of bool. Of course, we can define such a type by replacing Set by Prop above, but the proposition we arrive at is not very useful. It is logically equivalent to True, but it provides two indistinguishable primitive proofs, true and false. In the rest of the chapter, we will skip commenting on Curry-Howard versions of inductive definitions where such versions are not interesting.

# Simple Recursive Types

The natural numbers are the simplest common example of an inductive type that actually deserves the name.

Inductive nat : Set :=
| O : nat
| S : nat -> nat.

The constructor O is zero, and S is the successor function, so that 0 is syntactic sugar for O, 1 for S O, 2 for S (S O), and so on.
Pattern matching works as we demonstrated in the last chapter:

Definition isZero (n : nat) : bool :=
match n with
| O => true
| S _ => false
end.

Definition pred (n : nat) : nat :=
match n with
| O => O
| S n' => n'
end.

We can prove theorems by case analysis with destruct as for simpler inductive types, but we can also now get into genuine inductive theorems. First, we will need a recursive function, to make things interesting.

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

Recall that Fixpoint is Coq's mechanism for recursive function definitions. Some theorems about plus can be proved without induction.

Theorem O_plus_n : forall n : nat, plus O n = n.
intro; reflexivity.
Qed.

Coq's computation rules automatically simplify the application of plus, because unfolding the definition of plus gives us a match expression where the branch to be taken is obvious from syntax alone. If we just reverse the order of the arguments, though, this no longer works, and we need induction.

Theorem n_plus_O : forall n : nat, plus n O = n.
induction n.

Our first subgoal is plus O O = O, which is trivial by computation.

reflexivity.

Our second subgoal requires more work and also demonstrates our first inductive hypothesis.

n : nat
IHn : plus n O = n
============================
plus (S n) O = S n

We can start out by using computation to simplify the goal as far as we can.

simpl.

Now the conclusion is S (plus n O) = S n. Using our inductive hypothesis:

rewrite IHn.

...we get a trivial conclusion S n = S n.

reflexivity.

Not much really went on in this proof, so the crush tactic from the CpdtTactics module can prove this theorem automatically.

Restart.

induction n; crush.
Qed.

We can check out the induction principle at work here:

Check nat_ind.

nat_ind : forall P : nat -> Prop,
P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
Each of the two cases of our last proof came from the type of one of the arguments to nat_ind. We chose P to be (fun n : nat => plus n O = n). The first proof case corresponded to P O and the second case to (forall n : nat, P n -> P (S n)). The free variable n and inductive hypothesis IHn came from the argument types given here.
Since nat has a constructor that takes an argument, we may sometimes need to know that that constructor is injective.

Theorem S_inj : forall n m : nat, S n = S m -> n = m.
injection 1; trivial.
Qed.

The injection tactic refers to a premise by number, adding new equalities between the corresponding arguments of equated terms that are formed with the same constructor. We end up needing to prove n = m -> n = m, so it is unsurprising that a tactic named trivial is able to finish the proof. This tactic attempts a variety of single proof steps, drawn from a user-specified database that we will later see how to extend.
There is also a very useful tactic called congruence that can prove this theorem immediately. The congruence tactic generalizes discriminate and injection, and it also adds reasoning about the general properties of equality, such as that a function returns equal results on equal arguments. That is, congruence is a complete decision procedure for the theory of equality and uninterpreted functions, plus some smarts about inductive types.
We can define a type of lists of natural numbers.

Inductive nat_list : Set :=
| NNil : nat_list
| NCons : nat -> nat_list -> nat_list.

Recursive definitions over nat_list are straightforward extensions of what we have seen before.

Fixpoint nlength (ls : nat_list) : nat :=
match ls with
| NNil => O
| NCons _ ls' => S (nlength ls')
end.

Fixpoint napp (ls1 ls2 : nat_list) : nat_list :=
match ls1 with
| NNil => ls2
| NCons n ls1' => NCons n (napp ls1' ls2)
end.

Inductive theorem proving can again be automated quite effectively.

Theorem nlength_napp : forall ls1 ls2 : nat_list, nlength (napp ls1 ls2)
= plus (nlength ls1) (nlength ls2).
induction ls1; crush.
Qed.

Check nat_list_ind.

nat_list_ind
: forall P : nat_list -> Prop,
P NNil ->
(forall (n : nat) (n0 : nat_list), P n0 -> P (NCons n n0)) ->
forall n : nat_list, P n
In general, we can implement any "tree" type as an inductive type. For example, here are binary trees of naturals.

Inductive nat_btree : Set :=
| NLeaf : nat_btree
| NNode : nat_btree -> nat -> nat_btree -> nat_btree.

Here are two functions whose intuitive explanations are not so important. The first one computes the size of a tree, and the second performs some sort of splicing of one tree into the leftmost available leaf node of another.

Fixpoint nsize (tr : nat_btree) : nat :=
match tr with
| NLeaf => S O
| NNode tr1 _ tr2 => plus (nsize tr1) (nsize tr2)
end.

Fixpoint nsplice (tr1 tr2 : nat_btree) : nat_btree :=
match tr1 with
| NLeaf => NNode tr2 O NLeaf
| NNode tr1' n tr2' => NNode (nsplice tr1' tr2) n tr2'
end.

Theorem plus_assoc : forall n1 n2 n3 : nat, plus (plus n1 n2) n3 = plus n1 (plus n2 n3).
induction n1; crush.
Qed.

Hint Rewrite n_plus_O plus_assoc.

Theorem nsize_nsplice : forall tr1 tr2 : nat_btree, nsize (nsplice tr1 tr2)
= plus (nsize tr2) (nsize tr1).
induction tr1; crush.
Qed.

It is convenient that these proofs go through so easily, but it is still useful to look into the details of what happened, by checking the statement of the tree induction principle.

Check nat_btree_ind.

nat_btree_ind
: forall P : nat_btree -> Prop,
P NLeaf ->
(forall n : nat_btree,
P n -> forall (n0 : nat) (n1 : nat_btree), P n1 -> P (NNode n n0 n1)) ->
forall n : nat_btree, P n
We have the usual two cases, one for each constructor of nat_btree.

# Parameterized Types

We can also define polymorphic inductive types, as with algebraic datatypes in Haskell and ML.

Inductive list (T : Set) : Set :=
| Nil : list T
| Cons : T -> list T -> list T.

Fixpoint length T (ls : list T) : nat :=
match ls with
| Nil => O
| Cons _ ls' => S (length ls')
end.

Fixpoint app T (ls1 ls2 : list T) : list T :=
match ls1 with
| Nil => ls2
| Cons x ls1' => Cons x (app ls1' ls2)
end.

Theorem length_app : forall T (ls1 ls2 : list T), length (app ls1 ls2)
= plus (length ls1) (length ls2).
induction ls1; crush.
Qed.

There is a useful shorthand for writing many definitions that share the same parameter, based on Coq's section mechanism. The following block of code is equivalent to the above:

Section list.
Variable T : Set.

Inductive list : Set :=
| Nil : list
| Cons : T -> list -> list.

Fixpoint length (ls : list) : nat :=
match ls with
| Nil => O
| Cons _ ls' => S (length ls')
end.

Fixpoint app (ls1 ls2 : list) : list :=
match ls1 with
| Nil => ls2
| Cons x ls1' => Cons x (app ls1' ls2)
end.

Theorem length_app : forall ls1 ls2 : list, length (app ls1 ls2)
= plus (length ls1) (length ls2).
induction ls1; crush.
Qed.
End list.

Arguments Nil [T].

After we end the section, the Variables we used are added as extra function parameters for each defined identifier, as needed. With an Arguments command, we ask that T be inferred when we use Nil; Coq's heuristics already decided to apply a similar policy to Cons, because of the Set Implicit Arguments command elided at the beginning of this chapter. We verify that our definitions have been saved properly using the Print command, a cousin of Check which shows the definition of a symbol, rather than just its type.

Print list.

Inductive list (T : Set) : Set :=
Nil : list T | Cons : T -> list T -> list T
The final definition is the same as what we wrote manually before. The other elements of the section are altered similarly, turning out exactly as they were before, though we managed to write their definitions more succinctly.

Check length.

length
: forall T : Set, list T -> nat
The parameter T is treated as a new argument to the induction principle, too.

Check list_ind.

list_ind
: forall (T : Set) (P : list T -> Prop),
P (Nil T) ->
(forall (t : T) (l : list T), P l -> P (Cons t l)) ->
forall l : list T, P l
Thus, despite a very real sense in which the type T is an argument to the constructor Cons, the inductive case in the type of list_ind (i.e., the third line of the type) includes no quantifier for T, even though all of the other arguments are quantified explicitly. Parameters in other inductive definitions are treated similarly in stating induction principles.

# Mutually Inductive Types

We can define inductive types that refer to each other:

Inductive even_list : Set :=
| ENil : even_list
| ECons : nat -> odd_list -> even_list

with odd_list : Set :=
| OCons : nat -> even_list -> odd_list.

Fixpoint elength (el : even_list) : nat :=
match el with
| ENil => O
| ECons _ ol => S (olength ol)
end

with olength (ol : odd_list) : nat :=
match ol with
| OCons _ el => S (elength el)
end.

Fixpoint eapp (el1 el2 : even_list) : even_list :=
match el1 with
| ENil => el2
| ECons n ol => ECons n (oapp ol el2)
end

with oapp (ol : odd_list) (el : even_list) : odd_list :=
match ol with
| OCons n el' => OCons n (eapp el' el)
end.

Everything is going roughly the same as in past examples, until we try to prove a theorem similar to those that came before.

Theorem elength_eapp : forall el1 el2 : even_list,
elength (eapp el1 el2) = plus (elength el1) (elength el2).
induction el1; crush.

One goal remains:

n : nat
o : odd_list
el2 : even_list
============================
S (olength (oapp o el2)) = S (plus (olength o) (elength el2))
We have no induction hypothesis, so we cannot prove this goal without starting another induction, which would reach a similar point, sending us into a futile infinite chain of inductions. The problem is that Coq's generation of T_ind principles is incomplete. We only get non-mutual induction principles generated by default.

Abort.
Check even_list_ind.

even_list_ind
: forall P : even_list -> Prop,
P ENil ->
(forall (n : nat) (o : odd_list), P (ECons n o)) ->
forall e : even_list, P e
We see that no inductive hypotheses are included anywhere in the type. To get them, we must ask for mutual principles as we need them, using the Scheme command.

Scheme even_list_mut := Induction for even_list Sort Prop
with odd_list_mut := Induction for odd_list Sort Prop.

This invocation of Scheme asks for the creation of induction principles even_list_mut for the type even_list and odd_list_mut for the type odd_list. The Induction keyword says we want standard induction schemes, since Scheme supports more exotic choices. Finally, Sort Prop establishes that we really want induction schemes, not recursion schemes, which are the same according to Curry-Howard, save for the Prop/Set distinction.

Check even_list_mut.

even_list_mut
: forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
P ENil ->
(forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
(forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
forall e : even_list, P e
This is the principle we wanted in the first place.
The Scheme command is for asking Coq to generate particular induction schemes that are mutual among a set of inductive types (possibly only one such type, in which case we get a normal induction principle). In a sense, it generalizes the induction scheme generation that goes on automatically for each inductive definition. Future Coq versions might make that automatic generation smarter, so that Scheme is needed in fewer places. In a few sections, we will see how induction principles are derived theorems in Coq, so that there is not actually any need to build in any automatic scheme generation.
There is one more wrinkle left in using the even_list_mut induction principle: the induction tactic will not apply it for us automatically. It will be helpful to look at how to prove one of our past examples without using induction, so that we can then generalize the technique to mutual inductive types.

Theorem n_plus_O' : forall n : nat, plus n O = n.
apply nat_ind.
Here we use apply, which is one of the most essential basic tactics. When we are trying to prove fact P, and when thm is a theorem whose conclusion can be made to match P by proper choice of quantified variable values, the invocation apply thm will replace the current goal with one new goal for each premise of thm.
This use of apply may seem a bit too magical. To better see what is going on, we use a variant where we partially apply the theorem nat_ind to give an explicit value for the predicate that gives our induction hypothesis.

Undo.
apply (nat_ind (fun n => plus n O = n)); crush.
Qed.

From this example, we can see that induction is not magic. It only does some bookkeeping for us to make it easy to apply a theorem, which we can do directly with the apply tactic.
This technique generalizes to our mutual example:

Theorem elength_eapp : forall el1 el2 : even_list,
elength (eapp el1 el2) = plus (elength el1) (elength el2).

apply (even_list_mut
(fun el1 : even_list => forall el2 : even_list,
elength (eapp el1 el2) = plus (elength el1) (elength el2))
(fun ol : odd_list => forall el : even_list,
olength (oapp ol el) = plus (olength ol) (elength el))); crush.
Qed.

We simply need to specify two predicates, one for each of the mutually inductive types. In general, it is not a good idea to assume that a proof assistant can infer extra predicates, so this way of applying mutual induction is about as straightforward as we may hope for.

# Reflexive Types

A kind of inductive type called a reflexive type includes at least one constructor that takes as an argument a function returning the same type we are defining. One very useful class of examples is in modeling variable binders. Our example will be an encoding of the syntax of first-order logic. Since the idea of syntactic encodings of logic may require a bit of acclimation, let us first consider a simpler formula type for a subset of propositional logic. We are not yet using a reflexive type, but later we will extend the example reflexively.
A key distinction here is between, for instance, the syntax Truth and its semantics True. We can make the semantics explicit with a recursive function. This function uses the infix operator /\, which desugars to instances of the type family and from the standard library. The family and implements conjunction, the Prop Curry-Howard analogue of the usual pair type from functional programming (which is the type family prod in Coq's standard library).

Fixpoint pformulaDenote (f : pformula) : Prop :=
match f with
| Truth => True
| Falsehood => False
end.

This is just a warm-up that does not use reflexive types, the new feature we mean to introduce. When we set our sights on first-order logic instead, it becomes very handy to give constructors recursive arguments that are functions.

Inductive formula : Set :=
| Eq : nat -> nat -> formula
| And : formula -> formula -> formula
| Forall : (nat -> formula) -> formula.

Our kinds of formulas are equalities between naturals, conjunction, and universal quantification over natural numbers. We avoid needing to include a notion of "variables" in our type, by using Coq functions to encode the syntax of quantification. For instance, here is the encoding of forall x : nat, x = x:

Example forall_refl : formula := Forall (fun x => Eq x x).

We can write recursive functions over reflexive types quite naturally. Here is one translating our formulas into native Coq propositions.

Fixpoint formulaDenote (f : formula) : Prop :=
match f with
| Eq n1 n2 => n1 = n2
| Forall f' => forall n : nat, formulaDenote (f' n)
end.

We can also encode a trivial formula transformation that swaps the order of equality and conjunction operands.

Fixpoint swapper (f : formula) : formula :=
match f with
| Eq n1 n2 => Eq n2 n1
| And f1 f2 => And (swapper f2) (swapper f1)
| Forall f' => Forall (fun n => swapper (f' n))
end.

It is helpful to prove that this transformation does not make true formulas false.

induction f; crush.
Qed.

We can take a look at the induction principle behind this proof.

Check formula_ind.

formula_ind
: forall P : formula -> Prop,
(forall n n0 : nat, P (Eq n n0)) ->
(forall f0 : formula,
P f0 -> forall f1 : formula, P f1 -> P (And f0 f1)) ->
(forall f1 : nat -> formula,
(forall n : nat, P (f1 n)) -> P (Forall f1)) ->
forall f2 : formula, P f2
Focusing on the Forall case, which comes third, we see that we are allowed to assume that the theorem holds for any application of the argument function f1. That is, Coq induction principles do not follow a simple rule that the textual representations of induction variables must get shorter in appeals to induction hypotheses. Luckily for us, the people behind the metatheory of Coq have verified that this flexibility does not introduce unsoundness.
Up to this point, we have seen how to encode in Coq more and more of what is possible with algebraic datatypes in Haskell and ML. This may have given the inaccurate impression that inductive types are a strict extension of algebraic datatypes. In fact, Coq must rule out some types allowed by Haskell and ML, for reasons of soundness. Reflexive types provide our first good example of such a case; only some of them are legal.
Given our last example of an inductive type, many readers are probably eager to try encoding the syntax of lambda calculus. Indeed, the function-based representation technique that we just used, called higher-order abstract syntax (HOAS), is the representation of choice for lambda calculi in Twelf and in many applications implemented in Haskell and ML. Let us try to import that choice to Coq:

Inductive term : Set :=
| App : term -> term -> term
| Abs : (term -> term) -> term.
```Error: Non strictly positive occurrence of "term" in "(term -> term) -> term"
```
We have run afoul of the strict positivity requirement for inductive definitions, which says that the type being defined may not occur to the left of an arrow in the type of a constructor argument. It is important that the type of a constructor is viewed in terms of a series of arguments and a result, since obviously we need recursive occurrences to the lefts of the outermost arrows if we are to have recursive occurrences at all. Our candidate definition above violates the positivity requirement because it involves an argument of type term -> term, where the type term that we are defining appears to the left of an arrow. The candidate type of App is fine, however, since every occurrence of term is either a constructor argument or the final result type.
Why must Coq enforce this restriction? Imagine that our last definition had been accepted, allowing us to write this function:

Definition uhoh (t : term) : term :=
match t with
| Abs f => f t
| _ => t
end.
Using an informal idea of Coq's semantics, it is easy to verify that the application uhoh (Abs uhoh) will run forever. This would be a mere curiosity in OCaml and Haskell, where non-termination is commonplace, though the fact that we have a non-terminating program without explicit recursive function definitions is unusual.
For Coq, however, this would be a disaster. The possibility of writing such a function would destroy all our confidence that proving a theorem means anything. Since Coq combines programs and proofs in one language, we would be able to prove every theorem with an infinite loop.
Nonetheless, the basic insight of HOAS is a very useful one, and there are ways to realize most benefits of HOAS in Coq. We will study a particular technique of this kind in the final chapter, on programming language syntax and semantics.

# An Interlude on Induction Principles

As we have emphasized a few times already, Coq proofs are actually programs, written in the same language we have been using in our examples all along. We can get a first sense of what this means by taking a look at the definitions of some of the induction principles we have used. A close look at the details here will help us construct induction principles manually, which we will see is necessary for some more advanced inductive definitions.

Print nat_ind.

nat_ind =
fun P : nat -> Prop => nat_rect P
: forall P : nat -> Prop,
P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
We see that this induction principle is defined in terms of a more general principle, nat_rect. The rec stands for "recursion principle," and the t at the end stands for Type.

Check nat_rect.

nat_rect
: forall P : nat -> Type,
P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
The principle nat_rect gives P type nat -> Type instead of nat -> Prop. This Type is another universe, like Set and Prop. In fact, it is a common supertype of both. Later on, we will discuss exactly what the significances of the different universes are. For now, it is just important that we can use Type as a sort of meta-universe that may turn out to be either Set or Prop. We can see the symmetry inherent in the subtyping relationship by printing the definition of another principle that was generated for nat automatically:

Print nat_rec.

nat_rec =
fun P : nat -> Set => nat_rect P
: forall P : nat -> Set,
P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
This is identical to the definition for nat_ind, except that we have substituted Set for Prop. For most inductive types T, then, we get not just induction principles T_ind, but also recursion principles T_rec. We can use T_rec to write recursive definitions without explicit Fixpoint recursion. For instance, the following two definitions are equivalent:

Fixpoint plus_recursive (n : nat) : nat -> nat :=
match n with
| O => fun m => m
| S n' => fun m => S (plus_recursive n' m)
end.

Definition plus_rec : nat -> nat -> nat :=
nat_rec (fun _ : nat => nat -> nat) (fun m => m) (fun _ r m => S (r m)).

Theorem plus_equivalent : plus_recursive = plus_rec.
reflexivity.
Qed.

Going even further down the rabbit hole, nat_rect itself is not even a primitive. It is a functional program that we can write manually.

Print nat_rect.

nat_rect =
fun (P : nat -> Type) (f : P O) (f0 : forall n : nat, P n -> P (S n)) =>
fix F (n : nat) : P n :=
match n as n0 return (P n0) with
| O => f
| S n0 => f0 n0 (F n0)
end
: forall P : nat -> Type,
P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
The only new wrinkles here are, first, an anonymous recursive function definition, using the fix keyword of Gallina (which is like fun with recursion supported); and, second, the annotations on the match expression. This is a dependently typed pattern match, because the type of the expression depends on the value being matched on. We will meet more involved examples later, especially in Part II of the book.
Type inference for dependent pattern matching is undecidable, which can be proved by reduction from higher-order unification. Thus, we often find ourselves needing to annotate our programs in a way that explains dependencies to the type checker. In the example of nat_rect, we have an as clause, which binds a name for the discriminee; and a return clause, which gives a way to compute the match result type as a function of the discriminee.
To prove that nat_rect is nothing special, we can reimplement it manually.

Fixpoint nat_rect' (P : nat -> Type)
(HO : P O)
(HS : forall n, P n -> P (S n)) (n : nat) :=
match n return P n with
| O => HO
| S n' => HS n' (nat_rect' P HO HS n')
end.

We can understand the definition of nat_rect better by reimplementing nat_ind using sections.

Section nat_ind'.
First, we have the property of natural numbers that we aim to prove.

Variable P : nat -> Prop.

Then we require a proof of the O case, which we declare with the command Hypothesis, which is a synonym for Variable that, by convention, is used for variables whose types are propositions.

Hypothesis O_case : P O.

Next is a proof of the S case, which may assume an inductive hypothesis.

Hypothesis S_case : forall n : nat, P n -> P (S n).

Finally, we define a recursive function to tie the pieces together.

Fixpoint nat_ind' (n : nat) : P n :=
match n with
| O => O_case
| S n' => S_case (nat_ind' n')
end.
End nat_ind'.

Closing the section adds the Variables and Hypothesises as new fun-bound arguments to nat_ind', and, modulo the use of Prop instead of Type, we end up with the exact same definition that was generated automatically for nat_rect.
We can also examine the definition of even_list_mut, which we generated with Scheme for a mutually recursive type.

Print even_list_mut.

even_list_mut =
fun (P : even_list -> Prop) (P0 : odd_list -> Prop)
(f : P ENil) (f0 : forall (n : nat) (o : odd_list), P0 o -> P (ECons n o))
(f1 : forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) =>
fix F (e : even_list) : P e :=
match e as e0 return (P e0) with
| ENil => f
| ECons n o => f0 n o (F0 o)
end
with F0 (o : odd_list) : P0 o :=
match o as o0 return (P0 o0) with
| OCons n e => f1 n e (F e)
end
for F
: forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
P ENil ->
(forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
(forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
forall e : even_list, P e
We see a mutually recursive fix, with the different functions separated by with in the same way that they would be separated by and in ML. A final for clause identifies which of the mutually recursive functions should be the final value of the fix expression. Using this definition as a template, we can reimplement even_list_mut directly.

Section even_list_mut'.
First, we need the properties that we are proving.

Variable Peven : even_list -> Prop.
Variable Podd : odd_list -> Prop.

Next, we need proofs of the three cases.

Hypothesis ENil_case : Peven ENil.
Hypothesis ECons_case : forall (n : nat) (o : odd_list), Podd o -> Peven (ECons n o).
Hypothesis OCons_case : forall (n : nat) (e : even_list), Peven e -> Podd (OCons n e).

Finally, we define the recursive functions.

Fixpoint even_list_mut' (e : even_list) : Peven e :=
match e with
| ENil => ENil_case
| ECons n o => ECons_case n (odd_list_mut' o)
end
with odd_list_mut' (o : odd_list) : Podd o :=
match o with
| OCons n e => OCons_case n (even_list_mut' e)
end.
End even_list_mut'.

Even induction principles for reflexive types are easy to implement directly. For our formula type, we can use a recursive definition much like those we wrote above.

Section formula_ind'.
Variable P : formula -> Prop.
Hypothesis Eq_case : forall n1 n2 : nat, P (Eq n1 n2).
Hypothesis And_case : forall f1 f2 : formula,
P f1 -> P f2 -> P (And f1 f2).
Hypothesis Forall_case : forall f : nat -> formula,
(forall n : nat, P (f n)) -> P (Forall f).

Fixpoint formula_ind' (f : formula) : P f :=
match f with
| Eq n1 n2 => Eq_case n1 n2
| And f1 f2 => And_case (formula_ind' f1) (formula_ind' f2)
| Forall f' => Forall_case f' (fun n => formula_ind' (f' n))
end.
End formula_ind'.

It is apparent that induction principle implementations involve some tedium but not terribly much creativity.

# Nested Inductive Types

Suppose we want to extend our earlier type of binary trees to trees with arbitrary finite branching. We can use lists to give a simple definition.

Inductive nat_tree : Set :=
| NNode' : nat -> list nat_tree -> nat_tree.

This is an example of a nested inductive type definition, because we use the type we are defining as an argument to a parameterized type family. Coq will not allow all such definitions; it effectively pretends that we are defining nat_tree mutually with a version of list specialized to nat_tree, checking that the resulting expanded definition satisfies the usual rules. For instance, if we replaced list with a type family that used its parameter as a function argument, then the definition would be rejected as violating the positivity restriction.
As we encountered with mutual inductive types, we find that the automatically generated induction principle for nat_tree is too weak.

Check nat_tree_ind.

nat_tree_ind
: forall P : nat_tree -> Prop,
(forall (n : nat) (l : list nat_tree), P (NNode' n l)) ->
forall n : nat_tree, P n
There is no command like Scheme that will implement an improved principle for us. In general, it takes creativity to figure out good ways to incorporate nested uses of different type families. Now that we know how to implement induction principles manually, we are in a position to apply just such creativity to this problem.
Many induction principles for types with nested uses of list could benefit from a unified predicate capturing the idea that some property holds of every element in a list. By defining this generic predicate once, we facilitate reuse of library theorems about it. (Here, we are actually duplicating the standard library's Forall predicate, with a different implementation, for didactic purposes.)

Section All.
Variable T : Set.
Variable P : T -> Prop.

Fixpoint All (ls : list T) : Prop :=
match ls with
| Nil => True
| Cons h t => P h /\ All t
end.
End All.

It will be useful to review the definitions of True and /\, since we will want to write manual proofs of them below.

Print True.

Inductive True : Prop := I : True
That is, True is a proposition with exactly one proof, I, which we may always supply trivially.
Finding the definition of /\ takes a little more work. Coq supports user registration of arbitrary parsing rules, and it is such a rule that is letting us write /\ instead of an application of some inductive type family. We can find the underlying inductive type with the Locate command, whose argument may be a parsing token.

Locate "/\".

"A /\ B" := and A B : type_scope (default interpretation)

Print and.

Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
```  For conj: Arguments A, B are implicit
```
In addition to the definition of and itself, we get information on implicit arguments (and some other information that we omit here). The implicit argument information tells us that we build a proof of a conjunction by calling the constructor conj on proofs of the conjuncts, with no need to include the types of those proofs as explicit arguments.
Now we create a section for our induction principle, following the same basic plan as in the previous section of this chapter.

Section nat_tree_ind'.
Variable P : nat_tree -> Prop.

Hypothesis NNode'_case : forall (n : nat) (ls : list nat_tree),
All P ls -> P (NNode' n ls).

A first attempt at writing the induction principle itself follows the intuition that nested inductive type definitions are expanded into mutual inductive definitions.

Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
match tr with
| NNode' n ls => NNode'_case n ls (list_nat_tree_ind ls)
end

with list_nat_tree_ind (ls : list nat_tree) : All P ls :=
match ls with
| Nil => I
| Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
end.

Coq rejects this definition, saying
```  Recursive call to nat_tree_ind' has principal argument equal to "tr"
```
There is no deep theoretical reason why this program should be rejected; Coq applies incomplete termination-checking heuristics, and it is necessary to learn a few of the most important rules. The term "nested inductive type" hints at the solution to this particular problem. Just as mutually inductive types require mutually recursive induction principles, nested types require nested recursion.

Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
match tr with
| NNode' n ls => NNode'_case n ls
((fix list_nat_tree_ind (ls : list nat_tree) : All P ls :=
match ls with
| Nil => I
| Cons tr' rest => conj (nat_tree_ind' tr') (list_nat_tree_ind rest)
end) ls)
end.

We include an anonymous fix version of list_nat_tree_ind that is literally nested inside the definition of the recursive function corresponding to the inductive definition that had the nested use of list.
We can try our induction principle out by defining some recursive functions on nat_tree and proving a theorem about them. First, we define some helper functions that operate on lists.

Section map.
Variables T T' : Set.
Variable F : T -> T'.

Fixpoint map (ls : list T) : list T' :=
match ls with
| Nil => Nil
| Cons h t => Cons (F h) (map t)
end.
End map.

Fixpoint sum (ls : list nat) : nat :=
match ls with
| Nil => O
| Cons h t => plus h (sum t)
end.

Now we can define a size function over our trees.

Fixpoint ntsize (tr : nat_tree) : nat :=
match tr with
| NNode' _ trs => S (sum (map ntsize trs))
end.

Notice that Coq was smart enough to expand the definition of map to verify that we are using proper nested recursion, even through a use of a higher-order function.

Fixpoint ntsplice (tr1 tr2 : nat_tree) : nat_tree :=
match tr1 with
| NNode' n Nil => NNode' n (Cons tr2 Nil)
| NNode' n (Cons tr trs) => NNode' n (Cons (ntsplice tr tr2) trs)
end.

We have defined another arbitrary notion of tree splicing, similar to before, and we can prove an analogous theorem about its relationship with tree size. We start with a useful lemma about addition.

Lemma plus_S : forall n1 n2 : nat,
plus n1 (S n2) = S (plus n1 n2).
induction n1; crush.
Qed.

Now we begin the proof of the theorem, adding the lemma plus_S as a hint.

Hint Rewrite plus_S.

Theorem ntsize_ntsplice : forall tr1 tr2 : nat_tree, ntsize (ntsplice tr1 tr2)
= plus (ntsize tr2) (ntsize tr1).
We know that the standard induction principle is insufficient for the task, so we need to provide a using clause for the induction tactic to specify our alternate principle.

induction tr1 using nat_tree_ind'; crush.

One subgoal remains:
n : nat
ls : list nat_tree
H : All
(fun tr1 : nat_tree =>
forall tr2 : nat_tree,
ntsize (ntsplice tr1 tr2) = plus (ntsize tr2) (ntsize tr1)) ls
tr2 : nat_tree
============================
ntsize
match ls with
| Nil => NNode' n (Cons tr2 Nil)
| Cons tr trs => NNode' n (Cons (ntsplice tr tr2) trs)
end = S (plus (ntsize tr2) (sum (map ntsize ls)))

After a few moments of squinting at this goal, it becomes apparent that we need to do a case analysis on the structure of ls. The rest is routine.

destruct ls; crush.

We can go further in automating the proof by exploiting the hint mechanism.

Restart.

Hint Extern 1 (ntsize (match ?LS with Nil => _ | Cons _ _ => _ end) = _) =>
destruct LS; crush.
induction tr1 using nat_tree_ind'; crush.
Qed.

We will go into great detail on hints in a later chapter, but the only important thing to note here is that we register a pattern that describes a conclusion we expect to encounter during the proof. The pattern may contain unification variables, whose names are prefixed with question marks, and we may refer to those bound variables in a tactic that we ask to have run whenever the pattern matches.
The advantage of using the hint is not very clear here, because the original proof was so short. However, the hint has fundamentally improved the readability of our proof. Before, the proof referred to the local variable ls, which has an automatically generated name. To a human reading the proof script without stepping through it interactively, it was not clear where ls came from. The hint explains to the reader the process for choosing which variables to case analyze, and the hint can continue working even if the rest of the proof structure changes significantly.

It can be useful to understand how tactics like discriminate and injection work, so it is worth stepping through a manual proof of each kind. We will start with a proof fit for discriminate.

Theorem true_neq_false : true <> false.

We begin with the tactic red, which is short for "one step of reduction," to unfold the definition of logical negation.

red.

============================
true = false -> False
The negation is replaced with an implication of falsehood. We use the tactic intro H to change the assumption of the implication into a hypothesis named H.

intro H.

H : true = false
============================
False
This is the point in the proof where we apply some creativity. We define a function whose utility will become clear soon.

Definition toProp (b : bool) := if b then True else False.

It is worth recalling the difference between the lowercase and uppercase versions of truth and falsehood: True and False are logical propositions, while true and false are Boolean values that we can case-analyze. We have defined toProp such that our conclusion of False is computationally equivalent to toProp false. Thus, the change tactic will let us change the conclusion to toProp false. The general form change e replaces the conclusion with e, whenever Coq's built-in computation rules suffice to establish the equivalence of e with the original conclusion.

change (toProp false).

H : true = false
============================
toProp false
Now the righthand side of H's equality appears in the conclusion, so we can rewrite, using the notation <- to request to replace the righthand side of the equality with the lefthand side.

rewrite <- H.

H : true = false
============================
toProp true
We are almost done. Just how close we are to done is revealed by computational simplification.

simpl.

H : true = false
============================
True

trivial.
Qed.

I have no trivial automated version of this proof to suggest, beyond using discriminate or congruence in the first place.
We can perform a similar manual proof of injectivity of the constructor S. I leave a walk-through of the details to curious readers who want to run the proof script interactively.

Theorem S_inj' : forall n m : nat, S n = S m -> n = m.
intros n m H.
change (pred (S n) = pred (S m)).
rewrite H.
reflexivity.
Qed.

The key piece of creativity in this theorem comes in the use of the natural number predecessor function pred. Embodied in the implementation of injection is a generic recipe for writing such type-specific functions.
The examples in this section illustrate an important aspect of the design philosophy behind Coq. We could certainly design a Gallina replacement that built in rules for constructor discrimination and injectivity, but a simpler alternative is to include a few carefully chosen rules that enable the desired reasoning patterns and many others. A key benefit of this philosophy is that the complexity of proof checking is minimized, which bolsters our confidence that proved theorems are really true.