The last chapter highlighted a very heuristic approach to proving. In this chapter, we will study an alternative technique, proof by reflection . We will write, in Gallina, decision procedures with proofs of correctness, and we will appeal to these procedures in writing very short proofs. Such a proof is checked by running the decision procedure. The term reflection applies because we will need to translate Gallina propositions into values of inductive types representing syntax, so that Gallina programs may analyze them, and translating such a term back to the original form is called reflecting it.

Proving Evenness

Proving that particular natural number constants are even is certainly something we would rather have happen automatically. The Ltac-programming techniques that we learned in the last chapter make it easy to implement such a procedure.

even_256 =
Even_SS
  (Even_SS
     (Even_SS
        (Even_SS

...and so on. This procedure always works (at least on machines with infinite resources), but it has a serious drawback, which we see when we print the proof it generates that 256 is even. The final proof term has length super-linear in the input value. Coq's implicit arguments mechanism is hiding the values given for parameter n of Even_SS, which is why the proof term only appears linear here. Also, proof terms are represented internally as syntax trees, with opportunity for sharing of node representations, but in this chapter we will measure proof term size as simple textual length or as the number of nodes in the term's syntax tree, two measures that are approximately equivalent. Sometimes apparently large proof terms have enough internal sharing that they take up less memory than we expect, but one avoids having to reason about such sharing by ensuring that the size of a sharing-free version of a term is low enough. Superlinear evenness proof terms seem like a shame, since we could write a trivial and trustworthy program to verify evenness of constants. The proof checker could simply call our program where needed. It is also unfortunate not to have static typing guarantees that our tactic always behaves appropriately. Other invocations of similar tactics might fail with dynamic type errors, and we would not know about the bugs behind these errors until we happened to attempt to prove complex enough goals. The techniques of proof by reflection address both complaints. We will be able to write proofs like in the example above with constant size overhead beyond the size of the input, and we will do it with verified decision procedures written in Gallina. For this example, we begin by using a type from the MoreSpecif module (included in the book source) to write a certified evenness checker.

Inductive partial (P : Prop) : Set := Proved : P -> [P] | Uncertain : [P]

A partial P value is an optional proof of P. The notation [P] stands for partial P.

We bring into scope some notations for the partial type. These overlap with some of the notations we have seen previously for specification types, so they were placed in a separate scope that needs separate opening.

The function check_even may be viewed as a verified decision procedure, because its type guarantees that it never returns Yes for inputs that are not even. Now we can use dependent pattern-matching to write a function that performs a surprising feat. When given a partial P, this function partialOut returns a proof of P if the partial value contains a proof, and it returns a (useless) proof of True otherwise. From the standpoint of ML and Haskell programming, it seems impossible to write such a type, but it is trivial with a return annotation.

It may seem strange to define a function like this. However, it turns out to be very useful in writing a reflective version of our earlier prove_even tactic:

We identify which natural number we are considering, and we "prove" its evenness by pulling the proof out of the appropriate check_even call. Recall that the exact tactic proves a proposition P when given a proof term of precisely type P.

even_256' = partialOut (check_even 256)
     : isEven 256

We can see a constant wrapper around the object of the proof. For any even number, this form of proof will suffice. The size of the proof term is now linear in the number being checked, containing two repetitions of the unary form of that number, one of which is hidden above within the implicit argument to partialOut. What happens if we try the tactic with an odd number?

  prove_even_reflective.

User error: No matching clauses for match goal

Thankfully, the tactic fails. To see more precisely what goes wrong, we can run manually the body of the match.
  exact (partialOut (check_even 255)).

  Error: The term "partialOut (check_even 255)" has type
 "match check_even 255 with
  | Yes => isEven 255
  | No => True
  end" while it is expected to have type "isEven 255"

As usual, the type checker performs no reductions to simplify error messages. If we reduced the first term ourselves, we would see that check_even 255 reduces to a No, so that the first term is equivalent to True, which certainly does not unify with isEven 255.

Our tactic prove_even_reflective is reflective because it performs a proof search process (a trivial one, in this case) wholly within Gallina, where the only use of Ltac is to translate a goal into an appropriate use of check_even.

Reifying the Syntax of a Trivial Tautology Language

We might also like to have reflective proofs of trivial tautologies like this one:

true_galore =
fun H : True /\ True =>
and_ind (fun _ _ : True => or_introl (True /\ (True -> True)) I) H
     : True /\ True -> True \/ True /\ (True -> True)

As we might expect, the proof that tauto builds contains explicit applications of natural deduction rules. For large formulas, this can add a linear amount of proof size overhead, beyond the size of the input. To write a reflective procedure for this class of goals, we will need to get into the actual "reflection" part of "proof by reflection." It is impossible to case-analyze a Prop in any way in Gallina. We must reify Prop into some type that we can analyze. This inductive type is a good candidate:

We write a recursive function to reflect this syntax back to Prop. Such functions are also called interpretation functions, and we have used them in previous examples to give semantics to small programming languages.

It is easy to prove that every formula in the range of tautDenote is true.

To use tautTrue to prove particular formulas, we need to implement the syntax reification process. A recursive Ltac function does the job.

With tautReify available, it is easy to finish our reflective tactic. We look at the goal formula, reify it, and apply tautTrue to the reified formula.

We can verify that obvious solves our original example, with a proof term that does not mention details of the proof.

true_galore' =
tautTrue
  (TautImp (TautAnd TautTrue TautTrue)
     (TautOr TautTrue (TautAnd TautTrue (TautImp TautTrue TautTrue))))
     : True /\ True -> True \/ True /\ (True -> True)

It is worth considering how the reflective tactic improves on a pure-Ltac implementation. The formula reification process is just as ad-hoc as before, so we gain little there. In general, proofs will be more complicated than formula translation, and the "generic proof rule" that we apply here is on much better formal footing than a recursive Ltac function. The dependent type of the proof guarantees that it "works" on any input formula. This benefit is in addition to the proof-size improvement that we have already seen. It may also be worth pointing out that our previous example of evenness testing used a function partialOut for sound handling of input goals that the verified decision procedure fails to prove. Here, we prove that our procedure tautTrue (recall that an inductive proof may be viewed as a recursive procedure) is able to prove any goal representable in taut, so no extra step is necessary.

A Monoid Expression Simplifier

Proof by reflection does not require encoding of all of the syntax in a goal. We can insert "variables" in our syntax types to allow injection of arbitrary pieces, even if we cannot apply specialized reasoning to them. In this section, we explore that possibility by writing a tactic for normalizing monoid equations.

We add variables and hypotheses characterizing an arbitrary instance of the algebraic structure of monoids. We have an associative binary operator and an identity element for it. It is easy to define an expression tree type for monoid expressions. A Var constructor is a "catch-all" case for subexpressions that we cannot model. These subexpressions could be actual Gallina variables, or they could just use functions that our tactic is unable to understand.

Next, we write an interpretation function.

We will normalize expressions by flattening them into lists, via associativity, so it is helpful to have a denotation function for lists of monoid values.

The flattening function itself is easy to implement.

This function has a straightforward correctness proof in terms of our denote functions.

Now it is easy to prove a theorem that will be the main tool behind our simplification tactic.

We implement reification into the mexp type.

The final monoid tactic works on goals that equate two monoid terms. We reify each and change the goal to refer to the reified versions, finishing off by applying monoid_reflect and simplifying uses of mldenote. Recall that the change tactic replaces a conclusion formula with another that is definitionally equal to it.

We can make short work of theorems like this one:

  ============================
   a + (b + (c + (d + e))) = a + (b + (c + (d + e)))
 

Our tactic has canonicalized both sides of the equality, such that we can finish the proof by reflexivity.

It is interesting to look at the form of the proof.

t1 =
fun a b c d : A =>
monoid_reflect (Op (Op (Op (Var a) (Var b)) (Var c)) (Var d))
  (Op (Op (Var a) (Op (Var b) (Var c))) (Var d))
  (eq_refl (a + (b + (c + (d + e)))))
     : forall a b c d : A, a + b + c + d = a + (b + c) + d

The proof term contains only restatements of the equality operands in reified form, followed by a use of reflexivity on the shared canonical form.

Extensions of this basic approach are used in the implementations of the ring and field tactics that come packaged with Coq.

A Smarter Tautology Solver

Now we are ready to revisit our earlier tautology solver example. We want to broaden the scope of the tactic to include formulas whose truth is not syntactically apparent. We will want to allow injection of arbitrary formulas, like we allowed arbitrary monoid expressions in the last example. Since we are working in a richer theory, it is important to be able to use equalities between different injected formulas. For instance, we cannot prove P -> P by translating the formula into a value like Imp (Var P) (Var P), because a Gallina function has no way of comparing the two Ps for equality. To arrive at a nice implementation satisfying these criteria, we introduce the quote tactic and its associated library.

The type index comes from the Quote library and represents a countable variable type. The rest of formula's definition should be old hat by now. The quote tactic will implement injection from Prop into formula for us, but it is not quite as smart as we might like. In particular, it wants to treat function types specially, so it gets confused if function types are part of the structure we want to encode syntactically. To trick quote into not noticing our uses of function types to express logical implication, we will need to declare a wrapper definition for implication, as we did in the last chapter.

Now we can define our denotation function.

The varmap type family implements maps from index values. In this case, we define an assignment as a map from variables to Props. Our interpretation function formulaDenote works with an assignment, and we use the varmap_find function to consult the assignment in the Atomic case. The first argument to varmap_find is a default value, in case the variable is not found.

We define some shorthand for a particular variable being true, and now we are ready to define some helpful functions based on the ListSet module of the standard library, which (unsurprisingly) presents a view of lists as sets.

We define what it means for all members of an index set to represent true propositions, and we prove some lemmas about this notion.

Now we can write a function forward that implements deconstruction of hypotheses, expanding a compound formula into a set of sets of atomic formulas covering all possible cases introduced with use of Or. To handle consideration of multiple cases, the function takes in a continuation argument, which will be called once for each case. The forward function has a dependent type, in the style of Chapter 6, guaranteeing correctness. The arguments to forward are a goal formula f, a set known of atomic formulas that we may assume are true, a hypothesis formula hyp, and a success continuation cont that we call when we have extended known to hold new truths implied by hyp.

A backward function implements analysis of the final goal. It calls forward to handle implications.

A simple wrapper around backward gives us the usual type of a partial decision procedure.

Our final tactic implementation is now fairly straightforward. First, we intro all quantifiers that do not bind Props. Then we call the quote tactic, which implements the reification for us. Finally, we are able to construct an exact proof via partialOut and the my_tauto Gallina function.

A few examples demonstrate how the tactic works.

mt1 = partialOut (my_tauto (Empty_vm Prop) Truth)
     : True

We see my_tauto applied with an empty varmap, since every subformula is handled by formulaDenote.

mt2 =
fun x y : nat =>
partialOut
  (my_tauto (Node_vm (x = y) (Empty_vm Prop) (Empty_vm Prop))
     (Imp (Atomic End_idx) (Atomic End_idx)))
     : forall x y : nat, x = y --> x = y

Crucially, both instances of x = y are represented with the same index, End_idx. The value of this index only needs to appear once in the varmap, whose form reveals that varmaps are represented as binary trees, where index values denote paths from tree roots to leaves.

fun x y z : nat =>
partialOut
  (my_tauto
     (Node_vm (x < S y) (Node_vm (x < y) (Empty_vm Prop) (Empty_vm Prop))
        (Node_vm (y > z) (Empty_vm Prop) (Empty_vm Prop)))
     (Imp
        (Or (And (Atomic (Left_idx End_idx)) (Atomic (Right_idx End_idx)))
           (And (Atomic (Right_idx End_idx)) (Atomic End_idx)))
        (And (Atomic (Right_idx End_idx))
           (Or (Atomic (Left_idx End_idx)) (Atomic End_idx)))))
     : forall x y z : nat,
       x < y /\ y > z \/ y > z /\ x < S y --> y > z /\ (x < y \/ x < S y)

Our goal contained three distinct atomic formulas, and we see that a three-element varmap is generated. It can be interesting to observe differences between the level of repetition in proof terms generated by my_tauto and tauto for especially trivial theorems.

mt4 =
partialOut
  (my_tauto (Empty_vm Prop)
     (Imp
        (And Truth
           (And Truth
              (And Truth (And Truth (And Truth (And Truth Falsehood))))))
        Falsehood))
     : True /\ True /\ True /\ True /\ True /\ True /\ False --> False

mt4' =
fun H : True /\ True /\ True /\ True /\ True /\ True /\ False =>
and_ind
  (fun (_ : True) (H1 : True /\ True /\ True /\ True /\ True /\ False) =>
   and_ind
     (fun (_ : True) (H3 : True /\ True /\ True /\ True /\ False) =>
      and_ind
        (fun (_ : True) (H5 : True /\ True /\ True /\ False) =>
         and_ind
           (fun (_ : True) (H7 : True /\ True /\ False) =>
            and_ind
              (fun (_ : True) (H9 : True /\ False) =>
               and_ind (fun (_ : True) (H11 : False) => False_ind False H11)
                 H9) H7) H5) H3) H1) H
     : True /\ True /\ True /\ True /\ True /\ True /\ False -> False

The traditional tauto tactic introduces a quadratic blow-up in the size of the proof term, whereas proofs produced by my_tauto always have linear size.

Manual Reification of Terms with Variables

The action of the quote tactic above may seem like magic. Somehow it performs equality comparison between subterms of arbitrary types, so that these subterms may be represented with the same reified variable. While quote is implemented in OCaml, we can code the reification process completely in Ltac, as well. To make our job simpler, we will represent variables as nats, indexing into a simple list of variable values that may be referenced. Step one of the process is to crawl over a term, building a duplicate-free list of all values that appear in positions we will encode as variables. A useful helper function adds an element to a list, preventing duplicates. Note how we use Ltac pattern matching to implement an equality test on Gallina terms; this is simple syntactic equality, not even the richer definitional equality. We also represent lists as nested tuples, to allow different list elements to have different Gallina types.

Now we can write our recursive function to calculate the list of variable values we will want to use to represent a term.

We will also need a way to map a value to its position in a list.

The next building block is a procedure for reifying a term, given a list of all allowed variable values. We are free to make this procedure partial, where tactic failure may be triggered upon attempting to reify a term containing subterms not included in the list of variables. The type of the output term is a copy of formula where index is replaced by nat, in the type of the constructor for atomic formulas.

Note that, when we write our own Ltac procedure, we can work directly with the normal -> operator, rather than needing to introduce a wrapper for it.

Finally, we bring all the pieces together.

A quick test verifies that we are doing reification correctly.

Our simple tactic adds the translated term as a new variable:
f := Imp'
         (Or' (And' (Atomic' 2) (Atomic' 1)) (And' (Atomic' 1) (Atomic' 0)))
         (And' (Atomic' 1) (Or' (Atomic' 2) (Atomic' 0))) : formula'

More work would be needed to complete the reflective tactic, as we must connect our new syntax type with the real meanings of formulas, but the details are the same as in our prior implementation with quote.

Building a Reification Tactic that Recurses Under Binders

All of our examples so far have stayed away from reifying the syntax of terms that use such features as quantifiers and fun function abstractions. Such cases are complicated by the fact that different subterms may be allowed to reference different sets of free variables. Some cleverness is needed to clear this hurdle, but a few simple patterns will suffice. Consider this example of a simple dependently typed term language, where a function abstraction body is represented conveniently with a Coq function.

Here is a naive first attempt at a reification tactic.

Recall that a regular Ltac pattern variable ?X only matches terms that do not mention new variables introduced within the pattern. In our naive implementation, the case for matching function abstractions matches the function body in a way that prevents it from mentioning the function argument! Our code above plays fast and loose with the function body in a way that leads to independent problems, but we could change the code so that it indeed handles function abstractions that ignore their arguments. To handle functions in general, we will use the pattern variable form @?X, which allows X to mention newly introduced variables that are declared explicitly. A use of @?X must be followed by a list of the local variables that may be mentioned. The variable X then comes to stand for a Gallina function over the values of those variables. For instance:

Now, in the abstraction case, we bind E1 as a function from an x value to the value of the abstraction body. Unfortunately, our recursive call there is not destined for success. It will match the same abstraction pattern and trigger another recursive call, and so on through infinite recursion. One last refactoring yields a working procedure. The key idea is to consider every input to refl' as a function over the values of variables introduced during recursion.

Note how now even the addition case works in terms of functions, with @?X patterns. The abstraction case introduces a new variable by extending the type used to represent the free variables. In particular, the argument to refl' used type T to represent all free variables. We extend the type to T * nat for the type representing free variable values within the abstraction body. A bit of bookkeeping with pairs and their projections produces an appropriate version of the abstraction body to pass in a recursive call. To ensure that all this repackaging of terms does not interfere with pattern matching, we add an extra simpl reduction on the function argument, in the first line of the body of refl'. Now one more tactic provides an example of how to apply reification. Let us consider goals that are equalities between terms that can be reified. We want to change such goals into equalities between appropriate calls to termDenote.

  ============================
   termDenote
     (Abs
        (fun y : nat =>
         Abs (fun y0 : nat => Plus (Plus (Const y) (Const y0)) (Const 13)))) =
   termDenote (Abs (fun _ : nat => Abs (fun y0 : nat => Const y0)))

Our encoding here uses Coq functions to represent binding within the terms we reify, which makes it difficult to implement certain functions over reified terms. An alternative would be to represent variables with numbers. This can be done by writing a slightly smarter reification function that identifies variable references by detecting when term arguments are just compositions of fst and snd; from the order of the compositions we may read off the variable number. We leave the details as an exercise (though not a trivial one!) for the reader.