Truthiness

There are a couple of notions of truthiness in Coq. For example Prop and bool.

TODO Difference between bool and Prop

One advantage of Prop over bool is that Prop is essentially a built in which works well with tactics like rewrite in the case of equality.

TODO Difference between Type and Prop

Term erasing

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Proof Pitfalls

Induction

Introducing too much.

https://softwarefoundations.cis.upenn.edu/draft/lf-current/Tactics.html#lab139

By introducing a variable you are saying for a “particular” instance of that variable. Not for all such instances of the variable.

Thus, when introducing more than necessary prior to induction you may unintentionally weaken the induction hypothesis that you will get, making it impossible to prove your goal.

Destruct

Forgetting things

When you destruct a compound expression you lose what the original equation was. For instance:

destruct (beq_nat 3 n).

will give you two subgoals. One where (beq_nat 3 n) is replaced with true, and one where it is replaced with false.

Sometimes, however, you actually need the fact that (beq_nat 3 n) = true in that branch of the proof, or that (beq_nat 3 n) = false in the other branch of the proof. You can keep this information with:

destruct (beq_nat 3 n) eqn:Hbeq3.

Which will introduce in the context a hypothesis Hbeq3 which will be beq_nat 3 n = true in the true branch of the proof, and beq_nat 3 n = false in the false branch.

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Dependent Types and Matching in Coq

Induction

Working with dependent types in Coq has been causing me a number of headaches — which I have found somewhat surprising coming from Agda / Idris.

I believe part of this is syntactic, and that Agda for instance kind of has you working with terms in a dependently typed language in the raw, unlike Coq which often has you using tactics, which are a bit of a step away from what’s actually going on. But the other part seems to be that Agda includes this so called “axiom K” by default, and Coq does not.

What is axiom K?

Dependent pattern matching in Coq. The “convoy” pattern.

Dealing with absurd cases.

In Agda you might use the absurd pattern to deal with branches of a program which should never be executed due to a condition which will never hold. I.e., if the condition to enter the branch is true, then that condition implies False.

hd {X} (xs : list X) (length xs > 0) : X   

For instance, in the case of a hd function, which takes the first element of the list, you might have an argument that’s a proof that the length of the list is greater than 0. But, you still have to provide a case to the match on the list for when it’s empty.

Well, that’s okay. Because in Coq we can use an empty pattern match on False. So you might think you can do something like this:

Require Import List.

Lemma length_nil_not_gt_0 :
  forall {X}, @length X nil > 0 -> False.
Proof.
  intros X H. inversion H.
Qed.


Definition hd {X} (xs : list X) (pf : length xs > 0) : X :=
  match xs with
  | nil => match length_nil_not_gt_0 pf with end
  | h :: t => h
  end.

But this doesn’t quite work.

Error:
In environment
X : Type
xs : list X
pf : length xs > 0
The term "pf" has type "length xs > 0" while it is expected to have type
 "length nil > 0".

For some reason Coq seems to not be recognizing that in this branch xs = nil, and so it’s not replacing xs with nil when it’s typechecking.

So, how can we get around this? What seems to be happening is that pf is getting tied to the type length xs > 0 at the beginning, at the very top level.

The way to get around this is to make hd actually return a function, which takes a proof as an argument. This lets the pf have a different type in each branch.

Require Import List.

Lemma length_nil_not_gt_0 :
  forall {X}, @length X nil > 0 -> False.
Proof.
  intros X H. inversion H.
Qed.


Definition hd {X} (xs : list X) : (length xs > 0) -> X :=
  match xs with
  | nil => fun pf => match length_nil_not_gt_0 pf with end
  | h :: t => fun _ => h
  end.
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Coq Cheat Sheet

Tactics

intros

Introduces variables and hypothesis. This takes them out of the goal and puts them in the context.

You can automatically destruct variables with patterns like [x | y] which will split a sum type into its two constructors, and call the resulting variables x and y respectively, leading to two subgoals. A product type may also be destructed into its parts with [x y]. These may be combined in arbitrary ways such as [x | [x y] | [x | y z]], depending on the type of what is being introduced. The empty pattern [] may be used to destruct an impossible value, and automatically solve a branch, such as if you would be introducing False.

reflexivity

Does reflexivity. If a goal has x = y it will try to simplify (by normalization) both x and y. If their normal forms are syntactically identical, then this tactic will succeed and the goal will be completed.

E.g.,

1 + 1 = 2

Reflexivity will solve this because 1 + 1 will evaluate to 2, and 2 = 2 is obviously true because each side of the equality is syntactically identical.

simpl

Attempts to simplify the goal. This essentially just evaluates to weak head normal form. Useful for seeing next steps in a proof (hard to unwind definitions in your head sometimes), and it can be used to facilitate rewrites.

1 + 1 = 2

simpl will transform this to 2 = 2.

simpl sometimes won’t evaluate as much as you expect, because if it always evaluated as much as it possibly could you would end up with really long terms. So, simple has some heuristics to decide when to keep evaluating, and when not to.

rewrite

When we have x = y it can be used to replace x with y in a goal, or vice versa.

H : x = y
=========
x = y

Then rewrite H will leave us with the goal y = y, and rewrite <- H will leave us with x = x.

subst

If you have

H : x = y

You can call subst and it will substitute all variable equalities. This is a bit wild, but useful with other wild tactics like inversion

apply

Used to solve a goal by applying a theorem which has an identical conclusion to the current goal. Any hypothesis of the theorem will be added to the context.

This tactic can also be used on hypothesis in the context which then matches on the hypothesis of the theorem being applied and gives you a hypothesis in the context matching the goal of the theorem being applied.

H : x = y
=========
x = y

apply H will solve the goal.

H : x = y -> y = z
H0 : x = y
==================
y = z

You could solve this in two ways with apply:

  • apply H will use the theorem H to show that y = z if x = y, so it leaves you with x = y in your goal. This can then be solved by applying H0.
  • apply H in H0 will use H : x = y -> y = z to transform H0 : x = y to y = z. After this we can apply H0 to our goal.

assumption

This calls apply on a hypothesis in the context that matches the goal. I.e., it finds the hypothesis for you.

symmetry

Reverses an equality. Using symmetry will flip x = y to y = x.

This is useful for when you need to apply a theorem, but the goal is in a different order than the theorem.

destruct

The destruct tactic is used to perform case analysis in Coq. It will break a possible value into all cases for that type (one for each constructor of the type). This gives you multiple goals to prove; one for each constructor.

This can be used on variables, or compound expressions.

When using destruct on a compound expression it is also possible to save the original expression.

destruct (f (x + y)) eqn:Hfxy

induction

The induction tactic is very similar to destruct, except that it brings an induction hypothesis into the context for recursive data types.

inversion

inversion provides reasoning with constructors taking into account the fact that constructors are injective and disjoint.

  • Injective: for a constructor A, A x = A y means that x = y as well.
  • Disjoint: If you have two constructors A and B, then you know that A does not equal B

Thus the inversion tactic has several uses.

Injective usage

H : A x = A y
=============
x = y

When using inversion in H this gives us a new hypothesis, H1.

H : A x = A y
H1 : x = y
=============
y = y

It will also perform rewrites with the new hypothesis automatically, so our goal changed to y = y as well, since it rewrote x with y using the new hypothesis H1.

Inversion will apply this injective reasoning across multiple arguments in a constructor, and even recursively. So, if you have lists of three items that are equal you will get a hypothesis representing the equality of each item in the first list, with the equivalent item in the second list.

We can name the equations as well.

inversion in H as [Hxy] will give:

H : A x = A y
Hxy : x = y
=============
y = y

Disjoint usage

If we have values constructed with two different constructors A and B, then we know that the values must be different. So if we have a hypothesis in the context with disjoint constructors, like so:

H: A x = B y
============
false = true

We can use inversion H which will conclude that H is a false hypothesis, and since we have a contradiction in our set of assumptions we may conclude the goal via the principle of explosion.

generalize dependent

The generalize dependent tactic can be used to place a variable in the context back into the goal. This can be useful when you only want to introduce certain variables, like when you want to keep your induction hypothesis strong.

unfold

This tactic is used to expand a definition.

Definition square n := n * n
square (n * m) = square n * square m

unfold square will yield

(n * m) * (n * m) = (n * n) * (m * m)

which will actually display as

n * m * (n * m) = n * n * (m * m)

due to the associativity of the operators.

This tactic can be very useful when you can simplify a definition based on rewriting a term inside the definition. For instance if the definition contains a match like:

match test x with
  | true => some_stuff
  | false => other_stuff
end.

Then if you have test x = true in the context you can simplify this match to just some_stuff.

assert

This tactic will introduce a hypothesis into the context, and then split the goal into two subgoals. The first subgoal is to prove that this hypothesis is true, and the second subgoal is the original goal.

TODO ~replace~

split

Used to split a conjunction in a goal into two subgoals.

If the conjunction is in the context, and not the goal, then one would use destruct instead.

left and right

These tactics are used to pick a side of a disjunction in the goal to prove.

exfalso

Turns the goal into False. This is useful when working with negations.

exists

Used in proofs with existential quantifiers. For instance if our goal was

exists x : nat, x = 2 + 2

Then exists 4 would substitute 4 for x, removing the existential quantifier, and leaving us with:

4 = 2 + 2

Which can then be solved with reflexivity.

Miscellaneous

Applying theorems to arguments

Theorems have arguments, since they’re really just types of functions. For example:

plus_comm : forall n m : n + m = m + n

When applying a theorem normally Coq guesses what the arguments to the theorem will be based on the first suitable instance it finds. This can be annoying because sometimes it applies it to the wrong part of the expression we want to prove!

So, instead of having Coq guess what n and m should be we can provide it with arguments.

rewrite (plus_comm p).

Will rewrite a term where p is substituted for n in the plus_comm theorem.

plus_comm p : forall m : p + m = m + p

This works in pretty much any tactic, like destruct, since theorems are just first class objects in Coq.

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