Coq is an interactive theorem prover based on the Calculus of Inductive Constructions.
Coq is an interactive theorem prover based on the calculus of inductive constructions.
Questions tagged [coq]
66 questions
12
votes
1 answer
baz_num_elts exercise from Software Foundations
I'm at the following exercise in Software Foundations:
(** **** Exercise: 2 stars (baz_num_elts) *)
(** Consider the following inductive definition: *)
Inductive baz : Type :=
| x : baz -> baz
| y : baz -> bool -> baz.
(** How _many_…
Twernmilt
- 123
- 6
5
votes
2 answers
Proof of equality with destructuring let...in
I have some expression (f n in the example below) returning a tuple. I would like to prove that f n is equal to let (x, y) := f n in (x, y), which seems like it should be easy. What tactic should I use?
Definition f (n : nat) : nat * nat :=
…
Bruno Le Floch
- 151
- 5
5
votes
1 answer
How does this use of "apply" in Coq work?
I'm working my way through software foundations. In the Chapter titled "Tactics", I'm able to prove this theorem in Coq:
Theorem silly_ex :
(∀ n, evenb n = true → oddb (S n) = true) →
oddb 3 = true →
evenb 4 = true.
Proof.
intros…
Raiden Worley
- 245
- 1
- 4
3
votes
2 answers
I don't know how to prove a simple theorem used with fixpoint in Coq
I am a beginner in coq and want to prove the following theorem t1. First I used induction i and destruct j, but it got bogged down in the middle.
I would like some hints for this problem.
Function f takes two arguments of nat and returns Prop…
FUJII S.
- 33
- 5
3
votes
1 answer
For proof automation in Coq, when is it appropriate to use canonical structures or Equations instead of Ltac?
There are a few possible approaches to proof automation in modern Coq.
Writing proof scripts with Ltac. This is the approach described in http://adam.chlipala.net/cpdt/, which the author uses to great effect in projects like…
LogicChains
- 131
- 3
2
votes
1 answer
How to prove transitivity of < (Software Foundations exercise)?
I'm working through the "Properties of Relations" chapter of Software Foundations, but have got stuck on one of the exercises, lt_trans'':
Theorem lt_trans'' :
transitive lt.
Proof.
unfold lt. unfold transitive.
intros n m o Hnm Hmo.
…
Matt R
- 141
- 4
1
vote
1 answer
How to prove T Z = Z for binary representation of natural numbers in Coq
I have defined an Inductive in Coq for binary representation of natural numbers as follows:
Inductive bin : Type :=
| Z : bin
| T : bin -> bin
| M : bin -> bin.
and two recursive functions to convert between nat and bin types as:
Fixpoint…
Morin
- 25
- 4
1
vote
2 answers
Is there a tactic to help resolving existential quantifiers in Coq?
I am working on Software Foundations Volume 1 on my own it is its 2019 version by the way, and I have reached to its lesson Inductively Defined Propositions, and there, for almost one month I have been stuck on an exercise re_not_empty expressed…
Morin
- 25
- 4
1
vote
1 answer
Instantiating a class with a sig'd type in Coq
I can easily define a class that corresponds to the notion of a "monoidal structure" on a type M via
Definition associative {X:Type} (f : X -> X -> X) : Prop := forall x y z:X, f (f x y) z = f x (f y z).
Definition opId {X:Type} (f : X -> X -> X)…
Feryll
- 113
- 3
0
votes
0 answers
Coq: default values for vectors
Let's say there is a vector of length $n$:
Require Import Vector.
Variable T:Type.
Variable n:nat.
Variable v:t T n.
"list" gives a function "nth" that demands a default result that is returned when the list is longer than the index requested. This…
user7358
- 101
0
votes
0 answers
Coq stuck on proof for Theorem eqblist_true
I'm stuck on this proof for the theorem eqblist_true. So far what I have is:
Theorem eqblist_true : forall l1 l2,
eqblist l1 l2 = true -> l1 = l2.
Proof.
intros l1. induction l1 as [| l1'].
- intros l2 H. simpl in H. destruct l2 as [|…
Chloe
- 1