coq

I'm wondering, is there a commonly used library for vectors in coq, I.e. lists indexed by their length in their type. Some tutorials reference Bvector, but it's not found when I try to import it. There's Coq.Vectors.Vectordef, but the type defined there is just named t which makes me think it's intended for internal use. What is the best or most common practice for someone who doesn't want to roll their own library? Am I wrong about the vectors in the standard library? Or is there another Lib I'm missing? Or do people just use lists, paired with proofs of their length? There are generally

The simpl tactic unfolds expressions like 2 + a to "match trees" which doesn't seem simple at all. For example: Goal forall i:Z, ((fun x => x + i) 3 = i + 3). simpl. Leads to: forall i : Z, match i with | 0 => 3 | Z.pos y' => Z.pos match y' with | q~1 => match q with | q0~1 => (Pos.succ q0)~1 | q0~0 => (Pos.succ q0)~0 | 1 => 3 end~0 | q~0 => match q with | q0~1 => (Pos.succ q0)~0 | q0~0 => q0~1 | 1 => 2 end~1 | 1 => 4 end | Z.neg y' => Z.pos_sub 3 y' end = i + 3 How to avoid such complications with simpl tactic? This particular goal can be solved with omega , but if it is a bit more

I'm using CoqIDE to complete the exercises in the Software Foundations book about Coq. I can successfully compile Basics.v, resulting in Basics.vo and Basics.glob in my directory. When I try to run Induction.v, it works. When I try to compile it, it complains about tons of missing references, such as evenb and negb_involutive . If I copy Basics.v contents into Induction.v it compiles, but obviously this is not the way to go. This is not a duplicate of question Coq error: The reference evenb was not found in the current environment , as I have already done those things: Compile Basics.v. Check

is there anything like the tactic simpl for Program Fixpoint s? In particular, how can one proof the following trivial statement? Program Fixpoint bla (n:nat) {measure n} := match n with | 0 => 0 | S n' => S (bla n') end. Lemma obvious: forall n, bla n = n. induction n. reflexivity. (* I'm stuck here. For a normal fixpoint, I could for instance use simpl. rewrite IHn. reflexivity. But here, I couldn't find a tactic transforming bla (S n) to S (bla n).*) Obviously, there is no Program Fixpoint necessary for this toy example, but I'm facing the same problem in a more complicated setting where I

I am trying to define the Ackermann-Peters function in Coq, and I'm getting an error message that I don't understand. As you can see, I'm packaging the arguments a, b of Ackermann in a pair ab ; I provide an ordering defining an ordering function for the arguments. Then I use the Function form to define Ackermann itself, providing it with the ordering function for the ab argument. Require Import Recdef. Definition ack_ordering (ab1 ab2 : nat * nat) := match (ab1, ab2) with |((a1, b1), (a2, b2)) => (a1 > a2) \/ ((a1 = a2) /\ (b1 > b2)) end. Function ack (ab : nat * nat) {wf ack_ordering} : nat

Assume I have two hypotheses in the context, a_b : A -> B and a : A . I should be able to apply a_b to a to gain a further hypothesis, b : B . That is, given the following state: 1 subgoal A : Prop B : Prop C : Prop a_b : A -> B a : A ______________________________________(1/1) C There should be some tactic, foo (a_b a) , to transform this into the following state: 1 subgoal A : Prop B : Prop C : Prop a_b : A -> B a : A b : B ______________________________________(1/1) C But I don't know what foo is. One thing I can do is this: assert B as b. apply a_b. exact a. but this is rather long-winded,