# Questions tagged [tactic]

A tactic is a command or instruction for constructing a formal proof by applying a common proof technique. For questions about high-level techniques for constructing proofs, use the tag (strategy).

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### What's the difference between reflection and tactics?

Agda has a reflection mechanism (not equality reflection or reflexivity, but something related to metaprogramming based on goals and contexts to generate terms) and people have developed some ...
• 6,186
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### In Lean, why is the exact tactic necessary when the goal is the same as a hypothesis?

In Lean, when proving basic theorems, one runs into the following kind of thing: ...
• 373
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### Proving uniqueness of an instance of an indexed inductive type

Consider the simple indexed inductive type Inductive Single : nat -> Set := | single_O : Single O | single_S {n} : Single n -> Single (S n). Intuitively, I ...
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### Can Lean simp arguments be ordered?

I want to simplify the expression 0 * 1 * 1 * 1 * 0 using simp only [mul_zero, zero_mul]. I would like ...
212 views

### How to prove forall m n : nat, m == n -> m = n?

I am learning Coq with ssreflect. Just to understand things, I've proved forall a b : bool, a == b -> a = b but I can't figure out how to prove ...
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### How does Lean simp tactic work?

The doc at https://leanprover-community.github.io/extras/simp.html says about simp: all it does is repeatedly replace (or rewrite) subterms of the form A by B, for ...
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### How do we resolve metavariables that appear in hypotheses and targets in Lean?

There are two related questions that I expound on below. It might seem like these aren't quite related, but they are both about how to deal with meta-variables that appear when working through a ...
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### Examples of theories where tactic language is required for simple proofs

I was always under the impression, that separate tactic languages were generally considered to be vital for writing long proofs. I see tactic languages as a kind of interpreted DSL to generate ...
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### How to evaluate proof terms through opaque definitions?

Is there is a way to force computation over opaque terms, for the purposes of debugging/meta-analysis of proof scripts. I understand why Coq doesnāt do this by default, and guess it would probably ...
292 views

### Cannot discriminate 0 = 1

I am just practicing a bit with coq, doing some UniMath exercises and am trying to prove (0 = 1) -> empty. However, for some reason, I seem unable to reason ...
433 views

### Tactics for Array/List simplification in lean4

Because there are both Arrays and Lists in Lean4, sometimes you end up with code that has a mixture of Lists and Arrays interspersed with basic operations and conversions between the two. For example,...
464 views

### Simple Proof about String.split

I am new to lean, working on proving a simple lemma in lean4. lemma String.split_empty (c): String.split "" c = [""] I tried looking for ...
156 views

### Prove equality in a record type

I am trying to prove something about monoids an categories. This results in the following (partial) proof: ...
172 views

### Why does this trivial proof fail with structuring tacticals?

Given this: Inductive color := Black | White. Inductive point_state := | Occupied of color | Empty . this works: ...
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### Which proof assistants implement Church's rule?

Church's rule (CR) is one of the hallmarks of constructive mathematics, and is an admissible rule in a wide variety of constructive theories (you might consider CR to be a requirement for constructive ...
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### Doing case-by-case proofs about match statements in Lean4

In Lean4, I am stuck in a proof with a goal like this: ...
61 views

### Using CoqHammer from Ltac2

As it seems most likely to me, due to the special way arguments are evaluated in CoqHammer tactics (I tried to read the source code in OCaml but unfortunately I didn't understand much of it), it is ...
64 views

### Creating a tactic for 'destructing' a list by last element?

Sometimes, I have a context in which I have some l : list X, and I want to prove the goal by proving that (1) If l = [], the ...
194 views

### Selecting both a hypothesis and Goal while applying a tactic

I have a hypothesis H and some function foo. I want to simplify foo in both H and the ...
225 views

### Coq: can tauto be used to prove classical tautologies?

When I experiment, I get inconsistent results. Running the following code (with a proof included to double-check that it's provable) ...
125 views

### Lean4: How to construct an HEq between dependent functions?

I have an extremely simple goal to prove: HEq (fun px rd => match px, rd with | Sum.inr _ppos, dir => dir) fun x => id The reason the match ...
87 views

### Where is the discriminate tactic defined in Coq?

One can read the Coq documentation about discriminate tactic here. Were is this tactic actually defined?
195 views

### In Coq, is there a simpler tactic for introducing a disjunction and immediately destructing it?

Very often, I find myself writing some tactics like these: assert (delta = 1 \/ delta <> 1) as Hd by lia. destruct Hd. ...(proceed to work with two cases)... ...
145 views

### Applying custom tactic in hypothesis

To avoid tedious repetition I have a tactic that looks something like this: Ltac unfolds := try unfold foo; try unfold bar; try unfold baz; apply some_lemma. ...
55 views

### What's the idiomatic way to instantiate a tuple of evars in Ltac2?

Suppose that I have a local definition of a type ty in the context, and ty can be any nested tuple, e.g.: ...
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### What does induction ... in ... do in Coq?

I'm self-studying the Semantics course, and met the following proof script in the warmup directory: ...
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### Proving that equality is decidable on an Inductive Set

I've managed to prove that equality within a type is indeed decidable. ...
• 191
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### Debug autorewrite in Coq

I often meet proofs using autorewrite which Coq takes a while to process for some reason. (Setoid rewriting) I then manually figure out which rewrite rules were ...
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1 vote
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### Form of intros in Coq specifically for forall and explicitly for ->

Are there tactics in Coq that are more limited versions (subtactics?) of intros? I'm curious if there are any specifically for ...
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1 vote
114 views

### Rewriting inside quantified propositions in Coq?

Is there a simple way to use rewrites inside quantified Props? As an example, consider the following: ...
1 vote
101 views

### Proving that applicative functors compose

For simplicity, here an applicative functor means (in a proof assistant based on dependent type theory) the Haskellian applicative functor, bundled with its equational laws. This I can of course brute ...
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1 vote
95 views

### Question about the tactic "obtain"

I am having difficulty activating the tactic obtain. Is it part of mathlib and where is its exact location?
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### Specializing forall quantifiers in Coq

I have an inductively defined type of expressions: ...