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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Feeding rewrites and other hints into an omnibus tactic
How do I feed rewrites that I've marked as safe into a custom tactic?
I'm trying to write shorter Coq proofs with more of the easy stuff hidden. To that end, in the script below I proved that ...
- 3,095
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Coq equivalent of Lean's `nth_rewrite`
Does Coq have an equivalent of Lean's nth_rewrite? rewrite ... at ... appears to specialize at its first unification site ...
- 3,095
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Coq cannot `simple apply reflexivity` in custom tactic
The fast reflexivity tactic shown below is very interesting. It exposes some of the unification machinery by disabling it.
I'm planning on going back and using it in the first part of Software ...
- 3,095
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When is the lean 4 "by" required?
I am reading an introductory math/lean course here, and got confused about when the tactic/keyword by is required. It is used most of the time, but is occasionally ...
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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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Ltac, How to intro a fresh variable which may already have a good estiblished name given by a universal quantifier?
Context
I am currently self studying Coq following the Software Foundations book series which I am finding very approachable.
I have finally gotten round to ...
- 133
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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:
...
- 123
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Tactic to Propify a bool expression
Let's say I have bool expressions <bexp> consisting of true, false, variables, ...
- 1,502
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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 ...
- 41
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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 ...
- 1,502
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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.
...
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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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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 ...
- 1,502
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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 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 ...
- 313
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Specializing forall quantifiers in Coq
I have an inductively defined type of expressions:
...
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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 ...
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Rewriting inside quantified propositions in Coq?
Is there a simple way to use rewrites inside quantified Props? As an example, consider the following:
...
- 1,502
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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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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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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)...
...
- 1,502
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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:
...
- 257
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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,...
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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 ...
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154
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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.
...
- 170
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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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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)
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Where is the discriminate tactic defined in Coq?
One can read the Coq documentation about discriminate tactic here.
Were is this tactic actually defined?
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399
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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 doesn’t do this by default, and guess it would probably ...
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Prove equality in a record type
I am trying to prove something about monoids an categories. This results in the following (partial) proof:
...
- 265
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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 ...
- 265
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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 ...
- 213
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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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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 ...
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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 ...
- 3,095
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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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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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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 ...
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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:
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