Questions tagged [agda]
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20
questions
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`transp` In Agda
I'm still a bit confused about the transp operator in Agda:
transp : ∀ {ℓ} (A : I → Set ℓ) (r : I) (a : A i0) → A i1
Is found ...
- 227
1
vote
1
answer
145
views
Proving transitivity in an intuitionistic type theory without the K rule
In Agda, if I disable axiom $\mathbb{K}$ I'm not able to prove
$$
\forall\{A : \textbf{Set}\}\{a\ b : A\}\{p\ q : a \equiv b\} \to p \equiv q,
$$
which I guess is normal since the system does not ...
3
votes
1
answer
226
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Building non-classical logics in Agda & Coq
Is it possible to construct different systems of logic in Coq or Agda?
I ask because I'm interested in using a proof assistant to construct (and verify) theorems in things like many-valued logics, ...
2
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0
answers
52
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"Universe-shrinking" function in Agda
Agda does not allow datatypes in one universe to be indexed by, or non-trivially parametrized by a type in a larger universe (strangely, Coq does not appear to require this for propositional ...
- 316
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151
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Proving intuitionistic tautologies in Agda
I am to use Agda to prove some intuitionistic tautologies. One of them is the so called Weak Peirce's Law
$$
((((A \rightarrow B) \rightarrow A) \rightarrow A) \rightarrow B) \rightarrow B
$$
I ...
- 88
2
votes
1
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342
views
Which would be better for programming using Homotopy type theory Agda or Idris
I'm looking to model data inputs for an artificially intelligent system, which is affected by its internal parts and has feedback loops. I'd like to model it mathematically, using category theory or ...
- 21
2
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1
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77
views
Why does universe level restriction behave differently between inductive family and parameterized inductive type without axiom K in agda
An observation when defining List in agda with --without-K enabled:
The following parameterized inductive definition is accepted:...
- 191
9
votes
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382
views
Relationship between inductive families and type-returning functions
Dependently typed languages such as Agda support inductive families, also called indexed datatypes, which allow type parameters to vary between constructors. This can be used to define a set of ...
- 304
1
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2
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121
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Interpreting a proof of $2^\mathbb{N}$ being uncountable
Suppose I have the following proof:
...
- 175
3
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0
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114
views
Does Type:Type lead to inconsistency without general inductive types?
In e.g. Agda , it is possible to derive an element of the empty type by enabling the "type in type" option.
Every proof I have seen (and come up with) involves making a special inductive type ...
- 329
4
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96
views
Which inductive schemes can encode the following Agda definition?
Which induction schemes (e.g. induction-recursion by Dybjer and Setzer, "Irish" induction-recursion by McBride or induction-induction by Forsberg and Setzer or perhaps some simpler ones) allow one to ...
- 143
1
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0
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163
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Predecessor function with recursive types
I am defining the type Nat of natural numbers a recursive sum type:
$$ Nat = \mu X. Unit \oplus X$$
Now, I have defined zero ...
- 239
0
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1
answer
173
views
Is this statement of P = NP in Agda correct?
Looking for a self-contained statement of P = NP in type theory, I stumbled upon this short Agda formalization (a cleaned up version is reproduced below).
The statement here does seem to express the ...
- 101
2
votes
2
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150
views
Propositional truncation of excluded middle
It is clear to me that it should be impossible to prove :
exclMidl = isProp A → ((A) ⊎ (¬ A))
Because it would give deciding oracle for every Proposition.
My ...
- 125
3
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0
answers
210
views
Strict Positivity of Indexed Datatype in Agda
Agda is ruling out definitions like
data Bad : Set where
bad : (Bad → Bad) → Bad
Because "Non strictly-positive declarations are rejected because one can write ...
- 125
7
votes
1
answer
440
views
Proof that type does not have decidable equality in Agda
Can one create such function in Agda ?
ℕ→ℕ-undecidable : ¬ ( (f g : ℕ → ℕ ) → Dec (f ≡ g))
ℕ→ℕ-undecidable = ?
I am particularly interested in proof using ...
- 125
2
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1
answer
206
views
Can one get Turing-completeness without nontermination?
As I'm reading the movfuscator paper by Stephen Dolan, I encounter this claim:
In order to have Turing-completeness, we must allow for nontermination.
This seems like a reasonable statement. But I'...
- 121
2
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1
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387
views
Why this pattern matching fails in Agda?
Consider function wa'' (need natural number definition, either in stdlib or Agda.Builtin.Nat):
...
- 928
8
votes
2
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612
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In Agda's GADT, are "parameterized" and "indexed" different semantically?
I know they have different scoping:
data a (n : Set) : Set where
introA : a n
data b : Set -> Set where
introB : {n : Set} -> b n
That's not what I'...
- 928
5
votes
2
answers
467
views
What does it mean if we disable K-rule in Agda?
TL;DR: Can I say, "K-rule in Agda enables people to match $ \forall a.a \equiv a $ with $ \mathit{refl} $"?
In https://agda.readthedocs.io/en/v2.5.4.1/language/without-k.html#without-k, K-...
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