Questions tagged [inductive-type]
In terms of categorical semantics, an inductive type is a type whose interpretation is given by an initial algebra of an endofunctor. (from nLab)
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Categorical semantics of Agda
I would like to know the state of the art regarding the categorical semantics of the type theory implemented by Agda — or at least some approximation of that type theory that is amenable to ...
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What are the complex induction patterns supported by Agda?
A question was recently asked on the Coq-club mailing list on Coq rejecting a nastily nested inductive type. We encountered a similar difficulty while trying to port code from Agda to Coq: Agda ...
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Expressivity of mutual/nested inductives vs. regular inductives
This question is from @TaliaRinger:
Are mutually inductive types and plain old inductive types equally expressive? (Say, in Coq.) I assume yes, but the motive for induction will be a huge mess. But ...
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Is induction over mutually inductive coinductive types possible?
You can encode ordinals in Coq as
Inductive ord := O | S (n: ord) | Lim (s: nat -> ord).
Suppose you use the following encoding instead
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Rules for mutual inductive/coinductive types
Some proof assistants, like Agda and maybe Coq, allow families of mutually defined types, or nested definitions of types, in which some are inductive and others are coinductive. I have no idea what ...
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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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What are well-founded inductive types?
W-types are said to be "a well-founded inductive type" that is parameterized over arities and constructors, similar to a "tree of possible constructions". On nlab, it is said that ...
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Replacing (strict) positivity with monotonicity on propositions
When defining an inductive type, there is a famous "positivity" restriction on the constructor types. For example, an inductive type $\mathsf D$ has constructor $\mathsf c : F(\mathsf D) \to ...
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Are eliminators useful in practice, or are they only useful in the metatheory?
We can derive eliminators (as functions) for inductive types and translate (structurally) recursive functions into invocations of eliminators. However, eliminators seem to be essentially recursive ...
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Construction of inductive types "the hard way"
Most theorem provers simply axiomize inductive types (or equivalently W types) in the abstract which is fine.
But I'm curious about explicit constructions of inductive types within the theory.
I ...
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Strong eta-rules for functions on sum types
I am wondering whether a rule like the following is consistent with decidable conversion and type-checking for dependent type theory:
$$ \frac{f\, g : (x:\mathsf{bool}) \to C~x\quad f~\mathsf{tt} \...
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Inductive vs. recursive definitions
In Coq there are two ways to define a new type on an inductive type: Using Inductive and using Fixpoint. What are pros and cons ...
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How to prove in Lean that sums are distributive?
Assume we are given three types in Lean.
constants A B C : Type
There is a canonical map of the following form.
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Parameterized Datatypes in a Universe à la Tarski?
I'm wondering, is there a way to make a Universe à la Tarski that models all of the types in an open type theory, where there can be user defined parameterized inductive types?
For context, I'm trying ...
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Explain all the arguments to this rec eliminator
I defined this inductive type for representing JSON elements
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Comparing indexed induction to recursion
Many indexed inductive families can be "equivalently" defined by recursion over their indices. For instance, the type family of vectors:
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Pragmatic encodings of inductive inductive types
What's the most pragmatic encoding for inductive-inductive types such as for a universe of types?
In pseudo Coq syntax.
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Turning off some sProp checks
In Definitional Proof Irrelevance Without K, inductives in sProp need to satisfy three conditions to allow large elimination:
(1) Every non-forced argument must be in sProp.
(2) The return types of ...
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Making a finite graph type in Lean - introduction rule
I'm making a finite directed graph type in Lean. I know type theory from an abstract point of view, but I'm struggling to find the way Lean would produce a type playing the role of a "finite set&...
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Co-induction principle
It's known that Nat-ind = Nat-rec ⨯ Nat-initiality
Has someone figured out how to define a suitable Conat-coind
such that ...
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Why do record based inductive types with primitive projections lack an eta law?
In Coq there is ongoing work to shore up some weaknesses in subject reduction and coinductive types. Primitive projections are part of that effort for better behaviour.
I get why there might be ...
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Can you remove strict positivity from mere propositions?
You can encode induction impredicatively. Some minimalist theories dispense entirely with inductive/coinductive types for proof irrelevant impredicative sorts.
But in a practical theorem prover it's ...
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How do I define an induction principle for a type with a nested list of tuples?
I want to define an inductive type that describes records. The records are lists of elements, each element has a name and type. This requires nested recursion, so I've had to define an induction ...
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Uniqueness of proofs for inductively defined predicates
In Init/Peano.v the less-than-or-equal predicate is defined as follows:
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(In Lean), why cannot structural recursion on propositions be used?
Update: this has been fixed and the fix is available in the latest nightly release of Lean 4.
In Lean 4, structural recursion on functions on types is no issue.
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How to reason about and extract code for inductive types with negative occurrences in Coq?
I'm interested in proving correctness of the interpreter of Appel's compiler (appendix B), and compare it to the machine semantics given by Kennedy on his paper. The interpreter acts as a denotational ...
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Induction scheme on two arguments for custom type in Coq
I've been working on formalizing a Hilbert deductive system within Coq. I have the following definition for a term in first-order logic:
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Coq Induction on Hypothesis destroys the Hypothesis
I'm trying to prove something in coq
I have and Inductive prop type named in_order_merge which is a relation between three lists that shows third one is in_order merge of first two, here is the ...
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Defining Lists and Prove Associativity of Append [closed]
When I saw this question asking what is the "Hello, World!" for proof assistants I immediately thought of that exercise. Not a long time after this answer by Couchy was proposed.
Therefore, ...
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Elimination rules of inductive types
Why does the elimination rule of inductive types sometimes allow the target type to depend on the inductive type and sometimes not? I am confused by that. Is it correct that it makes no difference in ...
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Formalizing "finite or infinite" in Coq
In Coq, I am trying to formalize the notion of a finite or infinite sequence, e.g. of natural numbers: call it a run, and call ...
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How to apply constructor injectivity in the goal
Suppose I have a goal
Goal forall m n, S m = S n -> m = n.
intros m n H.
1 goal
m, n : nat
H : S m = S n
============================
m = n
I know ...
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Inductive types associated to instances of a structure in Lean
I am using the Lean computer proof assistant. I am using the combinatorial structure of a graph with an abelian operation on its edges as a learning example.
In it I have a structure Graph. I want to ...
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Heterogeneous lists, large indices
Recently I had cause to define a type of heterogeneous lists in Lean, and wrote
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how to inductively define paths from paths using unimath
I'd like to define a type of graph where given a set of edges, we can define another graph that has everything from graph 1 but extends the set of edges by adding higher level edges to parallel edges(...
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Why does Coq not allow constructor argument types to be strictly positive mutual inductive types?
Note: Apologies for the wicked mouthful of a title. I'm still getting acquainted with Coq terminology, so I might not have chosen the best words. If you have a better title suggestion, edits are more ...
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Can you always replace mutually recursive references with parameters?
This is a follow up to a question someone else previously posted: Expressivity of mutual/nested inductives vs. regular inductives.
pigworker answered that
Adopting Agda-ish notation, the basic ...
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Descriptions of heterogenous datatypes
When attempting to describe the datatype as appearing in my previous question, using indexed descriptions in style of The Gentle Art of Levitation to describe this datatype (using Agda for examples):
<...
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Why inductive types (or variants) are so rigid in terms of the set of constructors
An inductive type definition normally carries a set of constructors C, but I am not so sure why the set of constructors C is always once-for-all statically defined.
For instance:
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Help with strong induction
I have the following definition of divisibility by 3.
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