# Questions tagged [calculus-of-inductive-constructions]

Calculus of (co)Inductive Constructions is a pure type system (Coquand, Huet) equipped with addition types: arbitrary (co)inductive types implementing general (co)inductive schemes; universes as a cumulative hierarchy of predicative types of types; and an impredicative type of propositions.

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### What are the principal differences between Agda's core type theory and Coq's?

Agda is said to be based on Luo's unifying theory of dependent types while Coq is based on the Calculus of Inductive Constructions. Both of these as I understand it extend the impredicative ...
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### Universe inconsistency errors when using ZF model in Coq

I am trying to use a formal logic system I recently implemented in Coq to study ZF set theory. In order to do this, I need to define a type representing the domain in question, and then prove that ...
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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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### Can you help me lay out the different variations of CoC and their generalizations?

I am learning the typed lambda-calculus and looking into the Calculus of Constructions. It was going well until I was slapped in the face with variations, and now I'm confused about the layout of all ...
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### Calculus of (inductive) Constructions: Do inductive definitions increase proof strength?

Question Is CiC stronger than CoC, in terms of proof strength? Context To illustrate the kind of confusion I am in, and what I'd like to learn from the answer, here is part of my inner monologue: If I ...
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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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### What should be cited for "the Calculus of inductive Constructions"?

The history of dependent data types spans decades and is a bit confusing. I have seen some implausible claims about which documents present what. I would like to get it right for my own work without ...
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Timany and Sozeau's Predicative Calculus of Cumulative Inductive Constructions (pCuIC) [1] adds extra cumulativity to the inductive types of (Lee and Werner's [2], I think?) pCIC. But what does the &...
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### Proof-theoretic comparison table?

I read this CSTheory SE post, which suggests that it is often not clear what variant of MLTT or CIC is being referred to. But I would like to know the proof-theoretic strengths of the various ...
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### Is unguarded fixpoint reduction consistent?

In Coq, there are two restrictions on fixpoints to retain normalization: Recursive calls can only be done on structurally smaller arguments, enforced by a guard condition during type checking; and ...
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### How does Metamath Zero handle CIC as in Lean or Coq?

Metamath Zero (MM0) is a proof assistant developed by Mario Carneiro. It has a metalogic very similar to the metalogic of MetaMath, but it also borrows design choices from Lean (and maybe other ...
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### Are Logics Based on Dependent Types Stronger Than Ones Without?

There have been several times during I came across statements like Isabelle/HOL's logic is not rich enough to formalize X on various places online and in during personal discussions. Or similar ...
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