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The Calculus of Constructions
Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom
TLDR
A type theory in which it is possible to directly manipulate n-dimensional cubes based on an interpretation of dependenttype theory in a cubical set model that enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system.
Inductively defined types
A Model of Type Theory in Cubical Sets
TLDR
A model of type theory with dependent product, sum, and identity, in cubical sets is presented, and is a step towards a computational interpretation of Voevodsky's Univalence Axiom.
Pattern Matching with Dependent Types
For a more complete presentation of Martin-Lof's logical framework, which is implemented in ALF, we refer to the book \Programming in Martin-Lof's Type Theory" [16], chapter 19 and 20. We recall
On the Computational Content of the Axiom of Choice
TLDR
This work presents a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice and shows how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-Statements.
Constructions: A Higher Order Proof System for Mechanizing Mathematics
  • T. Coquand, G. Huet
  • Mathematics, Computer Science
    European Conference on Computer Algebra
  • 1 April 1985
We present an extensive set of mathematical propositions and proofs in order to demonstrate the power of expression of the theory of constructions.
An Analysis of Girard's Paradox
Infinite Objects in Type Theory
TLDR
According to this analysis, the proof expressions should have the same structure as the program expressions of a pure functional lazy language: variable, constructor, application, abstraction, case expressions, and local let expressions.
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