## Irrelevance in Type Theory with a Heterogeneous Equality Judgement

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Citations: | 8 - 1 self |

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@MISC{Abel_irrelevancein,

author = {Andreas Abel},

title = {Irrelevance in Type Theory with a Heterogeneous Equality Judgement},

year = {}

}

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### Abstract

Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance of not only the compiled program but also the type checker such static terms need to be erased as early as possible, preferably immediately after type checking. To this end, Pfenning’s type theory with irrelevant quantification, that models a distinction between static and dynamic code, is extended to universes and large eliminations. Novel is a heterogeneously typed implementation of equality which allows the smooth construction of a universal Kripke model that proves normalization, consistency and decidability.

### Citations

221 | Cayenne — a language with dependent types
- Augustsson
- 1998
(Show Context)
Citation Context ...fier of the proofs woven into the program. Program extraction takes type-checked terms and discards parts that are irrelevant for execution. Augustsson’s dependently typed functional language Cayenne =-=[Aug99]-=- erases types using a universe-based analysis. Coq’s extraction procedure has been designed by Letouzey [Let02] and discards not only types but also proofs. The erasure rests on Coq’s universe-based s... |

88 | On equivalence and canonical forms in the LF type theory
- Harper, Pfenning
- 2005
(Show Context)
Citation Context ...already observed that a heterogeneous treatment of algorith2mic equality solves the problem of defining a Kripke logical relation that shows completeness of algorithmic equality. Harper and Pfenning =-=[HP05]-=- hit the same problem, and their fix was to erase dependencies in types. In weak type theories like the logical framework erasure is possible, but it does not scale to large eliminations. Related to o... |

58 |
Synthesis of ML programs in the system Coq
- Paulin-Mohring, Werner
- 1993
(Show Context)
Citation Context ...t for execution. Augustsson’s dependently typed functional language Cayenne [6] erases types using a universe-based analysis. Coq’s extraction procedure has been designed by Paulin-Mohring and Werner =-=[21]-=- and Letouzey [14] and discards not only types but also proofs. The erasure rests on Coq’s universe-based separation between propositional (Prop) and computational parts (Set/Type). The rigid Prop/Set... |

54 |
A typed operational semantics for type theory
- Goguen
- 1994
(Show Context)
Citation Context ...es. In weak type theories like the logical framework erasure is possible, but it does not scale to large eliminations. Related to our present treatment of IITT is Goguen’s Typed Operational Semantics =-=[Gog94]-=-. He proves meta-theoretic properties such as normalization, subject reduction, and confluence by a Kripke logical predicate of well-typed terms. However, his notion of equality is based reduction and... |

53 |
A New Extraction for Coq
- Letouzey
- 2003
(Show Context)
Citation Context ...are irrelevant for execution. Augustsson’s dependently typed functional language Cayenne [Aug99] erases types using a universe-based analysis. Coq’s extraction procedure has been designed by Letouzey =-=[Let02]-=- and discards not only types but also proofs. The erasure rests on Coq’s universe-based separation between propositional (Prop) and computational parts (Set/Type). The rigid Prop/Set distinction has t... |

47 |
Proof Irrelevance in Modal Type Theory
- “Intensionality
- 2001
(Show Context)
Citation Context ... Besides whole-program analyses such as data flow, some type-based analyses have been put forward. One of them is Pfenning’s modal type theory of Intensionality, Extensionality, and Proof Irrelevance =-=[Pfe01]-=- which introduces functions with irrelevant arguments that play the role of proofs. Not only can these arguments be erased during extraction, they can also be disregarded in type conversion test durin... |

45 | An algorithm for type-checking dependent types
- Coquand
- 1996
(Show Context)
Citation Context ...X, which is ill-typed, since X cannot be equal to Set0. Contradiction! 5.5 Decidability To round off, we show that typing in IITT is decidable. Type checking algorithms such as bidirectional checking =-=[Coq96]-=- rely on injectivity of function type constructors, which is built into the definition of Ⓢ: Theorem 8 (Function type injectivity). If Γ ⊢ (x⋆U) → T : s = Γ ′ ⊢ (x⋆U ′ ) → T ′ : s ′ then Γ ⊢ U : s = Γ... |

30 | The Implicit Calculus of Constructions as a Programming Language with Dependent Types
- Barras, Bernardo
- 2008
(Show Context)
Citation Context ...levant for function execution but relevant during type conversion checking. Such arguments may only be erased in function application but not in the associated type instantiation. Barras and Bernardo =-=[BB08]-=- and Mishra-Linger and Sheard [MLS08] have build decidable type systems on top of ICC, but both have not fully integrated inductive types and types defined by recursion (large eliminations). Barras an... |

28 | Propositions as [types
- Awodey, Bauer
- 2004
(Show Context)
Citation Context ...rmalization, subject reduction, and confluence by a Kripke logical predicate of well-typed terms. However, his notion of equality is based on reduction and not a step-wise algorithm. Awodey and Bauer =-=[7]-=- give a categorical treatment of proof irrelevance which is very similar to Pfenning and Reed’s. However, they work in the setting of Extensional Type Theory with undecidable type checking, I could no... |

23 | A Brief Overview of Agda — A Functional Language with Dependent Types
- Bove, Dybjer, et al.
- 2009
(Show Context)
Citation Context ...Proof Irrelevance, Heterogeneously Typed Equality, Algorithmic Equality, Logical Relation, Universal Kripke Model. 1 Introduction and Related Work Dependently typed programming languages such as Agda =-=[BDN09]-=-, and Coq [INR08], and Epigram [CAM07] allow the programmer to express in one language programs, their types, rich invariants, and even proofs of these invariants. Besides code executed at run-time, d... |

20 |
The Implicit Calculus of Constructions
- Miquel
- 2001
(Show Context)
Citation Context ...ing’s type theory, proofs can never block computations even in open expressions (other than computations on proofs), thus, internal erasure is sound. Miquel’s Implicit Calculus of Constructions (ICC) =-=[Miq01a]-=- goes further than Pfenning and considers also parametric arguments as irrelevant. These are arguments which are irrelevant for function execution but relevant during type conversion checking. Such ar... |

20 |
Le Calcul des Constructions Implicite: Syntaxe et Sémantique
- Miquel
- 2001
(Show Context)
Citation Context ... to treat Pfenning’s notion of irrelevance, but our proof does not scale directly to parametric function types, a stronger notion of irrelevant function types called implicit quantification by Miquel =-=[Miq01b]-=-. 2 Two more type theories build on Miquel’s calculus [Miq01a], Barras and Bernardo’s ICC ∗ [BB08] and Mishra-Linger and Sheard’s Erasure Pure Type Systems (EPTS) [MLS08], but none has offered a satis... |

15 | MiniAgda: Integrating sized and dependent types
- Abel
- 2010
(Show Context)
Citation Context .... To this end, a type theory for irrelevance is needed that supports user-defined data types and functions and types defined by pattern matching. Experiments with my prototype implementation MiniAgda =-=[Abe10]-=- have revealed some issues when combining Miquel-style irrelevance with large eliminations. For instance, the equality checker ends up comparing expressions of different base types in some situations.... |

15 |
Epigram reloaded: A standalone typechecker for ETT
- Chapman, Altenkirch, et al.
- 2006
(Show Context)
Citation Context ...ed Equality, Algorithmic Equality, Logical Relation, Universal Kripke Model. 1 Introduction and Related Work Dependently typed programming languages such as Agda [BDN09], and Coq [INR08], and Epigram =-=[CAM07]-=- allow the programmer to express in one language programs, their types, rich invariants, and even proofs of these invariants. Besides code executed at run-time, dependently typed programs contain some... |

14 | T.: Erasure and polymorphism in pure type systems
- Mishra-Linger, Sheard
- 2008
(Show Context)
Citation Context ...levant during type conversion checking. Such arguments may only be erased in function application but not in the associated type instantiation. Barras and Bernardo [BB08] and Mishra-Linger and Sheard =-=[MLS08]-=- have build decidable type systems on top of ICC, but both have not fully integrated inductive types and types defined by recursion (large eliminations). Barras and Bernardo, as Miquel, have inductive... |

12 | Verifying a semantic βη-conversion test for Martin-Löf type theory
- Abel, Coquand, et al.
- 2008
(Show Context)
Citation Context ...e specification of type theory, and the universal Kripke model that yields all interesting meta-theoretic results at onve. The Kripke model is inspired by previous work on normalization by evaluation =-=[ACD08]-=-. There we have already observed that a heterogeneous treatment of algorith2mic equality solves the problem of defining a Kripke logical relation that shows completeness of algorithmic equality. Harp... |

9 | Untyped algorithmic equality for Martin-Löf’s logical framework with surjective pairs
- Abel, Coquand
- 2007
(Show Context)
Citation Context ...he function arguments may be completely unrelated. The heterogeneous definition of equality is more symmetric and has more degrees of liberty. Thus, closure under substitution is easier to prove (cf. =-=[13, 3]-=-). ⊢ 5In heterogeneous judgements such as equality, we maintain the invariant that the two contexts Γ and Γ ′ have the same shape, i. e., bind the same variables with the same irrelevance status. Onl... |

8 | On the strength of proof-irrelevant type theories
- Werner
- 2006
(Show Context)
Citation Context ...and it is also known as the type of proofs of (proposition) T [4, 8]. Using the extensions of Example 1, we can encode it as △T = ( ÷T ) × 1. 3 Some care is necessary for the type of Leibniz equality =-=[1, 25]-=-.7 Conclusions We have extended Pfenning’s notion of irrelevance to a type theory IITT with universes that accommodates types defined by recursion. A heterogeneous algorithmic equality judgement has ... |

7 | M.: A modular type-checking algorithm for type theory with singleton types and proof irrelevance
- Abel, Coquand, et al.
- 2011
(Show Context)
Citation Context ... proof t of T , then • is a new proof of T . This preserves provability while erasing the proof terms. Conservativity of this rule can be proven as in joint work of the author with Coquand and Pagano =-=[ACP09]-=-. Proof modality. Pfenning [Pfe01] suggests a modality △ formed by the rule Γ ⊢ t ÷ T Γ ⊢ t : △T . 1 Some care is necessary for the type of Leibniz equality [Abe09,Wer08]. 15which for instance allows... |

7 |
The Coq Proof Assistant Reference Manual. INRIA, version 8.4 edition, 2012. [Nor07] [Set12] [SP82] Ulf Norell. Towards a Practical Programming Language Based on Dependent Type Theory
- INRIA
(Show Context)
Citation Context ..., Heterogeneously Typed Equality, Algorithmic Equality, Logical Relation, Universal Kripke Model. 1 Introduction and Related Work Dependently typed programming languages such as Agda [BDN09], and Coq =-=[INR08]-=-, and Epigram [CAM07] allow the programmer to express in one language programs, their types, rich invariants, and even proofs of these invariants. Besides code executed at run-time, dependently typed ... |

7 | Proof irrelevance and strict definitions in a logical framework - Reed - 2002 |

7 |
Ulf Norell. A brief overview of Agda — a functional language with dependent types
- Bove, Dybjer
- 2009
(Show Context)
Citation Context ...proof irrelevance, heterogeneously typed equality, algorithmic equality, logical relation, universal Kripke model. 1 Introduction and Related Work Dependently typed programming languages such as Agda =-=[9]-=-, Coq [13], and Epigram [15] allow the programmer to express in one language programs, their types, rich invariants, and even proofs of these invariants. Besides code executed at run-time, dependently... |

6 | A model for impredicative type systems, universes, intersection types and subtyping
- Miquel
- 2000
(Show Context)
Citation Context ...iq01a], Barras and Bernardo’s ICC ∗ [BB08] and Mishra-Linger and Sheard’s Erasure Pure Type Systems (EPTS) [MLS08], but none has offered a satisfying account of large eliminations yet. Miquel’s model =-=[Miq00]-=- features data types only as impredicative encodings. For irrelevant, parametric, and recursive functions to coexist it seems like three different function types are necessary, e. g., in the style of ... |

2 |
Erasure in Type Theory
- Irrelevance
- 2008
(Show Context)
Citation Context ...ly integrated inductive types and types defined by recursion (large eliminations). Barras and Bernardo, as Miquel, have inductive types only in the form of their impredicative encodings, MishraLinger =-=[ML08]-=- gives introduction and elimination principles for inductive types by example, but does not show normalization or consistency. Our long-term goal is to equip Agda with internal and external erasure. T... |

2 | Extensional normalization in the logical framework with proof irrelevant equality
- Abel
- 2009
(Show Context)
Citation Context ...and it is also known as the type of proofs of (proposition) T [4, 8]. Using the extensions of Example 1, we can encode it as △T = ( ÷T ) × 1. 3 Some care is necessary for the type of Leibniz equality =-=[1, 25]-=-.7 Conclusions We have extended Pfenning’s notion of irrelevance to a type theory IITT with universes that accommodates types defined by recursion. A heterogeneous algorithmic equality judgement has ... |

2 |
Extending higher-order unification to support proof irrelevance
- Reed
- 2003
(Show Context)
Citation Context ...n to scale Pfenning’s notion of proof irrelevance up to inductive types. In this article, we start with the “extensionality and proof irrelevance” fragment of Pfenning’s type theory in Reed’s version =-=[23, 24]-=-. We extend it by a hierarchy of predicative universes, yielding Irrelevant Intensional Type Theory IITT (Sec. 2). Based on a heterogeneous algorithmic equality which compares two expressions, each in... |

1 |
A function argument is parametric if it is irrelevant for computing the function result while the type of the result may depend on it
- Comput
- 2004
(Show Context)
Citation Context ... normalization, subject reduction, and confluence by a Kripke logical predicate of well-typed terms. However, his notion of equality is based reduction and not a step-wise algorithm. Awodey and Bauer =-=[AB04]-=- give a categorical treatment of proof irrelevance which is very similar to Pfenning and Reed’s. However, they work in the setting of Extensional Type Theory with undecidable type checking, I could no... |

1 | 4962 of LNCS - FoSSaCS’08 - 2008 |

1 |
The view from the
- McBride, McKinna
- 2004
(Show Context)
Citation Context ...neously typed equality, algorithmic equality, logical relation, universal Kripke model. 1 Introduction and Related Work Dependently typed programming languages such as Agda [9], Coq [13], and Epigram =-=[15]-=- allow the programmer to express in one language programs, their types, rich invariants, and even proofs of these invariants. Besides code executed at run-time, dependently typed programs contain much... |