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Type Isomorphisms and Proof Reuse in Dependent Type Theory [chapter]

Gilles Barthe, Olivier Pons
2001 Lecture Notes in Computer Science  
We propose a theoretical foundation for proof reuse, based on the novel idea of a computational interpretation of type isomorphisms.  ...  Here we do not need extensionality because we enforce isomorphisms through new rewrite rules.  ...  Type isomorphisms provide a type-theoretical notion of equivalence between (generally simple) types: in a nutshell, a type isomorphism between A and B is a pair of expressions f : A → B and g : B → A that  ... 
doi:10.1007/3-540-45315-6_4 fatcat:3uepwxg6ujcffondm5pefvw4xq

Page 7598 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
In the proof of Theorem. 4.8 (<), it is unpleasant to go twice through almost the same argument (for basis and induction step), hence through two and a half pages.  ...  of simple functions including exponentiation.  ... 

Proof theory of higher-order equations: conservativity, normal forms and term rewriting

K. Meinke
2003 Journal of computer and system sciences (Print)  
Applied to extensional higher-order algebras with countable first-order carrier sets, the finite information topology is metric and second countable in every type. r  ...  We introduce a necessary and sufficient condition for the o-extensionality rule of higher-order equational logic to be conservative over first-order many-sorted equational logic for ground first-order  ...  We are also grateful to the anonymous referees for their careful work in pointing out errors and improvements in earlier versions of this paper.  ... 
doi:10.1016/s0022-0000(03)00048-5 fatcat:nwa4bmq2nreftc4fdkm5o3kgzi

Inductive Consequences in the Calculus of Constructions [chapter]

Daria Walukiewicz-Chrząszcz, Jacek Chrząszcz
2010 Lecture Notes in Computer Science  
However the positive answer can be recovered when the notion of inductive consequences is modified by allowing a certain form of functional extensionality.  ...  Now we investigate the question whether the remaining rules are inductive consequences of the basic subset. We show that the answer is positive for most practical rewrite systems.  ...  of a pure type system with inductive definitions and definitions by rewriting.  ... 
doi:10.1007/978-3-642-14052-5_31 fatcat:u7kk462xend3tb3q3tcntf74r4

Internalization of extensional equality [article]

Andrew Polonsky
2015 arXiv   pre-print
We give a type system in which the universe of types is closed by reflection into it of the logical relation defined externally by induction on the structure of types.  ...  This contribution is placed in the context of the search for a natural, syntactic construction of the extensional equality type (Tait [1995], Altenkirch [1999], Coquand [2011], Licata and Harper [2012]  ...  for any two sets, the set of isomorphisms between then), rather than a simple relation.  ... 
arXiv:1401.1148v3 fatcat:32vg73icqreqhe5eb7qzja4yoe

A Datastructure for Iterated Powers [chapter]

Ralph Matthes
2006 Lecture Notes in Computer Science  
For the time being, no induction principles for these datatypes are known.  ...  The article uses a generalization of Bushes to n-fold self-application and shows how to define elements of these types that have a number of data entries that is obtained by iterated raising to the power  ...  The next step yields the list of these lists where t 2 runs through l, and, finally, the list of lists of lists when t 1 runs through l is obtained.  ... 
doi:10.1007/11783596_18 fatcat:tvcvrcc4sfg6pfhowkdlway7ne

Confluent terminating extensional lambda-calculi with surjective pairing and terminal type [article]

Yohji Akama
2018 arXiv   pre-print
The rewriting system has (i) a rule that rewrites term of a terminal type rewrites to a term constant *, unless the term is not *, (ii) rewrite rules for the extensionality of function types and product  ...  types, and rewrite rules mediating the rewrite rules (i) and (ii).  ...  First they inductively defined the types "isomorphic to" the terminal type ⊤ and the canonical terms of such types.  ... 
arXiv:1805.02004v1 fatcat:7a52m3z2zfbajh4caa5xqkzg7u

An insertion operator preserving infinite reduction sequences

2008 Mathematical Structures in Computer Science  
of inductive types; and (2) with the representation of symmetric groups.  ...  Finally, this last lemma is applied in a peculiar and in a more general way to show the termination of some lambda-calculi with inductive types augmented with specific reductions dealing with: (1) copies  ...  Our purpose in Chemouil, 2004 Soloviev and Chemouil, 2004] was to add some other extensional rewrite rules on inductive types corresponding to some extensional relations between inductive types (see  ... 
doi:10.1017/s0960129508006816 fatcat:iadgwcvgbvcpxfqctf4lfaog4a

Special issue on 'Logical frameworks and metalanguages'

2003 Journal of functional programming  
of research, developing detailed knowledge about equational reasoning through canonical simplification (rewriting theory) and proofs by induction (following Boyer and Moore successful integration of primitive  ...  through ordinal notations, logical complexity, etc.  ...  Researchers moved to investigate definitional congruences significantly different from simple β-conversion from λ-calculus, notably with primitive inductive types, which cross-fertilised with term-rewriting  ... 
doi:10.1017/s0956796802004549 fatcat:zkmf6ncdjnhbvfhmshmsevmvt4

A comparative study of Coq and HOL [chapter]

Vincent Zammit
1997 Lecture Notes in Computer Science  
This paper illustrates the di erences between the style of theory mechanisation of Coq and of HOL.  ...  This comparative study is based on the mechanisation of fragments of the theory of computation in these systems.  ...  Acknowledgements I would like to thank my supervisor, Simon Thompson, for his support and encouragement as well as the anonymous referees for their comments and suggestions on an earlier draft of this  ... 
doi:10.1007/bfb0028403 fatcat:m5emdvn2qncudmbfy2hpe33sfe

Extensional Collapse Situations I: non-termination and unrecoverable errors [article]

Antonio Bucciarelli
2011 arXiv   pre-print
We consider a simple model of higher order, functional computation over the booleans.  ...  We show that the models so defined form a lattice when ordered by the extensional collapse situation relation, introduced in order to compare models with respect to the amount of "intensional information  ...  Thanks to an anonymous referee of a previous version of this work, for suggesting that pre-logical relations are the proper framework to deal with extensional collapse situations.  ... 
arXiv:1101.4465v1 fatcat:kjnwhlevhfb5xkejlaq2kj4dqi

Shallow Embedding of Type Theory is Morally Correct [article]

Ambrus Kaposi, András Kovács, Nicolai Kraus
2019 arXiv   pre-print
There are multiple ways to formalise the metatheory of type theory.  ...  We showcase our technique with very short formalisations of canonicity and parametricity for Martin-L\"of type theory.  ...  very simple constructions.  ... 
arXiv:1907.07562v1 fatcat:jbmoelwd6rahxobvvzc5anb3uq

Explicit Computational Paths in Type Theory

Arthur Freitas Ramos
2019 Bulletin of Symbolic Logic  
He has been in this journey with me for 5 years, since the end of my undergraduate studies. I have learned a great deal from him and he has shown me true wisdom.  ...  ACKNOWLEDGEMENTS This work would have not been possible without the help of many people. First, I can point the never-ending dedication of my parents, Neide and Valdez.  ...  rules are connected only to extensionality, we do not consider them as part of the basic rules of our rewriting system.  ... 
doi:10.1017/bsl.2019.2 fatcat:sqtv4xkq65ashdlwj7otpaeofu

Shallow Embedding of Type Theory is Morally Correct [chapter]

Ambrus Kaposi, András Kovács, Nicolai Kraus
2019 Lecture Notes in Computer Science  
very simple constructions.  ...  Rewrite rules must be syntax-directed and have a fixed direction of rewriting.  ... 
doi:10.1007/978-3-030-33636-3_12 fatcat:45etyokyhrconpefhajd3j2c34

Constructing a universe for the setoid model [chapter]

Thorsten Altenkirch, Simon Boulier, Ambrus Kaposi, Christian Sattler, Filippo Sestini
2021 Lecture Notes in Computer Science  
AbstractThe setoid model is a model of intensional type theory that validates certain extensionality principles, like function extensionality and propositional extensionality, the latter being a limited  ...  To this aim, we present the construction of a (non-univalent) universe of setoids within the setoid model, first as an inductive-recursive definition, which is then translated to an inductive-inductive  ...  For instance, whereas in cubical type theory equality of functions is isomorphic to pointwise equality, in SeTT the isomorphism is replaced by a definitional equality.  ... 
doi:10.1007/978-3-030-71995-1_1 fatcat:chq62pby25dabp5rleyvveztdi
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