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Formalizing Monotone Algebras for Certification of Termination and Complexity Proofs [chapter]

Christian Sternagel, René Thiemann
2014 Lecture Notes in Computer Science  
Monotone algebras are frequently used to generate reduction orders in automated termination and complexity proofs.  ...  To be able to certify these proofs, we formalized several kinds of interpretations in the proof assistant Isabelle/HOL.  ...  To this end we first need signature extensions for relative termination (see also [31] ). Theorem 4 (Signature Extensions Preserve Relative Termination).  ... 
doi:10.1007/978-3-319-08918-8_30 fatcat:g4ujbxaodzdyjcgti33xum5t44

An interpretation of dependent type theory in a model category of locally cartesian closed categories [article]

Martin E. Bidlingmaier
2021 arXiv   pre-print
The latter is our model of dependent type theory.  ...  The original interpretation in an individual lcc category can then be recovered by slicing.  ...  The covariant cwf Coa sLcc has an empty context and context extensions, and the forgetful functor Coa sLcc → sLcc preserves both. Proof.  ... 
arXiv:2007.02900v2 fatcat:2jjkpg7qmbc7xcydsqhgyoprs4

On the Formalization of Termination Techniques based on Multiset Orderings

René Thiemann, Guillaume Allais, Julian Nagele, Marc Herbstritt
2012 International Conference on Rewriting Techniques and Applications  
In order to integrate these techniques in a certifier for termination proofs, we have added them to the Isabelle Formalization of Rewriting.  ...  Multiset orderings are a key ingredient in certain termination techniques like the recursive path ordering and a variant of size-change termination.  ...  However, for the multiset extension dms it is essential that the signature F is finite as otherwise strong normalization is not necessarily preserved, cf. Ex. 5.12.  ... 
doi:10.4230/lipics.rta.2012.339 dblp:conf/rta/ThiemannAN12 fatcat:m6wwxsuzmbf6npqk2cym2s7noy

Automated theorem proving in a simple meta-logic for LF [chapter]

Carsten Schürmann, Frank Pfenning
1998 Lecture Notes in Computer Science  
We have used Twelf to automatically prove a number of non-trivial theorems, including type preservation for Mini-ML and the deduction theorem for intuitionistic propositional logic.  ...  A further extension of M 2 is the introduction of conjunction as formula and corresponding pairs as proof terms.  ...  The Logical Framework The type theory underlying the logical framework LF is an extension of the simplytyped λ-calculus by dependent types.  ... 
doi:10.1007/bfb0054266 fatcat:wkxymnxehzhrpb5rntixrlyikm

Algebras, Coalgebras, Monads and Comonads

Neil Ghani, Christoph Lüth, Federico de Marchi, John Power
2001 Electronical Notes in Theoretical Computer Science  
In developing these theories we strive to provide them with an associated notion of syntax.  ...  In the case of initial algebras and monads this corresponds to the standard notion of algebraic theories consisting of signatures and equations: models of such algebraic theories are precisely the algebras  ...  Since the terminal object does not depend upon any data, its arity is the empty category 0. Since the map ! X depends upon an object, its arity is the one object category 1.  ... 
doi:10.1016/s1571-0661(04)80905-8 fatcat:f6lif7mrpfhuvlpwhyw4ypwgti

Safety and conservativity of definitions in HOL and Isabelle/HOL

Ondřej Kunčar, Andrei Popescu
2017 Proceedings of the ACM on Programming Languages  
In particular, definitions should form a conservative extension of the pure logic.  ...  Some unique features of HOL, such as the requirement to give non-emptiness proofs when defining new types and the impossibility to unfold type definitions, make the proof of these properties, and also  ...  Another alternative is to define UNF(c zfun ) as an arbitrary element satisfying REL(zfun), via Choice, i.e., as ϵ REL(zfun).  ... 
doi:10.1145/3158112 dblp:journals/pacmpl/Kuncar018 fatcat:3hryowq7dzen7pnj3n2z7jrv7a

Uncurrying for Termination and Complexity

Nao Hirokawa, Aart Middeldorp, Harald Zankl
2012 Journal of automated reasoning  
In this article we present a transformation from untyped applicative term rewrite systems to functional term rewrite systems that preserves and reflects termination.  ...  To further increase the applicability of our transformation, we study the method for innermost rewriting and derivational complexity, and present a version for dependency pairs.  ...  Acknowledgements We thank the reviewers and Georg Moser for their helpful comments and René Thiemann for suggesting that the proof of [18, Theorem 33] contains a gap.  ... 
doi:10.1007/s10817-012-9248-3 fatcat:n7e255kqhjf6rdxucym2umaquu

A rationale for conditional equational programming

Nachum Dershowitz, Mitsuhiro Okada
1990 Theoretical Computer Science  
Some termination issues are also considered, including the case of rewriting with higher-order terms.  ...  Reasonable conditions for ensuring the completeness of narrowing as an operational mechanism for solving goals are provided; these results are then extended to handle rewriting with existentially quantified  ...  The preserv termination prope was proved in [44, 40, 4] . ese two preserva the preservation o he convergence property. reazu-Tannen an polymorphically typed A-calculus for those preservation res purpose  ... 
doi:10.1016/0304-3975(90)90064-o fatcat:rojkwf4kx5avrk7kcs23gkyogi

Generalized and Formalized Uncurrying [chapter]

Christian Sternagel, René Thiemann
2011 Lecture Notes in Computer Science  
Furthermore, we provide new results on uncurrying for relative termination and for dependency pairs.  ...  Uncurrying is a termination technique for applicative term rewrite systems.  ...  Uncurrying in the Dependency Pair Framework The DP framework [5] facilitates modular termination proofs.  ... 
doi:10.1007/978-3-642-24364-6_17 fatcat:wogzvsw2lrcsrd3dfqt2hn62qu

Equational completion in order-sorted algebras

Isabelle Gnaedig, Claude Kirchner, Héléne Kirchner
1990 Theoretical Computer Science  
This property is achieved through the hypothesis that rules are sort-dew xing and axioms arc sort-preserving.  ...  Finally a new technique, orderings for equational proofs, introduced in [3,2] allowed simpler and more intuitive correctness proofs of improved completion procedures.  ...  The E-completion procedure is expressed via the following inference rules. (1) Orienting a pair PI!  ... 
doi:10.1016/0304-3975(90)90034-f fatcat:zvosmzddsfew3e7xahbw2sxkdy

From Signatures to Monads in UniMath

Benedikt Ahrens, Ralph Matthes, Anders Mörtberg
2018 Journal of automated reasoning  
Altogether, we construct a framework that allows us to concisely specify, via a simple notion of binding signature, a language with variable binding.  ...  The UniMath system is a core dependent type theory, augmented by the univalence axiom.  ...  The authors are also very grateful to Peter LeFanu Lumsdaine for providing a simpler proof, with fewer hypotheses, that some functors are ω-cocontinuous.  ... 
doi:10.1007/s10817-018-9474-4 fatcat:4el4hfzeunhpvo22plhnvokvqi

A Mechanised Semantics for HOL with Ad-hoc Overloading [article]

Johannes Åman Pohjola, Arve Gengelbach
2020 arXiv   pre-print
In this paper, we present a mechanised proof that HOL with ad-hoc overloading is consistent. All our results have been formalised in the HOL4 theorem prover.  ...  preserves wellfoundedness.  ...  This termination proof is quite laborious, at 495 lines of proof script.  ... 
arXiv:2002.10212v2 fatcat:uz7qmnbbc5d2lbv5d2dzc7g5ne

Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

Arve Gengelbach, Johannes Åman Pohjola, Tjark Weber
2021 Electronic Proceedings in Theoretical Computer Science  
This work revises our earlier notion of model-theoretic conservative extension and generalises an earlier model construction.  ...  We prove that symbols that are independent of a new definition may keep their interpretation in a model extension.  ...  Alternatively, the problem as depicted in the example can be circumvented through extending the dependency relation with cross-dependencies.  ... 
doi:10.4204/eptcs.332.1 fatcat:cdcs4lxnova5xnjr3gc2tyejvq

Size-Change Termination for Term Rewriting [chapter]

René Thiemann, Jürgen Giesl
2003 Lecture Notes in Computer Science  
Moreover, we compare this approach with existing techniques for termination analysis of TRSs (such as recursive path orderings or dependency pairs).  ...  In this way, we obtain a new approach for automated termination proofs of TRSs which is more powerful than previous approaches.  ...  The tool is written in Java and termination proofs can be performed via a graphical user interface.  ... 
doi:10.1007/3-540-44881-0_19 fatcat:fyge57hmwzbmvksuz23v5p6as4

Strongly invertible knots, invariant surfaces, and the Atiyah-Singer signature theorem [article]

Antonio Alfieri, Keegan Boyle
2021 arXiv   pre-print
We use the G-signature theorem to define an invariant of strongly invertible knots analogous to the knot signature.  ...  In particular the equivariant signature of an equivariantly slice strongly invertible knot vanishes. Proof.  ...  We also show that for an alternating admissible diagram, the correction term e(∆, γ) is equal to minus the g-signature σ(K). Theorem 1.2.  ... 
arXiv:2109.09915v1 fatcat:2xw3d2pxsjhnroqfsmaysnk2hm
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