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Formalizing Monotone Algebras for Certification of Termination and Complexity Proofs
[chapter]
2014
Lecture Notes in Computer Science
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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]
2021
arXiv
pre-print
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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
2012
International Conference on Rewriting Techniques and Applications
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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]
1998
Lecture Notes in Computer Science
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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
2001
Electronical Notes in Theoretical Computer Science
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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
2017
Proceedings of the ACM on Programming Languages
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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
2012
Journal of automated reasoning
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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
1990
Theoretical Computer Science
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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]
2011
Lecture Notes in Computer Science
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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
1990
Theoretical Computer Science
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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
2018
Journal of automated reasoning
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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]
2020
arXiv
pre-print
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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
2021
Electronic Proceedings in Theoretical Computer Science
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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]
2003
Lecture Notes in Computer Science
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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]
2021
arXiv
pre-print
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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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